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http://lambda-the-ultimate.org/node/5049 | Extended Axiomatic Language
Axiomatic language is a formal system for specifying recursively enumerable sets of hierarchical symbolic expressions. But axiomatic language does not have negation. Extended axiomatic language is based on the idea that when one specifies a recursively enumerable set, one is simultaneously specifying the complement of that set (which may not be recursively enumerable). This complement set can be useful for specification. Extended axiomatic language makes use of this complement set and can be considered a form of logic programming negation. The web page defines the language and gives examples. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8313205242156982, "perplexity": 560.5462482810412}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348511950.89/warc/CC-MAIN-20200606062649-20200606092649-00334.warc.gz"} |
https://studysoup.com/tsg/993682/modern-physics-for-scientists-and-engineers-4-edition-chapter-13-problem-18 | Modern Physics For Scientists And Engineers - 4 Edition - Chapter 13 - Problem 18
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# Explain in your own words the origin of the names ofelements 103 through 108that is, who
Modern Physics for Scientists and Engineers | 4th Edition
Problem 18
Explain in your own words the origin of the names ofelements 103 through 108that is, who or what theelements were named after and the reasons for doingso.
Accepted Solution
Step-by-Step Solution:
Step 1 of 3
Electric Force • The electric force is the force exerted by the electric field. • The electric field is an aura that surrounds a charged particle. ,, ,, 0 E tells us how strong the electric force is at (x,y,z) when a charge (0) is placed there. Electric Force = 2 = 8.99 ∗ 109 = ℎ = ℎ
###### Chapter 13, Problem 18 is Solved
Step 2 of 3
Step 3 of 3
Unlock Textbook Solution | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8220689296722412, "perplexity": 1450.6551279045395}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500215.91/warc/CC-MAIN-20230205032040-20230205062040-00720.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/74121-flatness-criterion-modules.html | ## Flatness Criterion for Modules
This is exercise 25 in Chapter 10, Section 5 of Dummit and Foote.
Essentially we are trying to prove that A is a flat R-module iff for every finitely generated ideal I of R, the map from A tensor I -> A tensor R = A induced by the inclusion I into R is again injective. (equivalently A tensor I = AI contained in A).
First we need to prove that if A is flat then A tensor I -> A tensor R is injective. This isn't difficult.
The next step is very tricky for me:
If A tensor I -> A tensor R is injective for every finitely generated ideal I, prove that A tensor I -> A tensor R is injective for every ideal I.
Show that if K is any submodule of a finitely generated free module F then
A tensor K -> A tensor F is injective.
Then I have to decduce that the same is true for any free module F.
Any ideas? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9621797800064087, "perplexity": 402.04843725401525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398525032.0/warc/CC-MAIN-20151124205525-00019-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/im-stumped.93201/ | I'm stumped
1. Oct 9, 2005
godplayer
A 2.0 kg wood box slides down a vertical wood wall while you push on it at a 45 degree angle. What magnitude of force should you apply to cause the box to slide down at a constant speed? (Problem 51, Chapter 5, Knight)
I can't figure out this problem and do not have the text to refer to (alibris.com sent me the wrong book). I'm not having problems with FBD's. I tried to set the acceleration to zero so it travels at a constant speed but with zero acceleration there is zero force. I'm confused.
2. Oct 9, 2005
Physics Monkey
This is exactly right, you want zero net force. How else can the box move with constant velocity?
There are four forces in the problem, the force of gravity, the normal force, the friction force, and your applied force. Resolve these four into components and find the magnitude of the applied force necessary to have zero net force vertically.
Last edited: Oct 9, 2005
3. Oct 9, 2005
Pengwuino
Well remember, with constant speed, there is zero NET force.
4. Oct 9, 2005
godplayer
Oh. . . now I'm really confused.
5. Oct 9, 2005
godplayer
That seems to easy!
6. Oct 9, 2005
godplayer
I thought that if there was a zero NET force that the box would be at rest.
7. Oct 9, 2005
Physics Monkey
If there is zero net force, the acceleration is zero. But if the body is already moving then it keeps moving by Newton's 1st Law.
8. Oct 9, 2005
Pengwuino
Nope, it would just be moving at the same speed its been moving. Remember newton's first law. What you must imply however is that the box is already moving down that 45 degree plane when you start actually pushing back
9. Oct 9, 2005
godplayer
Thanks! That helps a lot!
I'm not sure how to find the normal force. I knoe its horizontal from the wall but I am unsure of how to find it. I need it to find the force of friction.
10. Oct 9, 2005
Physics Monkey
You can find the normal by balancing the forces in the horizontal direction (remember the applied force has a component in that direction). The box certainly isn't accelerating away from the wall.
11. Oct 9, 2005
godplayer
Yeah I know but there were know horizontal forces given in the problem.
Last edited: Oct 9, 2005
12. Oct 9, 2005
Physics Monkey
Since the applied force hits the block at a 45 degree angle, it has a component in the horizontal direction. In other words, you are pushing the box into the wall and the wall pushes back (normal force). There would be no normal force if no one was pushing the block into the wall.
13. Oct 9, 2005
godplayer
Thanks guys
Similar Discussions: I'm stumped | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9183416962623596, "perplexity": 834.6348724155173}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084892802.73/warc/CC-MAIN-20180123231023-20180124011023-00071.warc.gz"} |
http://laneas.com/publication/differential-feedback-scheme-exploiting-temporal-and-spectral-correlation | # A differential Feedback Scheme Exploiting the Temporal and Spectral Correlation
Journal Article
### Authors:
Mingxin Zhou; Leiming Zhang; Lingyang Song; Mérouane Debbah
### Source:
IEEE Transactions on Vehicular Technology , Volume 62, Issue 9, p.4701-4707 (2013)
### Abstract:
Channel state information (CSI) provided by a limited feed- back channel can be utilized to increase system throughput. However, in multiple-input–multiple-output (MIMO) systems, the signaling overhead realizing this CSI feedback can be quite large, whereas the capacity of the uplink feedback channel is typically limited. Hence, it is crucial to reduce the amount of feedback bits. Prior work on limited feedback compression commonly adopted the block-fading channel model, where only temporal or spectral correlation in a wireless channel is considered. In this paper, we propose a differential feedback scheme with full use of the temporal and spectral correlations to reduce the feedback load. Then, the minimal differential feedback rate over a MIMO time–frequency (or doubly)-selective fading channel is investigated. Finally, the analysis is verified by simulation results. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.800213098526001, "perplexity": 1138.8797331992025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743105.25/warc/CC-MAIN-20181116152954-20181116174954-00150.warc.gz"} |
https://www.physicsforums.com/threads/centrifugal-force-in-high-pressure-vessel.48663/ | # Centrifugal force in high pressure vessel
1. Oct 19, 2004
### drewman13
An enclosed cylindrical* (disc) vessel one inch thick and five inches in radius holding two pounds of fluid at ambient atmospheric pressure (14psi?)and being spun at 30,000 rpm's. Assume the centrifugal force acting on the fluid creates a pressure of 1,000 psi. The question is ....
"If the vessel is pre-pressurized, to say 3,000 psi, before it is spun up to 30,000 rpm's, will the centrifugal force add pressure to the already pre-pressurized (3,000 psi) vessel?"
Would the resultant force then be 4,000 psi? If not, what would the total psi be and why?
2. Oct 20, 2004
### aekanshchumber
Use Dalton's partial pressure law.
3. Oct 20, 2004
### Tide
He didn't say the vessel was pressurized with gas.
4. Oct 20, 2004
### aekanshchumber
If it is so, then the pressure of the gas will depend on the position from where it is being measured.
Due to rotations, the gas will be displaced toward the outer side. And the pressure of the gass will gradually increase when we move from ceter of rotation to the extreme of the daimeter.
5. Oct 20, 2004
### Clausius2
Yes, I'm with aekanshchumber, and I'll give him an additional support for his opinion writing a few equations:
Inside the vessel, and choosing a reference frame spinning with it, the Navier-Stokes equations yield:
$$\frac{\partial P}{\partial r}=\rho\omega^2 r$$ in Hydrostatic form.
Solving for P(r) you will obtain the pressure distribution. The constant resulting of the last equation has to depend on the initial value in the case of a closed vessel, where no boundary constraint is possible to satisfy.
Maybe, the unsteady process is harder [\B] of analyzing due to a progressive acceleration of the fluid is necessary. The kinematic field has to be solved firstly before doing any comment about the pressure at the very first instants. I don't think such a simple rule as drewman13 has stated would be correct for figuring the total pressure out.
6. Oct 20, 2004
### snbose
The pressure will surely increase, but it will not be 4000psi.
Remember that the pressure is proportional on the no of molecules. So for a fixed volume (of the cylinder, V) at 1.4psi if the Number of moecules be N, then at 3000 psi it will be around 2000N.
Now if N molecules generated a 1000psi pressure when gyroscoped @ 30000rpm, then 2000N molecules will generate 2000 x 1000psi at same rpm. I imagine they are proportional however they may not be directly proportional. But logically and from Classical kinetic theory they are likely to be directly proportional.
So you can imagine the net pressure to be humongous 2,000,000 psi.
7. Oct 20, 2004
### sal
The logic sounds good but the starting pressure was 14 psi, not 1.4 psi.
So, 3000 psi ---> 200N, not 2000N, and the final pressure will be 200,000 psi.
8. Oct 20, 2004
### snbose
OPPS..thanks for correcting me. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.863288938999176, "perplexity": 1990.3574954748037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540975.18/warc/CC-MAIN-20161202170900-00302-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://notendur.hi.is/eme1/skoli/edl_h05/masteringphysics/9_10/weightandwheel.htm | Weight and Wheel
Consider a bicycle wheel that initially is not rotating. A block of mass is attached to the wheel and is allowed to fall a distance . Assume that the wheel has a moment of inertia about its rotation axis.
Part A
Consider the case that the string tied to the block is attached to the outside of the wheel, at a radius
. Find , the angular speed of the wheel after the block has fallen a distance , for this case.
Hint A.1 Hint not displayed
Part A.2 Part not displayed
Part A.3 Part not displayed
Part A.4 Part not displayed
Express in terms of , , , , and . | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.927478551864624, "perplexity": 637.586672832914}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824345.69/warc/CC-MAIN-20160723071024-00225-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/power-series-representing-sinx-x.593197/ | # Power series representing ∫sinx/x
1. Apr 3, 2012
### Ryantruran
1. The problem statement, all variables and given/known data
Find the Power Series representing
g(x)=∫sin(x)/x
2. Relevant equations
sin(x)= x-(x^3/3!)+(x^5/5!)-(x^7/7!)
3. The attempt at a solution
I Havent attempted yet but was wondering if you start with the maclaurin series of sin(x)
then divide everything by x then integrate the entire summation
2. Apr 3, 2012
### Staff: Mentor
Yes, that's what you do.
Similar Discussions: Power series representing ∫sinx/x | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9090661406517029, "perplexity": 3818.774228332774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823605.33/warc/CC-MAIN-20171020010834-20171020030834-00169.warc.gz"} |
http://studyrankersonline.com/22363/which-of-following-is-not-example-of-redox-reaction | # Which of the following is not an example of redox reaction ?
4.4k views
Which of the following is not an example of redox reaction?
(a) CuO + H2 → Cu + H2O
(b) Fe2O+ 3CO → 2Fe + 3CO2
(c).2K + F2 →2KF
(d) BaCl+ H2SO4 →BaSO4 + 2FIC1
by (-1,017 points)
(d) BaCl2 + H2SO4 → BaSO4 + 2HCl is not a redox reaction. It is an example of double displacement reactions. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8744779229164124, "perplexity": 3972.727294815984}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141189030.27/warc/CC-MAIN-20201126230216-20201127020216-00046.warc.gz"} |
https://www.talkstats.com/search/2438530/ | # Search results
1. ### Justify the use of KS over Shapiro and Wilk normality test
Hello there. I am testing my data distribution for normality using both Shapiro Wilk and Kolmogorov and Smirnov tests in SPSS. S&W reveals that the data is not normally distributed whereas K&S shows its approximately normally distributed. How can i justify the use of K&S over S&W in order to...
2. ### Justify the use of parametric tests on non normally distributed data
Hello there. I have 5 Likert scale questionnaires that have been answered by the same people who have been randomly selected (N=24) and i have the following problem: I first test the data distribution of each scale in order to determine what statistical analysis tests should i employ... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8834019899368286, "perplexity": 1421.0796102451386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056752.16/warc/CC-MAIN-20210919065755-20210919095755-00347.warc.gz"} |
http://en.wikipedia.org/wiki/Tree_(set_theory) | # Tree (set theory)
For the concept in descriptive set theory, see Tree (descriptive set theory).
In set theory, a tree is a partially ordered set (T, <) such that for each tT, the set {sT : s < t} is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.
## Definition
A tree is a partially ordered set (poset) (T, <) such that for each tT, the set {sT : s < t} is well-ordered by the relation <. In particular, each well-ordered set (T, <) is a tree. For each tT, the order type of {sT : s < t} is called the height of t (denoted ht(tT)). The height of T itself is the least ordinal greater than the height of each element of T. A root of a tree T is an element of height 0. Frequently trees are assumed to have only one root.
Trees with a single root in which each element has finite height can be naturally viewed as rooted trees in the sense of graph-theory, or of theoretical computer science: there is an edge from x to y if and only if y is a direct successor of x (i.e., x<y, but there is no element between x and y). However, if T is a tree of height > ω, then there is no natural edge relation that will make T a tree in the sense of graph theory. For example, the set $\omega + 1 = \left\{0, 1, 2, \dots, \omega\right\}$ does not have a natural edge relationship, as there is no predecessor to ω.
A branch of a tree is a maximal chain in the tree (that is, any two elements of the branch are comparable, and any element of the tree not in the branch is incomparable with at least one element of the branch). The length of a branch is the ordinal that is order isomorphic to the branch. For each ordinal α, the α-th level of T is the set of all elements of T of height α. A tree is a κ-tree, for an ordinal number κ, if and only if it has height κ and every level has size less than the cardinality of κ. The width of a tree is the supremum of the cardinalities of its levels.
Single-rooted trees of height ≤ ω forms a meet-semilattice, where meet (common ancestor) is given by maximal element of intersection of ancestors, which exists as the set of ancestors is non-empty and finite well-ordered, hence has a maximal element. Without a single root, the intersection of parents can be empty (two elements need not have common ancestors), for example $\left\{a, b\right\}$ where the elements are not comparable; while if there are an infinite number of ancestors there need not be a maximal element – for example, $\left\{0, 1, 2, \dots, \omega_0, \omega_1\right\}$ where $\omega_0, \omega_1$ are not comparable.
## Properties
There are some fairly simply stated yet hard problems in infinite tree theory. Examples of this are the Kurepa conjecture and the Suslin conjecture. Both of these problems are known to be independent of Zermelo–Fraenkel set theory. König's lemma states that every ω-tree has an infinite branch. On the other hand, it is a theorem of ZFC that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as Aronszajn trees. A κ-Suslin tree is a tree of height κ which has no chains or antichains of size κ. In particular, if κ is singular (i.e. not regular) then there exists a κ-Aronszajn tree and a κ-Suslin tree. In fact, for any infinite cardinal κ, every κ-Suslin tree is a κ-Aronszajn tree (the converse does not hold).
The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of height ω1 has an antichain of cardinality ω1 or a branch of length ω1.
## Tree (automata theory)
Graphic illustration of the labeled tree described in the example
Following definition of a tree is slightly different from the above formalism. For example, each node of the tree is a word over set of natural numbers (ℕ), which helps this definition to be used in automata theory.
A tree is a set T ⊆ ℕ such that if t.cT, with t ∈ ℕ* and c ∈ ℕ, then tT and t.c1T for all 0 ≤ c1 < c. The elements of T are known as nodes, and the empty word ε is the (single) root of T. For every tT, the element t.cT is a successor of t in direction c. The number of successors of t is called its degree or arity, and represented as d(t). A node is a leaf if it has no successors. If every node of a tree has finitely many successors, then it is called a finitely, otherwise an infinitely branching tree. A path π is a subset of T such that ε ∈ π and for every tT, either t is a leaf or there exists a unique c ∈ ℕ such that t.c ∈ π. A path may be a finite or infinite set. If all paths of a tree are finite then the tree is called finite, otherwise infinite. A tree is called fully infinite if all its paths are infinite. Given an alphabet Σ, a Σ-labeled tree is a pair (T,V), where T is a tree and V: T → Σ maps each node of T to a symbol in Σ. A labeled tree formally defines a commonly used term tree structure. A set of labeled trees is called a tree language.
A tree is called ranked if there is an order among the successors of each of its nodes. Above definition of tree naturally suggests an order among the successors, which can be used to make the tree ranked. Sometimes, an extra function Ar: Σ → ℕ is defined. This function associates a fixed arity to each symbol of the alphabet. In this case, each tT has to satisfy Ar(V(t)) = d(t).
For example, above definition is used in the definition of an infinite tree automaton.
### Example
Let T = {0,1}* and Σ = {a,b}. We define a labeling function V as follows: the labeling for the root node is V(ε) = a and, for every other node t ∈ {0,1}*, the labellings for its successor nodes are V(t.0) = a and V(t.1) = b. It is clear from the picture that T forms a (fully) infinite binary tree. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9401699304580688, "perplexity": 424.35282990528776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273766.32/warc/CC-MAIN-20140728011753-00007-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://www.scholarpedia.org/article/Talk:Bowen-Margulis_measure | # Talk:Bowen-Margulis measure
The content of the article well describes the subject except for some minor points. My main criticism concerns the references and complete absence of the chronological timeline - which, I think, is indispensable for this kind of publication.
Let me list my comments paragraph by paragraph.
1. Maximizing entropy.
- It is not clear why in the case of topological entropy the reference to KH95 is accompanied with a reference to the original publication, and in the case of metric entropy it is not. I think that it would be reasonable in both cases just refer to the corresponding articles of Scholarpedia.
- There is no special article on "Variational principle". I think that in this case references to a contemoprary exposition in KH95 must be accompanied with references to original publications (and, of course, the names of the people who obtained the corresponding results).
- The last sentence is misleading. Bowen's construction does not shed any light on the inequality between the topological and metric entropies and the fact that the topological entropy is the supremum of metric ones. This part should be properly referenced (maybe, with a reference to the fact that the measure of maximal entropy need not exist in general).
2. Bowen measure.
- What exactly is the claim? It should say something like: for an expansive homeomeorphism the limit exists and is a measure of maximal entropy. What exactly is the scope of the definition of the Bowen measure? Is it still called this way when the maximal entropy is not unique?
3. Margulis measure.
- It might be better to choose the chronological sequence and put Margulis' measure first.
- Where for the first time appears the fact that Bowen's and Margulis' measures coincide? Was Bowen aware of Margulis' construction when writing his own paper?
- The definition of a conditional measure is misleading. However, it is not necessary for the construction of Margulis - as the latter goes in the opposite direction. Instead of decomposing a measure he first obtains his leafwise measures from a topological consideration (so that they are defined everywhere and not a.e. as would be the case with properly defined conditional measures), and then obtaines the global measure by "multiplying" these leafwise measures in stable and unstable directions.
4. The Patterson-Sullivan construction.
- No references are given whatsoever. I suggest that the original papers by Patterson and Sullivan (definition of this measure in the constant curvature case) and Kaimanovich (relation of the PS construction with the BM measure) should be quoted.
5. Hausdorff measures.
- Ones again, no references. The original paper by Hamenstadt should be quoted.
- I am not sure whether it is worth giving here the definition of the Hausdorff measure.
- Counting is just one particular application of the BM measure. I am not sure it is appropriate to discuss just one aspect in such a detailed way.
7. Generalizations of the Bowen construction.
- Again no references at all
- What about "generalizations of the Margulis construction". Why not mention Gibbs measures (measures with the prescribed Radon-Nikodym derivatives) which are a generalization of the Margulis construction in precisely the same as sense in which the author talks about generalizations of the Bowen construction. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9111971855163574, "perplexity": 557.3033341558348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156096.0/warc/CC-MAIN-20180919083126-20180919103126-00344.warc.gz"} |
https://scicomp.stackexchange.com/questions/35871/solving-system-of-nonlinear-vector-functions/35873 | # Solving system of nonlinear vector functions
I am trying to figure out how to implement a solver for a system of nonlinear equations of the form
\begin{align*} u_1 &= y_n + h\left(a_{1,1}f(t_n + c_1 h, u_1) + a_{1,2}f(t_n + c_2 h, u_2)\right) \\ u_2 &= y_n + h\left(a_{2,1}f(t_n + c_1 h, u_1) + a_{2,2}f(t_n + c_2 h, u_2)\right) \end{align*} where $$f: \mathbb{R}^m\to\mathbb{R}^m$$ can be any nonlinear function, and $$u_1, u_2 \in \mathbb{R}^m$$ are the only unknowns.
I know how to use Newton's method $$\vec{x}_{k+1} = \vec{x}_{k} - J^{-1}(F)F(\vec{x}_{k})$$ for a single vector function, but I am confused on how to adapt this for multiple. From the papers I have been reading, authors reference using a modified Newton method for block matrices created with the kronecker product, but when I do that it leaves me with a matrix in $$\mathbb{R}^{2m}$$ that I don't know what to do with. I have also seen authors define a Jacobian matrix that contains other Jacobians, but again, I don't know how to handle that on a computer. How should I go about creating an iterative method for a system like this? I am trying to implement this to use it with the Radau IIA methods.
• For the Newton method you'll need write your equations as $\vec{F}(\vec{x})=0$ where $\vec{x}$ is the vector of unknowns. Then this will be the familiar situation of a single vector function. Sep 5 '20 at 16:35
• @MaximUmansky I'm aware of that, but I am still confused on how to properly set up a system in $\mathbb{R}^{2m}$ Sep 5 '20 at 16:43
If you have a single vector equation $$\vec{F}(\vec{x})=0$$ then you solve it by representing that state vector $$\vec{x}$$ as a set of amplitudes $$[x_0,x_1,...,x_{n-1}]$$ after discretization by your favorite method (FD, FV, FEM, spectral); and we know how to solve it. If you also have a second equation $$\vec{G}(\vec{y})=0$$ then the full state vector is $$[x_0,x_1,...,x_{n-1}, y_0,y_1, ..., y_{n-1}]$$, otherwise the same thing. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9517812132835388, "perplexity": 173.45834450476679}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301217.83/warc/CC-MAIN-20220119003144-20220119033144-00592.warc.gz"} |
https://mathhelpforum.com/tags/nonempty/ | # nonempty
1. ### Equinumerous + A,B,C are nonempty sets
Let A,B,C are nonempty sets. If A~B, then A x C ~ B x C a) state the converse of this statement b) give an example showing that the converse is NOT always true. THANK YOU!
2. ### S is a nonempty set . A= LUB S. Let T={2x+1:x element in S}.Prove that LUB T=2A+1
Can some one please help me get started on this.I would really like to understand what I am doing. Suppose that S is a nonempty set such that A= LUB S. Let $T={2x+1:x \in S}$. Prove that $LUB T=2A+1$
3. ### To show that the image of a continuous function defined on a compact set is nonempty.
Given a continuous function f defined on a compact set A, to show that f(A) is nonempty, the author of my textbook simply says that "every continuous function defined on a compact set reaches a maximum." Can anyone explain it in more detail? Why is that? Or how should I prove its...
4. ### Sequence within a non-empty, bounded above set
Let S be a non-empty subset of R (real numbers) that is bounded above. Show that there exists a sequence (xn, n is a natural number), contained in S (that is, xn is an element of S for all n in the set of natural numbers) and which is convergent with limit equal to sup S. Any help would be...
5. ### At least two circles tangent to y axis with nonempty intersection
Hi. Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets \mathcal {Q} is a set of those circles in the plane such that for any x \in \mathbb{R} there exists a circle O \in \mathcal {Q} which intersects x axis in (x,0). \mathcal {T} is a set...
6. ### Complement of nonempty open set
I've tried to think this, help would be appreciated to get me going a gain. If we have two nonempty open set A,B \subset {R}^{n} so that A \cap B = \emptyset (two separate sets). How could I show that complement of union A and B is also nonempty, \complement (A \cup B) \neq \emptyset. I'm...
7. ### Revised: A sequence of non-empty, compact, nested sets converges to its intersection
Proposition 2.4.7 * S is a metric space with metric \rho . {A_n} is a sequence of descending, non-empty, compact sets. Then for \epsilon > 0,\ lim A_n = A = \bigcap A_n in the Hausdorff sense. In that sense, one must show that (1) A \subseteq N_\epsilon (A_n)...
8. ### A sequence of non-empty, compact, nested sets converges to its intersection.
A text* I am reading offers a proof that a sequence of non-empty, compact, nested sets converges to its intersection in the Hausdorff metric. I do not follow the second half of the proof which shows that, in the limit, a member of the sequence is contained in the intersection in the sense that...
9. ### Set Theory Problem
A is a convex, nonempty set. A is not bounded below and is bounded above. B* = sup A. Prove: Claim 1: (-inf, b*) is a subset of A Prove: Claim 2: A is a subset of (-inf, b*] Case 1: b* is not an element of A Prove: Claim 2.1: A = (-inf, b*) Case 2: b* is an element of A Prove: Claim 2.2: A =...
10. ### Showing that a non-empty subset M is an affine subset of R^4
Hi, I have to do a project on affine subsets and affine mappings, but I have no clue what they are... We are given only one clue and I can't find many notes on google. I would really appreciate it if someone could help me with this first problem (and if you could also give me a link to some good...
11. ### Non-empty, compact, disconnected and limit points
I am at the moment trying to get through some basic set theory and I'm getting very stuck with the proofs. This question is from a textbook I am studying from and as it is a prove question there is no solution in the back :( Any help with explanations would be very useful Let where E0 = [0...
12. ### Let B={-x : x ϵ A}, A a non-empty subset of R. Show B is bounded below
Hi, I have the following problem... I know the second bit of it, but I have no clue how to prove the first bit. I know B is bounded below (it is basically set A but with opposite signs, so the supremum of A will be the lowest number of set B, its infimum), but I don't know how to express it...
13. ### Prove that G acts transitively on normal subgroup orbits on nonempty finite set A
The Question is let G acts transitively on a nonempty finite set A let H be normal subgroup of G,Let Orbits of H on A be O_1,O_2,...,O_r prove that G acts transitively on O_1,O_2,...,O_r My work I want to prove that for any two oribts O_i,O_j there exist g\in G such that O_i = gO_j I...
14. ### Let ^ be a nonempty indexing set, let A* = {A subscript alpha such that alpha is an..
Let ^ be a nonempty indexing set, let A*={A (subscript alpha) such that alpha is an element of ^} be an indexing family of sets, and let B be a set. Use the results of the theorems for indeing sets and indexing families of sets to prove : B- ( Intersection of (alpha in the ^ <- below intersect...
15. ### Nonempty Subset of Q
Find a nonempty subset of Q (rational numbers) that is bounded above but has no least upper bound in Q. Justify your claim. Thanks for the help!
16. ### An open set is disconnected iff it is the union of two non-empty disjoint open sets
How do I proof this is the definition of disconnectedness that I am using that S is disconnected if there exists two sets S_{1}, S_{2} such that they are both non-emtpy, S_{1} \cup S_{2} = S and cl(S_{1})\cap S_{2} = cl(S_{2})\cap S_{1} = \emptyset Attempting to prove the first condition, that...
17. ### Any set of m positive integers contains a nonempty subset whose sum is ...
Any set of m positive integers contains a nonempty subset whose sum is a multiple of m. Proof. Suppose a set T has no nonempty subset with sum divisible by m. Look at the possible sums mod m of nonempty subsets of T. Adding a new element a to T will give at least one new sum mod m, namely the...
18. ### nonempty family sets
Let A be any NONEMPTY family sets. Prove \bigcap_{A \epsilon A} A \subset \bigcup_{A \epsilon A} A .
19. ### Closed, non-empty and bounded sets.
Suppose that for each n \in \mathbb{N} we have a non-empty closed and bounded set A_n \subset \mathbb{C} and A_1 \supseteq A_2 \supseteq ... \supseteq A_n \supseteq A_{n+1} \supseteq ... Prove that \bigcap_{n=1}^{\infty} A_n is non empty. [Hint: use Bolzano-Weierstrass] Solution...
20. ### Proof of non-empty subsets, glb and lub
Let S be a non-empty subset of R and Suppose there is a non-empty set of T subset of S. a) Prove: if S is bounded above, then T is bounded above and lub t< or = lub S. b) Prove: if S is bounded below, then T is bounded below and glbT is > or = glbS. I am having a really hard time with this. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9623278379440308, "perplexity": 448.46806884826407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671245.92/warc/CC-MAIN-20191122065327-20191122093327-00269.warc.gz"} |
http://math.stackexchange.com/users/28481/cardano | # Cardano
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2 How to prove that a continuous function is identically zero over $\mathbb{R}$? 0 limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$
# 36 Reputation
+35 How to prove that a continuous function is identically zero over $\mathbb{R}$?
# 0 Questions
This user has not asked any questions
# 5 Tags
2 analysis × 2 2 measure-theory 2 calculus × 2 0 limits 2 real-analysis × 2
# 3 Accounts
Stack Overflow 57 rep 3 Mathematics 36 rep 2 Cross Validated 8 rep 2 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8029948472976685, "perplexity": 2931.018183730435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678698356/warc/CC-MAIN-20140313024458-00025-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/53334/problems-with-tkz-linknodes | doesn't this example work because?
\LinkNodes doesn't work
\documentclass[a4paper,11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[upright]{fourier}
\usepackage{tikz}
\begin{document}
\begin{NodesList}
\begin{displaymath}
\begin{aligned}
\end{aligned}
\end{displaymath}
\LinkNodes{$3\cdot 2a$}%
\LinkNodes{$3\cdot(-5b)$}%
\LinkNodes{$-7a\cdot(2a)$}%
\LinkNodes{$-7a\cdot(3b)$}%
\LinkNodes{$5\cdot(a^2)$}%
\LinkNodes{$5\cdot(3b)$}%
\end{NodesList}
\end{document}
-
Could you clarify your question- at the moment it isn't very clear... – cmhughes Apr 25 '12 at 19:57
Seems to work here (I think). Did you compile twice? – Torbjørn T. Apr 25 '12 at 20:29
What is your problem exactly ? – Alain Matthes Apr 25 '12 at 21:59
You need to remove amsmath and tikz because tkz-linknodes loads these packages. You need to have a recent xkeyval package and the last thing, tkz-linknodes loads etex.sty. But it will be interesting to know what is going wrong ! – Alain Matthes Apr 26 '12 at 6:39
You need to compile twice Update : I added \AddNode[i] before the last &
\documentclass[a4paper,11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[upright]{fourier}
\usepackage{tikz}
\begin{document}
\begin{NodesList}
\begin{displaymath}
\begin{aligned}
\end{aligned}
\end{displaymath}
\tikzset{LabelStyle/.style = {left=0.1cm,pos=.5,text=red,fill=white}}
\LinkNodes[margin=3cm]{$3\cdot 2a$}%
\LinkNodes[margin=2cm]{$3\cdot(-5b)$}%
\LinkNodes[margin=1cm]{$-7a\cdot(2a)$}%
\LinkNodes[margin=0cm]{$-7a\cdot(3b)$}%
\LinkNodes[margin=-1cm]{$5\cdot(a^2)$}%
\LinkNodes[margin=-2cm]{$5\cdot(3b)$}%
\end{NodesList}
\end{document}
Another possibility :
\begin{NodesList}
\begin{displaymath}
\begin{aligned}
\end{aligned}
\end{displaymath}
\tikzset{LabelStyle/.style = {pos=0,above,text=red}}
\LinkNodes[margin=1.5cm]{$3\cdot (2a)$}%
\LinkNodes[margin=0cm]{$3\cdot(-5b)$}%
\LinkNodes[margin=-1.5cm]{$-7a\cdot(2a)$}%
\LinkNodes[margin=-3cm]{$-7a\cdot(3b)$}%
\LinkNodes[margin=-4.5cm]{$5\cdot(a^2)$}%
\LinkNodes[margin=-6cm]{$5\cdot(3b)$}%
\end{NodesList}
This is a little package to make simple tasks, perhaps I would be more easy for you to use TikZ directly to get exactly what you want.
-
the solution of the problem is this. correct? \tikzset{LabelStyle/.style = {left=0.1cm,pos=.5,text=red,fill=white}} – user13225 Apr 26 '12 at 5:34
@user13225 But what is your problem ? You can't get the same result ? – Alain Matthes Apr 26 '12 at 5:53
@user13225 No the style is only to avoid to mixt texts and lines. The most important is to add correct margin. – Alain Matthes Apr 26 '12 at 6:35
the writings in red, they were not correctly lined up, now all ok – user13225 Apr 26 '12 at 9:26
You want align the writings in red with what ? You can modify pos but here the different lines don't have the same length. – Alain Matthes Apr 26 '12 at 9:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8881543874740601, "perplexity": 1860.063435480626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049275429.29/warc/CC-MAIN-20160524002115-00030-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://everything.explained.today/Regular_cardinal/ | # Regular cardinal explained
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that
\kappa
is a regular cardinal if and only if every unbounded subset
C\subseteq\kappa
has cardinality
\kappa
. Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.
In the presence of the axiom of choice, any cardinal number can be well-ordered, and then the following are equivalent for a cardinal
\kappa
:
\kappa
is a regular cardinal.
1. If
\kappa=\sumiλi
and
λi<\kappa
for all
i
, then
|I|\ge\kappa
.
1. If
S=cupiSi
, and if
|I|<\kappa
and
|Si|<\kappa
for all
i
, then
|S|<\kappa
.
\operatorname{Set}<\kappa
of sets of cardinality less than
\kappa
and all functions between them is closed under colimits of cardinality less than
\kappa
.
\kappa
is a regular ordinal (see below)Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.
The situation is slightly more complicated in contexts where the axiom of choice might fail, as in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above equivalence holds for well-orderable cardinals only.
\alpha
is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a set has order type less than
\alpha
. A regular ordinal is always an initial ordinal, though some initial ordinals are not regular, e.g.,
\omega\omega
(see the example below).
## Examples
The ordinals less than
\omega
are finite. A finite sequence of finite ordinals always has a finite maximum, so
\omega
cannot be the limit of any sequence of type less than
\omega
whose elements are ordinals less than
\omega
, and is therefore a regular ordinal.
\aleph0
(aleph-null) is a regular cardinal because its initial ordinal,
\omega
, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.
\omega+1
is the next ordinal number greater than
\omega
. It is singular, since it is not a limit ordinal.
\omega+\omega
is the next limit ordinal after
\omega
. It can be written as the limit of the sequence
\omega
,
\omega+1
,
\omega+2
,
\omega+3
, and so on. This sequence has order type
\omega
, so
\omega+\omega
is the limit of a sequence of type less than
\omega+\omega
whose elements are ordinals less than
\omega+\omega
; therefore it is singular.
\aleph1
is the next cardinal number greater than
\aleph0
, so the cardinals less than
\aleph1
are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So
\aleph1
cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.
\aleph\omega
is the next cardinal number after the sequence
\aleph0
,
\aleph1
,
\aleph2
,
\aleph3
, and so on. Its initial ordinal
\omega\omega
is the limit of the sequence
\omega
,
\omega1
,
\omega2
,
\omega3
, and so on, which has order type
\omega
, so
\omega\omega
is singular, and so is
\aleph\omega
. Assuming the axiom of choice,
\aleph\omega
is the first infinite cardinal that is singular (the first infinite ordinal that is singular is
\omega+1
, and the first infinite limit ordinal that is singular is
\omega+\omega
). Proving the existence of singular cardinals requires the axiom of replacement, and in fact the inability to prove the existence of
\aleph\omega
in Zermelo set theory is what led Fraenkel to postulate this axiom.[1]
Uncountable (weak) limit cardinals that are also regular are known as (weakly) inaccessible cardinals. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily fixed points of the aleph function, though not all fixed points are regular. For instance, the first fixed point is the limit of the
\omega
-sequence
\aleph0,
\aleph \aleph0
,
\aleph
\aleph \aleph0
,...
and is therefore singular.
## Properties
If the axiom of choice holds, then every successor cardinal is regular. Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal. Some cardinalities cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see Easton's theorem). The continuum hypothesis postulates that the cardinality of the continuum is equal to
\aleph1
, which is regular assuming choice.
Without the axiom of choice, there would be cardinal numbers that were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore, only the aleph numbers can meaningfully be called regular or singular cardinals. Furthermore, a successor aleph need not be regular. For instance, the union of a countable set of countable sets need not be countable. It is consistent with ZF that
\omega1
be the limit of a countable sequence of countable ordinals as well as the set of real numbers be a countable union of countable sets. Furthermore, it is consistent with ZF that every aleph bigger than
\aleph0
is singular (a result proved by Moti Gitik).
If
\kappa
is a limit ordinal,
\kappa
is regular iff the set of
\alpha<\kappa
that are critical points of
\Sigma1
-elementary embeddings
j
with
j(\alpha)=\kappa
is club in
\kappa
.[2] | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9747832417488098, "perplexity": 486.3358511329478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499634.11/warc/CC-MAIN-20230128121809-20230128151809-00379.warc.gz"} |
https://courses.lumenlearning.com/calculus2/chapter/skills-review-for-other-strategies-for-integration/ | Skills Review for Other Strategies for Integration
Learning Outcome
• Use substitution to evaluate indefinite integrals
In the Other Strategies for Integration section, we will essentially use a table of integration formulas to evaluate integrals. The most important step when using an integration table is to build the exact integration table formula using the integral we are asked to evaluate. This can sometimes be tricky. Here we will review u-substitution techniques that can be useful when building the desired integration table formula.
Use Substitution to Evaluate Indefinite Integrals
Substitution is where we substitute part of the integrand with the variable $u$ and part of the integrand with du. It is also referred to as change of variables because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.
Substitution with Indefinite Integrals
Let $u=g(x),,$ where ${g}^{\prime }(x)$ is continuous over an interval, let $f(x)$ be continuous over the corresponding range of $g$, and let $F(x)$ be an antiderivative of $f(x).$ Then,
$\begin{array}{cc} {\displaystyle\int f\left[g(x)\right]{g}^{\prime }(x)dx}\hfill & = {\displaystyle\int f(u)du}\hfill \\ & =F(u)+C\hfill \\ & =F(g(x))+C.\hfill \end{array}$
The following steps should be followed when integrating by substitution:
1. Look carefully at the integrand and select an expression $g(x)$ within the integrand to set equal to $u$. Let’s select $g(x).$ such that ${g}^{\prime }(x)$ is also part of the integrand.
2. Substitute $u=g(x)$ and $du={g}^{\prime }(x)dx.$ into the integral.
3. We should now be able to evaluate the integral with respect to $u$. If the integral can’t be evaluated we need to go back and select a different expression to use as $u$.
4. Evaluate the integral in terms of $u$.
5. Write the result in terms of $x$ and the expression $g(x).$
Example: Evaluating an Indefinite Integral Using Substitution
Use substitution to find the antiderivative of $\displaystyle\int 6x{(3{x}^{2}+4)}^{4}dx.$
Example: Evaluating an Indefinite Integral Using Substitution
Use substitution to find the antiderivative of $\displaystyle\int z\sqrt{{z}^{2}-5}dz.$
Try It
Use substitution to find the antiderivative of $\displaystyle\int {x}^{2}{({x}^{3}+5)}^{9}dx.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9853328466415405, "perplexity": 234.3502058657031}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103271864.14/warc/CC-MAIN-20220626192142-20220626222142-00496.warc.gz"} |
https://routerunlock.com/copy-texts-scanned-documents-images-using-jocr-windows-1087/ | JOCR is another freeware tool that helps you in copying texts from images, PDF files, and scanned documents. Earlier, we had used Gttext to copy the text from images.
JOCR is a freeware software that enables you to capture the image from the screenshot and extract the texts from captured image. It also allows you to Copy Error Codes and Messages from Dialog Boxes. It is very useful if you want to revive the protected files whose text can not be copied. It can copy text from protected Web pages, PDF files, error messages etc as it offers several capture modes.
JOCR is not an independent tool like GetWindowText which copies texts from open Window. It requires Microsoft Office 2003 or higher version. In case JCOR does not work, you have to manually install “Microsoft Office Document Imaging” (MODI) that is a part of Microsoft Office. You can find MODI under “Office Tools” of the setup file. You can download JOCR from here.
Textify is an another free app that lets you copy Error Codes & Messages from dialog boxes in Windows 10/8/7. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8242629170417786, "perplexity": 3312.8126227915345}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719136.58/warc/CC-MAIN-20161020183839-00419-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://slideplayer.com/slide/4358138/ | # Frequency response I As the frequency of the processed signals increases, the effects of parasitic capacitance in (BJT/MOS) transistors start to manifest.
## Presentation on theme: "Frequency response I As the frequency of the processed signals increases, the effects of parasitic capacitance in (BJT/MOS) transistors start to manifest."— Presentation transcript:
Frequency response I As the frequency of the processed signals increases, the effects of parasitic capacitance in (BJT/MOS) transistors start to manifest The gain of the amplifier circuits is frequency dependent, usually decrease with the frequency increase of the input signals Computing by hand the exact frequency response of an amplifier circuits is a difficult and time-consuming task, therefore approximate techniques for obtaining the values of critical frequencies is desirable The exact frequency response can be obtained from computer simulations (e.g SPICE). However, to optimize your circuit for maximum bandwidth and to keep it stable one needs analytical expressions of the circuit parameters or at least know which parameter affects the required specification and how
Frequency response II The transfer function gives us the information about the behavior of a linear-time-invariant (LTI) circuit/system for a sinusoidal excitation with angular frequency This transfer function is nothing but the ratio between the Fourier transforms of the output and input signals and it is also called the frequency response of the LTI circuit. represents the gain magnitude of the frequency response of the circuit, whereas is the phase of frequency response. Often, it is more convenient to express gain magnitude in decibels. To amplify a signal without distortion, the amplifier gain magnitude must be the same for all of the frequency components
Frequency response of amplifers
A bode plot shows the the gain magnitude and phase in decibles versus frequency on logarithmic scale A few prerequisites for bode plot: Laplace transform and network transfer function Poles and zeros of transfer function Break frequencies Some useful rules for drawing high-order bode plots: Decompose transfer function into first order terms. Mark the break frequencies and represent them on the frequency axis the critical values for changes Make bode plot for each of the first order term For each first order term, keep the DC to the break frequency constant equal to the gain at DC After the break frequency, the gain magnitude starts to increase or decrease with a slope of 20db/decade if the term is in the numerator or denominator For phase plot, each first order term induces a 45 degrees of increase or decrease at the break frequency if the term is in the numerator or denominator Consider frequencies like one-tenth and ten times the break frequency and approximate the phase by 0 and 90 degrees if the frequency is with the numerator (or 0 and -90 degrees if in the denominator) Add all the first order terms for magnitude and phase response
Frequency response of RLC circuits
E.g. 1 E.g. 2 E.g. 3
The MOS Transistor Polysilicon Aluminum
The Gate Capacitance x L Polysilicon gate Top view Gate-bulk overlap
d L Polysilicon gate Top view Gate-bulk overlap Source n + Drain W t ox n + Cross section view L Gate oxide
Gate Capacitance Cut-off Resistive Saturation
Diffusion Capacitance
Channel-stop implant N 1 A Side wall Source W N D Bottom x Side wall j Channel L S Substrate N A
Frequency response of common source MOS amplifer
High-frequency MOS Small-signal equivalent circuit MOSFET common Source amplifier Small-signal equivalent circuit for the MOS common source amplifier Frequency response analysis shows that there are three break frequencies, and mainly the lowest one determines the upper half-power frequency, thus the -3db bandwidth
Exact frequency response of amplifiers
Exact frequency analysis of amplifier circuits is possible following the steps: Draw small-signal equivalent circuit (replace each component in the amplifier with its small-signal circuit) Write equations using voltage and current laws Find the voltage gain as a ratio of polynomial of laplace variable s Factor numerator and denominator of the polynomial to determine break frequencies Draw bode plot to approximate the frequency response
Graphs from Prentice Hall
The Miller Effect Consider the situation that an impedance is connected between input and output of an amplifier The same current flows from (out) the top input terminal if an impedance is connected across the input terminals The same current flows to (in) the top output terminal if an impedance is connected across the output terminal This is know as Miller Effect Two important notes to apply Miller Effect: There should be a common terminal for input and output The gain in the Miller Effect is the gain after connecting feedback impedance Graphs from Prentice Hall
Application of Miller Effect
If the voltage gain magnitude is large (say larger than 10) compared to unity, then we can perform an approximate analysis by assuming is equal to find the gain including loading effects of use the gain to find out Thus, using Miller Effect, gain calculation and frequency response characterization would be much simpler
Application of Miller Effect
If the feedback impedance is a capacitor ,then the Miller capacitance reflected across the input terminal is Therefore, connecting a capacitance from the input to output is equivalent to connecting a capacitance Due to Miller effect, a small feedback capacitance appears across the input terminals as a much larger equivalent capacitance with a large gain (e.g ). At high frequencies, this large capacitance has a low impedance that tends to short out the input signal
The model for low-frequency analysis
BJT small-signal models (for BJT amplifiers) The model for low-frequency analysis The model for high-frequency analysis The base-spreading resistance for the base region (very small) The dynamic resistance of the base emitter region The feedback resistance from collector to base (very large) Account for the upward slope of the output characteristic The depletion capacitance of the collector-to-base region The diffusion capacitance of the base-to-emitter junction Note: in the following analysis of the CE, EF and CB amplifier in the next three slides, we will assume for simplicity (though they still appear in the small signal models).
Miller Effect: common emitter amplifier I
Miller Effect: common emitter amplifier II
Assume the current flowing through is very small compared to , then the gain will be considering the input terminal of the amplifier at b’ to ground. Applying the miller effect for the amplifier, the following simplified circuit can be obtained: Thus, the total capacitance from terminal b’ to ground is given as follows (neglect the miller capacitance from output terminal c to ground): The break frequency, thus the -3db frequency is set by the RC lowpass filter (other voltage controlled current source, resistance does not contribute to the break frequency) is a main limiting factor for -3db bandwidth.
Emitter-follower amplifier
Using Miller Effect, we obtain the above equivalent circuit. If neglecting , It shows that the break frequency is
Common base amplifier What about amplifier that do not have capacitance connected directly from output to the input? For approximate analysis, we can neglect the simplified equivalent circuit can be shown in (c). Derive the transfer function for this circuit, it shows two break frequencies (with typical values, is approximately -3db bandwidth)
Fully-differential amplifier
X X To estimate the 3dB bandwidth of this one, note that the circuit if fully symmetric, so only the half circuit needs to be analyzed. The node voltage at X is 0 in small-signal analysis. Therefore we only need to analyze the right-hand side circuit, which is actually a common-emitter amplifier analyzed before. So the 3dB bandwidth of a common-emitter amplifier applies here.
Graphs from Prentice Hall
Cascode amplifier and differential amplifier The cascode amplifier can be viewed as a common-emitter amplifier with a common-base amplifier. Due to low input impedance of Q2, the voltage gain of Q1 is small. So, Miller effect on Q1 is small. The Emitter-coupled differential amplifier can be viewed as a emitter-follower amplifier cascaded with a common-base amplifier (both have wider bandwidth than common-emitter amplifier). Graphs from Prentice Hall
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Similar presentations | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473280310630798, "perplexity": 1612.9483203406792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651481.98/warc/CC-MAIN-20180325005509-20180325025509-00436.warc.gz"} |
http://mathhelpforum.com/differential-equations/90374-fourth-question.html | ## fourth question,
at this question i wish you give me the whole solution with steps i really have trouble with this kind of questions if you don't have enough time give me the way to solve the problem there are many problems like this i am trying to solve. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8110088109970093, "perplexity": 111.83548479813035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866965.84/warc/CC-MAIN-20180624141349-20180624161349-00025.warc.gz"} |
http://math.stackexchange.com/questions/206032/what-is-the-integral-of-1-x?answertab=oldest | # What is the integral of 1/x?
What is the integral of $1/x$? Do you get $\ln(x)$ or $\ln|x|$?
In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$?
Also, what is the derivative of $|f(x)|$? Is it $f'(x)$ or $|f'(x)|$?
-
@Potato Fair enough. – M Turgeon Oct 2 '12 at 15:19
This question is missing the domain of definition, when working in complex domain the restriction for $\ln x$ is not required – Arjang Feb 10 '13 at 2:05
In summary, the answer is not $\log x$, $\log |x|$, or "$\log |x| + C$". The answer is that $F'(x)=1/x$ on $\mathbb{R}$ implies that there are constants $C_1,C_2\in\mathbb{R}$ such that $F(x)=\log(x)+C_1$ for all $x>0$ and $F(x)=\log(-x)+C_2$ for all $x<0$. There is no such thing as "the integral of $1/x$". – wj32 Feb 10 '13 at 2:09
You have $$\int {1\over x}{\rm d}x=\ln|x|+C$$ (Note that the "constant" $C$ might take different values for positive or negative $x$. It is really a locally constant function.)
In the same way, $$\int {f'(x)\over f(x)}{\rm d}x=\ln|f(x)|+C$$ The last derivative is given by $${{\rm d}\over {\rm d}x}|f(x)|={\rm sgn}(f(x))f'(x)=\cases{f'(x) & if f(x)>0 \cr -f'(x) & if f(x)<0}$$
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9841824769973755, "perplexity": 217.7796825524362}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011025965/warc/CC-MAIN-20140305091705-00047-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/consistency-of-the-speed-of-light.227838/ | # Consistency of the speed of light
1. Apr 9, 2008
### aachenmann
Einstein's second postulate states that the speed of light is constant as viewed from any frame of reference. Most of the books on relativity that I have been reading usually ask the reader to accept that fact because proving it is behind the scope of the book. Can anyone help me understand the actual reason behind the second postulate?
thanks...
2. Apr 9, 2008
### Tachyonie
(if I may Ill add one more question here so I dont flood the forums)
Based on that assumption, isnt the speed of light defying the laws of relativity? Everything is relative BUT the speed of light, so shouldnt we be able to measure how fast we move no matter what frame of reference are we in beacause the speed of light is the same for all of us?
3. Apr 10, 2008
### MeJennifer
Actually it is the opposite, if the speed of light would not be equal for all intertial observers we could deduce a form of absolute motion with respect to light.
4. Apr 10, 2008
### meaw
>> Based on that assumption, isnt the speed of light defying the laws of relativity? Everything is relative BUT the speed of light
Wrong statement - "Everything is relative". The real thing is
1. All laws of nature are absolute - they are not relative to the speed of the observer.
2. observable quantities like mass, lenngth, time, etc are relative to the speed of the observer.
if there exists a law, which becomes relative to the speed of the observer - the law is wrong. it needs correction. When relativity was introduced, all the existing laws of physics needed a review in the light of relativity. In that sense relativity is a 'law of laws'. All theories of physics when through a change after relativity was taken into account - except maxwell's EM theory.
In 1905, the following theories existed in phusics
1. Newton's laws of motion F = m.a
2. Newotn's law of gravity.
3. Maxwell's EM theory
QM was not known in 1905, in the present form today. Statistical laws of nature were known - like thermodynamcs, but they didn't require a review.
1. Newtons' laws of motion was modified by the following replacements
m = m0 sqrt ( 1/ C^2 - v^2 ), similarly for time and lenght.
2. Newton's law of gravity was scrutinized in the light of relativity and it gave berth to General realtivity in 1915 or so
3. Maxwell's EM theory remained unaffected.
Motion of light doen't happen due to kinetic inertia, unlike a football. Motion of ligght if due to fluctuations in EM field and the formula for the same is C = sqrt ( 1/ Mu. Epsilon), derivable for EM theory.
Motion of light is a fundamental law of nature. More accurate answers can be given with higher dimensions, but those are not experimentally tested yet and of course, controversial.
5. Apr 10, 2008
### Staff: Mentor
We could say that it's because space-time apparently posseses Lorentz symmetry, but then that would lead to the question, "why does space-time possess Lorentz symmetry?", wouldn't it?
6. Apr 10, 2008
### HallsofIvy
The most fundamental "reason behind" the invariance of ligh speed is "that's what the experimental evidence shows"!
7. Apr 10, 2008
### DrGreg
The reasoning behind this has some subtle complexity that is not always appreciated. The short answer is that the speed of light is constant because we define distance and time in such a way that it is automatically true.
If you are a beginner in relativity, it might help to just accept the short answer and come back to this later when you have learnt more.
To measure the speed of something, you need to measure the distance it travels from A to B, the time when it passes A and the time when it passes B. To measure times we need two clocks that are stationary in the frame of reference, one at A and one at B. The reason we can't use a single clock that moves from A to B is that we already know, from the twin paradox, that motion affects clocks (or to be more precise, relative motion between clocks affects their synchronisation).
The problem arises, how do you synchronise the two clocks at A and B? You can't do it by transporting a clock from A to B, as already mentioned. The convention is, we use light signals sent from A to B and reflected back to A. Assuming the speed of light is the same in both directions, we can set the time at B to be half-way between the transmission and reception times at A.
However, as we have now used the constancy of the speed of light to sync the two clocks, if we now use those two clocks to measure the speed of light, it is inevitable that they will measure the same speed in both directions.
If we then combine this fact with the experimentally-verified fact of the constancy of the so-called "two-way speed of light" (its average speed when reflected A-B-A as above, and timed using a single clock at A) then we must conclude that the "one-way" (A-B) speed of light is constant in all reference frames.
Note that all of this depends on our choice of clock-synchronisation convention. In special relativity it is assumed that the "Einstein synchronisation convention", which I described above, is always used. It is possible to use other methods of synchronisation, and then the speed of light wouldn't be constant measured in those non-standard coordinates.
I have already said you can't sync two clocks by transporting a clock from A to B. But in fact, you can consider what happens as the transported clock moves slower and slower. If you extrapolate the results as the speed of the clock tends to zero (relative to our frame of reference), it can be proved that this "ultra slow clock transport" sync gives exactly the same result (in the limit) as the "Einstein sync" I described above. This lends credence to the proposition that "Einstein sync" is the natural way to sync, and thus that the speed of light should be constant.
8. Apr 11, 2008
### michael879
I dont think this is true.. Your basically suggesting SR is false and is simply a result of the fact that we cant properly synchronize clocks. If we were to synchronize clocks using sound rather than light (which has a constant speed in the rest frame of the earth), we would still find SR to be true...
9. Apr 11, 2008
### DrGreg
No, I'm saying we can sync clocks to make the one-way speed of light constant and so SR is true.
And you can sync clocks other ways, and then under those weird coords, the coord speed of light is no longer constant, but that doesn't mean SR is false, it just means you're using a weird coord system.
By the way, the problem of syncing with sound is that the "two way speed of sound" is not constant as can be proved by experiment. But there are weird ways of syncing where the "2-way speed of light" is constant but the "1-way speed" is not. See the very end of https://www.physicsforums.com/showpost.php?p=1685025&postcount=18"for an example!
Last edited by a moderator: Apr 23, 2017
10. Apr 11, 2008
### JustinLevy
Viewed from any inertial reference frame.
Much of learning relativity involves "retraining" your physical intuition. For this reason it is often best to ask many facts to be accepted outright at first, let the students play with the math and consequences, and then after they are able to form a better intuition it is possible to go back and relearn the foundations with much more precision.
There is an equivalent formulation of relativity (I saw it in Landau and Lif****z mechanics book if you happen to have that) which is basically just taking classical mechanics and adding the condition that there is a finite propagation speed of interactions.
In some sense, it is merely a coincidence that the speed of light is equal to this propagation speed limit (due to photons having zero invariant mass). It is not necessary for relativity to refer directly to light at all.
Some people find it much more "intuitive" that interactions between spatially separated objects are not 'instantaneous'. So some find the rephrasing of the postulate as something like "there is a finite propagation speed of interaction", to be more intuitive.
Does that help any?
I'm not entirely sure what you are looking for.
EDIT: Hahaha.. are you serious? Lifsh_itz is a censored word?
Last edited: Apr 11, 2008
11. Apr 11, 2008
### phyti
Einstein based his 2nd postulate on the results of the Michelson-Morley experiment. He did not attempt to explain it, but used it as a foundational element.
It's geometrical, and is explained in 'velocity measurement' http://info.awardspace.info" [Broken].
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http://mathhelpforum.com/algebra/49090-simple-problem.html | 1. ## Simple Problem
Q. Clock brass consists of 64.25% copper, 34% zinc and 1.75% lead. A manufacturer has an unlimited supply of copper and lead but has only 50kg of zinc. How much closs brass can he make?
Simple problem but I just can't get the answer.
Thanks.
2. I think you are missing a piece of information. What is the mass of each clock brass.
3. You will want to set up a proportion.
So $\frac {34}{100}$ = $\frac{50}{x}$
Then you cross multiply to get $\frac{54*100}{34}$= x
So x $\approx$158.82353 and there is your answer.
4. Originally Posted by JoshHJ
You will want to set up a proportion.
So $\frac {34}{100}$ = $\frac{50}{x}$
Then you cross multiply to get $\frac{54*100}{34}$= x
So x $\approx$158.82353 and there is your answer.
Edit:
you cross multiplied wrong
It should be
$\frac{50*100}{34}$= x
x=147
5. OK I get it now. Thanks.
6. Originally Posted by Linnus
Edit:
$\frac{50*100}{34}$= x | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.866884171962738, "perplexity": 1868.9728969919033}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280425.43/warc/CC-MAIN-20170116095120-00252-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://eng.libretexts.org/Bookshelves/Chemical_Engineering/Phase_Relations_in_Reservoir_Engineering_(Adewumi)/18%3A_Properties_of_Natural_Gas_and_Condensates_I/18.02%3A_Density | # 18.2: Density
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Density is customarily defined as the amount of mass contained in a unit volume of fluid. Density is the single-most important property of a fluid, once we realize that most other properties can be obtained or related to density. Both specific volume and density — which are inversely proportionally related to each other — tell us the story of how far apart the molecules in a fluid are from each other. For liquids, density is high — which translates to a very high molecular concentration and short intermolecular distances. For gases, density is low — which translates to low molecular concentrations large intermolecular distances.
The question then is: Given this, how can we obtain this all-important property called density? This takes us back to Equations of State (EOS). Since very early times, there have been correlations for the estimation of density of the liquids (oil, condensates) and gases/vapors (dry gases, wet gases). In modern times, equations of state (EOS) are a natural way of obtaining densities. The density of the fluid ‘f’ is calculated using its compressibility factor (Zf) as predicted by an appropriate equation of state. From the real gas law, the density can be expressed as:
$\rho_{f}=\frac{P}{R T}\left(\frac{M W_{f}}{Z_{f}}\right) \label{18.2}$
where: MWf is the molecular weight of fluid ‘f’. Expression \ref{18.2} is used for both the gas and liquid density. In either case, the proper value for MWf (either MWg or MWl) and Zf (either Zg or Zo) has to be used. This takes us back to the discussion of equations of state. From Equation \ref{18.2} it is clear that all that we need is the Z-factor.
The all-important parameter to calculate density is the Z-factor, both for the liquid and vapor phases. The relation between liquid behavior and Z-factor is not obvious, because Z-factor has been traditionally defined for gases. However, we can get “Z” for liquids. “Z” is, indeed, a measure of departure from the ideal gas behavior. Fair enough, for defining “Z” for liquids, we still measure the departure of liquid behavior from ideal gas behavior. A “liquid state” is a tremendous departure from ideal-gas conditions, and as such, “Z” for a liquid is always very far from unity. Typical values of “Z” for liquids are small.
Equations of State have proven very reliable for the estimation of vapor densities, but they do not do as good a job for liquid densities. There is actually a debate among different authors about the reliability of Z-factor estimations for liquids using EOS. In fact, people still believe the EOS are not reliable for liquid density predictions and that we should use correlations instead. However, Peng-Robinson EOS provides fair estimates for vapor and liquid densities as long as we are dealing with natural gas and condensate systems.
Empirical correlations for Z-factor for natural gases were developed before the advent of digital computers. Although their use is in decline, they can still be used for fast estimates of the Z-factor. The most popular of such correlations include those of Hall-Yarborough and Dranchuk-Abou-Kassem.
Chart look-up is another means of determining Z-factor of natural gas mixtures. These methods are invariably based on some type of corresponding states development. According to the theory of corresponding states, substances at corresponding states will exhibit the same behavior (and hence the same Z-factor). The chart of Standing and Katz is the most commonly used Z-factor chart for natural gas mixtures.
Methods of direct calculation using corresponding states have also been developed, ranging from correlations of chart values to sophisticated equation sets based on theoretical developments.
However, the use of equations of state to determine Z-factors has grown in popularity as computing capabilities have improved. Equations of state represent the most complex method of calculating Z-factor, but also the most accurate. A variety of equations of state have been developed to describe gas mixtures, ranging from the ideal EOS (which yields only one root for the vapor and poor predictions at high pressures and low temperatures), cubic EOS (which yields up to three roots, including one for the liquid phase), and more advanced EOS such as BWR and AGA8.
## References
Hall K., and Yarborough, L. (1973), “A New Equation of State for Z-factor Calculations”, Oil and Gas Journal, June 1973, pp. 82-92.
Dranchuk, P. and Abou-Kassem, J. (1975), “Calculation of Z-factors for Natural Gases Using Equations-of-State”, JCPT, July-September 1975, p. 34-36.
Standing, M. and Katz, D. (1942), “Density of Natural Gases”, Trans. AIME, v. 146, pp. 140-149. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8730373382568359, "perplexity": 1054.5282858586045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571869.23/warc/CC-MAIN-20220813021048-20220813051048-00120.warc.gz"} |
http://mathoverflow.net/questions/134773/homocyclic-primary-module-over-pid | # Homocyclic primary module over PID
I posed the question here, but get no answers yet.
Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following property?
Given any submodule $N$ of $M$ isomorphic to $R/\langle p^{s_1}\rangle\oplus\cdots\oplus R/\langle p^{s_r}\rangle$, $M/N$ is isomorphic to $R/\langle p^{s-s_1}\rangle\oplus\cdots\oplus R/\langle p^{s-s_r}\rangle$.
-
This is better suited for MSE. You could start with $r=1$, with $M=R/p^sR$, $N=R/p^{s_1}R$. – Dietrich Burde Jun 26 '13 at 8:23
Choose generators $n_1,\dotsc,n_r$ for $N$ with $p^{s_i}n_i=0$. It is not hard to see that the annihilator of $p^{s_i}$ on $M$ is $p^{s-s_i}M$, so we can choose $m_i$ with $p^{s-s_i}m_i=n_i$. If we can prove that the elements $m_i$ form a basis for $M$ over the ring $R/p^s$, then everything else is clear.
The given assumptions on $N$ imply that the elements $p^{s-1}m_i=p^{s_i-1}n_i$ are linearly independent over $R/p$ in the space $M[p]=\{m\in M:pm=0\}$, and by counting dimensions they must form a basis. Multiplication by $p^{s-1}$ gives an isomorphism $M/pM\to M[p]$, so the elements $m_i$ form a basis for $M/pM$ over $R/p$.
We can certainly write $m_i=\sum_ja_{ij}e_j$ for some matrix $A=(a_{ij})$ over $R$, where $e_1,\dotsc,e_r$ is the standard basis for $M$. The above shows that $\det(A)$ is invertible mod $p$. It follows easily that it is invertible mod $p^s$ as well, which proves the claim.
My argument refers to $p^{s_i-1}$ and so does not immediately work if $s_i=0$ for some $i$, but that can be cured with a few more steps.
Following is the way I can think of to work for the case when some $s_i=0$. Get $m_i$'s in your argument for nonzero $s_i$'s, and prove that they are $R$-linearly independent by showing a correspinding minor of $A$ is invertible mod $p^s$. Then extend these $m_i$'s to an $R$-basis of $M$ as $M$ is homocyclic. Is there any simpler way? – Binzhou Xia Jul 13 '13 at 13:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9694415926933289, "perplexity": 63.863800037012304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096738.25/warc/CC-MAIN-20150627031816-00104-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/247699/checking-random-effects-in-mixed-models-based-on-soundness | # Checking random effects in mixed models based on soundness
I am wondering if it should be possible to check if random effects should be included in a mixed model based on soundness (i.e. using ecological reasoning). The general idea is, that it should hopefully be possible to look at between subject variances and compare them to known values to see if a model without random effects would even make sense.
I don't have a dataset as an example, but I still would like to know if my hypothetical reasoning is correct. Let's just assume I have a bunch of reaction times (in reality not normal distributed, but we will just assume normality here) from various participants, each sampled 100 times. Now also assume, that I run a mixed model with a random intercept and I get a standard deviation of $\sigma_{between}=1000ms$ for the Intercept. As far as I understand, this means, that the intercept varies 1000ms between participants.
Lets call the $i$-th measurement from the $j$-th participant $X_{i,j}$ and the mean for participant $j$ be $\bar{X_j}$.
Now the idea I had was to check if an alternative H0 could be tested, that all participants have the same mean and the variation between participants I am observing is purely by chance. However, depending on the variance assumed under H0, I could observe any between subject variance, so there is nothing to test statistically here. What I could calculate, however, is the sd given H0. If all subsets are purely random around the same mean with an sd of $\sigma_0$ under H0, that would imply that the mean of random disjoint subsets of 100 Measurements would be distributed with the same mean and a variance of $\sigma_0^2/100$. Since we know the means of the sets to be distributed with $\sigma_{between}=1000ms$ we can solve $\sigma_0^2/100=(1000ms)^2$ and therefore $\sigma_0=1000ms*10=10s$
Now based on that, I may just conclude that a standard deviation of 10s between measurements is simply much too high to be true and therefore a random intercept must be included.
However, I am not sure that this reasoning is sound at all. Also, I can easily do the calculation of the variance for an intercept, because it relates to the mean. If this method makes sense at all, it would be interesting if a similar method could also be applied for random slopes.
NOTE:
If this reasoning makes sense, it would not be enough to simply estimate the sd or variance across the sample, because if H0 does not hold, the measurements are correlated, giving reduced estimates of the real (within subject) variance.
Code to test this idea
library(dplyr)
# Reproducibility
set.seed(327401)
N <- 100 # number of measurements
M <- 100 # number of participants / subsets
mu <- 0
sigma <- 10
# Generate some random data
df <- expand.grid(id=1:M, measurement=1:N)
df$value <- rnorm(nrow(df), mu, sigma) df %>% group_by(id) %>% summarize(MeanOfSet=mean(value)) %>% summarize(MeanOfMeans=mean(MeanOfSet), SdOfMeans=sd(MeanOfSet), VarOfMeans=var(MeanOfSet)) # Gives: # MeanOfMeans SdOfMeans VarOfMeans # -0.05118621 0.9692551 0.9394555 # i.e. the Sd and Var is reduced as expected ## Same test, but now with between subject variances around the ## same mean sigma.between <- 5 df <- data.frame(id=1:M, intercept=rnorm(M,mu,sigma.between)) df <- merge(df, expand.grid(id=1:M, measurement=1:N), by="id") df$value <- rnorm(nrow(df), 0, sigma) + df\$intercept
df %>% group_by(id) %>% summarize(MeanOfSet=mean(value)) %>%
summarize(MeanOfMeans=mean(MeanOfSet),
SdOfMeans=sd(MeanOfSet),
VarOfMeans=var(MeanOfSet))
# Gives:
# MeanOfMeans SdOfMeans VarOfMeans
# -0.1991563 5.317212 28.27275
# i.e. the expected between subject variances
df %>% summarize(Sd=sd(value), Var=var(value))
# Gives:
# Sd Var
# 11.29574 127.5936
# However, to see the given between subject variances by
# chance, sd should rather be 50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8956674933433533, "perplexity": 638.8105358552403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526153.35/warc/CC-MAIN-20190719074137-20190719100137-00345.warc.gz"} |
https://www.lessonplanet.com/teachers/how-we-see-things-science-4th-5th | # How We See Things
##### This resource also includes:
Students investigate how mirrors reflect light. In this reflection lesson, students draw the path of the light reflected from a mirror. Students construct a list of objects that are light sources.
Concepts
Resource Details | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8832710981369019, "perplexity": 1753.9492733370278}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822488.34/warc/CC-MAIN-20171017200905-20171017220905-00474.warc.gz"} |
https://scma.maragheh.ac.ir/article_23831.html | Document Type : Research Paper
Authors
1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2 Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.
Abstract
In this paper, we give some results on the common fixed point of self-mappings defined on complete $b$-metric spaces. Our results generalize Kannan and Chatterjea fixed point theorems on complete $b$-metric spaces. In particular, we show that two self-mappings satisfying a contraction type inequality have a unique common fixed point. We also give some examples to illustrate the given results.
Keywords
Main Subjects
###### ##### References
[1] H. Alsulami, E. Karapınar, and H. Piri, Fixed points of generalized $F$-Suzuki type contraction in complete $b$-metric spaces, Discrete Dyn. Nat. Soc., (2015), Art. ID 969726, 8 pp.
[2] H. Aydi, M.F. Bota, E. Karapinar, and S. Moradi, A common fixed point for weak $phi$-contractions on $b$-metric spaces, Fixed Point Theory, 13 (2012), no. 2, 337-346.
[3] I.A. Bakhtin, The contraction mapping principle in almost metric space, Functional analysis, (Russian), Ulýanovsk. Gos. Ped. Inst., Ulýanovsk, (1989) 26-37.
[4] M.F. Bota and E. Karapinar, A note on "Some results on multi-valued weakly Jungck mappings in b-metric space", Cent. Eur. J. Math., 11 (2013), No. 9, 1711-1712.
[5] M.F. Bota, E. Karapinar, and O. Mlesnite, Ulam-Hyers stability results for fixed point problems via $alpha - psi$-contractive mapping in $b$-metric space, Abstr. Appl. Anal. (2013), Art. ID 825293, 6 pp.
[6] S.K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972) 727-730.
[7] S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993) 5-11.
[8] S. Czerwik, Nonlinear Set-valued contraction mappings in $b$-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998) 263-276.
[9] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968) 71-76.
[10] M.A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73 (2010), no. 9, 3123-3129.
[11] M.S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), no. 1, 1-9.
[12] W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer, Cham, 2014.
[13] M.A. Kutbi, E. Karapinar, J. Ahmad, and A. Azam, Some fixed point results for multi-valued mappings in $b$-metric spaces, J. Inequal. Appl. 2014, (2014:126), 11 pp.
[14] C.S. Wong, Common fixed points of two mappings, Pacific J. Math., 48 (1973) 299-312. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8498963117599487, "perplexity": 4846.3939874321695}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303512.46/warc/CC-MAIN-20220121162107-20220121192107-00420.warc.gz"} |
https://proofwiki.org/wiki/Definition:Continuous_Real_Function | # Definition:Continuous Real Function
This page is about continuous mappings in the context of real analysis. For other uses, see Definition:Continuous Mapping.
## Definition
### Continuity at a Point
Let $A \subseteq \R$ be any subset of the real numbers.
Let $f: A \to \R$ be a real function.
Let $x \in A$ be a point of $A$.
Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \to x} f \left({y}\right)$ exists and:
$\displaystyle \lim_{y \to x} \ f \left({y}\right) = f \left({x}\right)$
### Continuous Everywhere
Let $f : \R \to \R$ be a real function.
Then $f$ is everywhere continuous if and only if $f$ is continuous at every point in $\R$.
### Continuity on a Subset of Domain
Let $A \subseteq \R$ be any subset of the real numbers.
Let $f: A \to \R$ be a real function.
Then $f$ is continuous on $A$ if and only if $f$ is continuous at every point of $A$.
## Continuity from One Side
### Continuity from the Left at a Point
Let $x_0 \in A$.
Then $f$ is said to be left-continuous at $x_0$ if and only if the limit from the left of $f \left({x}\right)$ as $x \to x_0$ exists and:
$\displaystyle \lim_{\substack{x \mathop \to x_0^- \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$
where $\displaystyle \lim_{x \mathop \to x_0^-}$ is a limit from the left.
### Continuity from the Right at a Point
Let $x_0 \in S$.
Then $f$ is said to be right-continuous at $x_0$ if and only if the limit from the right of $f \left({x}\right)$ as $x \to x_0$ exists and:
$\displaystyle \lim_{\substack{x \mathop \to x_0^+ \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$
where $\displaystyle \lim_{x \mathop \to x_0^+}$ is a limit from the right.
## Continuity on an Interval
Where $A$ is a real interval, it is considered as a specific example of continuity on a subset of the domain.
### Open Interval
This is a straightforward application of continuity on a set.
Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.
Then $f$ is continuous on $\left({a \,.\,.\, b}\right)$ if and only if it is continuous at every point of $\left({a \,.\,.\, b}\right)$.
### Closed Interval
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.
#### Definition 1
The function $f$ is continuous on $\left[{a \,.\,.\, b}\right]$ if and only if it is:
$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
$(2): \quad$ continuous on the right at $a$
$(3): \quad$ continuous on the left at $b$.
That is, if $f$ is to be continuous over the whole of a closed interval, it needs to be continuous at the end points.
Because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end points on which the function is defined.
#### Definition 2
The function $f$ is continuous on $\left[{a \,.\,.\, b}\right]$ if and only if it is continuous at every point of $\left[{a \,.\,.\, b}\right]$.
### Half Open Intervals
Similar definitions apply to half open intervals:
Let $f$ be a real function defined on a half open interval $\left({a \,.\,.\, b}\right]$.
Then $f$ is continuous on $\left({a \,.\,.\, b}\right]$ if and only if it is:
$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
$(2): \quad$ continuous on the left at $b$.
Let $f$ be a real function defined on a half open interval $\left[{a \,.\,.\, b}\right)$.
Then $f$ is continuous on $\left[{a \,.\,.\, b}\right)$ if and only if it is:
$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
$(2): \quad$ continuous on the right at $a$.
## Real-Valued Vector Function
Let $\R^n$ be the cartesian $n$-space.
Let $f: \R^n \to \R$ be a real-valued function on $\R^n$.
Then $f$ is continuous on $\R^n$ iff:
$\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: d \left({x, a}\right) < \delta \implies \left|{f \left({x}\right) - f \left({a}\right)}\right| < \epsilon$
where $d \left({x, a}\right)$ is the distance function on $\R^n$:
$\displaystyle d: \R^n \to \R: d \left({x, y}\right) := \sqrt {\left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)}\right)}$
where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right)$ are general elements of $\R^n$.
## Informal Definition
The concept of continuity makes precise the intuitive notion that a function has no "jumps" or "holes" at a given point.
Loosely speaking, a real function is continuous at a point if the graph of the function does not have a "break" at the point.
## Historical Note
The concept of a continuous real function was pioneered by the work of Carl Friedrich Gauss, Niels Henrik Abel and Augustin Louis Cauchy. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9828789830207825, "perplexity": 128.3014968284936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864648.30/warc/CC-MAIN-20180522073245-20180522093245-00614.warc.gz"} |
http://mathhelpforum.com/calculus/201260-differentiation-quotient-rule.html | Math Help - Differentiation - Quotient Rule.
1. Differentiation - Quotient Rule.
I'm working on an assignment and the probelm asks to differentiate h(x) = (x^1/3)/(x^3+1)
I keep getting (x^3 + 1 - 3x^(7/3)) / (3x^(2/3)(x^3 + 1)^2
However, the answer in the back of the book is: (1 - 8x^3) / (3x^(2/3)(x^3 + 1)^2
I've reworked this problem several times and i cannot figure out what I'm doing wrong. I would appreciate any help. Thanks!
2. Re: Differentiation - Quotient Rule.
Originally Posted by brandito239
I'm working on an assignment and the probelm asks to differentiate h(x) = (x^1/3)/(x^3+1)
${\left( {\frac{{{x^{\frac{1}{3}}}}}{{{x^3} + 1}}} \right)^\prime } = \frac{{\frac{1}{3}{x^{\frac{{ - 2}}{3}}}({x^3} + 1) - {x^{\frac{1}{3}}}(3{x^2})}}{{{{\left( {{x^3} + 1} \right)}^2}}} = \frac{{({x^3} + 1) - 3x(3{x^2})}}{{3{x^{\frac{2}{3}}}{{\left( {{x^3} + 1} \right)}^2}}}$
3. Re: Differentiation - Quotient Rule.
I still don't understand how you were able to simplify x^(1/3)(3x^2) to 3x(3x^2).
4. Re: Differentiation - Quotient Rule.
Originally Posted by brandito239
I still don't understand how you were able to simplify x^(1/3)(3x^2) to 3x(3x^2).
You need to study very basic algebra.
No one can do calculus, without a complete grounding in basic algebraic operations.
5. Re: Differentiation - Quotient Rule.
I certainly hope you're not representative of every MHF expert/helper on this forum, because if you are, you really make mathematicians look like horrible people. I do agree I need to brush up on all of my math skills. I know I'm far from being a math expert, which is why I came to this website. I was hoping someone who is an expert could offer some real guidance and not just give a condescending and rude response.
6. Re: Differentiation - Quotient Rule.
Originally Posted by brandito239
I still don't understand how you were able to simplify x^(1/3)(3x^2) to 3x(3x^2).
Multiply the top and bottom by \displaystyle \begin{align*} 3x^{\frac{2}{3}} \end{align*}.
7. Re: Differentiation - Quotient Rule.
Originally Posted by Prove It
Multiply the top and bottom by \displaystyle \begin{align*} 3x^{\frac{2}{3}} \end{align*}.
Thanks so much! I appreciate the help! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9531503319740295, "perplexity": 1653.8730864027625}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246658116.80/warc/CC-MAIN-20150417045738-00054-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/191701-finding-sample-size-print.html | # Finding sample size
• November 12th 2011, 03:55 AM
blueberryhill
Finding sample size
i need to find the size of 2 samples based on this information
The population variance of the distribution for X is 100. 2 random samples of size n1 and n2 are repeatedly taken from the population. the sample means for sample size n1 and sample size n2 have the follwing distributions:
X ~ N(1,10)
X ~ N(1,2.5)
(X should be X with a bar over it)
• November 12th 2011, 05:09 AM
CaptainBlack
Re: Finding sample size
Quote:
Originally Posted by blueberryhill
i need to find the size of 2 samples based on this information
The population variance of the distribution for X is 100. 2 random samples of size n1 and n2 are repeatedly taken from the population. the sample means for sample size n1 and sample size n2 have the follwing distributions:
X ~ N(1,10)
X ~ N(1,2.5)
(X should be X with a bar over it)
The sample mean for a sample of size $n$ is approximately $\sim N(\mu,\sigma^2/n)$ where $\mu$ is the population mean and $\sigma^2$ the population variance. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.922094464302063, "perplexity": 907.9872429147496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860125897.68/warc/CC-MAIN-20160428161525-00003-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/rank-of-the-commutator.597214/ | # Rank of the commutator.
1. Apr 16, 2012
### Hurin
I found this theorem on Prasolov's Problems and Theorems in Linear Algebra:
Let V be a $\mathbb{C}$-vector space and $A,B \in \mathcal{L}(V)$such that $rank([A,B])\leq 1$. Then $A$ and $B$ has a common eigenvector.
He gives this proof:
The proof will be carried out by induction on $n=dim(V)$. He states that we can assume that $ker(A)\neq \{0\}$, otherwise we can replace $A$ by$A - \lambda I$; doubt one: why can we assume that? For $n=1$ it's clear that the property holds, because $V = span(v)$ for some $v$. Supposing that holds for some $n$. Now he divides in to cases:
1. $ker(A)\subseteq ker(C)$; and
2. $ker(A)\not\subset ker(C)$.
Doubt two: the cases 1 and 2 come from (or is equivalent to) the division $rank([A,B])= 1$ or $rank([A,B])=0$?
After this division he continues for case one: $B(ker(A))\subseteq ker(A)$, since if $A(x) = 0$, then $[A,B](x) = 0$ and $AB(x) = BA(x) + [A,B](x) = 0$. Now, the doubt three is concerning the following step in witch is considered the restriction $B'$ of $B$ in $ker(A)$ and a selection of an eigenvector $v\in ker(A)$ of $B$ and the statment that $v$ is also a eigenvector of $A$. This proves the case 1.
Now, if $ker(A)\not\subset ker(C)$ then $A(x) = 0$ and $[A,B](x)\neq 0$ for some $x\in V$. Since $rank([A,B]) = 1$ then $Im([A,B]) = span(v)$, for some $v\in V$, where $v=[A,B](x)$, so that $y = AB(x) - BA(x) = AB(x) \in Im(A)$. It follows that $B(Im(A))\subseteq Im(A)$. Now, comes doubt four, that is similar to three: he takes the restrictions $A',B'$ of $A,B$ to $Im(A)$ and the states that $rank[A',B']\leq 1$ and therefor by the inductive hypothesis the operators $A'$ and $B'$ have a common eigenvector. And this proves the case 2, concluding the entire proof.
-Thanks | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9863872528076172, "perplexity": 117.09332645621383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589350.19/warc/CC-MAIN-20180716135037-20180716155037-00417.warc.gz"} |
https://www.storyofmathematics.com/comparing-fractions/ | # Comparing Fractions – According to the Denominators
## How to Compare Fractions?
Comparing fractions is actually the process telling if the one fraction is less than, greater than, or equal to another. Symbols for comparison similarly are used done with a comparison of whole numbers.
For instance, the following sentences can mathematically be represented as follows:
3 is less than 8 would be written as 3 < 8. 14 is greater than 2 would be written as 14 > 2.
17 is equal to 17 would be written as 17 = 17.
It, therefore, is possible to do the same thing with fractions. Let’s begin with fractions common denominators.
The standard method of comparing two fractions is by finding the equivalent fractions that have the same denominator. For example, to compare 1/2 and 1/3, multiply each fraction by the reciprocal of another’s denominator.
1/2 x 1/3= 3/6 and 1/3 x 1/2 = 2/6.
3/6 > 2/6. Therefore, 1/2 > 1/3
## Comparing Fractions with Different Denominators
There are several methods of comparing fractions when the denominators are different. These are:
1. Get the common denominators.
For example, to compare 4/5 and 2/9, these are the steps using the common denominator method:
Steps:
• Multiply numerator and denominator of each fraction by the denominator of another; 4/5 = 4/5 x 9/9 = 36/45 and 2/9 = 2/9 x 5/5 = 10/45.
• Now that the denominator is common, the numerators are compared.
• Since 36 > 10, therefore, 4/5 > 2/9 or 2/9 < 4/5.
2. Use of cross multiplication method
Compare 3/8 and 9/30.
Steps:
• Cross multiply 3/8 and 9/10 and make sure you write the product at the top of the fraction.
• 3/8 cross multiply with 9/10 = 3 x 10 = 30 and 8 x 9 =72.
• Now compare the products as: 30 < 72, and so, 3/8 < 9/10.
3. Simplification method
Compare 20/35 and 8/14.
These fractions can be compared after simplification as show below:
• 20/35 = (20 ÷ 5)/(35 ÷ 5) = 4/7 and 8/14 = (8 ÷ 2)/(14 ÷ 2) = 4/7.
• Both fractions have been simplified to an equivalent value, and therefore, 20/35 = 8/14.
4. Convert the Fractions to Decimals
By dividing the numerator by each fraction’s denominator, fractions can be converted to decimals, and comparisons are made.
Compare 3/4 and 4/5.
In this case, equivalent decimal fractions are:
• 3/4 = 0.75 and 4/5 = 0.8.
• Since 0.75 < 0.80, then 3/4 < 4/5.
Examples:
1. Which one is greater, 4/7 or 3/5?
Solution
Calculate the L.C.M. of the denominators 7 and 5 = 35
Divide both sides of the fractions by the L.C.M.
35 ÷ 7 = 5
35 ÷ 5 = 7
Multiply the denominator and numerator by the answer you get after division.
4 × 5/7 × 5 = 20/35
3 × 7/5 × 7 = 21/35
Since, 21/35 > 20/35
And so, 3/5 > 4/7
The above problem can be solved by cross multiplication method as show below:
4 × 5 = 20
3 × 7 = 21
And because, 21 > 20
Thus, 3/5 > 4/7
1. Compare the following fraction: 32/5 and 2 ¾.
Solution
First the mixed fraction into improper fraction.
2 ¾ = (4 × 2) + ¾ = 11/4
3 2/5 = (5 × 3) + 2/5 = 17/5
Now by cross multiplication of 11/4 and 17/5
11 × 5 = 55
17 × 4 = 68
Since 68 > 55.
Thus, 17/5 > 11/4
Or, 32/5 > 2 ¾
1. Compare the following fractions and put < or > sign between them accordingly:
a. 1/4 and 3/4
Solution
In this case,the denominator of each fraction 4. Therefore, the numerator 1 < 3 and thus,
1/4<3/4.
b. 2/3and 3/4
Solution
The LCM of denominator = 12
Therefore, 2/3 = 2/3 × 4/4 =8/12
And, 3/4 = 3/4×3/3 = 9/12
Since 8 < 9
Therefore, 2/3<3/4.
c. Compare: 3/5 and 5/3
Solution
Find the L.C.M. of 5 and 3 = 15
Therefore, 3/5 = 3/5 × 3= 9/15
5/3 = 25/15
Since, 9 < 25
Thus, 9/15 < 25/15.
### Practice Questions
1. $3/8 =\blacksquare/24$
What must the value of $\blacksquare$ be to make the equation true?
2. $4/9 =16/\blacksquare$
What must the value of $\blacksquare$ be to make the equation true?
3. $8/12 =24/\blacksquare$
What must the value of $\blacksquare$ be to make the equation true?
4. $2/9 =\blacksquare/ 36$
What must the value of $\blacksquare$ be to make the equation true?
5. $5/6 =25/\blacksquare$
What must the value of $\blacksquare$ be to make the equation true?
6. $4/7 = \blacksquare/35$
What must the value of $\blacksquare$ be to make the equation true?
7. $9/9 = \blacksquare/27$
What must the value of $\blacksquare$ be to make the equation true?
8. $1/4 = \blacksquare/36$
What must the value of $\blacksquare$ be to make the equation true?
9. Is the inequality, $\dfrac{6}{12} < \dfrac{3}{2}$, true or false?
10. Is the inequality, $\dfrac{3}{15} < \dfrac{1}{4}$, true or false?
11. Is the inequality, $\dfrac{12}{36} > \dfrac{3}{4}$, true or false?
12. Is the inequality, $\dfrac{8}{4} > \dfrac{9}{3}$, true or false?
13. Which of the following will make the statement, $\dfrac{1}{2}\,\underline{\,\,\,\,\,\,\,\,\,}\,\dfrac{3}{4}$, true?
14. Which of the following will make the statement, $\dfrac{16}{20}\,\underline{\,\,\,\,\,\,\,\,\,}\,\dfrac{3}{5}$, true?
15. Which of the following will make the statement, $\dfrac{2}{20}\,\underline{\,\,\,\,\,\,\,\,\,}\,\dfrac{1}{10}$, true?
16. Which of the following will make the statement, $\dfrac{20}{50}\,\underline{\,\,\,\,\,\,\,\,\,}\,\dfrac{1}{25}$, true?
17. Erick has $\dfrac{2}{5}$ of an orange and $\dfrac{3}{10}$ of an apple. Which type of fruit does he have the most?
18. Mohamed is supposed to read $\dfrac{3}{4}$ of history and $\dfrac{1}{3}$ of science chapters in a day. Which chapter does he read the most?
19. The teacher is dividing a bag of tennis balls to give his students. He gives $\dfrac{2}{9}$ of the balls to Mary, $\dfrac{1}{3}$ to Harish, $\dfrac{7}{27}$ to James, and keeps $\dfrac{5}{27}$ to himself. Who among them has the least number of balls?
20. The teacher is dividing a bag of tennis balls to give his students. He gives $\dfrac{2}{9}$ of the balls to Mary, $\dfrac{1}{3}$ to Harish, $\dfrac{7}{27}$ to James, and keeps $\dfrac{5}{27}$ to himself. Who among them has the largest number of balls?
21. Dustin and Brian have completed $\dfrac{7}{11}$ and $\dfrac{5}{8}$ of their homework, respectively. Who has completed less homework?
22. Last week, Pedro listened to $\dfrac{2}{3}$ of his favorite music while Adam listened to $\dfrac{3}{8}$ of his favorite songs. Who listened to a bigger fraction of his favorite music? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9361627101898193, "perplexity": 1272.6374087551428}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662527626.15/warc/CC-MAIN-20220519105247-20220519135247-00052.warc.gz"} |
https://www.physicsforums.com/threads/statistical-physics-basic-problem.300335/ | Statistical Physics basic problem
1. Mar 17, 2009
FourierX
1. The problem statement, all variables and given/known data
A box is separated by a partition which divides its volume in the ratio 3:1. The larger portion of the box contains 1000 molecules of Neon gas, the smaller one contains 100 molecules of He gas. A small hole is made in the partition, and one waits until equilibrium is obtained.
Find the mean number of molecules of each type on either side of the partition.
2. Relevant equations
Basic statistical and probability concept.
3. The attempt at a solution
At equilibrium, maintaining the volume ratio
the mean number of molecules of Ne in bigger partition = 750
the mean number of molecules of Ne in smaller partition = 250
the mean number of molecules of He in bigger partition = 25
the mean number of molecules of He in smaller partition = 75
That is what i ended up with. Any comments or suggestions will be greatly appreciated.
Gilchrist
2. Mar 17, 2009
lanedance
hi gilchrist
Ne makes sense, but I wonder why 75% of HE is in the small box, I would have thought everything would be distributed evenly at equilibirum...
3. Mar 17, 2009
FourierX
actually, you are right ! I typed the wrong information. My bad.
Thank you :)
Similar Discussions: Statistical Physics basic problem | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.920795202255249, "perplexity": 1051.1676586586518}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891543.65/warc/CC-MAIN-20180122213051-20180122233051-00695.warc.gz"} |
http://math.stackexchange.com/questions/221431/are-the-following-functions-even-odd-or-periodic | # Are the following functions even, odd or periodic?
For each of the following functions, state whether it is even or odd or periodic. If periodic, what is its smallest period?
1. $\sin ax \quad (a>0)$
2. $e^{ax} \quad (a>0)$
3. $x^m \quad (m= \text{ integer})$
4. $\tan x^2$
5. $|\sin (x/b)| \quad (b>0)$
6. $x\cos ax \quad (a>0)$
-
What are your thoughts? What did you try? Can you answer at least some of the questions for some of the functions? – commenter Oct 26 '12 at 9:15
well for the first one, it is periodic since sin is periodic function, Likewise (5) is also. The rest are not. I am really stuck on the periodicity – mary Oct 26 '12 at 9:21
That's correct: do you mean you can't prove your last comment and need hints on that? Or are you stuck on figuring out the smallest periods of (1) and (5)? Seeing if the functions are even or odd should be relatively easy: see if $f(-x) = f(x)$ or $f(-x) = -f(x)$. The first case is even and the second case is odd. – commenter Oct 26 '12 at 9:31
It makes sense. Can you answer the question? I have some ideas, but they will be lacking some nice detail that I want to make sure I have. – mary Oct 26 '12 at 9:33
The definition of an even function is $f(-x) = f(x)$. The definition of an odd function is $f(-x) = -f(x)$. A periodic function means that for a fixed number $P$, $f(x + P) = f(x)$.
Therefore, substituting $-x$ in for $x$, $\sin(-ax) = -\sin(ax)$. This matches the definition of an odd function, so $\sin(ax)$ is odd.
To find if it is periodic, draw a graph of $\sin(ax)$ (I used WolframAlpha) and see if the graph repeats itself at all. In this case it does, every $2\pi$. Therefore the period is $2\pi$.
I have corrected the $\LaTeX$ usage in your post; I will remark that mathematically it is incorrect, since the smallest period of $\sin(2x)$ would be $\pi$ and not $2\pi$; so the answer is actually $\frac{2\pi}a$. – Asaf Karagila Oct 26 '12 at 10:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9169124364852905, "perplexity": 223.79532224104165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207932737.93/warc/CC-MAIN-20150521113212-00210-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://deepai.org/publication/diversity-in-machine-learning | # Diversity in Machine Learning
Machine learning methods have achieved good performance and been widely applied in various real-world applications. It can learn the model adaptively and be better fit for special requirements of different tasks. Many factors can affect the performance of the machine learning process, among which diversity of the machine learning is an important one. Generally, a good machine learning system is composed of plentiful training data, a good model training process, and an accurate inference. The diversity could help each procedure to guarantee a total good machine learning: diversity of the training data ensures the data contain enough discriminative information, diversity of the learned model (diversity in parameters of each model or diversity in models) makes each parameter/model capture unique or complement information and the diversity in inference can provide multiple choices each of which corresponds to a plausible result. However, there is no systematical analysis of the diversification in machine learning system. In this paper, we systematically summarize the methods to make data diversification, model diversification, and inference diversification in machine learning process, respectively. In addition, the typical applications where the diversity technology improved the machine learning performances have been surveyed, including the remote sensing imaging tasks, machine translation, camera relocalization, image segmentation, object detection, topic modeling, and others. Finally, we discuss some challenges of diversity technology in machine learning and point out some directions in future work. Our analysis provides a deeper understanding of the diversity technology in machine learning tasks, and hence can help design and learn more effective models for specific tasks.
## Authors
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## I Introduction
Traditionally, machine learning methods can learn model’s parameters automatically with the training samples and thus it can provide models with good performances which can satisfy the special requirements of various applications. Actually, it has achieved great success in tackling many real-world artificial intelligence and data mining problems
[1], such as object detection [2, 3], natural image processing [5], autonomous car driving [6]
, urban scene understanding
[7], machine translation [4], and web search/information retrieval [8], and others. A success machine learning system often requires plentiful training data which can provide enough information to train the model, a good model learning process which can better model the data, and an accurate inference to discriminate different objects. However, in real-world applications, limited number of labelled training data are available. Besides, there exist large amounts of parameters in the machine learning model. These would make the ”over-fitting” phenomenon in the machine learning process. Therefore, obtaining an accurate inference from the machine learning model tends to be a difficult task. Many factors can help to improve the performance of the machine learning process, among which the diversity in machine learning plays an important role.
Diversity shows different concepts depending on context and application [39]. Generally, a diversified system contains more information and can better fit for various environments. It has already become an important property in many social fields, such as biological system, culture, products and so on. Particularly, the diversity property also has significant effects on the learning process of the machine learning system. Therefore, we wrote this survey mainly for two reasons. First, while the topic of diversity in machine learning methods has received attention for many years, there is no framework of diversity technology on general machine learning models. Although Kulesza et al. discussed the determinantal point processes (DPP) in machine learning which is only one of the measurements for diversity[39]. [11] mainly summarized the diversity-promoting methods for obtaining multiple diversified search results in the inference phase. Besides, [48, 68, 65]
analyzed several methods on classifier ensembles, which represents only a specific form of ensemble learning. All these works do not provide a full survey of the topic, nor do they focus on machine learning with general forms. Our main aim is to provide such a survey, hoping to induce diversity in general machine learning process. As a second motivation, this survey is also useful to researchers working on designing effective learning process.
Here, the diversity in machine learning works mainly on decreasing the redundancy between the data or the model and providing informative data or representative model in the machine learning process. This work will discuss the diversity property from different components of the machine learning process, including the training data, the learned model, and the inference. The diversity in machine learning tries to decrease the redundancy in the training data, the learned model as well as the inference and provide more information for machine learning process. It can improve the performance of the model and has played an important role in machine learning process. In this work, we summarize the diversification of machine learning into three categories: the diversity in training data (data diversification), the diversity of the model/models (model diversification) and the diversity of the inference (inference diversification).
Data diversification can provide samples with enough information to train the machine learning model. The diversity in training data aims to maximize the information contained in the data. Therefore, the model can learn more information from the data via the learning process and the learned model can be better fit for the data. Many prior works have imposed the diversity on the construction of each training batch for the machine learning process to train the model more effectively [9]. In addition, diversity in active learning can also make the labelled training data contain the most information [13, 14] and thus the learned model can achieve good performance with limited training samples. Moreover, in special unsupervised learning method by [38], diversity of the pseudo classes can encourage the classes to repulse from each other and thus the learned model can provide more discriminative features from the objects.
Model diversification comes from the diversity in human visual system. [16, 17, 18]
have shown that the human visual system represents decorrelation and sparseness, namely diversity. This makes different neurons in the human learning respond to different stimuli and generates little redundancy in the learning process which ensures the high effectiveness of the human learning. However, general machine learning methods usually perform the redundancy in the learned model where different factors model the similar features
[19]. Therefore, diversity between the parameters of the model (D-model) could significantly improve the performance of the machine learning systems. The D-model tries to encourage different parameters in each model to be diversified and each parameter can model unique information [20, 21]. As a result, the performance of each model can be significantly improved [22]. However, general machine learning model usually provides a local optimal representation of the data with limited training data. Therefore, ensemble learning, which can learn multiple models simultaneously, becomes another hot machine learning methods to provide multiple choices and has been widely applied in many real-world applications, such as the speech recognition [24, 25], and image segmentation [27]. However, general ensemble learning usually makes the learned multiple base models converge to the same or similar local optima. Thus, diversity among multiple base models by ensemble learning (D-models) , which tries to repulse different base models and encourages each base model to provide choice reflecting multi-modal belief [23, 27, 26], can provide multiple diversified choices and significantly improve the performance.
Instead of learning multiple models with D-models, one can also obtain multiple choices in the inference phase, which is generally called multiple choice learning (MCL). However, the obtained choices from usual machine learning systems presents similarity between each other where the next choice will be one-pixel shifted versions of others [28]. Therefore, to overcome this problem, diversity-promoting prior can be imposed over the obtained multiple choices from the inference. Under the inference diversification, the model can provide choices/representations with more complement information [29, 31, 30, 32]. This could further improve the performance of the machine learning process and provide multiple discriminative choices of the objects.
This work systematically covers the literature on diversity-promoting methods over data diversification, model diversification, and inference diversification in machine learning tasks. In particular, three main questions from the analysis of diversity technology in machine learning have arisen.
• How to measure the diversity of the training data, the learned model/models, and the inference and enhance these diversity in machine learning system, respectively? How do these methods work on the diversification of the machine learning system?
• Is there any difference between the diversification of the model and models? Furthermore, is there any similarity between the diversity in the training data, the learned model/models, and the inference?
• Which real-world applications can the diversity be applied in to improve the performance of the machine learning models? How do the diversification methods work on these applications?
Although all of the three problems are important, none of them has been thoroughly answered. Diversity in machine learning can balance the training data, encourage the learned parameters to be diversified, and diversify the multiple choices from the inference. Through enforcing diversity in the machine learning system, the machine learning model can present a better performance. Following the framework, the three questions above have been answered with both the theoretical analysis and the real-world applications.
The remainder of this paper is organized as Fig. 1 shows. Section II discusses the general forms of the supervised learning and the active learning as well as a special form of unsupervised learning in machine learning model. Besides, as Fig. 1 shows, Sections III, IV and V introduce the diversity methods in machine learning models. Section III outlines some of the prior works on diversification in training data. Section IV reviews the strategies for model diversification, including the D-model, and the D-models. The prior works for inference diversification are summarized in Section V. Finally, section VI introduces some applications of the diversity-promoting methods in prior works, and then we do some discussions, conclude the paper and point out some future directions.
## Ii General Machine Learning Models
Traditionally, machine learning consists of supervised learning, active learning, unsupervised learning, and reinforcement learning. For reinforcement learning, training data is given only as the feedback to the program’s actions in a dynamic environment, and it does not require accurate input/output pairs and the sub-optimal actions need not to be explicitly correct. However, the diversity technologies mainly work on the model itself to improve the model’s performance. Therefore, this work will ignore the reinforcement learning and mainly discuss the machine learning model as Fig.
2 shows. In the following, we’ll introduce the general forms of supervised learning and a representative form of active learning as well as a special form of unsupervised learning.
### Ii-a Supervised Learning
We consider the task of general supervised machine learning models, which are commonly used in real-word machine learning tasks. Fig. 2 shows the flowchart of general machine learning methods in this work. As Fig. 2 shows, the supervised machine learning model consists of data pre-processing, training (modeling), and inference. All of the steps can affect the performance of the machine learning process.
Let denote the set of training samples and is the corresponding label of , where ( is the set of class labels, is the number of the classes, and is the number of the labelled training samples). Traditionally, the machine learning task can be formulated as the following optimization problem [33, 34]:
maxWL(W|X) (1) s.t. g(W)≥0
where
represents the loss function and
is the parameters of the machine learning model. Besides, is the constraint of the parameters of the model. Then, the Lagrange multiplier of the optimization can be reformulated as follows.
L0=L(W|X)+ηg(W) (2)
where is a positive value. Therefore, the machine learning problem can be seen as the minimization of .
Figs. 3 and 4 show the flowchart of two special forms of supervised learning models, which are generally used in real-world applications. Among them, Fig. 3 shows the flowchart of a special form of supervised machine learning with a single model. Generally, in the data-preprocessing stage, the more diversification and balance each training batch has, the more effectiveness the training process is. In addition, it should be noted that the factors in the same layer of the model can be diversified to improve the representational ability of the model (which is called D-model in this paper). Moreover, when we obtain multiple choices from the model in the inference, the obtained choices are desired to provide more complement information. Therefore, some works focus on the diversification of multiple choices (which we call inference diversification). Fig. 4 shows the flowchart of supervised machine learning with multiple parallel base models. We can find that a best strategy to diversify the training set for different base models can improve the performance of the whole ensemble (which is called D-models). Furthermore, we can diversify these base models directly to enforce each base model to provide more complement information for further analysis.
### Ii-B Active Learning
Since labelling is always cost and time consuming, it usually cannot provide enough labelled samples for training in real world applications. Therefore, active learning, which can reduce the label cost and keep the training set in a moderate size, plays an important role in the machine learning model [35]. It can make use of the most informative samples and provide a higher performance with less labelled training samples.
Through active learning, we can choose the most informative samples for labelling to train the model. This paper will take the Convex Transductive Experimental Design (CTED) as a representative of the active learning methods [36, 37].
Denote as the candidate unlabelled samples for active learning, where represents the number of the candidate unlabelled samples. Then, the active learning problem can be formulated as the following optimization problem [37]:
A∗,b∗=argminA,b∥U−UA∥2F+N2∑i=1∑N2j=1a2ijbi+α∥b∥1 (3) s.t. bi≥0,i=1,2,⋯,N2
where is the -th entry of , and is a positive tradeoff parameter. represents the Frobenius norm (F-norm) which calculates the root of the quadratic sum of the items in a matrix. As is shown, CTED utilizes a data reconstruction framework to select the most informative samples for labelling. The matrix contains reconstruction coefficients and
is the sample selection vector. The
-norm makes the learned to be sparse. Then, the obtained is used to select samples for labelling and finally the training set is constructed with the selected training samples. However, the selected samples from CTED usually make similarity from each other, which leads to the redundancy of the training samples. Therefore, diversity property is also required in the active learning process.
### Ii-C Unsupervised Learning
As discussed in former subsection, limited number of the training samples will limit the performance of the machine learning process. Instead of the active learning, to solve the problem, unsupervised learning methods provide another way to train the machine learning model without the labelled training samples. This work will mainly discuss a special unsupervised learning process developed by [38], which is an end-to-end self-supervised method.
Denote as the center points which is used to formulate the pseudo classes in the training process where represents the number of the pseudo classes. Just as subsection II-B, represents the unlabelled training samples and denotes the number of the unsupervised samples. Besides, denote as the features of extracted from the machine learning model. Then, the pseudo label of the data can be defined as
zi=mink∈{1,2,⋯,Λ}∥ck−φ(ui)∥, (4)
Then, the problem can be transformed to a supervised one with the pseudo classes. As shown in subsection II-A, the machine learning task can be formulated as the following optimization [38]
maxW,ciL(W|U,zi)+ηg(W)+N2∑k=1∥czk−φ(uk)∥ (5)
where denotes the optimization term and
is used to minimize the intra-class variance of the constructed pseudo-classes.
demonstrates the constraints in Eq. 2. With the iteratively learning of Eq. 4 and Eq. 5, the machine learning model can be trained unsupervisedly.
Since the center points play an important role in the construction of the pseudo classes, diversifying these center points and repulsing the points from each other can better discriminate these pseudo classes. This would show positive effects on improving the effectiveness of the unsupervised learning process.
### Ii-D Analysis
As former subsections show, diversity can improve the performance of the machine learning process. In the following, this work will summarize the diversification in machine learning from three aspects: data diversification, model diversification, and inference diversification.
To be concluded, diversification can be used in supervised learning, active learning, and unsupervised learning to improve the model’s performance. According to the models in II-A and II-B, the diversification technology in machine learning model has been divided into three parts: data diversification (Section III), model diversification (Section IV), and inference diversification (Section V). Since the diversification in training batch (Fig. 3) and the diversification in active learning and unsupervised learning mainly consider the diversification in training data, we summarize the prior works in these diversification as data diversification in section III. Besides, the diversification of the model in Fig. 3 and the multiple base models in Fig. 4 mainly focus on the diversification in the machine learning model directly, and thus we summarize these works as model diversification in section IV. Finally, the inference diversification in Fig. 3 will be summarized in section V. In the following section, we’ll first introduce the data diversification in machine learning models.
## Iii Data Diversification
Obviously, the training data plays an important role in the training process of the machine learning models. For supervised learning in subsection II-A, the training data provides more plentiful information for the learning of the parameters. While for active learning in subsection II-B, the learning process would select the most informative and less redundant samples for labelling to obtain a better performance. Besides, for unsupervised learning in subsection II-C, the pseudo classes can be encouraged to repulse from each other and the model can provide more discriminative features unsupervisedly. The following will introduce the methods for these data diversification in detail.
### Iii-a Diversification in Supervised Learning
General supervised learning model is usually trained with mini-batches to accurately estimate the training model. Most of the former works generate the mini-batches randomly. However, due to the imbalance of the training samples under random selection, redundancy may occur in the generated mini-batches which shows negative effects on the effectiveness of the machine learning process. Different from classical stochastic gradient descent (SGD) method which relies on the uniformly sampling data points to form a mini-batch,
[9, 10] proposes a non-uniformly sampling scheme based on the determinantal point process (DPP) measurement.
A DPP is a distribution over subsets of a fixed ground set, which prefers a diverse set of data other than a redundant one [39]. Let denote a continuous space and the data . Then, the DPP denotes a positive semi-definite kernel function on ,
ϕ:Θ×Θ→R (6) P(X∈Θ)=det(ϕ(X))det(ϕ+I)
where denotes the kernel matrix and the pairwise is the pairwise correlation between the data and . denotes the determinant of matrix.
is an identity matrix. Since the space
is constant, is a constant value. Therefore, the corresponding diversity prior of transition parameter matrix modeled by DPP can be formulated as
P(X)∝det(ϕ(X)) (7)
In general, the kernel can be divided into the correlation and the prior part. Therefore, the kernel can be reformulated as
ϕ(xi,xj)=R(xi,xj)√π(xi)π(xj) (8)
where is the prior for the data and
denotes the correlation of these data. These kernels would always induce repulsion between different points and thus a diverse set of points tends to have higher probability. Generally, the vectors are supposed to be uniformly distributed variables. Therefore, the prior
is a constant value, and then, the kernel
ϕ(xi,xj)=R(xi,xj). (9)
The DPPs provide a probability measure over every configuration of subsets on data points. Based on a similarity matrix over the data and a determinant operator, the DPP assigns higher probabilities to those subsets with dissimilar items. Therefore, it can give lower probabilities to mini-batches which contain the redundant data, and higher probabilities to mini-batches with more diverse data [9]. This simultaneously balances the data and generates the stochastic gradients with lower variance. Moreover, [10] further regularizes the DPP (R-DPP) with an arbitrary fixed positive semi-definite matrix inside of the determinant to accelerate the training process.
Besides, [12] generalizes the diversification of the mini-batch sampling to arbitrary repulsive point processes, such as the Stationary Poisson Disk Sampling (PDS). The PDS is one type of repulsive point process. It can provide point arrangements similar to DPP but with much more efficiency. The PDS indicates that the smallest distance between each pair of sample points should be at least with respect to some distance measurement [12], such as the Euclidean distance and the heat kernel. The measurement can be formulated as
Euclidean distance:
D(xi,xj)=∥xi−xj∥2 (10)
Heat kernel:
D(xi,xj)=e∥xi−xj∥2σ (11)
where is a positive value. Given a new mini-batch , and the algorithm of PDS can work as follows in each iteration.
• Randomly select a data point .
• If , throw out the point; otherwise add in batch .
The computational complexity of PDS is much lower than that of the DPP.
Under these diversification prior, such as the DPP and the PDS, each mini-batch consists of the training samples with more diversity and information, which can train the model more effectively, and thus the learned model can exact more discriminative features from the objects.
### Iii-B Diversification in Active Learning
As section II-B shows, active learning can obtain a good performance with less labelled training samples. However, some selected samples with CTED are similar to each other and contain the overlapping and redundant information. The highly similar samples make the redundancy of the training samples, and this further decreases the training efficiency, which requires more training samples for a comparable performance.
To select more informative and complement samples with the active learning method, some prior works introduce the diversity in the selected samples obtained from CTED (Eq. 3) [14, 13]. To promote diversity between the selected samples, [14] enhances CTED with a diversity regularizer
minA,b∥U−UA∥2F+N2∑i=1∑N21a2ijbi+α∥b∥1+γbTSb (12) s.t. bi≥0,i=1,2,⋯,N2
where , represents the F-norm, and the similarity matrix is used to model the pairwise similarities among all the samples, such that larger value of demonstrates the higher similarity between the th sample and the th one. Particularly, [14]
chooses the cosine similarity measurement to formulate the diversity term. And the diversity term can be formulated as
sij=ai(aj)T∥ai∥∥aj∥. (13)
As [22] introduces, tends to be zero when and tends to be uncorrelated.
Similarly, [13] denotes the diversity term in active learning with the angular of the cosine similarity to obtain a diverse set of training samples. The diversity term can be formulated as
sij=π2−arccos(ai(aj)T∥ai∥∥aj∥). (14)
Obviously speaking, when the two vectors become vertical, the vectors tend to be uncorrelated. Therefore, under the diversification, the selected samples would be more informative.
Besides,[15] takes advantage of the well-known RBF kernel to measure the diversity of the selected samples, the diversity term can be calculated by
sij=∥ai−aj∥2σ2 (15)
where is a positive value. Different from Eqs. 13 and Eq. 14 which measure the diversity from the angular view, Eq. 15 calculates the diversity from the distance view. Generally, given two data, if they are similar to each other, the term will have a large value.
Through adding diversity regularization over the selected samples by active learning, samples with more information and less redundancy would be chosen for labelling and then used for training. Therefore, the machine learning process can obtain comparable or even better performance with limited training samples than that with plentiful training samples.
### Iii-C Diversification in Unsupervised Learning
As subsection II-C shows, the unsupervised learning in [38] is based on the construction of the pseudo classes with the center points. By repulsing the center points from each other, the pseudo classes would be further enforced to be away from one another. If we encourage the center points to be diversified and repulse from each other, the learned features from different classes can be more discriminative. Generally, the Euclidean distance can be used to calculate the diversification of the center points. The pseudo label of is also calculated by Eq. 4. Then, the unsupervised learning method with the diversity-promoting prior can be formulated as
maxW,ciL(W|U,zi)+ηg(W)+N2∑k=1∥czk−uk∥+γ∑j≠k∥cj−ck∥ (16)
where is a positive value which denotes the tradeoff between the optimization term and the diversity term. Under the diversification term, in the training process, the center points would be encouraged to repulse from each other. This makes the unsupervised learning process be more effective to obtain discriminative features from samples in different classes.
## Iv Model Diversification
In addition to the data diversification to improve the performance with more informative and less redundant samples, we can also diversify the model to improve the representational ability of the model directly. As introduction shows, the machine learning methods aim to learn parameters by the machine itself with the training samples. However, due to the limited and imbalanced training samples, highly similar parameters would be learned by general machine learning process. This would lead to the redundancy of the learned model and negatively affect the model’s representational ability.
Therefore, in addition to the data diversification, one can also diversify the learned parameters in the training process and further improve the representational ability of the model (D-model). Under the diversification prior, each parameter factor can model unique information and the whole factors model a larger proportional of information [22]. Another method is to obtain diversified multiple models (D-models) through machine learning. Traditionally, if we train the multiple models separately, the obtained representations from different models would be similar and this would lead to the redundancy between different representations. Through regularizing the multiple base models with the diversification prior, different models would be enforced to repulse from each other and each base model can provide choices reflecting multi-modal belief [27]. In the following subsections, we’ll introduce the diversity methods for D-model and D-models in detail separately.
### Iv-a D-Model
The first method tries to diversify the parameters of the model in the training process to directly improve the representational ability of the model. Fig. 5 shows the effects of D-model on improving the performance of the machine learning model. As Fig. 5 shows, under the D-model, each factor would model unique information and the whole factors model a larger proportional of information and then the information will be further improved. Traditionally, Bayesian method and posterior regularization method can be used to impose diversity over the parameters of the model. Different diversity-promoting priors have been developed in prior works to measure the diversity between the learned parameter factors according to the special requirements of different tasks. This subsection will mainly introduce the methods which can enforce the diversity of the model and summarize these methods occurred in prior works.
#### Iv-A1 Bayesian Method
Traditionally, diversity-promoting priors can be used to measure the diversification of the model. The parameters of the model can be calculated by the Bayesian method as
W∝P(W|X)=P(X|W)×P(W) (17)
where denotes the parameters in the machine learning model, is the number of the parameters, represents the likelihood of the training set on the constructed model and stands for the prior knowledge of the learned model. For the machine learning task at hand, describes the diversity-promoting prior. Then, the machine learning task can be written as
W∗=argmaxWP(W|X)=argmaxWP(X|W)×P(W) (18)
The log-likelihood of the optimization can be formulated as
W∗=argmaxW(logP(X|W)+logP(W)) (19)
Then, Eq. 19 can be written as the following optimization
maxWlogP(X|W)+logP(W) (20)
where represents the optimization objective of the model, which can be formulated as in subsection II-A. the diversity-promoting prior aims to encourage the learned factors to be diversified. With Eq. 20, the diversity prior can be imposed over the parameters of the learned model.
#### Iv-A2 Posterior Regularization Method
In addition to the former Bayesian method, posterior regularization methods can be also used to impose the diversity property over the learned model [138]. Generally, the regularization method can add side information into the parameter estimation and thus it can encourage the learned factors to possess a specific property. We can also use the posterior regularization to enforce the learned model to be diversified. The diversity regularized optimization problem can be formulated as
maxWL0+γf(W) (21)
where stands for the diversity regularization which measures the diversity of the factors in the learned model. represents the optimization term of the model which can be seen in subsection II-A. demonstrates the tradeoff between the optimization and the diversification term.
From Eqs. 20 and 21, we can find that the posterior regularization has the similar form as the Bayesian method. In general, the optimization (20) can be transformed into the form (21). Many methods can be applied to measure the diversity property of the learned parameters. In the following, we will introduce different diversity priors to realize the D-model in detail.
#### Iv-A3 Diversity Regularization
As Fig. 5
shows, the diversity regularization encourages the factors to repulse from each other or to be uncorrelated. The key problem with the diversity regularization is the way to calculate the diversification of the factors in the model. Prior works mainly impose the diversity property into the machine learning process from six aspects, namely the distance, the angular, the eigenvalue, the divergence, the
, and the DPP. The following will introduce the measurements and further discuss the advantages and disadvantages of these measurements.
Distance-based measurements. The simplest way to formulate the diversity between different factors is the Euclidean distance. Generally, enlarging the distances between different factors can decrease the similarity between these factors. Therefore, the redundancy between the factors can be decreased and the factors can be diversified. [40, 41, 42] have applied the Euclidean distance as the measurements to encourage the latent factors in machine learning to be diversified.
In general, the larger of the Euclidean distance two vectors have, the more difference the vectors are. Therefore, we can diversify different vectors through enlarging the pairwise Euclidean distances between these vectors. Then, the diversity regularization by Euclidean distance from Eq. 21 can be formulated as
f(W)=K∑i≠j∥wi−wj∥2 (22)
where is the number of the factors which we intend to diversify in the machine learning model. Since the Euclidean distance uses the distance between different factors to measure the similarity of these factors , generally the regularizer in Eq. 22 is variant to scale due to the characteristics of the distance. This may decrease the effectiveness of the diversity measurement and cannot fit for some special models with large scale range.
Another commonly used distance-based method to encourage diversity in the machine learning is the heat kernel [2, 3, 143]. The correlation between different factors is formulated through Gaussian function and it can be calculated as
f(wi,wj)=−e−∥wi−wj∥2σ (23)
where is a positive value. The term measures the correlation between different factors and we can find that when and are dissimilar, tends to zero. Then, the diversity-promoting prior by the heat kernel from Eq. 20 can be formulated as
P(W)=e−γ∑Ki≠je−∥wi−wj∥2σ (24)
The corresponding diversity regularization form can be formulated as
f(W)=−K∑i≠je−∥wi−wj∥2σ (25)
where is a positive value. Heat kernel takes advantage of the distance between the factors to encourage the diversity of the model. It can be noted that the heat kernel has the form of Gaussian function and the weight of the diversity penalization is affected by the distance. Thus, the heat kernel presents more variance with the penalization and shows better performance than general Euclidean distance.
All the former distance-based methods encourage the diversity of the model by enforcing the factors away from each other and thus these factors would show more difference. However, it should be noted that the distance-based measurements can be significantly affected by scaling which can limit the performance of the diversity prior over the machine learning.
Angular-based measurements. To make the diversity measurement be invariant to scale, some works take advantage of the angular to encourage the diversity of the model. Among these works, the cosine similarity measurement is the most commonly used [22, 20]. Obviously, the cosine similarity can measure the similarity between different vectors. In machine learning tasks, it can be used to measure the redundancy between different latent parameter factors [22, 20, 44, 19, 43]. The aim of cosine similarity prior is to encourage different latent factors to be uncorrelated, such that each factor in the learned model can model unique features from the samples.
The cosine similarity between different factors and can be calculated as [45, 46]
cij=∥wi∥∥wj∥,i≠j,1≤i,j≠K (26)
Then, the diversity-promoting prior of generalized cosine similarity measurement from Eq. 20 can be written as
P(W)∝e−γ(∑i≠jcpij)1p (27)
It should be noted that when is set to 1, the diversity-promoting prior over different vectors by cosine similarity from Eq. 20 can be formulated as
P(W)∝e−γ∑i≠jcij (28)
where is a positive value. It can be noted that under the diversity-promoting prior in Eq. 28, the is encouraged to be 0. Then, and tend to be orthogonal and different factors are encouraged to be uncorrelated and diversified. Besides, the diversity regularization form by the cosine similarity measurement from Eq. 21 can be formulated as
f(W)=−K∑i≠j∥wi∥∥wj∥ (29)
However, there exist some defects in the former measurement where the measurement is variant to orientation. To overcome this problem, many works use the angular of cosine similarity to measure the diversity between different factors [21, 19, 47].
Since the angular between different factors is invariant to translation, rotation, orientation and scale, [21, 19, 47]
develops the angular-based diversifying method for Restricted Boltzmann Machine (RBM). These works use the variance and mean value of the angular between different factors to formulate the diversity of the model to overcome the problem occurred in cosine similarity. The angular between different factors can be formulated as
Γij=arccos∥wi∥∥wj∥ (30)
Since we do not care about the orientation of the vectors just as [21], we prefer the angular to be acute or right. From the mathematical view, two factors would tend to be uncorrelated when the angular between the factors enlarges. Then, the diversity function can be defined as [48, 21, 50, 49]
f(W)=Ψ(W)−Π(W) (31)
where
Ψ(W)=1K2∑i≠jΓij,
Π(W)=1K2∑i≠j(Γij−Ψ(W))2.
In other words, denotes the mean of the angular between different factors and represents the variance of the angular. Generally, a larger indicates that the weight vectors in are more diverse. Then, the diversity promoting prior by the angular of cosine similarity measurement can be formulated as
P(W)∝eγf(W) (32)
The prior in Eq. 32 encourages the angular between different factors to approach , and thus these factors are enforced to be diversified under the diversification prior. Moreover, the measurement is invariant to scale, translation, rotation, and orientation.
Another form of the angular-based measurements is to calculate the diversity with the inner product [43, 51]. Different vectors present more diversity when they tend to be more orthogonal. The inner product can measure the orthogonality between different vectors and therefore it can be applied in machine learning models for more diversity. The general form of diversity-promoting prior by inner product measurement can be written as [43, 51]
P(W)=e−γ∑Ki≠j. (33)
Besides, [63] uses the special form of the inner product measurement, which is called exclusivity. The exclusivity between two vectors and is defined as
χ(wi,wj)=∥wi⊙wj∥0=m∑k=1wi(k)⋅wj(k) (34)
where denotes the Hadamard product, and denotes the norm. Therefore, the diversity-promoting prior can be written as
P(W)=e−γ∑Ki≠j∥wi⊙wj∥0 (35)
Due to the non-convexity and discontinuity of norm, the relaxed exclusivity is calculated as [63]
χr(wi,wj)=∥wi⊙wj∥1=m∑k=1|wi(k)|⋅|wj(k)| (36)
where denotes the norm. Then, the diversity-promoting prior based on relaxed exclusivity can be calculated as
P(W)=e−γ∑Ki≠j∥wi⊙wj∥1 (37)
The inner product measurement takes advantage of the characteristics among the vectors and tries to encourage different factors to be orthogonal to enforce the learned factors to be diversified. It should be noted that the measurement can be seen as a special form of cosine similarity measurement. Even though the inner product measurement is variant to scale and orientation, in many real-world applications, it is usually considered first to diversify the model since it is easier to implement than other measurements.
Instead of the distance-based and angular-based measurements, the eigenvalues of the kernel matrix can also be used to encourage different factors to be orthogonal and diversified. Recall that, for an orthogonal matrix, all the eigenvalues of the kernel matrix are equal to 1. Here, we denote
as the kernel matrix of . Therefore, when we constrain the eigenvalues to 1, the obtained vectors would tend to be orthogonal [52, 53]. Three ways are generally used to encourage the eigenvalues to approach constant 1, including the submodular spectral diversity (SSD) measurement, the uncorrelation and evenness measurement, and the log-determinant divergence (LDD). In the following, the two form of the eigenvalue-based measurements will be introduced in detail.
Eigenvalue-based measurements. As the former denotes, stands for the kernel matrix of the latent factors. Two commonly used methods to promote diversity in the machine learning process based on the kernel matrix would be introduced. The first method is the submodular spectral diversity (SSD), which is based on the eigenvalues of the kernel matrix. [54]
introduces the SSD measurement in the process of feature selection, which aims to select a diverse set of features. Feature selection is a key component in many machine learning settings. The process involves choosing a small subset of features in order to build a model to approximate the target concept well.
The SSD measurement uses the square distance to encourage the eigenvalues to approach 1 directly. Define as the eigenvalues of the kernel matrix. Then, the diversity-promoting prior by SSD from Eq. 20 can be formulated as [54]
P(W)=e−γ∑Ki=1(λi(κ(W))−1)2 (38)
where is also a positive value. From Eq. 21, the diversity regularization can be formulated as
f(W)=−K∑i=1(λi(κ(W))−1)2 (39)
This measurement regularizes the variance of the eigenvalues of the matrix. Since all the eigenvalues are enforced to approach 1, the obtained factors tend to be more orthogonal and thus the model can present more diversity.
Another diversity measurement based on the kernel matrix is the uncorrelation and evenness [55]. This measurement encourages the learned factors to be uncorrelated and to play equally important roles in modeling data. Formally, this amounts to encouraging the kernel matrix of the vectors to have more uniform eigenvalues. The basic idea is to normalize the eigenvalues into a probability simplex and encourage the discrete distribution parameterized by the normalized eigenvalues to have small Kullback-Leibler (KL) divergence with the uniform distribution [55]. Then, the diversity-promoting prior by uniform eigenvalues from Eq. 20 is formulated as
P(W)=e−γ(tr((1dκ(W))log(1dκ(W)))tr(1dκ(W))−logtr(1dκ(W))) (40)
subject to ( is positive definite matrix) and , where is the kernel matrix. Besides, the diversity-promoting uniform eigenvalue regularizer (UER) from Eq. 21 is formulated as
f(W)=−[tr((1dκ(W))log(1dκ(W)))tr(1dκ(W))−logtr(1dκ(W))] (41)
where is the dimension of each factor.
Besides, [53] takes advantage of the log-determinant divergence (LDD) to measure the similarity between different factors. The diversity-promoting prior in [53] combines the orthogonality-promoting LDD regularizer with the sparsity-promoting regularizer. Then, the diversity-promoting prior from Eq. 20 can be formulated as
P(W)=e−γ(tr(κ(W))−logdet(κ(W))+τ|W|1) (42)
where denotes the matrix trace. Then, the corresponding regularizer from Eq. 21 is formulated as
f(W)=−(tr(κ(W))−logdet(κ(W))+τ|W|1)). (43)
The LDD-based regularizer can effectively promote nonoverlap [53]. Under the regularizer, the factors would be sparse and orthogonal simultaneously.
These eigenvalue-based measurements calculate the diversity of the factors from the kernel matrix view. They not only consider the pairwise correlation between the factors, but also take the multiple correlation into consideration. Therefore, they generally present better performance than the distance-based and angular-based methods which only consider the pairwise correlation. However, the eigenvalue-based measurements would cost more computational sources in the implementation. Moreover, the gradient of the diversity term which is used for back propagation would be complex to compute and usually requires special processing methods, such as projected gradient descent algorithm [55] for the uncorrelation and evenness.
DPP measurement. Instead of the eigenvalue-based measurements, another measurement which takes the multiple correlation into consideration is the determinantal point process (DPP) measurement. As subsection III-A shows, the DPP on the parameter factors has the form as
P(W)∝det(ϕ(W)). (44)
Generally, it can encourage the learned factors to repulse from each other. Therefore, the DPP-based diversifying prior can obtain machine learning models with a diverse set of the learned factors other than a redundant one. Some works have shown that the DPP prior is usually not arbitrarily strong for some special case when applied into machine learning models [60]. To encourage the DPP prior strong enough for all the training data, the DPP prior is augmented by an additional positive parameter . Therefore, just as section III-A, the DPP prior can be reformulated as
P(W)∝det(ϕ(W))γ (45)
where denotes the kernel matrix and demonstrates the pairwise correlation between and . The learned factors are usually normalized, and thus the optimization for machine learning can be written as
maxWlogP(X|W)+γlog(det(ϕ(W))) (46)
where represents the diversity term for machine learning. It should be noted that different kernels can be selected according to the special requirements of different machine learning tasks [61, 62]. For example, in [62], the similarity kernel is adopted for the DPP prior which can be formulated as
ϕ(wi,wj)=∥wi∥∥wj∥. (47)
When we set the cosine similarity as the correlation kernel , from geometric interpretation, the DPP prior can be seen as the volume of the parallelepiped spanned by the columns of [39]. Therefore, diverse sets are more probable because their feature vectors are more orthogonal, and hence span larger volumes. It should be noted that most of the diversity measurements consider the pairwise correlation between the factors and ignore the multiple correlation between three or more factors. While the DPP measurement takes advantage of the merits of the DPP to make use of the multiple correlation by calculating the similarity between multiple factors.
measurement. While all the former measurements promote the diversity of the model from the pairwise or multiple correlation view, many prior works prefer to use the for diversity since can take advantage of the group-wise correlation and obtain a group-wise sparse representation of the latent factors [56, 57, 58, 59].
It is well known that the -norm leads to the group-wise sparse representation of . can also be used to measure the correlation between different parameter factors and diversify the learned factors to improve the representational ability of the model. Then, the prior from Eq. 20 can be calculated as
P(W)=e−γ∑Ki(∑nj|wi(j)|)2 (48)
where means the th entry of . The internal norm encourages different factors to be sparse, while the external norm is used to control the complexity of entire model. Besides, the diversity term based on from Eq. 21 can be formulated as
f(W)=−K∑i(n∑j|wi(j)|)2 (49)
where is the dimension of each factor . The internal norm encourages different factors to be sparse, while the external norm is used to control the complexity of entire model.
In most of the machine learning models, the parameters of the model can be looked as the vectors and diversity of these factors can be calculated from the mathematical view just as these former measurements. When the norm of the vectors are constrained to constant 1, we can also take these factors as the probability distribution. Then, the diversity between the factors can be also measured from the Bayesian view.
Divergence measurement. Traditionally, divergence, which is generally used Bayesian method to measure the difference between different distributions, can be used to promote diversity of the learned model [40].
Each factor is processed as a probability distribution firstly. Then, the divergence between factors and can be calculated as
D(wi∥wj)=n∑k=1(wi(k)logwi(k)wj(k)−wi(k)+wj(k)) (50)
subject to .
The divergence can measure the dissimilarity between the learned factors, such that the diversity-promoting regularization by divergence from Eq. 21 can be formulated as [40]
f(W)= K∑i≠jD(wi∥wj) (51) = K∑i≠jn∑k=1(wi(k)logwi(k)wj(k)−wi(k)+wj(k))
The measurement takes advantage of the characteristics of the divergence to measure the dissimilarity between different distributions. However, the norm of the learned factors need to satisfy which limits the application field of the diversity measurement.
In conclusion, there are numerous approaches to diversify the learned factors in machine learning models. A summary of the most frequently encountered diversity methods is shown in Table I. Although most papers use slightly different specifications for their diversification of the learned model, the fundamental representation of the diversification is similar. It should also be noted that the thing in common among the studied diversity methods is that the diversity enforced in a pairwise form between members strikes a good balance between complexity and effectiveness [63]. In addition, different applications should choose the proper diversity measurements according to the specific requirements of different machine learning tasks.
#### Iv-A4 Analysis
These diversity measurements can calculate the similarity between different vectors and thus encourage the diversity of the machine learning model. However, there exists the difference between these measurements. The details of these diversity measurements can be seen in Table II. It can be noted from the table that all these methods take advantage of the pairwise correlation except which uses the group-wise correlation between different factors. Moreover, the determinantal point process, submodular spectral diversity, and uncorrelation and evenness can also take advantage of correlation among three or more factors.
Another property of these diversity measurement is scale invariant. Scale invariant can make the diversity of the model be invariant w.r.t. the norm of these factors. The cosine similarity measurement calculates the diversity via the angular between different vectors. As a special case for DPP, the cosine similarity can be used as the correlation term in DPP and thus the DPP measurement is scale invariant. Besides, for divergence measurement, since the factors are constrained with , the measurement is scale invariant.
These measurements can encourage diversity within different vectors. Generally, the machine learning models can be looked as the set of latent parameter factors, which can be represented as the vectors. These factors can be learned and used to represent the objects. In the following, we’ll mainly summarize the methods to diversify the ensemble learning (D-models) for better performance of machine learning tasks.
### Iv-B D-Models
The former subsection introduces the way to diversify the parameters in single model and improve the representational ability of the model directly. Much efforts have been done to obtain the highest probability configuration of the machine learning models in prior works. However, even when the training samples are sufficient, the maximum a (MAP) solution could also be sub-optimal. In many situations, one could benefit from additional representations with multiple models. As Fig. 4 shows, ensemble learning (the way for training multiple models) has already occurred in many prior works. However, traditional ensemble learning methods to train multiple models may provide representations that tend to be similar while the representations obtained from different models are desired to provide complement information. Recently, many diversifying methods have been proposed to overcome this problem. As Fig. 6 shows, under the model diversification, each base model of the ensemble can produce different outputs reflecting multi-modal belief. Therefore, the whole performance of the machine learning model can be improved. Especially, the D-models play an important role in structured prediction problems with multiple reasonable interpretations, of which only one is the groundtruth [27].
Denote and as the parameters and the inference from the th model where is the number of the parallel base models. Then, the optimization of the machine learning to obtain multiple models can be written as
maxW1,W2,⋯,Wss∑i=1L(Wi|Xi) (52)
where represents the optimization term of the th model and denotes the training samples of the th model. Traditionally, the training samples are randomly divided into multiple subsets and each subset trains a corresponding model. However, selecting subsets randomly may lead to the redundancy between different representations. Therefore, the first way to obtain multiple diversified models is to diversify these training samples over different base models, which we call sample-based methods.
Another way to encourage the diversification between different models is to measure the similarity between different base models with a special similarity measurement and encourage different base models to be diversified in the training process, which is summarized as the optimization-based methods. The optimization of these methods can be written as
maxW1,W2,⋯,Wss∑i=1L(Wi|X)+γΓ(W1,W2,⋯,Ws) (53)
where measures the diversification between different base models. These methods are similar to the methods for D-model in former subsection.
Finally, some other methods try to obtain large amounts of models and select the top- models as the final ensemble, which is called the ranking-based methods in this work. In the following, we’ll summarize different methods for diversifying multiple models from the three aspects in detail.
#### Iv-B1 Optimization-Based Methods
Optimization-based methods are one of the most commonly used methods to diversify multiple models. These methods try to obtain multiple diversified models by optimizing a given objective function as Eq. 53 shows, which includes a diversity measurement. Just as the diversity of D-model in prior subsection, the main problem of these methods is to define diversity measurements which can calculate the difference between different models.
Many prior works [65, 64, 66, 67, 68, 48] have summarized some pairwise diversity measurements, such as Q-statistics measure [69, 48], correlation coefficient measure [69, 48], disagreement measure [70, 64, 127], double-fault measure [71, 64, 127], statistic measure [72], Kohavi-Wolpert variance [65, 127], inter-rater agreement [65, 127], the generalized diversity [65] and the measure of ”Difficult” [65, 127]. Recently, some more measurements have also been developed, including not only the pairwise diversity measurement [26, 69, 66] but also the measurements which calculate the multiple correlation and others [58, 75, 74, 73]. This subsection will summarize these methods systematically.
Bayesian-based measurements. Similar to D-model, Bayesian methods can also be applied in D-models. Among these Bayesian methods, divergence is the most commonly used one. As former subsection shows, the divergence can measure the difference between different distributions. The way to formulate the diversity-promoting term by the divergence method over the ensemble learning is to calculate the divergence between different distributions from the inference of different models, respectively [26, 69]. The diversity-promoting term by divergence from Eq. 53 can be formulated as
Γ(W1,W2,⋯, Ws)= (54) s∑i,jn∑k=1(P(Wi (k))logP(Wi(k))P(Wj(k))−P(Wi(k))+P(Wj(k)))
where represents the th entry in . denotes the distributions of the inference from the th model. The former diversity term can increase the difference between the inference obtained from different models and would encourage the learned multiple models to be diversified.
In addition to the divergence measurements, Renyi-entropy which measures the kernelized distances between the images of samples and the center of ensemble in the high-dimensional feature space can also be used to encourage the diversity of the learned multiple models [76]. The Renyi-entropy is calculated based on the Gaussian kernel function and the diversity-promoting term from Eq. 53 can be formulated as
Γ(W1,W2,⋯,Ws) = (55) −log[1s2s∑i=1s∑j=1 G(P(Wi)−P(Wj),2σ2)]
where is a positive value and represents the Gaussian kernel function, which can be calculated as
G(Wi−Wj, 2σ2)= (56) 1(2π)d2σd exp{−(P(Wi)−P(Wj))T(P(Wi)−P(Wj))2σ2}
where denotes the dimension of . Compared with the divergence measurement, the Renyi-entropy measurement can be more fit for the machine learning model since the difference can be adapted for different models with different value . However, the Renyi-entropy would cost more computational sources and the update of the ensemble would be more complex.
Another measurement which is based on the Bayesian method is the cross entropy measurement[77, 78, 124]. The cross entropy measurement uses the cross entropy between pairwise distributions to encourage two distributions to be dissimilar and then different base models could provide more complement information. Therefore, the cross-entropy between different base models can be calculated as
Γ(wi,wj)= 1nn∑k=1(Pk(wi)logPk(wj) (57) +(1−Pk(wi))log(1−Pk(wj)))
where is the inference of the th model and is the probability of the sample belonging to the th class. According to the characteristics of the cross entropy and the requirement of the diversity regularization, the diversity-promoting regularization of the cross entropy from Eq. 53 can be formulated as
Γ(w1,w2,⋯,wK)= 1ns∑i,jn∑k=1(Pk(wi)logPk(wj) (58) +(1−Pk(wi))log(1−Pk(wj)))
We all know that the larger the cross entropy is, the more difference the distributions are. Therefore, under the cross entropy measurement, different models can be diversified and provide more complement information. Most of the former Bayesian methods promote the diversity in the learned multiple base models by calculating the pairwise difference between these base models. However, these methods ignore the correlation among three or more base models.
To overcome this problem, [75]
proposes a hierarchical pair competition-based parallel genetic algorithm (HFC-PGA) to increase the diversity among the component neural networks. The HFC-PGA takes advantage of the average of all the distributions from the ensemble to calculate the difference of each base model. The diversity term by HFC-PGA from Eq.
53 can be formulated as
Γ(W1,W2,⋯,Ws)=s∑j=1(1ss∑i=1P(Wi)−P(Wj))2 (59)
It should be noted that the HFC-PGA takes advantage of multiple correlation between the multiple models. However, the HFC-PGA method uses the fix weight to calculate the mean of the distributions and further calculate the covariance of the multiple models which usually cannot fit for different tasks. This would limit the performance of the diversity promoting prior.
To deal with the shortcomings of the HFC-PGA, negative correlation learning (NCL) tries to reduce the covariance among all the models while the variance and bias terms are not increased [82, 73, 74, 83]. The NCL trains the base models simultaneously in a cooperative manner that decorrelates individual errors. The penalty term can be designed in different ways depending on whether the models are trained sequentially or parallelly. [82] uses the penalty to decorrelate the current learning model with all previously learned models
Γ(W1,W2,⋯,Ws)=s∑k=1(P(Wk)−l)k−1∑j=1(P(Wj)−l) (60)
where represents the target function which is a desired output scalar vector. Besides, define where . Then, the penalty term can also be defined to reduce the correlation mutually among all the learned models by using the actual distribution obtained from each model instead of the target function [73, 74, 68].
Γ(W1,W2,⋯,Ws)=s∑k=1(P(Wk)−¯¯¯¯P)k−1∑j=1(P(Wj)−¯¯¯¯P) (61)
This measurement uses the covariance of the inference results obtained from the multiple models to reduce the correlation mutually among the learned models. Therefore, the learned multiple models can be diversified. In addition, [84] further combines the NCL with sparsity. The sparsity is purely pursued by the norm regularization without considering the complementary characteristics of the available base models.
Most of the Bayesian methods promote diversity in ensemble learning mainly by increasing the difference between the probability distributions of the inference of different base models. There exist other methods which can promote diversity over the parameters of each base model directly.
Cosine similarity measurement. Different from the Bayesian methods which promote diversity from the distribution view, [66] introduces the cosine similarity measurements to calculate the difference between different models from geometric view. Generally, the diversity-promoting term from Eq. 53 can be written as
Γ(W1,W2,⋯,Ws)=−s∑i≠j∥Wi∥∥Wj∥. (62)
In addition, as a special form of angular-based measurement, a special form of inner product measurement, termed as exclusivity, has been proposed by [63] to obtain diversified models. It can jointly suppress the training error of ensemble and enhance the diversity between bases. The diversity-promoting term by exclusivity (see Eq. 37 for details) from Eq. 53 can be written as
Γ(W1,W2,⋯,Ws)=−s∑i≠j∥Wi⨀Wj∥1 (63)
These measurements try to encourage the pairwise models to be uncorrelated such that each base model can provide more complement information.
measurement. Just as the former subsection, norm can also be used as the diversification of multiple models[58]. the diversity-promoting regularization by from Eq. 53 can be formulated as
Γ(W1,W2,⋯,Ws)=−s∑i(K∑j|Wi(j)|)2 (64)
The measurement uses the group-wise correlation between different base models and favors selecting diverse models residing in more groups.
Some other diversity measurements have been proposed for deep ensemble. [85] reveals that it may be better to ensemble many instead of all of the neural networks at hand. The paper develops an approach named Genetic Algorithm based Selective Ensemble (GASEN) to obtain different weights of each neural network. Then, based on the obtained weights, the deep ensemble can be formulated. Moreover, [86] also encourages the diversity of the deep ensemble by defining a pair-wise similarity between different terms.
These optimization-based methods utilize the correlation between different models and try to repulse these models from one another. The aim is to enforce these representations which are obtained from different models to be diversified and thus these base models can provide outputs reflecting multi-modal belief.
#### Iv-B2 Sample-Based Methods
In addition to diversify the ensemble learning from the optimization view, we can also diversify the models from the sample view. Generally, we randomly divide the training set into multiple subsets where each base model corresponds to a specific subset which is used as the training samples. However, there exists the overlapping between the representations of different base models. This may cause the redundancy and even decrease the performance of the ensemble learning due to the reduction of the training samples over each model by the division of the whole training set.
To overcome this problem and provide more complement information from different models, [27] develops a novel method by dividing the training samples into multiple subsets by assigning the different training samples into the specified subset where the corresponding learned model shows the lowest predict error. Therefore, each base model would focus on modeling the features from specific classes. Besides, clustering is another popular method to divide the training samples for different models [87]. Although diversifying the obtained multiple subsets can make the multiple models provide more complement information, the less of training samples by dividing the whole training set will show negative effects over the performance.
To overcome this problem, another way to enforce different models to be diversified is to assign each sample with a specified weight [88]. By training different base models with different weights of samples, each base model can focus on complement information from the samples. The detailed steps in [88] are as follows:
• Define the weights over each training sample randomly, and train the model with the given weights;
• Revise the weights over each training sample based on the final loss from the obtained model, and train the second model with the updated weights;
• Train models with the aforementioned strategies.
The former methods take advantage of the labelled training samples to enforce the diversity of multiple models. There exists another method, namely Unlabeled Data to Enhance Ensemble (UDEED) [89], which focuses on the unlabelled samples to promote diversity of the model. Unlike the existing semi-supervised ensemble methods where error-prone pseudo-labels are estimated for unlabelled data to enlarge the labelled data to improve accuracy. UDEED works by maximizing accuracies of base models on labelled data while maximizing diversity among them on unlabelled data. Besides, [90] combines the different initializations, different training sets and different feature subsets to encourage the diversity of the multiple models.
The methods in this subsection process on the training sets to diversify different models. By training different models with different training samples or samples with different weights, these models would provide different information and thus the whole models could provide a larger proportional of information.
#### Iv-B3 Ranking-Based Methods
Another kind of methods to promote diversity in the obtained multiple models is ranking-based methods. All the models is first ranked according to some criterion, and then the top- are selected to form the final ensemble. Here, [91] focuses on pruning techniques based on forward/backward selection, since they allow a direct comparison with the simple estimation of accuracy from different models.
Cluster can be also used as ranking-based method to enforce diversity of the multiple models [92]. In [92], each model is first clustered based on the similarity of their predictions, and then each cluster is then pruned to remove redundant models, and finally the remaining models in each cluster are finally combined as the base models.
In addition to the former mentioned methods, [23] provides multiple diversified models by selecting different sets of multiple features. Through multi-scale or other tricks, each sample will provide large amounts of features, and then choose top- multiple features from the all the features as the base features (see [23] for details). Then, each base feature from the samples is used to train a specific model, and the final inference can be obtained through the combination of these models.
In summary, this paper summarizes the diversification methods for D-models from three aspects: optimization-based methods, sample-based methods, and ranking-based methods. The details of the most frequently encountered diversity methods is shown in Table III. Optimization-based methods encourage the multiple models to be diversified by imposing diversity regularization between different base models while optimizing these models. In contrary, sample-based methods mainly obtain diversified models by training different models with specific training sets. Most of the prior works focus on diversifying the ensemble learning from the two aspects. While the ranking-based methods try to obtain the multiple diversified models by choosing the top- models. The researchers can choose the specific method for D-models based on the special requirements of the machine learning tasks.
## V Inference Diversification
The former section summarizes the methods to diversify different parameters in the model or multiple base models. The D-model focuses on the diversification of parameters in the model and improves the representational ability of the model itself while D-models tries to obtain multiple diversified base models, each of which focus on modeling different features from the samples. These works improve the performance of the machine learning process in the modeling stage (see Fig. 2 for details). In addition to these methods, there exist some other works focusing on obtaining multiple choices in the inference of the machine learning model. This section will summarize these diversifying methods in the inference stage. To introduce the methods for inference diversification in detail, we choose the graph model as the representation of the machine learning models.
We consider a set of discrete random variables
, each taking value in a finite label set . Let (, ) describe a graph defined over these variable. The set denotes a Cartesian product of sets of labels corresponding to the subset of variables. Besides, denote , () as the functions which define the energy at each node and edge for the labelling of variables in scope. The goal of the MAP inference is to find the labelling of the variables that minimizes this real-valued energy function:
y∗=argminyE(y)=argminy∑A∈V∪EθA(y) (65)
However, usually converges to the sub-optimal results due to the limited representational ability of the model and the limited training samples. Therefore, multiple choices, which can provide complement information, are desired from the model for the specialist. Traditional methods to obtain multiple choices try to solve the following optimization:
ym=argminyE(y)=argminy∑A∈V⋃EθA(y) (66) s.t. ym≠yi,i=1,2,⋯,m−1
However, the obtained second-best choice will typically be one-pixel shifted versions of the best [28]. In other words, the next best choices will almost certainly be located on the upper slope of the peak corresponding with the most confident detection, while other peaks may be ignored entirely.
To overcome this problem, many methods, such as diversified multiple choice learning (D-MCL), submodular, M-Modes, M-NMS, have been developed for inference diversification in prior works. These methods try to diversify the obtained choices (do not overlap under a user-defined criteria) while obtaining high score on the optimization term. Fig. 7 shows some results of image segmentation from [31]. Under the inference diversification, we can obtain multiple diversified choices, which represent the different optima of the data. Actually, there also exist many methods which focus on providing multiple diversified choices in the inference phase. In this work, we summarize the diversification in these works as inference diversification. The following subsections will introduce these works in detail.
### V-a Diversity-Promoting Multiple Choice Learning (D-MCL)
The D-MCL tries to find a diverse set of highly probable solutions under a discrete probabilistic model. Given a dissimilarity function measuring similarity between the pairwise choices, our formulation involves maximizing a linear combination of the probability and the dissimilarity to the previous choices. Even if the MAP solution alone is of poor quality, a diverse set of highly probable hypotheses might still enable accurate predictions. The goal of D-MCL is to produce a diverse set of low-energy solutions.
The first method is to approach the problem with a greedy algorithm, where the next choice is defined as the lowest energy state with at least some minimum dissimilarity from the previously chosen choices. To do so, a dissimilarity function is defined first. In order to find the diverse, low energy, labellings , the method proceeds by solving a sequence of problems of the form [29, 31, 95, 96, 144]
ym=argminy(E(y)−γm−1∑i=1Δ(y,yi)) (67)
for , where determines a trade-off between diversity and energy, is the MAP-solution and the function defines the diversity of two labels. In other words, takes a large value if and are diverse, and a small value otherwise. For special case, the M-Best MAP is obtained when is a 0-1 dissimilarity (i.e. ). The method considers the pairwise dissimilarity between the obtained choices. More importantly, it is easy to understand and implement. However, under the greedy strategy, each new labelling is obtained based on the previously found solutions, and ignores the upcoming labellings [94].
Contrary to the former form, the second method formulate the -best diverse problem in form of a single energy minimization problem [94]. Instead of the greedy sequential procedure in (67), this method suggests to infer all labellings jointly, by minimizing
EM(y1,y2,⋯,yM)=M∑i=1E(yi)−γΔM(y1,y2,⋯,yM) (68)
where defines the total diversity of any labellings. To achieve this, let us first create copies of the initial model. Three specific different diversity measures are introduced. The split-diversity measure is written as the sum of pairwise diversities, i.e. those penalizing pairs of labellings [94]
ΔM(y1,y2,⋯,yM)=M∑i=2i−1∑j=1Δ(yi,yj) (69)
The node-diversity measure is defined as [94]
ΔM(y1,y2,⋯,yM)=∑v∈VΔv(y1v,y2v,⋯,yMv) (70)
Finally, the special case of the split-diversity and node-diversity measures is the node-split-diversity measure [94]
ΔM(y1,y2,⋯,yM)=∑v∈VM∑i=2i−1∑j=1Δv(yiv,yjv) (71)
The D-MCL methods try to find multiple choices with a dissimilarity function. This can help the machine learning model to provide choices with more difference and show more diversity. However, the obtained choices may not be the local extrema and there may exist other choices which could better represent the objects than the obtained ones.
### V-B Submodular for Diversification
The problem of searching for a diverse but high-quality subset of items in a ground set of items has been studied in information retrieval [99], web search [98], social networks [103], sensor placement [104], observation selection problem [102], set cover problem [105]
[100, 101], and others. In many of these works, an effective, theoretically-grounded and practical tool for measuring the diversity of a set are submodular set functions. Submodularity is a property that comes from marginal gains. A set function is submodular when its marginal gains are decreasing: for all and . In addition, if is monotone, i.e. whenever , then a simple greedy algorithm that iteratively picks the element with the largest marginal gain to add to the current set , achieves the best possible approximation bound of [107]. This result has presented significant practical impact. Unfortunately, if the number of items is exponentially large, then even a single linear scan for greedy augmentation is simply infeasible. The diversity is measured by a monotone, nondecreasing and normalized submodular function .
Denote as the set of choices. The diversification is measured by a monotone, nondecreasing and normalized submodular function . Then, the problem can be transformed to find a maximizing configurations for the combined score [97, 98, 99, 100, 101]
F(S)=E(S)+γD(S) (72)
The optimization can be solved by the greedy algorithm that starts out with , and iteratively adds the best term:
ym= argmaxyF(y|Sm−1) (73) = argmaxy{E(y)+γD(y|Sm−1)}
where . The selected choice is within a factor of of the optimal solution :
F(Sm)≥(1−1e)F(S∗). (74)
The submodular takes advantage of the maximization of marginal gains to find multiple choices which can provide the maximum of complement information.
### V-C M-Nms
Another way to obtain multiple diversified choices is the non-maximum suppression (M-NMS) [150, 111]. The M-NMS is typically defined in an algorithmic way: starting from the MAP prediction one goes through all labellings according to an increasing order of the energy. A labelling becomes part of the predicted set if and only if it is more than away from the ones chosen before, where is the threshold defined by user to judge whether two labellings are similar. The M-NMS guarantee the choices to be apart from each other. The M-NMS is typically implemented by greedy algorithm [32, 120, 110, 111].
A simple greedy algorithm for instantiating multiple choices are used: Search over the exponentially large space of choices for the maximally scoring choice, instantiate it, remove all choices with overlapping, and repeat. The process is repeated until the score for the next-best choice is below a threshold or M choices have been instantiated. However, the general implementation of such an algorithm would take exponential time.
The M-NMS method tries to find M-best choices by throwing away the similar choices from the candidate set. To be concluded, the D-MCL, submodular, and M-NMS have the similar idea. All of them tries to find the M-best choices under a dissimilarity function or the ones which can provide the most complement information.
### V-D M-modes
Even though the former three methods guarantee the obtained multiple choices to be apart from each other, the choices are typically not local extrema of the probability distribution. To further guarantee both the local extrema and the diversification of the obtained multiple choices simultaneously, the problem can be transformed to the M-modes. The M-modes have multiple possible applications, because they are intrinsically diverse.
For a non-negative integer , define the -neighborhood of a labelling to be as the set of labellings whose distances from is no more than , where | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.812349796295166, "perplexity": 778.4927210909709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057131.88/warc/CC-MAIN-20210921011047-20210921041047-00224.warc.gz"} |
https://mathoverflow.net/questions/235893/does-there-exist-a-bijection-of-mathbbrn-to-itself-such-that-the-forward-m/309671 | # Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ is connected.
Question: Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the connectedness of (the induced power set map) $f$ imply that of $f^{-1}$?
Full disclosure: I originally asked this on Math.SE a year and a half ago; there were some discussion in the comments as well as a few answer attempts that were unfortunately flawed. Remarks 3 and 5 below capture the essences of many of those attempts.
Various remarks
1. If we remove the bijection requirement then the answer is clearly "no". For example, $f(x) = \sin(x)$ when $n = 1$ is a map whose forward map preserves connectedness but the inverse map does not.
2. With the bijection, it holds true in $n = 1$. But this is using the order structure of $\mathbb{R}$: a bijection that preserves connectedness on $\mathbb{R}$ must be monotone.
3. As a result of the invariance of domain theorem if either $f$ or $f^{-1}$ is continuous, we must have that $f$ is a homeomorphism, which would imply that both $f$ and $f^{-1}$ must be connected. (See https://math.stackexchange.com/q/949168/1543 which inspired this question for more about this.)
Invariance of domain, in fact, asserts a positive answer to the following question which is very similar in shape and spirit to the one I asked above:
Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the fact that $f$ is an open map imply that $f^{-1}$ is open?
4. In fact, it is a Theorem of Tanaka's (see my answer here) that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection such that both $f$ and $f^{-1}$ are connected, then $f$ is a homeomorphism. So an equivalent formulation of the question is
Equivalent Question: Does there exist a bijection $f:\mathbb{R}^n\to\mathbb{R}^n$ such that $f$ is connected but discontinuous?
5. Some properties of $\mathbb{R}^n$ must factor in heavily in the answer. If we replace the question and consider, instead of self-maps of $\mathbb{R}^n$ with the standard topology to itself, by self-maps of some arbitrary topological space, it is easy to make the answer go either way.
• For example, if the topological space $(X,\tau)$ is such that there are only finitely many connected subsets of $X$ (which would be the case if $X$ itself is a finite set), then by cardinality argument we have that the answer is yes, $f^{-1}$ must also be connected.
• On the other hand, one can easily cook up examples where the answer is no; a large number of examples can be constructed as variants of the following: let $X = \mathbb{Z}$ equipped with the topology generated by $$\{\mathbb{N}\} \cup \{ \{k\} \mid k \in \mathbb{Z} \setminus \mathbb{N} \}$$ then the map $k \mapsto k+\ell$ for any $\ell > 0$ maps connected sets to connected sets, but its inverse $k\mapsto k-\ell$ can map connected sets to disconnected ones.
• I noticed some subtleties that might help prove/disprove the conjecture. If you can prove that the image of a disconnected set in $X$ must be disconnected in $Y$, then you would be done. But attempting to do this, I run into a problem: I can get separate connected sets in $Y$, but I do not actually know if their union is disconnected. I have attempted to get around this by considering their closures and using the fact that $\mathbb{R}^n$ is normal to get the separation, but then I can't ensure that the intersection of their closures is empty (and if it isn't empty, what goes wrong?) – Justin Benfield Apr 21 '16 at 2:39
• @JustinBenfield: I don't think that can work in general. Let $X = \mathbb{R}\setminus \{0\}$ and $f$ be the identity map. The closures of images of the two connected components of $X$ under $f(X)$ have non-empty intersection. But perhaps I misunderstand what you intend. – Willie Wong Apr 21 '16 at 13:40
• I am aware that the closures can be nonempty, and that's where the problem I haven't solved is: what to do w/ those cases. My idea was to show that the preimage of the overlap in $Y$ would lie in the boundary of both sets in $X$, implying that the union of those closures (in $Y$) would therefore be connected. But I can't quite seem to show that. – Justin Benfield Apr 21 '16 at 20:56
• @Farewell: the question asks for $f$ defined on exactly $\mathbb{R}^n$, not some fancy subset of it. – Willie Wong Jun 15 '16 at 4:09
• You could ask the same question more generally for a bijection $\mathbb{R}^m\to\mathbb{R}^n$: are there some pairs $(m,n)$ other than $(1,1)$ for which you know the answer? (I expect that such a bijection can be constructed by transfinite induction for $m=1$ and $n>1$, but I didn't give it too much thought.) – Gro-Tsen Jan 26 '17 at 10:11
This is only a partial answer; but it's too long for a comment, and I believe the following facts, which come close to the question being asked and might provide a clue to answering it, are worth pointing out.
Let me show that there exists a bijection $\mathbb{R} \to \mathbb{R}^2$ such that the forward map is connected but the inverse is not.
Let me call a subset $E$ of the plane "hyperdense" when it meets every perfect set of the plane (perfect = nonempty, closed, with no isolated point). In particular, it meets every nonempty open set and every uncountable closed set (by Cantor-Bendixson). Let me reproduce the argument given here to show that every hyperdense set of the plane is connected: if not, there would be $U,V\subseteq\mathbb{R}^2$ open such that $E \subseteq U\cup V$, $E$ meets both $U$ and $V$ and does not meet $U\cap V$; but since $E$ is hyperdense, not meeting the open set $U\cap V$ is possible only if $U\cap V = \varnothing$, so $U\cup V$ is disconnected, so the closed set $\mathbb{R}^2 \setminus (U\cup V)$ must be uncountable (because the complement of a countable set in the plane is connected), so $E$ must meet it, a contradiction.
Now I construct a bijection $\mathbb{R} \to \mathbb{R}^2$ such that $f(I)$ is hyperdense for every nonempty open interval $I$. Let $\mathfrak{c} = 2^{\aleph_0}$ (seen as the smallest ordinal of that cardinality); let $(I_\beta,P_\beta)_{\beta<\mathfrak{c}}$ be an enumeration of pairs consisting of a nonempty open interval $I \subseteq \mathbb{R}$ and a perfect set $P \subseteq \mathbb{R}^2$ (recall that there are precisely continuum-many perfect sets), and let $(z_\beta)_{\beta<\mathfrak{c}}$ and $(t_\beta)_{\beta<\mathfrak{c}}$ be an enumeration of $\mathbb{R}$ and $\mathbb{R}^2$ respectively. Construct $(x_\alpha,y_\alpha) \in \mathbb{R} \times \mathbb{R}^2$ by induction on $\alpha<\mathfrak{c}$ such that $x_\alpha$ is different from all the (previously constructed) $x_{\alpha'}$ for $\alpha'<\alpha$ and $y_\alpha$ is different from all $y_{\alpha'}$ for $\alpha'<\alpha$, and additionally: (A) if $\alpha=2\beta$ is even, then choose $x_\alpha \in I_\beta$ and $y_\alpha \in P_\beta$ (these conditions can be met because $I_\beta$ and $P_\beta$ have cardinality $2^{\aleph_0}$ and there are $<2^{\aleph_0}$ previously constructed $x_{\alpha'}$ and $y_{\alpha'}$ to be avoided); (B₁) if $\alpha=4\beta+1$ then take $x_\alpha=z_\beta$ unless $z_\beta$ is already among the previous $x_{\alpha'}$ (otherwise, the choice is unconstrained), and (B₂) if $\alpha=4\beta+3$ then take $y_\alpha=t_\beta$ unless $t_\beta$ is already among the previous $y_{\alpha'}$ (otherwise, the choice is unconstrained; conditions (B₁) and (B₂) are only there to guarantee that $f$ will be defined everywhere and surjective). Clearly the choice can be made at every stage, so the sequence $(x_\alpha,y_\alpha)_{\alpha<\mathfrak{c}}$ exists. By construction, the $x_\alpha$ are distinct and by (B₁) every real $z_\beta$ is equal to some $x_\alpha$, so we can define $f(x_\alpha) = y_\alpha$, giving a function $\mathbb{R} \to \mathbb{R}^2$, which is injective because the $y_\alpha$ are distinct and surjective by (B₂). Now if $I$ is any nonempty open interval and $P$ is some perfect set, we can write $(I,P) = (I_\beta,P_\beta)$ and if $\alpha = 2\beta$ we have $f(x_\alpha) = y_\alpha \in P$ for $x_\alpha \in I$, which shows $f(I) \cap P \neq \varnothing$. So $f(I)$ is always hyperdense.
Now if $I \subseteq \mathbb{R}$ is connected, either it is empty or a singleton, in which case $f(I)$ is trivially connected, or it is an interval with nonzero length, but then it contains a nonempty open interval, so $f(I)$ is hyperdense, so it is connected as explained above. This shows that the forward map of $f$ is connected.
On the other hand, the inverse map of $f$ cannot be connected, because if it were, the inverse image of an open disk in $\mathbb{R}^2$ would be connected, hence an interval, which cannot be trivial since $f$ is bijective, so it must contain an open interval, so $f$ is continuous; but $f$ is certainly not continuous since it is bounded on no interval.
In a related vein, let me show that there exists a bijection $f\colon \mathbb{R}^2 \to \mathbb{R}^2$ such that the forward image of every PATH-connected set is connected and such that the inverse map is not connected.
The construction above easily gives a bijection $f\colon \mathbb{R}^2 \setminus\Delta \to \mathbb{R}^2 \setminus Z$, where $\Delta$ is the diagonal and $Z = (0,1)\times\{0\}$, such that $f(Q) \cap P \neq\varnothing$ for every perfect set $Q$ in $\mathbb{R}^2 \setminus\Delta$ and every perfect set $P$ in $\mathbb{R}^2 \setminus Z$, i.e., $f(Q)$ is hyperdense in $\mathbb{R}^2 \setminus Z$ for every perfect $Q$ in $\mathbb{R}^2 \setminus\Delta$.
Extend $f$ to a bijection $\mathbb{R}^2 \to \mathbb{R}^2$ by using a homeomorphism between $\Delta$ and $Z$.
Clearly, the inverse map of $f$ is not connected, since $f^{-1}(\mathbb{R}^2 \setminus Z) = \mathbb{R}^2 \setminus\Delta$ and $\mathbb{R}^2 \setminus\Delta$ is not connected whereas $\mathbb{R}^2 \setminus Z$ is.
Now consider the forward image of a path-connected set $A \subseteq \mathbb{R}^2$. If $A \subseteq \Delta$ then $f(A)$ is connected because $f$ restricts to a homeomorphism between $\Delta$ and $Z$. On the other hand, if $A \not\subseteq \Delta$, clearly $A\setminus\Delta$ is not a single point, so $A$ must contain a path between two points not in $\Delta$, and at least part of this path is not in $\Delta$, so it contains a perfect set $Q \subseteq \mathbb{R}^2 \setminus\Delta$. So $f(Q)$ is hyperdense in $\mathbb{R}^2 \setminus Z$; in particular, it is dense in $\mathbb{R}^2$, and connected (the same argument showing that hyperdense sets in $\mathbb{R}^2$ are connected still works for $\mathbb{R}^2\setminus Z$ since the complement of a countable set in the latter is connected). Now $f(A)$ contains $f(Q)$ which is dense in $\mathbb{R}^2$ connected, and anything containing a dense connected set is still connected (because if $D$ is dense connected and $D \subseteq B$, then any continuous $B\to\{0,1\}$ has a constant restriction to the dense subset $D$ so it is constant). So $f(A)$ is connected.
It is conceivable that the same kind of arguments give a positive answer to the original question, provided we can find something to play the role of the intervals $I$ in the first part and the perfect sets $Q$ in the second. But the naïve approach fails: indeed, one might ask the following question
Question: Does there exists $2^{\aleph_0}$ subsets $(H_\gamma)$ of the plane $\mathbb{R}^2$, each of cardinality $2^{\aleph_0}$, such that every connected $A \subseteq \mathbb{R}^2$ that is not a singleton contains one of the $H_\gamma$?
(Then we could define a bijection $f\colon\mathbb{R}^2\to\mathbb{R}^2$ by transfinite induction as above so that $f(H_\gamma)$ is hyperdense for each $H_\gamma$. Its forward map would be connected, but it wouldn't be continuous so this would answer the "equivalent question" cited in the original post.)
However, the answer to this question must be "no", because if such $(H_\gamma)$ existed, we could also use them to construct a bijection $f\colon\mathbb{R}^2\to\mathbb{R}^2$ such that $f(H_\gamma)$ and $f^{-1}(H_\gamma)$ are both hyperdense for each $H_\gamma$, which would make both the forward and inverse maps of $f$ connected, but $f$ would not be continuous (say we include all perfect sets among the $H_\gamma$ to be sure), contradicting the result of Tanaka cited by Willie Wong's question. (Probably there exists a more direct proof that the question has a negative answer. [EDIT: Will Brian gives such a proof in the comments]) So either this approach is doomed or we must be smarter in how to use it.
Incidentally, does ZFC prove that connected subsets of the plane satisfy the continuum hypothesis? [EDIT: again, Will Brian gave a (positive) answer in the comments]
• In fact, Tanaka's theorem which I alluded to in my question does not require that the dimensions be equal: in fact he worked at the level of biconnected mappings between topological spaces (with some mild local connectivity assumption). So a corollary of Tanaka's theorem + Invariance of domain is that there does not exist biconnected bijections from $\mathbb{R}^n \to \mathbb{R}^m$ when $n\neq m$. So in this case the demonstration boils down to the existence of a forward-connected bijection, which you showed. – Willie Wong Jan 26 '17 at 14:28
• Very interesting! Let me point out that the answer to your highlighted question at the end is no. To see this, suppose otherwise. Using transfinite recursion, you can construct a Bernstein set whose complement contains a point of every $H_\gamma$. (And, as you point out, Bernstein sets are connected.) As for your second question (Do nontrivial connected subsets of the plane have cardinality c?) the answer is yes. If $X$ is a set of smaller cardinality, pick two distinct points in $X$ and observe that one of some family of (continuum many) disjoint parallel lines between them separates $X$. – Will Brian Jan 26 '17 at 14:29
• @WillBrian: Ah, thanks for the term "Bernstein set", I probably read this somewhere already, but I couldn't remember it. I had also concluded that my question must have a negative answer, but your explanation is clearer — I guess this means this approach cannot succeed for the original question. – Gro-Tsen Jan 26 '17 at 14:39
This is not an answer to the original question of Willie Wong but a partial answer to the following related
Problem. Recognize pairs of topological spaces $$X,Y$$ for which every Darboux injection $$f:X\to Y$$ is continuous.
A function $$f:X\to Y$$ between topological spaces is called Darboux for any connected subspace $$C\subset X$$ its image $$f(C)$$ is connected. An injection is any injective function.
The following theorem proved in this preprint gives a partial answer to the above general problem:
Theorem. A Darboux injective function $$f:X\to Y$$ between connected metrizable spaces is continuous if one of the following conditions is satisfied:
1) $$Y$$ is a 1-manifold and $$X$$ is compact;
2) $$Y$$ is a 2-manifold and $$X$$ is a closed $$n$$-manifold for some $$n\ge 2$$;
3) $$Y$$ is a 3-manifold and $$X$$ is a closed $$n$$-manifold of dimension $$n\ge 3$$ with finite first homology group $$H_1(X)$$.
Corollary. For $$n\le 3$$ any Darboux bijection $$f:S^n\to S^n$$ of the $$n$$-dimensional sphere is a homeomorphism.
The proof of the first statement in Theorem 1 is more-or-less elementary, but 2 and 3 use heavy machinery of Algebraic Topology (Alexander Duality between homologies and Cech cohomologies, long exact sequences of Cech cohomology groups and Mayer-Vietoris exact sequence for singular homologies). I do not know if the Theorem can be generalized to higher dimensions $$n>3$$.
So, for example the following problem seem to be open.
Problem 1. Is any Darboux permutation of the 4-dimensional sphere continuous?
• I look forward to seeing the proof! – Willie Wong Sep 4 '18 at 14:44
• @WillieWong The update of the preprint with the proof has already appeared on arxiv: arxiv.org/pdf/1809.00401.pdf – Taras Banakh Sep 6 '18 at 4:57
• Thanks for letting me know, I didn't see that you updated your answer. My topology is weak enough that your proof will take me a long time to digest. – Willie Wong Sep 6 '18 at 16:03
• @WillieWong For me this was also not easy -- Algebraic Topology is not my area of expertise (but nonetheless). – Taras Banakh Sep 6 '18 at 16:38
This is a partial answer, that significantly narrows possible counterexamples.
Want to show: If $D\subset X$ is a disconnected set, then it its image, $f(D)\subset Y$ is a disconnected set.
Attempted proof: Let $\mathfrak{U}$ denote the set of connected components of $D$, then for each $U\in\mathfrak{U}$, we have, by $f$ connected, that $f(U)$ is connected. Let $f(U),f(V)$ be images of distinct connected components. Then by $f$ bijective, we have that $f(U)\cap f(V)=\emptyset$. If $\overline{f(U)}\cap\overline{f(V)}=\emptyset$, then by $\mathbb{R}^n$ normal, we have a separation of $\overline{f(U)}$ and $\overline{f(V)}$ by disjoint open sets. Those same sets separate $f(U)$ from $f(V)$, hence $f(U)\cup f(V)$ is disconnected.
The problem: What happens if $\overline{f(U)}\cap\overline{f(V)}\neq\emptyset$?
Thoughts: My suspicion is that in order for $f$ to be a connected map, it would have to be the case that the pre-image of that intersection must lie in the boundary of both $U$ and $V$, but I haven't quite been able to show that. I thought about picking an arbitrary point in the intersection and finding sequences in $f(U)$ and $f(V)$ that converge to it and then using the pre-images of those sequences, which live in $U$ and $V$ respectively to show that the pre-image of that point was indeed in the boundary of both sets, but we don't know that $f^{-1}$ preserves convergence. What happens in $Y$ if it doesn't? If I knew that the pre-image of the sequences were bounded I could perhaps use compactness to get a separation of the tail from the point, and image those to get a disconnection between the tail and the point it converges to (a contradiction), but I don't know that the pre-images of the sequences are even bounded.
• Remark: in the case of $\mathbb{R}$, your "thoughts" is basically solved by appealing to the order structure; there are some structural similarities of your proof to the statement that connected bijections are monotone on $\mathbb{R}$. – Willie Wong Apr 22 '16 at 14:10
• Remark 2: I fixed a small typo in your thoughts. And yes, your thoughts is also tied to my remarks 3 and 4 in the question. The preservation of limits, for example, would mean that $f^{-1}$ is continuous. And you can see from Tanaka's proof that your main difficulty is basically equivalent to the original question. // Still, thanks for thinking about it! – Willie Wong Apr 22 '16 at 14:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9540352821350098, "perplexity": 142.08084721995309}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202688.89/warc/CC-MAIN-20190322180106-20190322202106-00223.warc.gz"} |
https://docs.microsoft.com/en-us/windows-server/networking/technologies/netsh/netsh-contexts | # Netsh Command Syntax, Contexts, and Formatting
Applies To: Windows Server 2016
You can use this topic to learn how to enter netsh contexts and subcontexts, understand netsh syntax and command formatting, and how to run netsh commands on local and remote computers.
Netsh is a command-line scripting utility that allows you to display or modify the network configuration of a computer that is currently running. Netsh commands can be run by typing commands at the netsh prompt and they can be used in batch files or scripts. Remote computers and the local computer can be configured by using netsh commands.
Netsh also provides a scripting feature that allows you to run a group of commands in batch mode against a specified computer. With netsh, you can save a configuration script in a text file for archival purposes or to help you configure other computers.
## Netsh contexts
Netsh interacts with other operating system components by using dynamic-link library (DLL) files.
Each netsh helper DLL provides an extensive set of features called a context, which is a group of commands specific to a networking server role or feature. These contexts extend the functionality of netsh by providing configuration and monitoring support for one or more services, utilities, or protocols. For example, Dhcpmon.dll provides netsh with the context and set of commands necessary to configure and manage DHCP servers.
### Obtain a list of contexts
You can obtain a list of netsh contexts by opening either command prompt or Windows PowerShell on a computer running Windows Server 2016 or Windows 10. Type the command netsh and press ENTER. Type /?, and then press ENTER.
Following is example output for these commands on a computer running Windows Server 2016 Datacenter.
``````PS C:\Windows\system32> netsh
netsh>/?
The following commands are available:
Commands in this context:
.. - Goes up one context level.
? - Displays a list of commands.
branchcache - Changes to the `netsh branchcache' context.
bridge - Changes to the `netsh bridge' context.
bye - Exits the program.
commit - Commits changes made while in offline mode.
delete - Deletes a configuration entry from a list of entries.
dhcpclient - Changes to the `netsh dhcpclient' context.
dnsclient - Changes to the `netsh dnsclient' context.
dump - Displays a configuration script.
exec - Runs a script file.
exit - Exits the program.
firewall - Changes to the `netsh firewall' context.
help - Displays a list of commands.
http - Changes to the `netsh http' context.
interface - Changes to the `netsh interface' context.
ipsec - Changes to the `netsh ipsec' context.
ipsecdosprotection - Changes to the `netsh ipsecdosprotection' context.
lan - Changes to the `netsh lan' context.
namespace - Changes to the `netsh namespace' context.
netio - Changes to the `netsh netio' context.
offline - Sets the current mode to offline.
online - Sets the current mode to online.
popd - Pops a context from the stack.
pushd - Pushes current context on stack.
quit - Exits the program.
ras - Changes to the `netsh ras' context.
rpc - Changes to the `netsh rpc' context.
show - Displays information.
trace - Changes to the `netsh trace' context.
unalias - Deletes an alias.
wfp - Changes to the `netsh wfp' context.
winhttp - Changes to the `netsh winhttp' context.
winsock - Changes to the `netsh winsock' context.
The following sub-contexts are available:
advfirewall branchcache bridge dhcpclient dnsclient firewall http interface ipsec ipsecdosprotection lan namespace netio ras rpc trace wfp winhttp winsock
To view help for a command, type the command, followed by a space, and then
type ?.
``````
### Subcontexts
Netsh contexts can contain both commands and additional contexts, called subcontexts. For example, within the Routing context, you can change to the IP and IPv6 subcontexts.
To display a list of commands and subcontexts that you can use within a context, at the netsh prompt, type the context name, and then type either /? or help. For example, to display a list of subcontexts and commands that you can use in the Routing context, at the netsh prompt (that is, netsh>), type one of the following:
routing /?
routing help
To perform tasks in another context without changing from your current context, type the context path of the command you want to use at the netsh prompt. For example, to add an interface named "Local Area Connection" in the IGMP context without first changing to the IGMP context, at the netsh prompt, type:
routing ip igmp add interface "Local Area Connection" startupqueryinterval=21
## Running netsh commands
To run a netsh command, you must start netsh from the command prompt by typing netsh and then pressing ENTER. Next, you can change to the context that contains the command you want to use. The contexts that are available depend on the networking components that you have installed. For example, if you type dhcp at the netsh prompt and press ENTER, netsh changes to the DHCP server context. If you do not have DHCP installed, however, the following message appears:
## Formatting Legend
You can use the following formatting legend to interpret and use correct netsh command syntax when you run the command at the netsh prompt or in a batch file or script.
• Text in Italic is information that you must supply while you type the command. For example, if a command has a parameter named -UserName, you must type the actual user name.
• Text in Bold is information that you must type exactly as shown while you type the command.
• Text followed by an ellipsis (...) is a parameter that can be repeated several times in a command line.
• Text that is between brackets [ ] is an optional item.
• Text that is between braces { } with choices separated by a pipe provides a set of choices from which you must select only one, such as `{enable|disable}`.
• Text that is formatted with the Courier font is code or program output.
## Running Netsh commands from the command prompt or Windows PowerShell
To start Network Shell and enter netsh at the command prompt or in Windows PowerShell, you can use the following command.
### netsh
Netsh is a command-line scripting utility that allows you to, either locally or remotely, display or modify the network configuration of a currently running computer. Used without parameters, netsh opens the Netsh.exe command prompt (that is, netsh>).
#### Syntax
netsh[ -a AliasFile] [ -c Context ] [-r RemoteComputer] [ -u [ DomainName\ ] UserName ] [ -p Password | *] [{NetshCommand | -f ScriptFile}]
#### Parameters
`-a`
Optional. Specifies that you are returned to the netsh prompt after running AliasFile.
`AliasFile`
Optional. Specifies the name of the text file that contains one or more netsh commands.
`-c`
Optional. Specifies that netsh enters the specified netsh context.
`Context`
Optional. Specifies the netsh context that you want to enter.
`-r`
Optional. Specifies that you want the command to run on a remote computer.
##### Important
When you use some netsh commands remotely on another computer with the netsh –r parameter, the Remote Registry service must be running on the remote computer. If it is not running, Windows displays a “Network Path Not Found” error message.
`RemoteComputer`
Optional. Specifies the remote computer that you want to configure.
`-u`
Optional. Specifies that you want to run the netsh command under a user account.
`DomainName\\`
Optional. Specifies the domain where the user account is located. The default is the local domain if DomainName\ is not specified.
`UserName`
Optional. Specifies the user account name.
`-p`
Optional. Specifies that you want to provide a password for the user account.
`Password`
Optional. Specifies the password for the user account that you specified with -u UserName.
`NetshCommand`
Optional. Specifies the netsh command that you want to run.
`-f`
Optional. Exits netsh after running the script that you designate with ScriptFile.
`ScriptFile`
Optional. Specifies the script that you want to run.
`/?`
Optional. Displays help at the netsh prompt.
##### Note
If you specify `-r` followed by another command, netsh runs the command on the remote computer and then returns to the Cmd.exe command prompt. If you specify `-r` without another command, netsh opens in remote mode. The process is similar to using set machine at the Netsh command prompt. When you use `-r`, you set the target computer for the current instance of netsh only. After you exit and reenter netsh, the target computer is reset as the local computer. You can run netsh commands on a remote computer by specifying a computer name stored in WINS, a UNC name, an Internet name to be resolved by the DNS server, or an IP address.
Typing parameter string values for netsh commands
Throughout the Netsh command reference there are commands that contain parameters for which a string value is required.
In the case where a string value contains spaces between characters, such as string values that consist of more than one word, it is required that you enclose the string value in quotation marks. For example, for a parameter named interface with a string value of Wireless Network Connection, use quotation marks around the string value:
`interface="Wireless Network Connection"` | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8645802736282349, "perplexity": 4957.258851269569}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608004.38/warc/CC-MAIN-20170525063740-20170525083740-00119.warc.gz"} |
https://math.stackexchange.com/questions/2687788/what-is-the-sum-of-sum-n-1-infty-left-frac2n1n42n3n2-right | # What is the sum of $\sum_{n=1}^\infty\left(\frac{2n+1}{n^4+2n^3+n^2}\right) =$?
$$\sum_{n=1}^\infty\left(\frac{2n+1}{n^4+2n^3+n^2}\right)=\sum_{n=1}^\infty\left(\frac{2n+1}{n^2}\frac1{{(n+1)}^2}\right)$$ I assume that I should get a telescoping sum in some way, but I'm couldn't find it yet.
then we have, for an integer $N>1$ $$\sum_{n=1}^N\left(\frac{2n+1}{n^4+2n^3+n^2}\right)=1-\frac1{(N+1)^2}$$
Hint: $2n+1=(n+1)^2-n^2$, and telescope.
$$\frac{2n+1}{n^2}\frac{1}{(n+1)^2}=\frac{n+n+1}{n^2(n+1)^2}$$ $$\frac{2n+1}{n^2}\frac{1}{(n+1)^2}=\frac{1}{n(n+1)^2}+\frac{1}{n^2(n+1)}$$ And of course your telescoping element: $$\frac{1}{n(n+1)}=\left( \frac{1}{n}-\frac{1}{n+1}\right)$$ Therefore: $$\frac{2n+1}{n^2}\frac{1}{(n+1)^2}=\frac{1}{n(n+1)}-\frac{1}{(n+1)^2}+\frac{1}{n^2}-\frac{1}{n(n+1)}$$ $$\frac{2n+1}{n^2}\frac{1}{(n+1)^2}=-\frac{1}{(n+1)^2}+\frac{1}{n^2}$$ And the result is simply 1. As would [WolframAlpha confirm ]( https://www.wolframalpha.com/input/?i=sum((2*n%2B1)%2F((n%5E2(n%2B1)%5E2) ) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8943186402320862, "perplexity": 606.2413187394708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255837.21/warc/CC-MAIN-20190520081942-20190520103942-00085.warc.gz"} |
http://physics.stackexchange.com/questions/14431/equation-of-motion-in-a-non-inertial-rotating-frame | # Equation of Motion in a Non-Inertial (Rotating) Frame
Let me preface my question by informing you this is for an assignment, so I would rather not have explicit answers but rather be given guidance in arriving at the correct solution.
The question is thus; I have a mass on a frictionless rotating turntable. I need to adopt a co-rotating frame to find the equation of motion for this mass. I am expected to solve this using only Newtonian mechanics.
As we are restricted to a plane, there is no component of motion in the $\vec{k}$ direction. The only forces acting on our mass (in the rotating frame) are the Centrifugal force, and the Coriolis force.
My attempt at a solution;
$\vec{F} = m\vec{a}$
$\vec{F_{cent}} = -m\vec{\Omega}$ x $\vec{\Omega}$ x $\vec{r}$
$\vec{F_{cor}} = -2m\vec{\Omega}$ x $\vec{v}$
$m\vec{a} = -m\vec{\Omega}$ x $\vec{\Omega}$ x $\vec{r} - 2m\vec{\Omega}$ x $\vec{v}$
As I'm working in the rotating frame, I don't see any reason to not use cartaesian coordinates (please note this is my first time using latex, so in this case $\vec{i},\vec{j},\vec{k}$ are unit vectors, and x,y,z are magnitudes in the respective directions);
$\vec{\Omega} = \Omega\vec{k}$
$\vec{r} = x\vec{i} + y\vec{j}$
$\vec{v} = \vec{\dot{r}} = \dot{x}\vec{i} + \dot{y}\vec{j}$
Substituting this in, evaluating the cross-products and simplifying yields;
$\ddot{x}\vec{i} + \ddot{y}\vec{j} = \Omega^2x\vec{i} + \Omega^2y\vec{j} - 2\Omega\dot{x}\vec{j} + 2\Omega\dot{y}\vec{i}$
And so we have two coupled second order differential equations;
$\ddot{x} = \Omega^2x + 2\Omega\dot{y}$
$\ddot{y} = \Omega^2y - 2\Omega\dot{x}$
A method we had used previously in class to solve coupled equations was to set $\Omega = 0$ and solve, then substitute this solution back in for $\dot{x} and \dot{y}$. I attempted this, however it yielded two cubic equations. The solution I am told this system has, for the initial conditions $(x(0),y(0)) = (x_0,0)$, is a spiral when mapped parametrically, namely;
$x(t) = (x_0 + v_{x0}t)\cos\Omega t + (v_{y0} + \Omega x_0)t\cos\Omega t$
$y(t) = -(x_0 + v_{x0}t)\sin\Omega t + (v_{y0} + \Omega x_0)t\sin\Omega t$
To me these appeared to be solutions gained from solving the non-homogenous linear second-order differential equation, however this did not work either.
Is my derivation of the original vector equation of motion correct? If not, where did I go wrong? If so, what method should I use to solve these equations to find the appropriate solutions?
-
Instead of $\vec{i}$, you should probably write $\widehat{i}$ (similarly for $\vec{j}$ and $\vec{k}$) to indicate that these vectors are unit vectors. Furthermore, it is my personal preference to use $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$ over $\widehat{i}$, $\widehat{j}$, and $\widehat{k}$ because of confusion with quaternions. – Jonathan Gleason Sep 8 '11 at 17:19
Also, the centrifugal force term is $-m\mathbf{\Omega}\times \left( \mathbf{\Omega}\times \mathbf{r}\right)$. Without the parenthesis, your equation implies that the centrifugal force is $-m\left( \mathbf{\Omega}\times \mathbf{\Omega}\right) \times \mathbf{r}$, which can't possibly be the case because this is always $\mathbf{0}$. Remember, the cross-product is nonassociative. Furthermore, this is just another personal preference, but you might want to use boldface to denote vectors in \LaTeX because it makes them stand out more. – Jonathan Gleason Sep 8 '11 at 17:23
I couldn't find the command in LaTeX for the hats, hence my reliance on the i,j,k notation. So for future reference, what is the command? And thank you for the advice on the parentheses, unfortunately my lecturer never wrote it out with them (which lead to some confusion). – Daniel Blay Sep 9 '11 at 5:27
\widehat{}. \hat{} should also work. – Jonathan Gleason Sep 10 '11 at 2:10
My hint is to solve the problem using polar coodriantes along with the handy equations founds here. We have that $$\mathbf{a}=\ddot{\mathbf{r}}=\left( \ddot{r}-r\dot{\theta}^2\right) \widehat{\mathbf{r}}+\left( 2\dot{r}\dot{\theta}+r\ddot{\theta}\right) \widehat{\mathbf{\theta}}=-\Omega ^2r\widehat{\mathbf{z}}\times \widehat{\mathbf{\theta}}-2\Omega \widehat{\mathbf{z}}\times \left( \dot{r}\widehat{\mathbf{r}}+r\dot{\theta}\widehat{\mathbf{\theta}}\right)$$ Thus, $$\left( \ddot{r}-r\dot{\theta}^2\right) \widehat{\mathbf{r}}+\left( 2\dot{r}\dot{\theta}+r\ddot{\theta}\right) \widehat{\mathbf{\theta}}=\left( \Omega ^2r+2\Omega r\dot{\theta}\right) \widehat{\mathbf{r}}-2\Omega \dot{r}\widehat{\mathbf{\theta}}.$$ Thus, $$\ddot{r}-r\dot{\theta}^2=\Omega ^2r+2\Omega r\dot{\theta}$$ and $$2\dot{r}\dot{\theta}+r\ddot{\theta}=-2\Omega \dot{r}.$$ This might seem difficult to solve, but you easily show that $\dot{\theta}=-\Omega$ (just think about how $\theta$ and $\Omega$ are defiend), and so these just reduce to $$\ddot{r}=0.$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9192142486572266, "perplexity": 175.61170908051523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246634331.38/warc/CC-MAIN-20150417045714-00134-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/356781/a-series-of-rational-number-converges-to-an-irrational-number | # A series of rational number converges to an irrational number
This problem is an example from Putnam and Beyond, page 118:
Let$(n_k)_{k\geq1}$ be a strictly increasing sequence of positive integers with the property that
$$\displaystyle\lim_{k \to \infty} \frac{n_k}{n_1n_2 \ldots n_{k-1}} = \infty$$
Prove that the series $\sum_{k>1}\frac{1}{n_k}$ is convergent and the its sum is an irrational number.
The proof claims that $n_{k+1} \geq 3 n_k$ for $k\geq 3$ (Edit: for $k$ sufficiently large) and then ratio test to prove the convergence of $\sum_{k>1}\frac{1}{n_k}$.
My question: If I define $n_k$ as $n_1 = 1, n_2 = 2$ and $n_{2m+1} = m\times \prod_{i=1}^{2m} n_i, n_{2m+2}=n_{2m+1}+1$ for $m \geq1$ so that $\lim\sup \frac{n_k}{n_1n_2 \ldots n_{k-1}} = \infty$ and $\lim\inf \frac{n_k}{n_1n_2 \ldots n_{k-1}} = 0$, then does it satisfy the condition?
If we use $\{n_k\}$ constructed above, I can not see why $n_{k+1} \geq 3 n_k$ for $k$ sufficiently large.
Edit: My $\epsilon-\delta$ interpretation of $\lim a_k = \infty$ is that $\forall N \in \mathbb{R}^{+} \text{ and } \forall n \in \mathbb{N}, \exists n_0 \ge n$ such that $a_{n_0} \geq N$.
-
In your last statement, the $\forall n\in\mathbb N$ isn't meaningful - there isn't an $n$ in the rest of the statement. – Greg Martin Apr 10 '13 at 16:33
@GregMartin: typo fixed – Lei Lei Apr 10 '13 at 18:19
With your definition, it's clearly true that $\liminf \frac{n_k}{n_1n_2\cdots n_{k-1}} \le 1$. But that doesn't seem relevant to the stated problem, given its hypothesis.
It's not necessarily true that $n_{k+1} \ge 3n_k$ for $k\ge3$ (the sequence can start with $n_k=k$ for as many terms in a row as you like), but there does exist $k_0$ such that it's true for $k\ge k_0$.
-
probably the proof says that without loss of generality you can ignore the initial terms that don't respect the condition – suissidle Apr 10 '13 at 6:27
So $n_k$ from my example does not satisfies the hypothesis of the stated problem? – Lei Lei Apr 10 '13 at 7:23
Definitely not. – Greg Martin Apr 10 '13 at 16:34
@GregMartin: I guess I know why. I interpret $\lim a_k = \infty$ as $\{a_k\}_{k \geq 1}$ as a divergent sequence, not as a convergent sequence in extended real number system. – Lei Lei Apr 10 '13 at 18:18
Good diagnosis: the symbols $\lim a_k=\infty$ indicates a sequence that diverges in a very specific way. – Greg Martin Apr 10 '13 at 18:28
let $\varepsilon=3$ then with the definition of limit there must be a $n_0\in N$ such that for any $n>n_0$ there has $n_{k+1}>3n_kn_{k-1}n_{k-2}...n_2n_1>3n_k$
and $n_0 = 3$ has no effects on how solving this problem
-
that is the definition for convergent sequence. I think for divergent sequence, it is $\forall n \exists n_0 > n$ such that $n_{k+1} > 3n_k n_{k-1} \ldots n_2 n_1$. – Lei Lei Apr 10 '13 at 7:39
you can consider infinity as a number .but in fact we say a same thing – Xiaolang Apr 10 '13 at 8:00
your definition fails for the example I constructed, but my definition still works. – Lei Lei Apr 10 '13 at 16:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9762715101242065, "perplexity": 339.20054624518326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644065828.38/warc/CC-MAIN-20150827025425-00077-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.jiskha.com/display.cgi?id=1322411675 | Saturday
April 19, 2014
# Homework Help: physics
Posted by alaa on Sunday, November 27, 2011 at 11:34am.
A 5.30 kg package slides 1.57 m down a long ramp that is inclined at 11.8^\circ below the horizontal. The coefficient of kinetic friction between the package and the ramp is mu_k = 0.320
Calculate the work done on the package by the normal force.
If the package has a speed of 2.17 m/s at the top of the ramp, what is its speed after sliding the distance 1.57 m down the ramp?
• physics - drwls, Sunday, November 27, 2011 at 12:42pm
Why the name changes?
First Name: School Subject: Answer:
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Members | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9071294665336609, "perplexity": 2418.6154343315757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00282-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/math-challenge-problems/167007-parabola-property-print.html | # A Parabola Property
• Dec 28th 2010, 04:08 AM
Soroban
A Parabola Property
Triangle $PQR$ is inscribed in parabola $y \,=\,ax^2.$
Tangents at $P,Q,R$ intersect at $P'\!,Q'\!,R',$ forming triangle $P'Q'R'.$
Show that the areas of the two triangles are in the ratio $2:1.$
• Dec 28th 2010, 11:16 AM
Opalg
Quote:
Originally Posted by Soroban
Triangle $PQR$ is inscribed in parabola $y \,=\,ax^2.$
Tangents at $P,Q,R$ intersect at $P'\!,Q'\!,R',$ forming triangle $P'Q'R'.$
Show that the areas of the two triangles are in the ratio $2:1.$
I'll do this for the parabola $y^2=4ax$ since I'm more familiar with that. But the proof applies to any parabola.
Take the three points to be $(ar^2,2ar),\ (as^2,2as,\ (at^2,2at).$ The area of the triangle PQR is half the absolute value of $\begin{vmatrix}ar^2&2ar&1\\ as^2&2as&1\\ at^2&2at&1\end{vmatrix}.$ Using elementary row operations and expanding down the right-hand column, this comes out to be
$\begin{vmatrix}ar^2&2ar&1\\ a(s^2-r^2)&2a(s-r)&0\\ a(t^2-r^2)&2a(-r)t&0\end{vmatrix} = 2a^2(s-r)(t-r)\begin{vmatrix}s+r&1\\t+r&1\end{vmatrix} = 2a^2(s-r)(t-r)(s-t).$
The tangent at R is $x-ty+at^2 = 0$. It meets the tangent at S at the point $P' = (ast,a(s+t))$. Similarly, $Q' = (atr,a(t+r))$ and $R' = (ars,a(r+s)).$ The area of the triangle P'Q'R' is half the absolute value of
$\begin{vmatrix}ast&a(s+t)&1\\ atr&a(t+r)&1\\ ars&a(r+s)&1\end{vmatrix} = \begin{vmatrix}ast&a(s+t)&1\\ at(r-s)&a(r-s)&0\\ as(r-t)&a(r-t)&0\end{vmatrix} = a^2(r-s)(r-t)\begin{vmatrix}t&1\\s&1\end{vmatrix} = a^2(r-s)(r-t)(t-s).$
Thus area(PQR) = 2area(P'Q'R').
• Dec 28th 2010, 11:20 AM
red_dog
Let $P(x_1,ax_1^2), \ Q(x_2,ax_2^2), \ Q(x_3,ax_3^2)$
The slopes of the tangents at $P, \ Q, \ R$ are:
$m_1=2ax_1, \ m_2=2ax_2, \ m_3=2ax_3$
The equations of the tangents are:
$y-ax_1^2=2ax_1(x-x_1)$
$y-ax_2^2=2ax_1(x-x_2)$
$y-ax_3^2=2ax_1(x-x_3)$
Solving the systems formed by two of three equations we find the coordinates of the points $P', \ Q', \ R'$:
$P'\left(\dfrac{x_2+x_3}{2},ax_2x_3\right), \ Q'\left(\dfrac{x_1+x_3}{2},ax_1x_3\right), \ R'\left(\dfrac{x_1+x_2}{2},ax_1x_2\right)$
The area of the triangle $P'Q'R'$ is $A_{P'Q'R'}=\dfrac{1}{2}|\Delta '|$
where $\Delta '=\begin{vmatrix}\dfrac{x_1+x_2}{2} & ax_1x_2 & 1\\
\dfrac{x_2+x_3}{2} & ax_2x_3 & 1\\
\dfrac{x_1+x_3}{2} & ax_1x_3 & 1\end{vmatrix}=\dfrac{a}{2}\begin{vmatrix}x_1+x_2 & x_1x_2 & 1\\
x_2+x_3 & x_2x_3 & 1\\
x_1+x_3 & x_1x_3 & 1\end{vmatrix}=\dfrac{a}{2}(x_1-x_2)(x_3-x_2)(x_3-x_1)$
Then $A_{P'Q'R'}=\dfrac{1}{4}|a(x_1-x_2)(x_2-x_3)(x_3-x_1)|$
But $A_{PQR}=\dfrac{1}{2}|\Delta|$
where $\Delta=\begin{vmatrix}x_1 & ax_1^2 & 1\\
x_2 & ax_2^2 & 1\\
x_3 & ax_3^2 & 1\end{vmatrix}=a\begin{vmatrix}1 & 1 & 1\\x_1 & x_2 & x_3\\x_1^2 & x_2^2 & x_3^2\end{vmatrix}=a(x_3-x_2)(x_3-x_1)(x_2-x_1)$
Then $A_{PQR}=\dfrac{1}{2}|a(x_1-x_2)(x_2-x_3)(x_3-x_1)|$
and $A_{P'Q'R'}=\dfrac{1}{2}A_{PQR}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 37, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9205737709999084, "perplexity": 450.41668734499973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123484.45/warc/CC-MAIN-20170423031203-00397-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/schools-out-there-uses-maple.43183/ | # Schools out there uses maple
1. Sep 14, 2004
### CellCoree
dont know how many schools out there uses maple, but let me ask my question anyway. maple is a program that does math calculation and other stuff, it's pretty impressive. you can use it to do integrals and stuff.
here's my question using maple:
1.) Let f(x) =x^2 and g(x) = x*e^(-x/2). Find where these graphs intesect and determine the area enclosed in the intersectng region.
this has to be found using maple.
ok i tried these commands using maples:
f:=x^2;
g:=x*e^(-x/2);
plot({f,g},x=-10..10);
then i get this error "Warning, unable to evaluate 1 of the 2 functions to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct" and it only displays the f(x) graph. i checked the maple book, and i typed the extact commands, i dont know why it doesnt work.
2. Sep 15, 2004
### nrqed
I am not a Maple expert (far from it!) but an obvious thing to try would be to use exp(-x/2) instead of e^(-x/2). I am not sure but it might be that Maple treats "e" as just a variable and does know that you mean e = 2.718...
That's just a wild guess.
Pat
3. Sep 15, 2004
### CellCoree
thanks alot, works perfectly. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.941440224647522, "perplexity": 1783.8426211720382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718034.77/warc/CC-MAIN-20161020183838-00044-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/resistance-and-voltage-switch-diagram.804066/ | Resistance and Voltage, switch diagram
1. Mar 19, 2015
vysero
1. The problem statement, all variables and given/known data
2. Relevant equations
V=IR
3. The attempt at a solution
I said: A>D=F>B=C>G>H=E
H=E>C=B>G>D>F>A
Can someone explain to me why this is? As you can see H=E which combined gives a total resistance of (2/3)ohms is, according to the answers, going to give the largest volt meter reading. Why is this true?
Last edited by a moderator: Mar 19, 2015
2. Mar 19, 2015
SammyS
Staff Emeritus
Why assume that? The volt-meter only measures the voltage drop across that one resistor. Its resistance is fixed, so the voltage is proportional to the current through that resistor. Any current supplied by the battery must pass through that resistor, some how is that current related to the overall resistance of the circuit?
How do you get any particular resistance value?
No resistance value is given for any of the resistors. Are we to assume that they all have the same resistance?
3. Mar 19, 2015
vysero
OH geesh sorry I feel like a dork I forgot a part of the question here is the explanation:
4. Mar 19, 2015
SammyS
Staff Emeritus
You could type some of this out.
Part of the explanation has been given. We need a response from you. You have over 100 posts, so you should know how things work here.
How did you arrive at your order? (in addition to having voltage behavior reversed.)
5. Mar 19, 2015
vysero
I am not sure how I had voltage behavior reversed. V = IR so and increase in resistance will increase voltage. If I = 1 and R = 4 then V = 4 if I = 1 and R = 10 then V = 10. 1<10
However, I did not take into account that the measurement of V was across only one resistor. I am currently trying to work out how to get the V for that one resistor in each configuration.
6. Mar 19, 2015
vysero
Thank you for the insight I understand now!
Draft saved Draft deleted
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https://www.math.rutgers.edu/academics/graduate-program/1279-syllabus-of-the-spp-algebra-program | ## Syllabus of the SPP Algebra Program
To provide an in-depth review and to fill in gaps in some background material in Abstract Linear Algebra, which is often presumed in standard first year graduate courses. The material to be covered is also part of the syllabus of the qualifying exams on Algebra.
Below is a tentative list of topics to be covered; the actural coverage may vary depending on the instructor.
• Vector spaces, isomorphism, linear transformations: basis, dimension, quotient spaces, direct sums, rank and nullity. Coordinatization.
• Examples from various places: geometry, linear ODE, quantum mechanics, graph theory, etc.
• Similarity, eigenvalues, diagonalization, Jordan canonical form, application to ODE's and other areas, Rational canonical form.
• Role of the ground field (or extended ground field): In particular applications involving linear operators on vector spaces over the complex field (E.g. Jordan canonical form)
• Bilinear forms, sesquilinear forms, nondegeneracy, Euclidean and Unitary inner products.
• Some detailed study of Hermitian and unitary matrices, in particular, diagonalization involving Hermitian and unitary matrices. Basic properties of orthogonal and unitary groups. Self-adjoint linear transformations.
• Duality, esp. finite-dimensional case.
• Additional topics, if time permits: tensor product defined by naming basis, symmetric and wedge square, higher powers, determinants, Kronecker product, $$V^*\otimes W$$, differential forms, Schur duality.
## Contacts
Departmental Chair
Michael Saks | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9057779908180237, "perplexity": 3274.650864387563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651780.99/warc/CC-MAIN-20180325025050-20180325045050-00005.warc.gz"} |
http://link.springer.com/article/10.1007%2Fs10773-008-9764-4 | , Volume 47, Issue 12, pp 3288-3297
Date: 03 Jun 2008
# Bulk Viscous Bianchi Type I Cosmological Models with Time-Dependent Cosmological Term Λ
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## Abstract
Bianchi type I cosmological models with time-varying cosmological constant Λ and bulk viscous fluid are investigated. Cosmic matter is chosen to obey a barotropic equation of state. Exact solutions of Einstein’s field equations are obtained assuming the volume expansion θ proportional to the eigen values of shear tensor σ ij . Physical and kinematical properties of the models are discussed considering bulk viscosity to be a power function of matter density. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9345292448997498, "perplexity": 1690.545167491027}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500835488.52/warc/CC-MAIN-20140820021355-00406-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://projecteuclid.org/euclid.die/1356059714 | ## Differential and Integral Equations
### Subcritical pseudodifferential equation on a half-line with nonanalytic symbol
Elena I. Kaikina
#### Abstract
We study nonlinear pseudodifferential equations on a half-line with a nonanalytic symbol \begin{equation*} \left\{ \begin{array}{c} \partial _{t}u+\mathbb{K}u=\lambda \left\vert u\right\vert ^{\sigma }u,\text{ }x\in \mathbf{R}^{+},\text{ }t>0, \\ u\left( 0,x\right) =u_{0}\left( x\right) \text{, }x\in \mathbf{R}^{+}, \end{array} \right. \end{equation*} where $0<$ $\sigma <1,$ $\lambda \in \mathbf{R}$ and \begin{equation*} \mathbb{K}u=\frac{1}{2\pi i}\theta (x)\int_{-i\infty }^{i\infty }e^{px}K(p) \widehat{u}(t,p)dp,\qquad K(p)=\frac{p^{2}}{p^{2}-1}. \end{equation*} The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem and to find the main term of the asymptotic representation of solutions in subcritical case, when the nonlinear term of equation has the time decay rate less than that of the linear terms.
#### Article information
Source
Differential Integral Equations, Volume 18, Number 12 (2005), 1341-1370.
Dates
First available in Project Euclid: 21 December 2012 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9919674396514893, "perplexity": 2127.0626910696733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158001.44/warc/CC-MAIN-20180922005340-20180922025740-00374.warc.gz"} |
http://mathhelpforum.com/trigonometry/8084-trig-question-finding-all-solutions.html | # Math Help - Trig question finding all solutions
1. ## Trig question finding all solutions
what is the best way to find all the solutions of sin x-1=0, with the answer in A+Bkpi, where A=? with 0<A<pi
where B=? and k is any integer
Thankx
Keith Stevens
2. Originally Posted by kcsteven
what is the best way to find all the solutions of sin x-1=0, with the answer in A+Bkpi, where A=? with 0<A<pi
where B=? and k is any integer
Thankx
Keith Stevens
There is no such x, $0 < x < \pi$.
$sin(x) - 1 = 0$
$sin(x) = 1$
For what values of x is sin(x) = 1?
$x = 0, \pi$
Neither of which is in your indicated domain.
-Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9423078894615173, "perplexity": 3767.2164345819438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802769709.84/warc/CC-MAIN-20141217075249-00163-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/easy-domain.225092/ | # Easy Domain
1. Mar 29, 2008
### Calixto
What would the domain of y = sqrt(cosx) be in mathematical terms. I know that it is all the reals that lie in the first and fourth quadrant of the unit circle, but how would you express that in mathematical terms?
2. Mar 29, 2008
### nicksauce
Something like
$$D=\{x|x\in[\frac{(4n-1)\pi}{2},\frac{(4n+1)\pi}{2}]\}$$
where n is any integer
3. Mar 30, 2008
### unplebeian
You first need to define the domain of x.
If x is +60 for example then cos(x)>0 but if it is -60, then it is <0. sqrt of that is an imaginary number which is not defined.
So basically for x>0 domain is (0,1)
I would like to know how Calixto got his answer as I may be wrong.
Similar Discussions: Easy Domain | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9262188673019409, "perplexity": 1305.1971115656427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320669.83/warc/CC-MAIN-20170626032235-20170626052235-00524.warc.gz"} |
http://mathhelpforum.com/number-theory/67606-eulers-totient-function.html | 1. Eulers totient function
Hi,
I need to show that $$11^{40} + 33^{40} + 51^{40} + 97^{40}$$ is 4 in $$Z_{100}$$ using the totient function. The problem as i see it, is that 100 is not a prime number. i could of course go the long way $$(11^{2})^{20} = (21^{2})^{10}$$and so forth but that's a bit tedious...
/Jones
2. Note, $\phi (100) = 40$ thus if $\gcd(a,100) = 1$ then $a^{\phi(100)} = a^{40} \equiv 1 (\bmod 100)$.
Then, $\gcd(11,100) = \gcd(33,100) = \gcd(51,100) = \gcd(97,100) = 1$.
Thus, $11^{40}+33^{40}+51^{40} + 97^{40} \equiv 1+1+1+1 = 4(\bmod 100)$.
3. It doesnt have to be prime if you know the zeta function, the two numbers just have to have a gcd of 1.
4. Originally Posted by ThePerfectHacker
Note, $\phi (100) = 40$ thus if $\gcd(a,100) = 1$ then $a^{\phi(100)} = a^{40} \equiv 1 (\bmod 100)$.
Then, $\gcd(11,100) = \gcd(33,100) = \gcd(51,100) = \gcd(97,100) = 1$.
Thus, $11^{40}+33^{40}+51^{40} + 97^{40} \equiv 1+1+1+1 = 4(\bmod 100)$.
Thank you, how would this be different if we had for example $$25^{40}$$ when the gcd is not 1
5. You would have to treat numbers like 2, 5, 25, 26, ... as special cases and work them out by hand. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9390052556991577, "perplexity": 165.05576931134289}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549429548.55/warc/CC-MAIN-20170727222533-20170728002533-00681.warc.gz"} |
http://math.stackexchange.com/questions/88916/correspondences-between-borel-algebras-and-topological-spaces/89344 | # Correspondences between Borel algebras and topological spaces
Though tangentially related to another post on MathOverflow (here), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...
Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.
Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?
If the answer to the previous question is affirmative, it makes sense to ask for the following too:
Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?
-
A comment on Question 1: If $\kappa_1<\kappa_2$ are uncountable cardinals, $X$ is a set with cardinality $\kappa_2$, and $\Sigma$ is the set of subsets $S$ of $X$ such that $S$ or $X\setminus S$ has cardinality at most $\kappa_1$, then can $\Sigma$ be the $\sigma$-algebra generated by a topology? – Jonas Meyer Dec 7 '11 at 4:02
@Jonas: Can’t you simply take $\tau$ to be $\{\varnothing\}\cup\{V\subseteq X:|X\setminus V|\le\kappa_1\}$? – Brian M. Scott Dec 7 '11 at 7:13
OK, elephant in the room: assuming choice, the Lebesgue sigma-algebra on $\mathbb{R}$ is probably not a Borel sigma-algebra. Of course this makes problem 1 into one that depends on your set theory since word on the street is there exist models of ZF in which all subsets of $\mathbb{R}$ are measureable. – Mike F Dec 7 '11 at 8:03
@Mike, I posted an answer arguing that the Lebesgue algebra is not a counterexample. – JDH Dec 7 '11 at 19:56
George, can you post a proof that your counterexample is a counterexample? I don't quite see it yet... – JDH Dec 8 '11 at 0:07
This is a great question. I am posting not an answer to the question, but an answer to the comment posted by Mike about a proposed counterexample, the elephant in the room.
Theorem. The elephant in the room is not a counterexample. That is, the $\sigma$-algebra of Lebesgue measurable sets is the Borel algebra of the topology consisting of all sets of the form $O-N$, where $O$ is open in the usual topology and $N$ has measure $0$.
Proof. Note that the empty set and the whole of $\mathbb{R}$ have the desired form. Next, note that sets of that form are closed under finite intersection, since $(O-N)\cap(U-M)=(O\cap U)-(N\cup M)$. Next, I claim that they are closed under countable unions. This is because $\bigcup_i (O_i-N_i) = (\bigcup_i O_i)-N$, where $N$ is a certain subset of $\bigcup_iN_i$, which is still measure $0$ since this is a countable union. Next, note that it is closed under arbitrary unions in the case where the open set is the same, $\bigcup_i (O-N_i)$, since this is just the same as $O-(\bigcap_i N_i)$, which is $O$ minus a smaller null set. Finally, since we have a countable basis for the usual topology, using rational intervals, say, it follows now that my sets are closed under arbitrary unions, since for any sized union, we may rewrite it using basic open sets minus null sets, and then group together all the terms arising for each basic open set as a single term, thereby reducing the entire union to a countable union, which we've argued still has the desired form.
Thus, the sets of the form $O-N$ do indeed form a topology, and this topology is clearly contained within the Lebesgue algebra. Furthermore, the Borel algebra generated by my collection of sets includes all measure zero sets, as well as all open sets, and so it is the same as the algebra of all Lebesgue measurable sets. QED
-
Thanks for the explaining this. I'm not sure how exactly this came about, but I've been walking around for a few years now thinking that the Lebesgue $\sigma$-algebra was somehow bigger than the completion of the Borel $\sigma$-algebra (wrt Lebesgue measure) - but I guess they are the same thing! – Mike F Dec 8 '11 at 6:42
..yet somehow I was also aware that, when $S \subset \mathbb{R}$ is measureable, there exist $\mathscr{G}_\delta$ $G$ and $\mathscr{F}_\sigma$ $F$ such that $F \subset S \subset G$ and $G - F$ has measure zero. A couple of contradictory beliefs which somehow never bumped into each other. – Mike F Dec 8 '11 at 6:49
@JDH. It's your answer that is great. – Salvo Tringali Dec 9 '11 at 12:58
I think that I can answer the second question. For each point $p \in \mathbb{R}$, let $\tau_p$ be the topology on $\mathbb{R}$ consisting of $\varnothing$ together with all the standard open neighbourhoods of $p$. Unless I've made some mistake, the Borel sigma-algebra generated by $\tau_p$ is the standard one. However, $\bigcap_{p \in \mathbb{R}} \tau_p$ is the indiscrete topology on $\mathbb{R}$.
-
I can't see why the Borel sets would be the same as the standard one. – Asaf Karagila Dec 6 '11 at 20:45
Asaf, it's because $\{p\}$ is Borel in $\tau_p$, arising from the intersection of nested intervals centered on $p$. Once you've got $\{p\}$, then you can get all the open sets not containing $p$ simply by subtracting, and so you get all the ordinary open sets. Thus, the Borel sets of $\tau_p$ include all the usual Borel sets. (And of course, it is included in this, so they are equal.) – JDH Dec 6 '11 at 21:13
@JDH: Ah, I figured it was something related to that. I did not keep in mind that we have the standard topology in the background; rather than just a continuum-sized set. – Asaf Karagila Dec 6 '11 at 21:17
@Mike. So nice! I'm editing the OP to update it and add that Q2 was replied in the negative. – Salvo Tringali Dec 6 '11 at 21:41
@JDH: Well $\{p\}$ isn't quite the whole story since subtracting that only gives you standard open sets having $p$ as a 2-sided limit point. But you can also get any closed interval containing $p$ by intersecting nested open intervals - and subtracting these should give the rest of the standard open sets (or at least the open intervals omitting $p$ which is enough). – Mike F Dec 6 '11 at 23:48
For Q1. How about this. $X = \{0,1\}^A$ for uncountable $A$, and $\Sigma$ is the product $\sigma$-algebra. So each element of $\Sigma$ depends on only countably many coordinates.
Now we just need a proof that this cannot be the Borel algebra of any topology.
-
That is indeed true. I was about to post something along these lines and, in fact, I think you can convert it to a sigma algebra on A itself rather than on $2^A$. – George Lowther Dec 7 '11 at 21:38
I think you can show that there must exist an $x\in X$ such that $\{x\}$ is closed. But, $\{x\}\not\in\Sigma$. – George Lowther Dec 7 '11 at 21:49
@George: Of course this needs more work to show that... Our space is $T_0$ but not $T_1$. – GEdgar Dec 7 '11 at 22:19
Yes, I was too quick there. What I said is not true in the case where A is countably infinite, because $2^A$ is isomorphic (as a measurable space) to the reals with Borel sigma-algebra generated by the topology $\{(x,\infty):x\in R\}$, for which no $\{x\}$ is closed. – George Lowther Dec 8 '11 at 1:24
EDIT: It's best just to ignore this nonsensical post.
The product space mentioned in the answer by Gerald Edgar does not provide a counterexample. The product $\sigma$-algebra is determined by countably many coordinate, the product topology by finitely many coordinates. A set determined by countably many coordiantes is a countable intersection of sets determined by finitely many coordinates.
Remark: This answer contained a flawed "proof" that the answer to the first question is yes. I've edited it.
-
If you still claim my space $X = \{0,1\}^A$ is not a counterexample, you need more explanation. Certainly that product sigma-algebra is not the Borel algebra for the product topology. But is it the Borel algebra for some other topology? That is the question! – GEdgar Dec 24 '11 at 17:16
You are right of course, the Borel sigma-algebra is too large. I'm sorry. Explaining what my confusion was would take more than the number of characters left... – Michael Greinecker Dec 27 '11 at 0:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9577801823616028, "perplexity": 196.93584913182028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398446500.34/warc/CC-MAIN-20151124205406-00232-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.molpro.net/info/2015.1/doc/manual/node480.html | ## 37.3 DFT-SAPT
It is of crucial importance to account for the intramolecular correlation effects of the individual SAPT terms since Hartree-Fock theory often yields poor first- and second-order electrostatic properties. While this can be done using many-body perturbation theory [1] (in a double perturbation theory ansatz) a more efficient way is to use static and time-dependent DFT theory. This variant of SAPT, termed as DFT-SAPT [2-6], has in contrast to Hartree-Fock-SAPT the appealing feature that the polarisation terms ( , , ) are potentially exact, i.e. they come out exactly if the exact exchange-correlation (xc) potential and the exact (frequency-dependent) xc response kernel of the monomers were known. On the other hand, this does not hold for the exchange terms since Kohn-Sham theory can at best give a good approximation to the exact density matrix of a many-body system. It has been shown [6] that this is indeed the the case and therefore DFT-SAPT has the potential to produce highly accurate interaction energies comparable to high-level supermolecular many-body perturbation or coupled cluster theory. However, in order to achieve this accuracy, it is of crucial importance to correct the wrong asymptotic behaviour of the xc potential in current DFT functionals [3-5]. This can be done by using e.g.:
{ks,lda; asymp,<shift>}
which activates the gradient-regulated asymptotic correction approach of Grüning et al. (J. Chem. Phys. 114, 652 (2001)) for the respective monomer calculation. The user has to supply a shift parameter ( ) for the bulk potential which should approximate the difference between the HOMO energy ( ) obtained from the respective standard Kohn-Sham calculation and the (negative) ionisation potential of the monomer ():
(57)
This method accounts for the derivative discontinuity of the exact xc-potential and that is missing in approximate ones. Note that this needs to be done only once for each system. (See also section 37.7.2 for an explicit example).
Concerning the more technical parameters in the DFT monomer calculations it is recommended to use lower convergence thresholds and larger intergration grids compared to standard Kohn-Sham calculations.
molpro@molpro.net 2019-03-18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9085679054260254, "perplexity": 898.9586677401452}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201953.19/warc/CC-MAIN-20190319093341-20190319115341-00322.warc.gz"} |
http://mathhelpforum.com/calculus/118588-evaluate-definite-integral-print.html | # Evaluate the definite Integral
• December 5th 2009, 12:51 AM
zorro
Evaluate the definite Integral
Question : Evaluate
$
\int_{-1}^{1} \ \frac{3 + sin^3 x}{1 + x^2} \ dx
$
• December 5th 2009, 01:08 AM
flyingsquirrel
Quote:
Originally Posted by zorro
Question : Evaluate
$
\int_{-1}^{1} \ \frac{3 + sin^3 x}{1 + x^2} \ dx
$
$
\int_{-1}^{1} \frac{3 + \sin^3 x}{1 + x^2} \ \mathrm{d}x=3\int_{-1}^{1} \frac{1}{1 + x^2} \ \mathrm{d}x+\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x
$
Hint : note that $x\mapsto \frac{\sin^3 x}{1 + x^2}$ is odd.
• December 8th 2009, 12:27 AM
zorro
Stuck again !!!
Quote:
Originally Posted by flyingsquirrel
$
\int_{-1}^{1} \frac{3 + \sin^3 x}{1 + x^2} \ \mathrm{d}x=3\int_{-1}^{1} \frac{1}{1 + x^2} \ \mathrm{d}x+\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x
$
Hint : note that $x\mapsto \frac{\sin^3 x}{1 + x^2}$ is odd.
I am able to integrate the first term but not the last one
= $3 \int_{-1}^{1} \frac{1}{1 + x^2} + \int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$
= $I_1 + I_2$
$I_1$ = $3 \left[ tan^{-1} (1) - tan^{-1} (-1) \right] + c_1$
= $3 \left[ tan^{-1} (1) - tan^{-1} (-1) \right] + c_1$
= $3 \left[ \frac{\pi}{4} +\frac{\pi}{4} \right] + c_1$
= $3 \left[ \frac{2 \pi}{4} \right] + c_1$
= $\left[ \frac{6 \pi}{4} \right] + c_1$
$I_2$ = $\int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$ $dx_2$
= $\int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$ $dx_2$
I am stuck here!!!!!......
• December 8th 2009, 01:59 AM
dedust
Quote:
Originally Posted by flyingsquirrel
Hint : note that $x\mapsto \frac{\sin^3 x}{1 + x^2}$ is odd.
so $\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x
= 0$
• December 8th 2009, 02:03 AM
flyingsquirrel
Quote:
Originally Posted by zorro
$I_1$ = $3 \left[ tan^{-1} (1) - tan^{-1} (-1) \right] + c_1$
There is no need to add a constant since $\int_{-1}^1\frac{1}{1+x^2}\,\mathrm{d}x$ is a definite integral.
Quote:
Originally Posted by zorro
= $\left[ \frac{6 \pi}{4} \right] + c_1$
$\frac{6\pi}{4}=\frac{3\times 2\pi}{2\times 2}=\frac{3\pi}{2}$
Quote:
Originally Posted by zorro
$I_2$ = $\int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$ $dx_2$
$I_2 = \int_{-1}^0\frac{\sin^3x}{1+x^2}\,\mathrm{d}x + \int_0^1\frac{\sin^3x}{1+x^2}\,\mathrm{d}x$
Substitute $x=-t$ ( $\implies t=-x$ and $\mathrm{d}t=-\mathrm{d}x$) in the first integral :
$\int_{-1}^0\frac{\sin^3x}{1+x^2}\,\mathrm{d}x =-\int_{1}^0\frac{\sin^3(-t)}{1+(-t)^2}\,\mathrm{d}t=\int_0^1\frac{-\sin^3t}{1+t^2}\,\mathrm{d}t=-\int_0^1\frac{\sin^3t}{1+t^2}\,\mathrm{d}t$
Thus
$I_2 = -\int_0^1\frac{\sin^3t}{1+t^2}\,\mathrm{d}t+\int_0^ 1\frac{\sin^3x}{1+x^2}\,\mathrm{d}x=0.$
This is a special case of a more general result: Let $f:\mathbb{R}\to\mathbb{R}$ be an odd function and let $A$ be a real number. Then
$\int_{-A}^Af(t)\,\mathrm{d}t=0.$
Edit :
Quote:
Originally Posted by dedust
so $\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x
= 0$
Yes !
• December 8th 2009, 02:17 AM
zorro
Thanks u very much
Thank u all very much for helping me(Bow)
• December 8th 2009, 05:48 AM
Krizalid
as for that property about the odd function, we also require that $f$ must be integrable there.
• December 8th 2009, 12:01 PM
zorro
what do u mean by that
Quote:
Originally Posted by Krizalid
as for that property about the odd function, we also require that $f$ must be integrable there.
could u please explain it in non mathematical terms ,as i m not that bright??
• December 21st 2009, 03:30 PM
zorro
Got it
Quote:
Originally Posted by zorro
could u please explain it in non mathematical terms ,as i m not that bright??
Thanks u all for helping (Beer) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 41, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9349957704544067, "perplexity": 2237.988376845269}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1433195035744.13/warc/CC-MAIN-20150601214355-00036-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://quant.stackexchange.com/questions/55516/confusion-about-optimal-choices-with-exotic-options | # Confusion about optimal choices with exotic options
With exotic options, holders usually face choices at certain times. In my understanding, the price of the option is determined by assuming the optimal choice is taken and computing the discounted expectation of the payoff under the risk neutral measure with backward induction. My question is if I hold such options, how does taking the optimal choice benefit me? Such optimal choices are optimal under the risk neutral measure, so how does taking such choices imply/guarantee (even probabilistically) anything under the actual measure? If I take the optimal choice as determined by the pricing model, that would maximize the arbitrage-free price of the option that I am holding, but does that bear any optimality in the actual world? Does that tend to lead to a higher P/L? Say the option has intrinsic value $$x$$, extrinsic value $$y$$, and I just happen to hold this option, not for hedging purposes and without any particular view on the underlying. How can I benefit from the optionality/extrinsic value? I understand that I would lose some of the extrinsic value if I don't follow the optimal choice, but that's just a theoretical construct, where would I see the manifestation of this "loss of value"?
Consider a vanilla european option. The optimal strategy in the risk neutral measure is to exercise if $$S(T)>K$$. (because in the risk neutral world, I value my payoff at its expectation, which at that point is the payoff itself. At time T, if $$S(T)-K$$ is positive, I will take it over not exercising, which is a payoff of 0.
This is of course also optimal in the real world - more money is better than less money. So my 'optimal exercise strategy' matches.
The manifestation of loss in value is a PnL leak. Consider the same example as above. Say $$S(T)>K$$ but you don't exercise your option at all: so you've just paid something for the option, but sub-optimal exercise strategy means that you've got nothing. This is an extreme example of course but you see the idea.
The same idea extends to exotics. Consider a Bermudan with 2 exercise dates. Denote by E the immediate exercise value at the first date, and by C the continuation value. Denote by $$w$$ the state of the world, and $$T$$ denotes the 1st exercise date, and $$T_ex$$ denote the date I exercise my option on (which is random).
Let $$A={w:E(w)>C(w)}$$. Then $$Pr[(T_ex=T)|A]=1$$ in the risk neutral measure. Since real world and risk neutral measure are equivalent (they agree on what's possible and not), we get $$Pr[(T_ex=T)|A]=1$$ in the real world. This tells you that you will exercise in the real world exactly when you exercise in the 'risk neutral measure'. Which tells you that the 'optimal strategy' is exactly the same.
Appendix:
The proof of conditional probabilities being the same in equivalent measures is better thought of by considering (assume $$Pr(A)>0$$ and $$Pr(B)>0$$) say $$Pr(B|A)=0$$ in the R.N. measure. Then $$Pr(B,A)=0 => Pr(B,A)=0$$ in the real world measure. As $$Pr(A)>0$$, we must have $$Pr(B|A)=0$$ in the real world measure too. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.940972089767456, "perplexity": 554.8916297896764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487616657.20/warc/CC-MAIN-20210615022806-20210615052806-00346.warc.gz"} |
http://mathhelpforum.com/calculus/92311-trigonometric-limit-print.html | # Trigonometric limit?
• Jun 9th 2009, 08:20 AM
fardeen_gen
Trigonometric limit?
Evaluate:
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x}$
• Jun 9th 2009, 11:14 AM
sinewave85
Quote:
Originally Posted by fardeen_gen
Evaluate:
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x}$
It would seem that:
$\lim_{x\rightarrow 0+} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} = (+\infty)^{0} = 1$
and
$\lim_{x\rightarrow 0-} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} = (-\infty)^{0} = 1$
so
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} = 1$
• Jun 9th 2009, 11:32 AM
Moo
Quote:
Originally Posted by sinewave85
It would seem that:
$\lim_{x\rightarrow 0+} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} = (+\infty)^{0} = 1$
But this is not defined (Worried)
Quote:
and
$\lim_{x\rightarrow 0-} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} = (-\infty)^{0} = 1$
How can something positive have a negative limit ?
Quote:
so
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} = 1$
...
• Jun 9th 2009, 11:36 AM
sinewave85
Quote:
Originally Posted by Moo
But this is not defined (Worried)
How can something positive have a negative limit ?
...
Very sorry -- I forgot that $\infty^{0}$ was undefined. Thanks for catching that, Moo. I would hate to lead someone astray.
• Jun 9th 2009, 09:34 PM
fardeen_gen
How do we deal with $\infty^{0}$ forms? This problem has been giving me a headache.
• Jun 10th 2009, 03:55 AM
Ruun
$\frac{\sin^2(x)}{\sin^2(x)} (\frac{1}{\sin^2(x)}+\frac{1}{2\sin^2(x)}+ \mbox(...) +\frac{1}{n \sin^2(x)})^{\sin^2(x)}=\frac{1}{\sin^2(x)}(H_n)^{ \sin^2(x)}=L$
$ln (L) = ln(H_n)$
Harmonic number - Wikipedia, the free encyclopedia
• Jun 10th 2009, 04:32 AM
Isomorphism
Quote:
Originally Posted by fardeen_gen
Evaluate:
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x}$
Using the fact that $\sin x \approx x$, for small x, we can write:
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} =$
$\lim_{t \rightarrow \infty} \left(1^{t^2} + 2^{t^2} + 3^{t^2} + \mbox{...} + n^{t^2}\right)^{\frac1{t^2}}$
Taking logs and apply LHospital's rule, we have to evaluate $\lim_{t \rightarrow \infty} \frac{\log \left(1^{t^2} + 2^{t^2} + 3^{t^2} + \mbox{...} + n^{t^2}\right)}{t^2}$ to get $\log n!$, Finally the answer is n!
• Jun 11th 2009, 11:40 AM
Random Variable
Quote:
Originally Posted by Isomorphism
Using the fact that $\sin x \approx x$, for small x, we can write:
$\lim_{x\rightarrow 0} \left(1^{\frac{1}{\sin^2 x}} + 2^{\frac{1}{\sin^2 x}} + 3^{\frac{1}{\sin^2 x}} + \mbox{...} + n^{\frac{1}{\sin^2 x}}\right)^{\sin^2 x} =$ $\lim_{t \rightarrow \infty} \left(1^{t^2} + 2^{t^2} + 3^{t^2} + \mbox{...} + n^{t^2}\right)^{\frac1{t^2}}$
Taking logs and apply LHospital's rule, we have to evaluate $\lim_{t \rightarrow \infty} \frac{\log \left(1^{t^2} + 2^{t^2} + 3^{t^2} + \mbox{...} + n^{t^2}\right)}{t^2}$ to get $\log n!$, Finally the answer is n!
After one application of L'Hospital's rule, I get
$\lim_{t \to \infty} \frac {\ln(1)1^{t^{2}} + \ln(2)2^{t^{2}} + \ln(3)3^{t^{2}} + ... + \ln(n)n^{t^{2}}}{1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}}}$
then it would appear you did the following
$\lim_{t \to \infty} \frac {\big(\ln(1) +\ln(2) + \ln(3) + ... + \ln(n)\big)(1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}})}{ 1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}}}$ ?
$= \lim_{t \to \infty} \ln(1*2*3*...*n)$ which would be correct if the previous step were correct
What am I missing?
• Jun 11th 2009, 09:38 PM
Isomorphism
Quote:
Originally Posted by Random Variable
After one application of L'Hospital's rule, I get
$\lim_{t \to \infty} \frac {\ln(1)1^{t^{2}} + \ln(2)2^{t^{2}} + \ln(3)3^{t^{2}} + ... + \ln(n)n^{t^{2}}}{1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}}}$
then it would appear you did the following
$\lim_{t \to \infty} \frac {\big(\ln(1) +\ln(2) + \ln(3) + ... + \ln(n)\big)(1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}})}{ 1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}}}$ ?
$= \lim_{t \to \infty} \ln(1*2*3*...*n)$ which would be correct if the previous step were correct
What am I missing?
(Tmi) You are not missing anything...Its wrong and was hastily written(I applied $t \to 0$)
Here is the solution:
$\lim_{t \to \infty} \frac {\ln(1)1^{t^{2}} + \ln(2)2^{t^{2}} + \ln(3)3^{t^{2}} + ... + \ln(n)n^{t^{2}}}{1^{t^{2}} + 2^{t^{2}} + 3^{t^{2}} + ... + n^{t^{2}}}$
$= \lim_{t \to \infty} \dfrac {0 + \ln(2)\left(\frac2{n}\right)^{t^{2}} + \ln(3)\left(\frac3{n}\right)^{t^{2}} + ... + \ln(n)}{\left(\frac1{n}\right)^{t^{2}} + \left(\frac2{n}\right)^{t^{2}} + \left(\frac3{n}\right)^{t^{2}} + ... + 1}$
$=\ln(n)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 33, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9888268709182739, "perplexity": 3372.0109967496346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891811795.10/warc/CC-MAIN-20180218081112-20180218101112-00522.warc.gz"} |
https://asmedigitalcollection.asme.org/GT/proceedings-abstract/GT2019/58615/V04AT04A053/1066744?redirectedFrom=PDF | ## Abstract
Modern gas turbines usually adopt very lean premixed flames to meet the current strict law restrictions on nitric oxides emissions. In such devices, strong combustion instabilities and blow-off susceptibility often prevent from achieving a stable flame in leaner conditions. Numerical models to predict the lean blow-off in turbulent flames are essential to prevent such instabilities, but the simulation of blow-off still represents a challenge, requiring the appropriate modelling for the turbulence-chemistry interactions and the highly transient behaviour of the flame near the extinction limit.
The present work explores the capabilities of the widely-used Flamelet Generated Manifold model in predicting the lean blow-off of a turbulent swirl-stabilized premixed flame within LES framework. An atmospheric premixed methane-air flame, experimentally studied at the University of Cambridge, is firstly analyzed in three operating conditions approaching blow-off to validate the numerical setup. An extended Turbulent Flame Closure (TFC) model, implemented within the FGM framework in Fluent to introduce the effect of stretch and heat loss on the flame, reproduces the evolution of the key flame characteristics. Then, the chosen setup is used to study the blow-off inception and the dynamics in two conditions with different flow rate. An accelerated numerical procedure with progressive step reductions of equivalence ratio is used to trigger the blow-off. The extinction equivalence ratio is predicted quite accurately, showing that the Extended TFC is suitable for the study of the blow-off, without an increase in computational cost. The validity of the model could be extended, allowing the study of lean blow-off in realistic conditions and complex flames of gas turbine combustors.
This content is only available via PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8106809258460999, "perplexity": 2899.357992053838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335190.45/warc/CC-MAIN-20220928082743-20220928112743-00012.warc.gz"} |
https://www.h2knowledgecentre.com/content/conference746 | 1900
# Blast Wave from Bursting Enclosure with Internal Hydrogen-air Deflagration
### Abstract
Most studies on blast waves generated by gas explosions have focused on gas explosions occurring in open spaces. However, accidental gas explosions often occur in confined spaces and the blast wave generates from a bursting vessel as a result of an increase in pressure caused by the gas explosion. In this study, blast waves from bursting plastic vessels in which gas explosions occurred are investigated. The flammable mixtures used in the experiments were hydrogen-air mixtures at several equivalence ratios and a stoichiometric methane-air mixture. The overpressures of the blast waves were generated by venting high-pressure gas in the enclosure and volumetric expansion with a combustion reaction. The measured intensities of the blast waves were greater than the calculated values resulting from high-pressure bursting without a combustion reaction. The intensities of the blast waves resulting from the explosions of hydrogen-air mixtures were much greater than those of the methane-air mixture.
Keywords:
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2015-10-19
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#### Blast wave from bursing enclosure with internal hydrogen-air deflagration
This is a required field | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9012472629547119, "perplexity": 3541.3679363052192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00115.warc.gz"} |
https://api-project-1022638073839.appspot.com/questions/how-do-you-integrate-int-x-5-x-3-x-7-x-5-using-partial-fractions | Calculus
Topics
# How do you integrate int (x+5)/((x+3)(x-7)(x-5)) using partial fractions?
int (x+5)/((x+3)(x-7)(x-5) $\mathrm{dx}$
$= \frac{1}{40} \cdot \ln \left(x + 3\right) + \frac{3}{5} \cdot \ln \left(x - 7\right) - \frac{5}{8} \cdot \ln \left(x - 5\right) + K$
for a definite integral, a constant K must be added.
#### Explanation:
from the given $\int \frac{x + 5}{\left(x + 3\right) \left(x - 7\right) \left(x - 5\right)}$ $\mathrm{dx}$ =
$\int \frac{A}{x + 3}$$\mathrm{dx}$ + $\frac{B}{x - 7}$$\mathrm{dx}$ + $\frac{C}{x - 5}$*$\mathrm{dx}$
our equation becomes
(x+5)/((x+3)(x-7)(x-5)= $\frac{A}{x + 3} + \frac{B}{x - 7} + \frac{C}{x - 5}$
it follows;
(x+5)/((x+3)(x-7)(x-5)=
$\frac{A \left(x - 7\right) \left(x - 5\right) + B \left(x + 3\right) \left(x - 5\right) + C \left(x + 3\right) \left(x - 5\right)}{\left(x + 3\right) \left(x - 5\right) \left(x - 7\right)}$
by using only the numerators;
x+5=A(x^2-12x+35)+B(x^2-2x-15)+C(x^2-4x-21)
collecting the like terms
$x + 5 = \left(A + B + C\right) {x}^{2} + \left(- 12 A - 2 B - 4 C\right) x + \left(35 A - 15 B - 21 C\right)$
because the left side of the equation means
$0 \cdot {x}^{2} + 1 \cdot x + 5$
and the right side means
$\left(A + B + C\right) {x}^{2} + \left(- 12 A - 2 B - 4 C\right) x + \left(35 A - 15 B - 21 C\right)$
the equations are now formed;
$A + B + C = 0$ first equation
$- 12 A - 2 B - 4 C = 1$ second equation
$35 A - 15 B - 21 C = 5$ third equation
using your skills in solving 3 equations with 3 unknowns A, B, C
the values are $A = \frac{1}{40}$, $B = \frac{3}{5}$, and $C = - \frac{5}{8}$
Now, go back to the first line of the explanation to do the integration procedures.
$\int \frac{A}{x + 3}$$\mathrm{dx}$ + $\frac{B}{x - 7}$$\mathrm{dx}$ + $\frac{C}{x - 5}$*$\mathrm{dx}$
$= \int \frac{\frac{1}{40}}{x + 3}$$\mathrm{dx}$ + $\int \frac{\frac{3}{5}}{x - 7}$$\mathrm{dx}$ + $\int \frac{- \frac{5}{8}}{x - 5}$$\mathrm{dx}$
int (x+5)/((x+3)(x-7)(x-5) $\mathrm{dx}$
$= \frac{1}{40} \cdot \ln \left(x + 3\right) + \frac{3}{5} \cdot \ln \left(x - 7\right) - \frac{5}{8} \cdot \ln \left(x - 5\right) + K$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 40, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9983240365982056, "perplexity": 2813.8384603081076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585199.76/warc/CC-MAIN-20211018062819-20211018092819-00349.warc.gz"} |
https://mathoverflow.net/questions/194775/groupoid-cardinality-of-dm-stack-and-point-counting-on-coarse-moduli-spaces | # Groupoid cardinality of DM stack and point counting on coarse moduli spaces
Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which conditions is $\# X(k) = \# X_c(k)$?
Here we take the groupoid cardinality/stacky sum/mass formula to define $\#X(k)$: $$\# X(k) := \sum_{x\in X(k)} \frac{1}{\# Aut(x)}.$$
I think this equality holds if $X$ is a separated finite type DM stack whose coarse moduli space is a scheme (and not just an algebraic space).
Does it hold more generally, ie, only assuming the existence of $X_c$ and not any other properties?
• How do you prove it in the case you mention? – Mattia Talpo Feb 5 '15 at 7:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9321514964103699, "perplexity": 299.92745908408193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578721468.57/warc/CC-MAIN-20190425134058-20190425160058-00489.warc.gz"} |
https://www.physicsforums.com/threads/math-log-proof.21341/ | # Math log proof
1. Apr 18, 2004
### gnome
Would someone please show me why
$${ a^{log_cb} = b^{log_ca}$$
for all a, b and c.
2. Apr 18, 2004
### deltabourne
I don't know if this is really a proof.. but just take the log of both sides:
$${ a^{log_cb} = b^{log_ca}$$
$$log_c { a^{log_cb} = log_c b^{log_ca}$$
$$({log_cb})({log_ca}) = ({log_ca})({log_cb})$$
using the property that:
$${log_ca^r} = r{log_ca}$$
3. Apr 18, 2004
### matt grime
it is a proof. perhaps to make it appear more rigorous, you could write $$a^{log_cb}=c^{log_c(a^{log_cb})}$$ and similarly for the rhs, and say that the only way for c^xto be equal to c^y is if x=y.
4. Apr 18, 2004
### gnome
Got it!
Thanks, folks.
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https://rojefferson.blog/2019/10/27/interior-operators-in-ads-cft/ | In a previous post, I mentioned that the firewall paradox could be phrased as a question about the existence of interior operators that satisfy the correct thermal correlation functions, namely
$\displaystyle \langle\Psi|\mathcal{O}(t,\mathbf{x})\tilde{\mathcal{O}}(t',\mathbf{x}')|\Psi\rangle =Z^{-1}\mathrm{tr}\left[e^{-\beta H}\mathcal{O}(t,\mathbf{x})\mathcal{O}(t'+i\beta/2,\mathbf{x}')\right]~, \ \ \ \ \ (1)$
where ${\tilde{\mathcal{O}}}$ and ${\mathcal{O}}$ operators inside and outside the black hole, respectively; cf. eqn. (2) here. In this short post, I’d like to review the basic argument leading up to this statement, following the original works [1,2].
Consider the eternal black hole in AdS as depicted in the following diagram, which I stole from [1]:
The blue line connecting the two asymptotic boundaries is the Cauchy slice on which we’ll construct our states, denoted ${\Sigma_I}$ in exterior region ${I}$ and ${\Sigma_{III}}$ in exterior region ${III}$. Note that, modulo possible UV divergences at the origin, either half serves as a complete Cauchy slice if we restrict our inquiries to the associated exterior region. But if we wish to reconstruct states in the interior (henceforth just ${II}$, since we don’t care about ${IV}$), then we need the entire slice. Pictorially, one can see this from the fact that only the left-moving modes from region ${I}$, and the right-moving modes from region ${III}$, cross the horizon into region ${II}$, but we need both left- and right-movers to have a complete mode decomposition.
To expand on this, imagine we proceed with the quantization of a free scalar field in region ${I}$. We need to solve the Klein-Gordon equation,
$\displaystyle \left(\square-m^2\right)\phi=\frac{1}{\sqrt{-g}}\,\partial_\mu\left( g^{\mu\nu}\sqrt{-g}\,\partial_\nu\phi\right)-m^2\phi=0 \ \ \ \ \ (2)$
on the AdS black brane background,
$\displaystyle \mathrm{d} s^2=\frac{1}{z^2}\left[-h(z)\mathrm{d} t^2+\mathrm{d} z^2+\mathrm{d}\mathbf{x}^2\right]~, \quad\quad h(z)\equiv1-\left(\frac{z}{z_0}\right)^d~. \ \ \ \ \ (3)$
where, in Poincaré coordinates, the asymptotic boundary is at ${z\!=\!0}$, and the horizon is at ${z\!=\!z_0}$. We work in ${(d+1)-}$spacetime dimensions, so ${\mathbf{x}}$ is a ${(d\!-\!1)}$-dimensional vector representing the transverse coordinates. Note that we’ve set the AdS radius to 1. Substituting the usual plane-wave ansatz
$\displaystyle f_{\omega,\mathbf{k}}(t,\mathbf{x},z)=e^{-i\omega t+i\mathbf{k}\mathbf{x}}\psi_{\omega,\mathbf{k}(z)} \ \ \ \ \ (4)$
into the Klein-Gordon equation results in a second order ordinary differential equation for the radial function ${\psi_{\omega,\mathbf{k}}(z)}$, and hence two linearly independent solutions. As usual, we then impose normalizable boundary conditions at infinity, which leaves us with a single linear combination for each ${(\omega,\mathbf{k})}$. Note that we do not impose boundary conditions at the horizon. Naïvely, one might have thought to impose ingoing boundary conditions there; however, as remarked in [1], this precludes the existence of real ${\omega}$. More intuitively, I think of this as simply the statement that the black hole is evaporating, so we should allow the possibility for outgoing modes as well. (That is, assuming a large black hole in AdS, the black hole is in thermal equilibrium with the surrounding environment, so the outgoing and ingoing fluxes are precisely matched, and it maintains constant size). The expression for ${\psi_{\omega,\mathbf{k}}(z)}$ is not relevant here; see [1] for more details.
We thus arrive at the standard expression of the (bulk) field ${\phi}$ in terms of creation and annihilation operators,
$\displaystyle \phi_I(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{\sqrt{2\omega}(2\pi)^d}\,\bigg[ a_{\omega,\mathbf{k}}\,f_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathrm{h.c.}\bigg]~, \ \ \ \ \ (5)$
where the creation/annihilation operators for the modes may be normalized with respect to the Klein-Gordon norm, so that
$\displaystyle [a_{\omega,\mathbf{k}},a^\dagger_{\omega',\mathbf{k}'}]=\delta(\omega-\omega')\delta^{d-1}(\mathbf{k}-\mathbf{k}')~. \ \ \ \ \ (6)$
Of course, a similar expansion holds for region ${III}$:
$\displaystyle \phi_{III}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{\sqrt{2\omega}(2\pi)^d}\,\bigg[\tilde a_{\omega,\mathbf{k}}\,g_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathrm{h.c.}\bigg]~, \ \ \ \ \ (7)$
where the mode operators ${\tilde a_{\omega,\mathbf{k}}}$ commute with all ${a_{\omega,\mathbf{k}}}$ by construction.
Now, what of the future interior, region ${II}$? Unlike the exterior regions, we no longer have any boundary condition to impose, since every Cauchy slice which crosses this region is bounded on both sides by a future horizon. Consequently, we retain both the linear combinations obtained from the Klein-Gordon equation, and hence have twice as many modes as in either ${I}$ or ${III}$—which makes sense, since the interior receives contributions from both exterior regions. Nonetheless, it may be a bit confusing from the bulk perspective, since any local observer would simply arrive at the usual mode expansion involving only a single set of creation/annihilation operators, and I don’t have an intuition as to how ${a_{\omega,\mathbf{k}}}$ and ${\tilde a_{\omega,\mathbf{k}}}$ relate vis-à-vis their commutation relations in this shared domain. However, the entire framework in which the interior is fed by two exterior regions is only properly formulated in AdS/CFT, in which — it is generally thought — the interior region emerges from the entanglement structure between the two boundaries, so I prefer to uplift this discussion to the CFT before discussing the interior region in detail. This avoids the commutation confusion above — since the operators live in different CFTs — and it was the next step in our analysis anyway. (Incidentally, appendix B of [1] performs the mode decomposition in all three regions explicitly for the case of Rindler space, which provides a nice concrete example in which one can get one’s hands dirty).
So, we want to discuss local bulk fields from the perspective of the boundary CFT. From the extrapolate dictionary, we know that local bulk operators become increasingly smeared over the boundary (in both space and time) the farther we move into the bulk. Thus in region ${I}$, we can construct the operator
$\displaystyle \phi^I_{\mathrm{CFT}}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{(2\pi)^d}\,\bigg[\mathcal{O}_{\omega,\mathbf{k}}\,f_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathcal{O}^\dagger_{\omega,\mathbf{k}}f^*_{\omega,\mathbf{k}}(t,\mathbf{x},z)\bigg]~, \ \ \ \ \ (8)$
which, while a non-local operator in the CFT (constructed from local CFT operators ${\mathcal{O}_{\omega,\mathbf{k}}}$ which act as creation operators of light primary fields), behaves like a local operator in the bulk. Note that from the perspective of the CFT, ${z}$ is an auxiliary coordinate that simply parametrizes how smeared-out this operator is on the boundary.
As an aside, the critical difference between (8) and the more familiar HKLL prescription [3-5] is that the former is formulated directly in momentum space, while the latter is defined in position space as
$\displaystyle \phi_{\mathrm{CFT}}(t,\mathbf{x},z)=\int\!\mathrm{d} t'\mathrm{d}^{d-1}\mathbf{x}'\,K(t,\mathbf{x},z;t',\mathbf{x}')\mathcal{O}(t',\mathbf{x}')~, \ \ \ \ \ (9)$
where the integration kernel ${K}$ is known as the “smearing function”, and depends on the details of the spacetime. To solve for ${K}$, one performs a mode expansion of the local bulk field ${\phi}$ and identifies the normalizable mode with the local bulk operator ${\mathcal{O}}$ in the boundary limit. One then has to invert this relation to find the bulk mode operator, and then insert this into the original expansion of ${\phi}$. The problem now is that to identify ${K}$, one needs to swap the order of integration between position and momentum space, and the presence of the horizon results in a fatal divergence that obstructs this maneuver. As discussed in more detail in [1] however, working directly in momentum space avoids this technical issue. But the basic relation “smeared boundary operators ${\longleftrightarrow}$ local bulk fields” is the same.
Continuing, we have a similar bulk-boundary relation in region ${III}$, in terms of operators ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ living in the left CFT:
$\displaystyle \phi^{III}_{\mathrm{CFT}}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{(2\pi)^d}\,\bigg[\tilde{\mathcal{O}}_{\omega,\mathbf{k}}\,f_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\tilde{\mathcal{O}}^\dagger_{\omega,\mathbf{k}}f^*_{\omega,\mathbf{k}}(t,\mathbf{x},z)\bigg]~. \ \ \ \ \ (10)$
Note that even though I’ve used the same coordinate labels, ${t}$ runs backwards in the left wedge, so that ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ plays the role of the creation operator here. From the discussion above, the form of the field in the black hole interior is then
$\displaystyle \phi^{II}_{\mathrm{CFT}}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{(2\pi)^d}\,\bigg[\mathcal{O}_{\omega,\mathbf{k}}\,g^{(1)}_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\tilde{\mathcal{O}}_{\omega,\mathbf{k}}g^{(2)}_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathrm{h.c}\bigg]~, \ \ \ \ \ (11)$
where ${\mathcal{O}_{\omega,\mathbf{k}}}$ and ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ are the (creation/annihilation operators for the) boundary modes in the right and left CFTs, respectively. The point is that in order to construct a local field operator behind the horizon, both sets of modes — the left-movers ${\mathcal{O}_{\omega,\mathbf{k}}}$ from ${I}$ and the right-movers ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ from ${III}$ — are required. In the eternal black hole considered above, the latter originate in the second copy of the CFT. But in the one-sided case, we would seem to have only the left-movers ${\mathcal{O}_{\omega,\mathbf{k}}}$, hence we arrive at the crucial question: for a one-sided black hole — such as that formed from collapse in our universe — what are the interior modes ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$? Equivalently: how can we represent the black hole interior given access to only one copy of the CFT?
To answer this question, recall that the thermofield double state,
$\displaystyle |\mathrm{TFD}\rangle=\frac{1}{\sqrt{Z_\beta}}\sum_ie^{-\beta E_i/2}|E_i\rangle\otimes|E_i\rangle~, \ \ \ \ \ (12)$
is constructed so that either CFT appears exactly thermal when tracing out the other side, and that this well-approximates the late-time thermodynamics of a large black hole formed from collapse. That is, the exterior region will be in the Hartle-Hawking vacuum (which is to Schwarzschild as Rindler is to Minkowski), with the temperature ${\beta^{-1}}$ of the CFT set by the mass of the black hole. This implies that correlation functions of operators ${\mathcal{O}}$ in the pure state ${|\mathrm{TFD}\rangle}$ may be computed as thermal expectation values in their (mixed) half of the total Hilbert space, i.e.,
$\displaystyle \langle\mathrm{TFD}|\mathcal{O}(t_1,\mathbf{x}_1)\ldots\mathcal{O}(t_n,\mathbf{x}_n)|\mathrm{TFD}\rangle =Z^{-1}_\beta\mathrm{tr}\left[e^{-\beta H}\mathcal{O}(t_1,\mathbf{x}_1)\ldots\mathcal{O}(t_n,\mathbf{x}_n)\right]~. \ \ \ \ \ (13)$
The same fundamental relation remains true in the case of the one-sided black hole as well: given the Hartle-Hawking state representing the exterior region, we can always obtain a purification such that expectation values in the original, thermal state are equivalent to standard correlators in the “fictitious” pure state, by the same doubling formalism that yielded the TFD. (Of course, the purification of a given mixed state is not unique, but as pointed out in [2] “the correct way to pick it, assuming that expectation values [of the operators] are all the information we have, is to pick the density matrix which maximizes the entropy.” That is, we pick the purification such that the original mixed state is thermal, i.e., ${\rho\simeq Z^{-1}_\beta e^{-\beta H}}$ up to ${1/N^2}$ corrections. The reason this is the “correct” prescription is that it’s the only one which does not impose additional constraints.) Thus (13) can be generally thought of as the statement that operators in an arbitrary pure state have the correct thermal expectation values when restricted to some suitably mixed subsystem (e.g., the black hole exterior dual to a single CFT).
Now, what if we wish to compute a correlation function involving operators across the horizon, e.g., ${\langle\mathcal{O}\tilde{\mathcal{O}}\rangle}$? In the two-sided case, we can simply compute this correlator in the pure state ${|\mathrm{TFD}\rangle}$. But in the one-sided case, we only have access to the thermal state representing the exterior. Thus we’d like to know how to compute the correlator using only the available data in the CFT corresponding to region ${I}$. In order to do this, we re-express all operators ${\tilde{\mathcal{O}}}$ appearing in the correlator with analytically continued operators ${\mathcal{O}}$ via the KMS condition, i.e., we make the replacement
$\displaystyle \tilde{\mathcal{O}}(t,\mathbf{x}) \longrightarrow \mathcal{O}(t+i\beta/2,\mathbf{x})~. \ \ \ \ \ (14)$
This is essentially the usual statement that thermal Green functions are periodic in imaginary time; see [1] for details. This relationship allows us to express the desired correlator as
$\displaystyle \langle\mathrm{TFD}|\mathcal{O}(t_1,\mathbf{x}_1)\ldots\tilde{\mathcal{O}}(t_n,\mathbf{x}_n)|\mathrm{TFD}\rangle =Z^{-1}_\beta\mathrm{tr}\left[e^{-\beta H}\mathcal{O}(t_1,\mathbf{x}_1)\ldots\mathcal{O}_{\omega,\mathbf{k}}(t_n+i\beta/2,\mathbf{x}_n)\right]~, \ \ \ \ \ (15)$
which is precisely eqn. (2) in our earlier post, cf. the two-point function (1) above. Note the lack of tilde’s on the right-hand side: this thermal expectation value can be computed entirely in the right CFT.
If the CFT did not admit operators which satisfy the correlation relation (15), it would imply a breakdown of effective field theory across the horizon. Alternatively, observing deviations from the correct thermal correlators would allow us to locally detect the horizon, in contradiction to the equivalence principle. In this sense, this expression may be summarized as the statement that the horizon is smooth. Thus, for the CFT to represent a black hole with no firewall, it must contain a representation of interior operators ${\tilde{\mathcal{O}}}$ with the correct behaviour inside low-point correlators. This last qualifier hints at the state-dependent nature of these so-called “mirror operators”, which I’ve discussed at length elsewhere [6].
References
[1] K. Papadodimas and S. Raju, “An Infalling Observer in AdS/CFT,” JHEP 10 (2013) 212,arXiv:1211.6767 [hep-th].
[2] K. Papadodimas and S. Raju, “State-Dependent Bulk-Boundary Maps and Black Hole Complementarity,” Phys. Rev. D89 no. 8, (2014) 086010, arXiv:1310.6335 [hep-th].
[3] A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev. D74 (2006) 066009, arXiv:hep-th/0606141 [hep-th].
[4] A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Local bulk operators in AdS/CFT: A Boundary view of horizons and locality,” Phys. Rev. D73 (2006) 086003,arXiv:hep-th/0506118 [hep-th].
[5] A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Local bulk operators in AdS/CFT: A Holographic description of the black hole interior,” Phys. Rev. D75 (2007) 106001,arXiv:hep-th/0612053 [hep-th]. [Erratum: Phys. Rev.D75,129902(2007)].
[6] R. Jefferson, “Comments on black hole interiors and modular inclusions,” SciPost Phys. 6 no. 4, (2019) 042, arXiv:1811.08900 [hep-th].
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https://physics.stackexchange.com/questions/136667/observing-a-particle-over-a-certain-domain | # Observing a particle over a certain domain
I was just thinking: in Quantum Mechanics, we start out with that whole collapsing business by observing the x position of a particle. I was thinking: why do we have to do that? What if we only observe a particle over a specific domain? Let's be more specific. I have an electron trapped in a tube. This tube has very, very thin walls. The potential in these walls is quite large but not infinite. Therefore, there is a probability the particle will quantum tunnel through these walls to the outside. If we let the particle go in the tunnel and wait a short time, this is the sort of wave function we would expect: (or should I say, the wave function's probability distribution)
Now, we shoot a photon down the tube between the two barriers (let's assume with 100% certainty the light particle will stay within the walls). There are two outcomes: the light particle keeps on going and doesn't hit the particle (the particle was observed to be outside the walls), or the photon bounces back containing information about the particle's position (the particle was observed to be inside the walls). Now, here's my real question: does the particle's wave function collapse if the particle is outside the boundary? If the particle was observed inside the walls, then it will obviously collapse (the photon contains the definite position of the particle). However, things are more messy outside the tube. Let's imagine the particle does collapse. How can we pluck apart that argument? Well, I didn't say the particle HAS to be let go inside the tube. For all that matters, this could have been the initial wave function:
For all practical purposes, the electron could be centered on the moon: so long as there is a probability of observing the particle in the tube, we will collapse it. Furthermore, if there are two particles with similar wave functions (some probability for either to be in the tube), then we will collapse them both. In conclusion, once this experiment is done, every particle in the universe (technically on the light-cone) will simultaneously be collapsed. Doesn't sound very physical, does it? Just to determine whether the 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001% chance that the particle might be observed in the tube isn't a very good reason to collapse every wave function in the universe. So, that can't be the case. Then, there is another alternative: we don't collapse the wave function (assuming the particle was not found inside the tube). How can that be the case either? If we didn't collapse the wave function in the first place, how could we know whether the particle was inside the tube or outside it? This is essentially what the problem boils down to: we must collapse the wave function in order to tell whether we need to collapse the wave function. What's the solution here? It looks hopeless either way! Thanks in advance.
P.S. There is an easy test to see what's happening, were anyone actually bothered enough to try. Let's assume the photon went right through the tube and showed the particle to be outside. Then, measure the momentum. Do this again and again. If we get a wide range of answers, we know the particle got collapsed each time. Since the particle must have a well defined position, we can't know anything about the momentum. However, if the momentum is well localized (around zero), we know the particle did not collapse. Of course, how localized "well localized" is depends on the time between measuring position and measuring momentum. Thanks again!
You cannot measure "where the particle is not at" in this way.
When you send a photon it either interacts with the electron, collapsing its wave function, or it doesn't.
If it interacts with the electron, then you can say, with some certainty related to the photon energy, where the electron was.
But if not, there's nothing you can say about it. It does not mean the electron is outside just that the particles did not interact, for some reason.
For example, a photon will never interact with the ground electron in a Hydrogen atom if it does not have at least 13.6 eV. Doesn't mean it is not there.
By the way, this "pipe' of yours is about the same as the Hydrogen atom example, the walls are like the binding forces between the nucleus and the electron, and likewise it has specific energy levels permitted for the electron.
You could argue that, "ok, but my photon has the required energy and any other quantum number required for the interaction", then comes in that probability distribution you drew. When they interact, and only then, you will find the electron at some point within the tube with that probability, but the photon could still totally "miss" it. Don't forget the photon also has a wave associated with it, it is not everywhere within the tube with the same probability. (Or maybe it's because of the "intrinsic randomness of quantum mechanics". That one always works too. :)
But the fact is, when you get the photon "on the other side", it has not interacted with the electron, therefore it has not collapsed its wave function. But also, there's nothing you can say about where the electron is.
Furthermore, if there are two particles with similar wave functions (some probability for either to be in the tube), then we will collapse them both.
No. Quantum mechanical solutions are exact and deterministic. Their square determines completely the probabilities of observation and each solution has definite boundary conditions that generate it as a model for physical observations. So, 'similar wave functions" has no mathematical meaning , similar is different, as far as QM is concerned.
So, we set up a problem, your potential and electron in this case. This is described by a definite wave function , squared giving the probability of finding that electron in a certain (x,y,z,t) , even outside the walls. One electron, one measurement. For example a scatter with another electron, measuring both tracks would give the location of the vertex, which would be the location for this electron at that time. Scattering with a photon needs complicated photon detectors but the same holds: one measurement.
To get a probability distribution the experiment should be repeated enough times with different electrons to find the distribution in your plot.
If we didn't collapse the wave function in the first place, how could we know whether the particle was inside the tube or outside it?
I dislike this collapse business which makes beautiful mathematical structures sound like balloons. We only know what happens to one particle with one measurement at a time, and then get the distribution for a probability plot.
This is essentially what the problem boils down to: we must collapse the wave function in order to tell whether we need to collapse the wave function.
Meaning? We have to take many measurements with the same boundary conditions to be able to get the probability distribution. There may be need for measurements if the exercise requires it, but the question is not whether one needs a measurement. That is for the experimenter to decide.
Maybe you should read similar links to this to get some words to search for, that wills how that tunneling is a well defined phenomenon and even useful in applications of nanotechnology. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9263358116149902, "perplexity": 324.9722163039499}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178364008.55/warc/CC-MAIN-20210302125936-20210302155936-00399.warc.gz"} |
https://socratic.org/questions/is-entropy-state-function-how-prove-it | Chemistry
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# Is entropy a state function? How? Prove it?
Jan 16, 2017
Essentially, this shows a derivation of entropy and that a state function can be written as a total derivative, $\mathrm{dF} \left(x , y\right) = {\left(\frac{\partial F}{\partial x}\right)}_{y} \mathrm{dx} + {\left(\frac{\partial F}{\partial y}\right)}_{x} \mathrm{dy}$.
From the first law of thermodynamics:
$\mathrm{dU} = \delta {q}_{\text{rev" + deltaw_"rev}}$,
where $q$ is the heat flow, $w$ is the work (which we define as $- \int P \mathrm{dV}$), and $\delta$ indicates that heat flow and work are inexact differentials (path functions).
Solving for $\delta {q}_{\text{rev}}$ gives:
$\delta {q}_{\text{rev" = dU - delw_"rev}} = {C}_{V} \left(T\right) \mathrm{dT} + P \mathrm{dV}$,
since ${\left(\frac{\partial U}{\partial T}\right)}_{V} = {C}_{V}$, the constant-volume heat capacity. For an ideal gas, we'd get:
$\delta {q}_{\text{rev}} \left(T , V\right) = {C}_{V} \left(T\right) \mathrm{dT} + \frac{n R T}{V} \mathrm{dV}$
It can be shown that this is an inexact total derivative, indicative of a path function. Euler's reciprocity relation states that for the total derivative
$\boldsymbol{\mathrm{dF} \left(x , y\right) = M \left(x\right) \mathrm{dx} + N \left(y\right) \mathrm{dy}}$,
where $M \left(x\right) = {\left(\frac{\partial F}{\partial x}\right)}_{y}$ and $N \left(y\right) = {\left(\frac{\partial F}{\partial y}\right)}_{x}$,
a differential is exact if ${\left(\frac{\partial M}{\partial y}\right)}_{x} = {\left(\frac{\partial N}{\partial x}\right)}_{y}$. If this is the case, this would indicate that we have a state function.
Let $M \left(T\right) = {\left(\frac{\partial {q}_{\text{rev}}}{\partial T}\right)}_{V} = {C}_{V} \left(T\right)$, $N \left(V\right) = {\left(\frac{\partial {q}_{\text{rev}}}{\partial V}\right)}_{T} = \frac{n R T}{V}$, $x = T$, and $y = V$. If we use our current expression for $\delta {q}_{\text{rev}}$, we obtain:
((delC_V(T))/(delV))_T stackrel(?" ")(=) ((del(nRT"/"V))/(delT))_V
But since ${C}_{V} \left(T\right)$ is only a function of $T$ for an ideal gas, we have:
$0 \ne \frac{n R}{V}$
However, if we multiply through by $\frac{1}{T}$, called an integrating factor, we would get a new function of $T$ and $V$ which is an exact differential:
$\textcolor{g r e e n}{\frac{\delta {q}_{\text{rev}} \left(T , V\right)}{T} = \frac{{C}_{V} \left(T\right)}{T} \mathrm{dT} + \frac{n R}{V} \mathrm{dV}}$
Now, Euler's reciprocity relation works:
${\left(\frac{\partial \left[{C}_{V} \left(T\right) \text{/"T])/(delV))_T stackrel(?" ")(=) ((del(nR"/} V\right)}{\partial T}\right)}_{V}$
$0 = 0$ color(blue)(sqrt"")
Therefore, this new function, $\frac{{q}_{\text{rev}} \left(T , V\right)}{T}$ can be defined as the state function $S$, entropy, which in this case is a function of $T$ and $V$:
$\textcolor{b l u e}{\mathrm{dS} \left(T , V\right) = \frac{\delta {q}_{\text{rev}}}{T}}$
and it can be shown that for the definition of the total derivative of $S$:
$\mathrm{dS} = {\left(\frac{\partial S}{\partial T}\right)}_{V} \mathrm{dT} + {\left(\frac{\partial S}{\partial V}\right)}_{T} \mathrm{dV}$
$= {\left(\frac{\partial S}{\partial T}\right)}_{V} \mathrm{dT} + {\left(\frac{\partial P}{\partial T}\right)}_{V} \mathrm{dV}$
(where we've used a cyclic relation in the Helmholtz free energy Maxwell relation)
which for an ideal gas is:
$= \frac{{C}_{V}}{T} \mathrm{dV} + \frac{n R}{V} \mathrm{dV}$
##### Impact of this question
7714 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 39, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9888355135917664, "perplexity": 339.0471466382192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672431.45/warc/CC-MAIN-20191016235542-20191017023042-00470.warc.gz"} |
https://www.preprints.org/manuscript/201801.0089/v1 | Preprint Article Version 1 This version not peer reviewed
# A New Extension of Extended Caputo Fractional Derivative Operator
Version 1 : Received: 9 January 2018 / Approved: 10 January 2018 / Online: 10 January 2018 (09:37:35 CET)
How to cite: Rahman, G.; Nisar, K.S.; Mubeen, S. A New Extension of Extended Caputo Fractional Derivative Operator. Preprints 2018, 2018010089 (doi: 10.20944/preprints201801.0089.v1). Rahman, G.; Nisar, K.S.; Mubeen, S. A New Extension of Extended Caputo Fractional Derivative Operator. Preprints 2018, 2018010089 (doi: 10.20944/preprints201801.0089.v1).
## Abstract
Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.
## Subject Areas
hypergeometric function; beta function; extended hypergeometric function; mellin transform; fractional derivative; caputo fractional derivative; appell's function; generating relation | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8586416840553284, "perplexity": 2570.823178374086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814140.9/warc/CC-MAIN-20180222160706-20180222180706-00563.warc.gz"} |
https://latexref.xyz/bm.html | 16.2.2.1 bm: Individual bold math symbols
Specifying \boldmath is the best method for typesetting a whole math expression in bold. But to typeset individual symbols within an expression in bold, the bm package provided by the LaTeX Project team is better. Its usage is outside the scope of this document (see its documentation at https://ctan.org/pkg/bm or in your installation) but the spacing in the output of this small example will show that it is an improvement over \boldmath within an expression:
\usepackage{bm} % in preamble
...
we have $\bm{v} = 5\cdot\bm{u}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9231868386268616, "perplexity": 1712.7936705622053}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500035.14/warc/CC-MAIN-20230202165041-20230202195041-00555.warc.gz"} |
https://chem.libretexts.org/Courses/Pacific_Union_College/Kinetics/06%3A_The_Distribution_of_Gas_Velocities/6.02%3A_Probability_Density_Functions_for_Velocity_Components_in_Spherical_Coordinates | 6.2: Probability Density Functions for Velocity Components in Spherical Coordinates
We introduce the idea of a three-dimensional probability-density function by showing how to find it from data referred to a Cartesian coordinates system. The probability density associated with a particular molecular velocity is just a number—a number that depends only on the velocity. Given a velocity, the probability density associated with that velocity must be independent of our choice of coordinate system. We can express the three-dimensional probability density using any coordinate system. We turn now to expressing velocities and probability density functions using spherical coordinates.
Just as we did for the Cartesian velocity components, we deduce the cumulative probability functions $$f_v\left(v\right)$$, $$f_{\theta }\left(\theta \right)$$, and $$f_{\varphi }\left(\varphi \right)$$ for the spherical-coordinate components. Our deduction of $$f_v\left(v\right)$$ from the experimental data uses $$v$$-values that are associated with all possible values of $$\theta$$ and $$\varphi$$. Corresponding statements apply to our deductions of $$f_{\theta }\left(\theta \right)$$, and $$f_{\varphi }\left(\varphi \right)$$. We also obtain their derivatives, the probability-density functions $${df_v\left(v\right)}/{dv}$$, $${df_{\theta }\left(\theta \right)}/{d\theta }$$, and $${{df}_{\varphi }\left(\varphi \right)}/{d\varphi }$$. From the properties of probability-density functions, we have
$\int^{\infty }_0{\left(\frac{{df}_v\left(v\right)}{dv}\right)}dv=\int^{\pi }_0{\left(\frac{{df}_{\theta }\left(\theta \right)}{d\theta }\right)}d\theta =\int^{2\pi }_0{\left(\frac{{df}_{\varphi }\left(\varphi \right)}{d\varphi }\right)}d\varphi =1$
Let $$\textrm{ʋ}\prime$$ be the arbitrarily small increment of volume in velocity space in which the $$v$$-, $$\theta$$-, and $$\varphi$$-components of velocity lie between $$v$$ and $$v+dv$$, $$\theta$$ and $$\theta +d\theta$$, and $$\varphi$$ and $$\varphi +d\varphi$$. Then the probability that the velocity of a randomly selected molecule lies within $$\textrm{ʋ}\prime$$ is
$dP\left(\textrm{ʋ}\prime \right)=\left(\frac{df_v\left(v\right)}{dv}\right)\left(\frac{df_{\theta }\left(\theta \right)}{d\theta }\right)\left(\frac{{df}_{\varphi }\left(\varphi \right)}{d\varphi }\right)dvd\theta d\varphi$
Note that the product
$\left(\frac{df_v\left(v\right)}{dv}\right)\left(\frac{df_{\theta }\left(\theta \right)}{d\theta }\right)\left(\frac{{df}_{\varphi }\left(\varphi \right)}{d\varphi }\right)$
is not a three-dimensional probability density function. This is most immediately appreciated by recognizing that $$dvd\theta d\varphi$$ is not an incremental “volume” in velocity space. That is, $$\textrm{ʋ}\prime \neq \ dvd\theta d\varphi$$
We let $$\rho \left(v,\ \theta ,\varphi \right)$$ be the probability-density function for the velocity vector in spherical coordinates. When $$v$$, $$\theta$$, and $$\varphi$$ specify the velocity, $$\rho \left(v,\ \theta ,\varphi \right)$$ is the probability per unit volume at that velocity. We want to use $$\rho \left(v,\ \theta ,\varphi \right)$$ to express the probability that an arbitrarily selected molecule has a velocity vector whose magnitude lies between$$\ v$$ and $$v+dv$$, while its $$\theta$$-component lies between $$\theta$$ and $$\theta +d\theta$$, and its $$\varphi$$-component lies between $$\varphi$$ and $$\varphi +d\varphi$$. This is just $$\rho \left(v,\ \theta ,\varphi \right)$$ times the velocity-space “volume” included by these ranges of $$v$$, $$\theta$$, and $$\varphi$$.
When we change from Cartesian coordinates, $$\mathop{v}\limits^{\rightharpoonup}=\left(v_x,v_y,v_z\right)$$, to spherical coordinates, $$\mathop{v}\limits^{\rightharpoonup}=\left(v,\theta ,\varphi \right)$$, the transformation is $$v_x=v{\mathrm{sin} \theta \ }{\mathrm{cos} \varphi \ }$$, $$v_y=v{\mathrm{sin} \theta \ }{\mathrm{sin} \varphi \ }$$, $$v_z=v{\mathrm{cos} \theta \ }$$. (See Figure 1.) As sketched in Figure 2, an incremental increase in each of the coordinates of the point specified by the vector $$\left(v,\ \theta ,\varphi \right)$$ advances the vector to the point $$\left(v+dv,\theta +d\theta ,\varphi +d\varphi \right)$$. When $$dv$$, $$d\theta$$, and $$d\varphi$$ are arbitrarily small, these two points specify the diagonally opposite corners of a rectangular parallelepiped, whose edges have the lengths $$dv$$, $$vd\theta$$, and $$v{\mathrm{sin} \theta \ }d\varphi$$. The volume of this parallelepiped is $$v^2{\mathrm{sin} \theta \ }dvd\theta d\varphi$$. Hence, the differential volume elementdifferential volume element in Cartesian coordinates, $$dv_xdv_ydv_z$$, becomes $$v^2{\mathrm{sin} \theta \ }dvd\theta d\varphi$$ in spherical coordinates.
Mathematically, this conversion is obtained using the absolute value of the Jacobian, $$J\left(\frac{v_x,v_y,v_z}{v,\theta ,\varphi }\right)$$, of the transformation. That is,
$dv_xdv_ydv_z=\left|J\left(\frac{v_x,v_y,v_z}{v,\theta ,\varphi }\right)\right|dvd\theta d\varphi$
where the Jacobian is a determinate of partial derivatives
$J\left(\frac{v_x,v_y,v_z}{v,\theta ,\varphi }\right)=\left| \begin{array}{ccc} {\partial v_x}/{\partial v} & {\partial v_x}/{\partial \theta } & {\partial v_x}/{\partial \varphi } \\ {\partial v_y}/{\partial v} & {\partial v_y}/{\partial \theta } & {\partial v_y}/{\partial \varphi } \\ {\partial v_z}/{\partial v} & {\partial v_z}/{\partial \theta } & {\partial v_z}/{\partial \varphi } \end{array} \right|$ ${=v}^2{\mathrm{sin} \theta \ }$
Since the differential unit of volume in spherical coordinates is $$v^2{\mathrm{sin} \theta \ }$$$$dvd\theta d\varphi$$, the probability that the velocity components lie within the indicated ranges is
$dP\left(\textrm{ʋ}\prime \right)=\rho \left(v,\theta ,\varphi \right)v^2{\mathrm{sin} \theta \ }dvd\theta d\varphi$
We can develop the next step in Maxwell’s argument by taking his assumption to mean that the three-dimensional probability density function is expressible as a product of three one-dimensional functions. That is, we take Maxwell’s assumption to assert the existence of independent functions $${\rho }_v\left(v\right)$$, $${\rho }_{\theta }\left(\theta \right)$$, and $${\rho }_{\varphi }\left(\varphi \right)$$ such that $$\rho \left(v,\ \theta ,\varphi \right)={\rho }_v\left(v\right){\rho }_{\theta }\left(\theta \right){\rho }_{\varphi }\left(\varphi \right)$$. The probability that the $$v$$-, $$\theta$$-, and $$\varphi$$-components of velocity lie between $$v$$ and $$v+dv$$, $$\theta$$ and $$\theta +d\theta$$, and $$\varphi$$ and $$\varphi +d\varphi$$ becomes
\begin{aligned} dP\left(\textrm{ʋ}\prime \right) & =\left(\frac{df_v\left(v\right)}{dv}\right)\left(\frac{df_{\theta }\left(\theta \right)}{d\theta }\right)\left(\frac{{df}_{\varphi }\left(\varphi \right)}{d\varphi }\right)dvd\theta d\varphi \\ ~ & =\rho \left(v,\theta ,\varphi \right)v^2{\mathrm{sin} \theta dvd\theta d\varphi \ } \\ ~ & ={\rho }_v\left(v\right){\rho }_{\theta }\left(\theta \right){\rho }_{\varphi }\left(\varphi \right)v^2{\mathrm{sin} \theta dvd\theta d\varphi \ } \end{aligned}
Since $$v$$, $$\theta$$, and $$\varphi$$ are independent, it follows that
$\frac{df_v\left(v\right)}{dv}=v^2{\rho }_v\left(v\right)$
$\frac{df_{\theta }\left(\theta \right)}{d\theta }={\rho }_{\theta }\left(\theta \right){\mathrm{sin} \theta \ }$
$\frac{df_{\varphi }\left(\varphi \right)}{d\varphi }={\rho }_{\varphi }\left(\varphi \right)$
Moreover, the assumption that velocity is independent of direction means that $${\rho }_{\theta }\left(\theta \right)$$ must actually be independent of $$\theta$$; that is, $${\rho }_{\theta }\left(\theta \right)$$ must be a constant. We let this constant be $${\alpha }_{\theta }$$; so$$\ {\rho }_{\theta }\left(\theta \right)={\alpha }_{\theta }$$. By the same argument, we set $${\rho }_{\varphi }\left(\varphi \right)={\alpha }_{\varphi }$$. Each of these probability-density functions must be normalized. This means that
$1=\int^{\infty }_0{v^2{\rho }_v\left(v\right)}dv$
$1=\int^{\pi }_0{{\alpha }_{\theta }{\mathrm{sin} \theta \ }d\theta }=2{\alpha }_{\theta }$
$1=\int^{2\pi }_0{{\alpha }_{\varphi }d\varphi }=2\pi {\alpha }_{\varphi }$
from which we see that $${\rho }_{\theta }\left(\theta \right)={\alpha }_{\theta }={1}/{2}$$ and $${\rho }_{\varphi }\left(\varphi \right)={\alpha }_{\varphi }={1}/{2}\pi$$. It is important to recognize that, while $${\rho }_x\left(v_x\right)$$, $${\rho }_y\left(v_y\right)$$, and $${\rho }_z\left(v_z\right)$$ are probability density functions, $${\rho }_{\theta }\left(\theta \right)$$ and $${\rho }_v\left(v\right)$$ are not. (However, $${\rho }_{\varphi }\left(\varphi \right)$$ is a probability density function.) We can see this by noting that, if $${\rho }_{\theta }\left(\theta \right)$$ were a probability density, its integral over all possible values of $$\theta$$ $$\left(0<\theta <\pi \right)$$would be one. Instead, we find
$\int^{\pi }_0 \rho_{\theta} \left(\theta \right)d\theta =\int^{\pi }_0 d\theta /2= \pi /2$
Similarly, when we find $${\rho }_v\left(v\right)$$, we can show explicitly that
$\int^{\infty }_0{{\rho }_v\left(v\right)dv\neq 1}$
Our notation now allows us to express the probability that an arbitrarily selected molecule has a velocity vector whose magnitude lies between $$v$$ and $$v+dv$$, while its $$\theta$$-component lies between $$\theta$$ and$$\ \theta +d\theta$$, and its $$\varphi$$-component lies between $$\varphi$$ and $$\varphi +d\varphi$$ using three equivalent representations of the probability density function:
$dP\left(\textrm{ʋ}\prime \right)=\rho \left(v,\theta ,\varphi \right)v^2{\mathrm{sin} \theta dvd\theta d\varphi \ } - {\rho }_v\left(v\right){\rho }_{\theta }\left(\theta \right){\rho }_{\varphi }\left(\varphi \right)v^2{\mathrm{sin} \theta dvd\theta d\varphi \ }=\left(\frac{1}{4\pi }\right){\rho }_v\left(v\right)v^2{\mathrm{sin} \theta \ }dvd\theta d\varphi$
The three-dimensional probability-density function in spherical coordinates is
$\rho \left(v,\ \theta ,\varphi \right)={\rho }_v\left(v\right){\rho }_{\theta }\left(\theta \right){\rho }_{\varphi }\left(\varphi \right)=\frac{{\rho }_v\left(v\right)}{4\pi }$ This shows explicitly that $$\rho \left(v,\ \theta ,\varphi \right)$$ is independent of $$\theta$$ and $$\varphi$$; if the speed is independent of direction, the probability density function that describes velocity must be independent of the coordinates, $$\theta$$ and $$\varphi$$, that specify its direction. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9966093897819519, "perplexity": 194.87363314931937}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00050.warc.gz"} |
https://bookdown.org/yihui/bookdown/publishers.html | ## 6.3 Publishers
Besides publishing your book online, you can certainly consider publishing it with a publisher. For example, this book was published with Chapman & Hall/CRC, and there is also a free online version at https://bookdown.org/yihui/bookdown/ (with an agreement with the publisher). Another option that you can consider is self-publishing (https://en.wikipedia.org/wiki/Self-publishing) if you do not want to work with an established publisher.
It will be much easier to publish a book written with bookdown if the publisher you choose supports LaTeX. For example, Chapman & Hall provides a LaTeX class named krantz.cls, and Springer provides svmono.cls. To apply these LaTeX classes to your PDF book, set documentclass in the YAML metadata of index.Rmd to the class filename (without the extension .cls).
The LaTeX class is the most important setting in the YAML metadata. It controls the overall style of the PDF book. There are often other settings you want to tweak, and we will show some details about this book below.
The YAML metadata of this book contains these settings:
documentclass: krantz
lot: yes
lof: yes
fontsize: 12pt
monofont: "Source Code Pro"
monofontoptions: "Scale=0.7"
The field lot: yes means we want the List of Tables, and similarly, lof means List of Figures. The base font size is 12pt, and we used Source Code Pro as the monospaced (fixed-width) font, which is applied to all program code in this book.
In the LaTeX preamble (Section 4.1), we have a few more settings. First, we set the main font to be Alegreya, and since this font does not have the Small Capitals feature, we used the Alegreya SC font.
\setmainfont[
UprightFeatures={SmallCapsFont=AlegreyaSC-Regular}
]{Alegreya}
The following commands make floating environments less likely to float by allowing them to occupy larger fractions of pages without floating.
\renewcommand{\textfraction}{0.05}
\renewcommand{\topfraction}{0.8}
\renewcommand{\bottomfraction}{0.8}
\renewcommand{\floatpagefraction}{0.75}
Since krantz.cls provided an environment VF for quotes, we redefine the standard quote environment to VF. You can see its style in Section 2.1.
\renewenvironment{quote}{\begin{VF}}{\end{VF}}
Then we redefine hyperlinks to be footnotes, because when the book is printed on paper, readers are not able to click on links in text. Footnotes will tell them what the actual links are.
\let\oldhref\href
\renewcommand{\href}[2]{#2\footnote{\url{#1}}}
We also have some settings for the bookdown::pdf_book format in _output.yml:
bookdown::pdf_book:
includes:
before_body: latex/before_body.tex
after_body: latex/after_body.tex
keep_tex: yes
dev: "cairo_pdf"
latex_engine: xelatex
citation_package: natbib
template: null
pandoc_args: "--chapters"
toc_unnumbered: no
toc_appendix: yes
quote_footer: ["\\VA{", "}{}"]
highlight_bw: yes
All preamble settings we mentioned above are in the file latex/preamble.tex. In latex/before_body.tex, we inserted a few blank pages required by the publisher, wrote the dedication page, and specified that the front matter starts:
\frontmatter
Before the first chapter of the book, we inserted
\mainmatter
so that LaTeX knows to change the page numbering style from Roman numerals (for the front matter) to Arabic numerals (for the book body).
We printed the index in latex/after_body.tex (Section 2.9).
The graphical device (dev) for saving plots was set to cairo_pdf so that the fonts are embedded in plots, since the default device pdf does not embed fonts. Your copyeditor is likely to require you to embed all fonts used in the PDF, so that the book can be printed exactly as it looks, otherwise certain fonts may be substituted and the typeface can be unpredictable.
The quote_footer field was to make sure the quote footers were right-aligned: the LaTeX command \VA{} was provided by krantz.cls to include the quote footer.
The highlight_bw option was set to true so that the colors in syntax highlighted code blocks were converted to grayscale, since this book will be printed in black-and-white.
The book was compiled to PDF through xelatex to make it easier for us to use custom fonts.
All above settings except the VF environment and the \VA{} command can be applied to any other LaTeX document classes.
In case you want to work with Chapman & Hall as well, you may start with the copy of krantz.cls in our repository (https://github.com/rstudio/bookdown/tree/master/inst/examples) instead of the copy you get from your editor. We have worked with the LaTeX help desk to fix quite a few issues with this LaTeX class, so hopefully it will work well for your book if you use bookdown. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8808004856109619, "perplexity": 2692.4851063405436}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612283.85/warc/CC-MAIN-20170529111205-20170529131205-00383.warc.gz"} |
http://mathhelpforum.com/calculus/43284-taylor-series-remainder.html | 1. ## Taylor series remainder
could anyone help me out with this? thankx
use taylor approximation theorem to estimate e with error less than 10^-6
2. Hi
Originally Posted by pc31
use taylor approximation theorem to estimate e with error less than 10^-6
Let $f:x\mapsto \exp x$. $f\in\mathcal{C}^{\infty}(\mathbb{R})$ hence for any integer $n$ and for all $a,\,b\in \mathbb{R}$, Taylor's theorem states that
$f(b)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(b-a)^k+R_n$ with $R_n=\frac{f^{(n+1)}(c)}{(n+1)!}(b-a)^{n+1}$ for some $c\in[a,b]$.
As we want to evaluate $\exp 1$ we choose $b=1$. $a$ can be any real number, let's choose $a=0$, it'll make things easier. For any integer $k$ and for any real number $x$, $f^{(k)}(x)=\exp x$ hence the previous equality becomes
$\exp 1=\sum_{k=0}^n\frac{\exp 0}{k!}(1-0)^k+\frac{\exp c}{(n+1)!}(1-0)^{n+1}=\underbrace{\sum_{k=0}^n\frac{1}{k!}}_{\t ext{approximation}}+\underbrace{\frac{\exp c}{(n+1)!}}_{\text{error}}$ for some $c\in[0,1]$.
As we want an error $<10^{-6}$, you have to find $n$ such that $\frac{\exp c}{(n+1)!} <10^{-6}$. Can you take it from here ? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9947939515113831, "perplexity": 147.13669228893838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662507.79/warc/CC-MAIN-20160924173742-00214-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://www.konradvoelkel.com/2012/11/homotopy-limits/ | ### Homotopy limits
Tuesday, November 06th, 2012 | Author:
In this short posting, I want to give some intuitive idea on homotopy limits. Homotopy (co)limits appear whenever one has a notion of homotopy equivalence or weak equivalence between objects and one doesn't want to have constructions that distinguish between equivalent objects. The most prominent settings are, of course, classical homotopy theory and homological algebra. Although not necessary for the definition of homotopy (co)limits, I also talk about model categories.
First, let us recall what a limit is:
Given a small category I (let's think of a diagram like $\cdot \rightarrow \cdot \leftarrow \cdot$) and a functor $D : I \to \mathcal{C}$ (let's think of a diagram in $\mathcal{C}$ that has the prescribed shape), we look at all cones over $D$. A cone over $D$ is just an object $X$ of $\mathcal{C}$ together with morphisms, one for each object $i$ of $I$, from $X$ to $D(i)$, such that these morphisms commute with the morphisms in the image of $D$. This thing is called cone because we can imagine the diagram $D$ to be planar (on a blackboard) and the object $X$ hovers above it, the morphisms down to the diagram look like a cone. It is clear what a morphism between cones should be: a morphism of the objects $X$ that commutes with the morphisms down to the diagram. This yields a category of cones over $D$, conveniently called $Cone(D)$. We call a terminal object in $Cone(D)$ a limit of $D$. It is important to notice that the morphisms down to the diagram in $\mathcal{C}$ are part of the limit, not only the object itself. So, to state it briefly, limits are terminal cones.
Homotopy limits are quite similar. They are terminal homotopy cones. Let's see what that means. I will just tell you now what it doesn't mean: homotopy limits are not just limits-up-to-weak equivalence. Homotopy limits are also not just limits computed in the homotopy category. But: Homotopy limits are only well-defined up to weak equivalence (unless you specify to use a particular computational recipe). Technically, that means there is not a unique holim-functor, but we can safely ignore that for a first approximation.
We first suppose that we have a category $\mathcal{C}$ equipped with a lluf subcategory $W$, where lluf just means all objects of $\mathcal{C}$ are also in $W$, we just may have fewer morphisms. We call the morphisms in $W$ the weak equivalences of $\mathcal{C}$. We define $Ho(\mathcal{C}) := [W^{-1}]\mathcal{C}$, the homotopy category is the localization along the weak equivalences.
Homotopy limits solve the following problem: Suppose I have a diagram in $\mathcal{C}$ but I don't care about replacing the objects by weakly equivalent ones, then what is the terminal cone (up to weak equivalence) not depending on these choices? The difference to a limit in the homotopy category is that we look at honest maps, not morphisms in the homotopy category.
So, now we have a vague idea about homotopy limits. How to compute them? That is where model categories appear in the story. In principle, model categories are not necessary to define homotopy limits, and homotopy limits don't depend on a model structure - they depend only on the class of weak equivalences. But model categories allow to compute certain homotopy limits.
The default approach: Suppose we have a category $\mathcal{C}$ with a class of weak equivalences as before. Now construct a model category $(W,C,F)$ around. Then suppose the index category $I$ for the diagram you want to compute a homotopy limit of is a Reedy category, which means you can assign a degree to each object and all morphisms can be uniquely factorized in one that lowers degree and one that raises degree (a technical condition, satisfied by most small interesting diagrams). Then there is a convenient model structure on the diagram category $\mathcal{C}^I$ and we can define the homotopy limit as the fibrant replacement in $\mathcal{C}^I$ followed by the ordinary limit. This is a classical derived-functor definition.
The crux is: How does the fibrant replacement look like? This question is as hard as any other "how does the injective resolution look like"-type question. Only in particular cases one can really compute a homotopy limit, like for pullbacks.
If you have a diagram $X \to Z \leftarrow Y$, it turns out that you can compute the homotopy pullback (in any proper simplicial model category, like topological spaces) by replacing one of the maps by a fibration (that means, replacing $X$ with a weakly equivalent $\tilde{X}$ such that $X \to Z$ factors over $\tilde{X}$ by the weak equivalence $X \to \tilde{X}$ and a fibration $\tilde{X} \to Z$) and then computing the ordinary pullback.
I would recommend to read in Dugger's exposition of homotopy colimits to learn more. There is some nice geometric intuition possible for homotopy colimits, which you shouldn't miss!
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https://www.gradesaver.com/textbooks/math/calculus/university-calculus-early-transcendentals-3rd-edition/chapter-9-section-9-5-absolute-convergence-the-ratio-and-root-tests-exercises-page-515/7 | ## University Calculus: Early Transcendentals (3rd Edition)
Consider $a_n=(-1)^n\dfrac{n^2(n+2)}{ n! 3^{2n}}$ Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)^2(n+1+2)}{ (n+1)! 3^{2n+2}}}{\dfrac{n^2(n+2)}{ n! 3^{2n}}}|$ Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{(n+1)^2 (n+3)}{(9) n^2 (n+1)}|=\dfrac{1}{9} \lt 1$ Hence, the series Converges absolutely by the ratio test. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987297058105469, "perplexity": 380.13132906199365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371886991.92/warc/CC-MAIN-20200410043735-20200410074235-00124.warc.gz"} |
http://piglix.com/piglix-tmpl.php?title=Sea_ice_concentration | ## Sea ice concentration
• Sea ice concentration is a useful variable for climate scientists and nautical navigators. It is defined as the area of sea ice relative to the total at a given point in the ocean. This article will deal primarily with its determination from remote sensing measurements.
Sea ice concentration helps determine a number of other important climate variables. Since the albedo of ice is much higher than that of water, ice concentration will regulate insolation in the polar oceans. When combined with ice thickness, it determines several other important fluxes between the air and sea, such as salt and fresh-water fluxes between the polar oceans (see for instance bottom water) as well as heat transfer between the atmosphere. Maps of sea ice concentration can be used to determine ice area and ice extent, both of which are important markers of climate change.
Ice concentration charts are also used by navigators to determine potentially passable regions—see icebreaker.
Measurements from ships and aircraft are based on simply calculating the relative area of ice versus water visible within the scene. This can be done using photographs or by eye. In situ measurements are used to validate remote sensing measurements.
Both synthetic aperture radar and visible sensors (such as Landsat) are normally high enough resolution that each pixel is simply classified as a distinct surface type, i.e. water versus ice. The concentration can then be determined by counting the number of ice pixels in a given area which is useful for validating concentration estimates from lower resolution instruments such as microwave radiometers. Since SAR images are normally monochrome and the backscatter of ice can vary quite considerably, classification is normally done based on texture using groups of pixels—see pattern recognition.
${\displaystyle {\vec {T}}_{b}={\vec {T}}_{b0}+\sum _{i=1}^{n}({\vec {T}}_{bi}-{\vec {T}}_{b0})C_{i}}$
Wikipedia | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8576441407203674, "perplexity": 1202.9632507511922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125944982.32/warc/CC-MAIN-20180421032230-20180421052230-00262.warc.gz"} |
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## anonymous 3 years ago I am having a problem in arriving at the correct answer to a second order inhomogeneous differential equation (below): Delete Cancel Submit
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1. anonymous
• 3 years ago
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$\frac{ d ^{2}y }{ dx ^{2} }-6\frac{ dy }{ dx }+8y=8e^{4x}$
2. anonymous
• 3 years ago
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I know that the complimentary function is $Ae^{4x}+Be^{2x}$ however i seem to arrive at an incorrect answer for the particular integral.
3. anonymous
• 3 years ago
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use the undetermined coefficients method
4. anonymous
• 3 years ago
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For the particular integral i get $16Ce^{4x}-24Ce^{4x}+8Ce^{4x}=8e^{4x}$ Based on derivatives of the general function $Ce^{4x}$ I therefore get $e^{4x}(16C-24C+8C)=8e^{4x}$ Therefore $0=8$ Which means there is no solution. The answer in my book for the particular integral however is $4e^{4x}$. Where did i go wrong?
5. anonymous
• 3 years ago
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What is the undetermined coefficients method?
6. anonymous
• 3 years ago
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Oh, i believe that is what i am doing
7. anonymous
• 3 years ago
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your should assume the particular$y = Ce^{kx}$then you solve for C and k
8. anonymous
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thats what i tried, but i get C = 0
9. anonymous
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I believe 'k' is actually 4 in this instance.
10. anonymous
• 3 years ago
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Can you see where i went wrong in working through the answer? Or do you think it is an error in my book?
11. anonymous
• 3 years ago
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wait I am working on it too, if Ce^(kx) won't work, try Cxe^(kx)
12. anonymous
• 3 years ago
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Oh... i'll try that only i thought you only did that when there was already a $e^{kx}$ on the LHS.
13. anonymous
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try$y = Cxe^{4x}$it works
14. anonymous
• 3 years ago
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Just trying that now...
15. anonymous
• 3 years ago
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Awesome, as you said, that works. I didn't realize that you could use that method when there is no solution of a coefficient. I thought it was only when an element of the general solution appeared in the LHS. Good to know, i've learned a new trick =)
16. anonymous
• 3 years ago
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Oh and thanks heaps... =)
17. anonymous
• 3 years ago
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lol, you are welcome
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998489618301392, "perplexity": 3828.625016025764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661449.76/warc/CC-MAIN-20160924173741-00158-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/425811/excluding-columns-with-csvsimple-autotables | # Excluding columns with csvsimple & autotables
first my apologies for not providing a full minimum working example.
Description:
I have a large number of (mainly large) tables in the csv format that I wish to include in a LaTeX document.
The job itself is easily achieved with the csvsimple package, calling:
\usepackage{longtable}
\usepackage{csvsimple}
I include my tables using this command:
{
\small
\begingroup\catcode"=9
\csvautolongtable[respect underscore=true]{table_file.csv}
\endgroup
}
Problem:
Some of my tables are too wide to fit onto an A4 page, even in landscape mode. However, I have a number of columns that I do not need to include in the document. (They are nice to have, but not absolutely necessary.)
Question:
Can I exclude specific columns while retaining the use of the \cvsautotable command?
In case it helps, this is my header, consistent across the files. The columns I would want to drop are H to Br. Writing a custom header might work, but then I guess it needs a customization of the function to read in the table? (The header includes spaces as well...):
Name,dHf ref.(kJ/mol),DFT E0(Eh),DLPNO E0(Eh),ZPVE(J/mol),Htot(J/mol),H,C,N,O,F,Si,P,S,Cl,Br
(Yes, I saw the manual - and unfortunately I didn't extract a solution from it...)
Many thanks.
Thanks to a post from tex.stackexchange ( Importing CSV file as a table in Latex but file too long ), the following approach works:
Rather than excluding columns, only the desired columns are picked. Referring to columns by name would be difficult (due to use of the space character for example), so numbers can be used instead.
The "autotable" function is replaced with a manually defined layout, specifying a longtable and the header. Columns 1 to 6 are given a name and the name is used to specify the columns printed.
\csvreader[longtable=lr|rrrr,%
table head= Name & $\Delta Hf_{ref}$ (\si{kJ/mol}) & %
DFT $E_0$ (\si{E_h}) & DLPNO $E_0$ (\si{E_h}) & $ZPVE$ (\si{J/mol}) & $Htot$ (\si{J/mol}) \\ \hline\endhead,%
late after line=\\,%
,respect underscore=true]{csv_data_file.csv}
{1=\Name,2=\dHfref,3=\EzeroDFT,4=\EzeroDLPNO,5=\ZPVE,6=\Htot}%
{\Name & \dHfref & \EzeroDFT & \EzeroDLPNO & \ZPVE & \Htot}%
`
Not the most elegant solution, but it works. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8443189859390259, "perplexity": 2380.7979929180246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488286726.71/warc/CC-MAIN-20210621151134-20210621181134-00038.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-10-problems-page-399/40 | College Physics (4th Edition)
The maximum speed of vibration is $2.5\times 10^{-6}~m/s$
We can find the angular frequency $\omega$: $\omega = 2\pi~f = (2\pi)(4000~Hz) = 8000\pi~rad/s$ We can find the maximum speed: $v_m = A~\omega$ $v_m = (0.10\times 10^{-9}~m)~(8000\pi~rad/s)$ $v_m = 2.5\times 10^{-6}~m/s$ The maximum speed of vibration is $2.5\times 10^{-6}~m/s$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9562850594520569, "perplexity": 386.4340937614768}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337415.12/warc/CC-MAIN-20221003101805-20221003131805-00765.warc.gz"} |
https://www.livmathssoc.org.uk/cgi-bin/sews_diff.py?Axiom | Most recent change of Axiom
Edit made on January 08, 2013 by ColinWright at 15:08:22
Deleted text in red / Inserted text in green
WW WM
An axiom (or postulate) is any mathematical statement that is the starting point from which other statements (often called theorems) are logically derived.
Axioms are considered self-evident requiring neither proof nor justification.
Many areas of mathematics are based on axioms: set theory, geometry, number theory, probability, /etc./
At the start of 20th Century Mathematicians showed great confidence in the development of the axiomatic foundations of Mathematics. However this confidence was shattered by the work of Kurt Godel. Goedel.
One requirement of an axiomatic system is that it is consistent i.e. does not lead to contradictory theorems - thus creating a paradox (see Russell's Paradox).
----
[[[>50
Here are the axioms of Euclidean Geometry
* Any two points can be joined by a straight line.
* Any straight line segment can be extended indefinitely in a straight line.
* Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
* All right angles are congruent.
* If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
From these, all the rich and interesting theorems of school geometry can be derived.
]]]
It is possible to have different and conflicting axiomatic systems which lead to consistent but different areas of Mathematical study.
In Euclidian Geometry the 5th (or parallel) postulate states:
* If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
IMG:ParallelAxiom.svg.png
However Nikolai Lobachevsky (1792 - 1856) and János Bolyai (1802 - 1860) considered this axiom was not self-evident. When they investigated the effect of substituting an alternative axiom such that lines do not meet, however far extended, found no resulting contradictions thus formulating the first non-Euclidian Geometry. This Geometry (called hyperbolic geometry) has many features different from Euclidean Geometry. For example, the sum of the angles of a triangle is less than 180 degrees and the greater the area of the triangle the smaller the sum. Poincare's Disc is a model of such a geometry.
Different axioms lead to different geometries. Mathematics thus supplies a number of competing descriptions of Space, the correct interpretation being found experimentally. The descriptions of Space by Albert Einstein and others requires Space to be non-Euclidean. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8770624995231628, "perplexity": 340.99203211421}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104204514.62/warc/CC-MAIN-20220702192528-20220702222528-00433.warc.gz"} |
https://www.physicsforums.com/threads/wronskian-determinants-help.196298/ | # Wronskian Determinants help
1. Nov 5, 2007
### sapiental
1. The problem statement, all variables and given/known data
Hi, could someone please confirm my results. I just put my answers because the procedure is so long. let me know if you get the same results.
1) Wronskian(e^x, e^-x, sinh(x)) = 0
2) Wronskian(cos(ln(x)), sin(ln(x)) = 1/x * [cos^2(ln(x)) + sin^2(ln(x))] = 1/x
2. Nov 5, 2007
### Dick
The one is super easy. sinh(x) is a linear combination of e^x and e^(-x). So 0. You don't even have to compute anything. And you are right on the second one as well.
Have something to add?
Similar Discussions: Wronskian Determinants help | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8394523859024048, "perplexity": 2764.4684211179483}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738660916.37/warc/CC-MAIN-20160924173740-00205-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://etna.ricam.oeaw.ac.at/volumes/2001-2010/vol37/abstract.php?vol=37&pages=413-436 | ## Analysis of a non-standard finite element method based on boundary integral operators
Clemens Hofreither, Ulrich Langer, and Clemens Pechstein
### Abstract
We present and analyze a non-standard finite element method based on element-local boundary integral operators that permits polyhedral element shapes as well as meshes with hanging nodes. The method employs elementwise PDE-harmonic trial functions and can thus be interpreted as a local Trefftz method. The construction principle requires the explicit knowledge of the fundamental solution of the partial differential operator, but only locally, i.e., in every polyhedral element. This allows us to solve PDEs with elementwise constant coefficients. In this paper we consider the diffusion equation as a model problem, but the method can be generalized to convection-diffusion-reaction problems and to systems of PDEs such as the linear elasticity system and the time-harmonic Maxwell equations with elementwise constant coefficients. We provide a rigorous error analysis of the method under quite general assumptions on the geometric properties of the elements. Numerical results confirm our theoretical estimates.
Full Text (PDF) [292 KB], BibTeX
### Key words
Finite elements, boundary elements, BEM-based FEM, Trefftz methods, error estimates, polyhedral meshes.
65N30, 65N38 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9235100746154785, "perplexity": 684.5957679918787}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540527010.70/warc/CC-MAIN-20191210070602-20191210094602-00335.warc.gz"} |
http://www.boundaryvalueproblems.com/content/2014/1/29 | Research
# On solvability of a nonlocal problem for the Laplace equation with the fractional-order boundary operator
Moldir A Muratbekova*, Kanat M Shinaliyev and Batirkhan K Turmetov
Author Affiliations
Department of Mathematics, Akhmet Yasawi International Kazakh-Turkish University, B. Sattarkhanov Street 29, Turkistan, 161200, Kazakhstan
For all author emails, please log on.
Boundary Value Problems 2014, 2014:29 doi:10.1186/1687-2770-2014-29
Received: 15 August 2013 Accepted: 9 January 2014 Published: 30 January 2014
This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
### Abstract
In the present work, we study properties of some integro-differential operators of the Hadamard-Marchaud type in the class of harmonic functions. As an application of these properties, we consider the question of the solvability of a nonlocal boundary value problem for the Laplace equation in the unit ball.
MSC: 35J05, 35J25, 26A33.
##### Keywords:
Hadamard-Marchaud operator; fractional derivative; nonlocal problem
### 1 Introduction
Let be the unit ball, . A paradigm in the theory of elliptic partial differential equations and harmonic functions is the Laplace equation
(1)
If we prescribe the values of the solution at the boundary Ω of Ω, then we can solve equation (1) uniquely. Of course, one can consider many other boundary conditions such as Neumann’s boundary conditions.
In some applied problems of hydrodynamics [1], it is necessary to prescribe the value of a fractional derivative of the solution on the boundary. Fractional differential equations and boundary value problems involving fractional derivatives appear in many applied problems ranging from the spring-pot model [2] to geology [3] or from nonlinear circuits [4] to alternative models to differential equations [5].
Hence, in this paper we study the Laplace equation concentrating on some conditions on the boundary involving derivatives of fractional order.
Note that numerous works of authors [6-13] were dedicated to the research questions of the solvability of boundary value problems for partial differential equations with boundary operators of high (whole and fractional) order. In the paper of A.N. Tikhonov [14] boundary value problems with boundary conditions containing derivatives of higher order have been investigated for the heat equation. Research questions as regards the solvability of similar problems for higher-order equations with boundary operators of whole and fractional order were carried out in [15,16]. Later in [17], these results were generalized for partial differential equations of fractional order. In [18-20] questions about the solvability of boundary value problems with boundary operators of high order were studied for the Laplace equation. In the studies of these authors the exact conditions for the solvability have been established and the integral representations of solutions of the studied problems have been found. The cycle of studies by the authors [21-26] is devoted to the study of the existence and smoothness of solutions of boundary value problems for the second-order elliptic equations with boundary operators of fractional order. In the paper mentioned above local boundary value problems with boundary operators of fractional order in the Riemann-Liouville or Caputo sense are studied. In this paper we study nonlocal problems with boundary operators of fractional-order derivatives of Hadamard type. Definitions of Hadamard operators, a statement of the main problems, and the history of the questions on this topic are in Section 3.
The organization of this paper is as follows. In Section 2, we present the operators of integration and differentiation in the Hadamard sense and some modifications. In the third section we provide a formulation of the basic problem of this paper and some historical information as regards nonlocal boundary value problems. In the fourth section we study the properties of integral and differential Hadamard-Marchaud operators in the class of harmonic functions in the ball. In Section 5 we provide some auxiliary propositions. Finally, Section 6 is devoted to the study of the fundamental problem, where we formulate and prove the main statement of the paper.
### 2 Definition of Hadamard operators of integration and differentiation and some modifications
In this section, we give a statement on the operators of fractional differentiation in the sense of Hadamard, Hadamard-Marchaud, and their modifications.
For any positive α, fractional integrals and derivatives of the order α in the sense of Hadamard are defined by the following formulas [27]:
(2)
(3)
where is the Dirac operator, is the integral part of α.
If , then, in the class of sufficiently ‘good’ functions, operator (3) can be reduced to the following form [27]:
(4)
This operator is said to be the differentiation operator of order α in the sense of Hadamard-Marchaud.
In [28], the following modification of the Hadamard-Marchaud operator was considered:
(5)
In [18], in the class of harmonic functions in a ball, the properties and applications of the operators in the form of
(6)
are considered. Here , , , and is a differential operator in the form of .
Let be a harmonic function in the domain Ω, and let , be arbitrary real numbers. Let us consider a modification of the Bavrin operator (6).
Introduce the operators
If , , then we obtain the Bavrin operator .
### 3 Statement of the problem
Let , , , be continuous mappings, and let be continuous functions satisfying the condition
(7)
We assume that the series (7) converges uniformly on Ω.
Further, let , , and , i.e.α and β are not equal to zero simultaneously.
Consider the following boundary value problem:
(8)
(9)
where .
A harmonic function from the class , such that and condition (9) is realized in the classical sense, will be called a solution of problem (8)-(9).
The above-mentioned problem is a simple generalization of Bitsadze-Samarskii’s nonlocal problem [29]. For convenience of the reader, we formulate the Bitsadze-Samarskii problem.
Let D be a finite simply-connected domain of the plane of complex variables with the smooth boundary , and let be a closed simple smooth curve lying in D.
We denote by , , a diffeomorphism between S and .
Formulation of the problem: We are to find a harmonic function in D, which is continuous in and satisfies the boundary condition
where is a given function.
Similar problems with operators of integer order were considered in [30-32], and for operators of fractional order with fractional-order derivatives in the sense of Riemann-Liouville and Caputo in [33-41]. It should also be noted that some questions of solvability of nonlocal problems for fractional-order equations in the one-dimensional case were studied in [42-44].
### 4 Properties of operators and
In this section, we study some properties of the operators and in the class of harmonic functions. Further, for convenience, we shall take everywhere .
Lemma 1Let, , andbe a homogeneous harmonic polynomial of the power. Then the following equalities are correct:
(10)
(11)
Proof Let . Then, using homogeneity of the polynomial , we obtain
The value of the last integral can easily be calculated with the help of the change of variables . In fact,
The equality (10) is proved.
Further, note that the relation
(12)
holds for the operator .
Now, let us study actions of the operator to the functions . Using the definition of and the homogeneity of , we have
Denoting and integrating by parts, we get
After the change of variables , as in the proof of equality (10), we easily obtain
which implies .
Further, taking into account fulfilling of equality (12), we obtain in the general case for :
The lemma is proved. □
Lemma 2Let, , andbe a harmonic function in the ball Ω. Then the functionsandare also harmonic in Ω.
Proof Let be a harmonic function in the ball Ω. Then it is known [45] that the function is represented in the form of the series
(13)
where is a complete system of homogeneous harmonic polynomials of power k, and are coefficients of the expansion (13). Applying formally the operator to the series (13) and taking into account equality (11), we obtain
(14)
Now let us check convergence of the series (13) and (14). The following asymptotical estimate is valid for :
Moreover, the series (13) converges absolutely and uniformly by x at , hence, for any and any , the equalities hold. Since , we have for and
Therefore, the series (14) converges absolutely and uniformly by x at , where , , and its sum is a harmonic function. By virtue of the arbitrariness of and , the function is defined in the whole ball Ω.
Let us study the function . Applying formally the operator to the series (13), taking into account equality (10), we obtain
Convergence of this series can be checked as in the case of series (14), and that is why is a harmonic function in the ball Ω. The lemma is proved. □
Now we show that the function can be represented in terms of the function .
Lemma 3Let, , andbe a harmonic function in the domain Ω. Then for anythe equality
is valid.
Proof Let . Represent a harmonic function in the form of the series (13) and transform it to the form of
(15)
Further, taking into account equalities (10)-(11), and the absolute and uniform convergence of the series (15) by x at , it can be reduced to the form of
The lemma is proved. □
One can similarly prove the following lemma.
Lemma 4Let, , andbe a harmonic function in the domain Ω. Then for anythe equality
(16)
is valid.
Lemma 5Let, , andbe a harmonic function in the domain Ω. Then the following equalities hold:
Proof Let . Applying the operator to the function , we obtain
By virtue of Lemma 3, the value of the last integral is equal to , i.e..
To prove the second equality, apply the operator to the function . We get
Then, in the general case,
The lemma is proved. □
### 5 Some auxiliary propositions
Let and satisfy the conditions from Section 2.
Consider the following problem in the domain Ω:
(17)
(18)
where , , and , i.e.α and β are not equal to zero simultaneously, .
A harmonic function from the class , satisfying condition (18) in the classical case, will be called a solution of problem (17)-(18).
It should be noted that problem (17)-(18) was investigated for the case of in [30].
Let us investigate uniqueness for the solution of problem (17)-(18). The following statement holds.
Lemma 6Let, , , , be continuous functions satisfying the condition
(19)
and let a solution of problem (17)-(18) exist.
Then:
(1) If
(20)
then the solution of problem (17)-(18) is unique.
(2) If
(21)
then the solution of problem (17)-(18) is unique up to a constant summand.
Proof Let be the solution of problem (17)-(18) at .
Denote , .
Then if , then, by virtue of the maximum principle for harmonic functions [46], the inequality holds for any .
The boundary condition (18) at implies
Further, since , , , then , and for any , . Therefore .
Hence,
If now condition (19) is realized , then , and we obtain from this the contradiction .
Hence, if condition (19) holds, it is necessary that . Since , substituting the function into the boundary condition (18), for we have
The last equality is equivalent to the equality
We obtain from this the result that either or .
Thus, if conditions (19) and (20) are fulfilled, we obtain , i.e..
If the conditions (21) are fulfilled, then any constant is a solution of the homogeneous problem (17)-(18). In fact, substituting into equation (18), we obtain
The lemma is proved. □
Now investigate existence of a solution of problem (17)-(18). Let and let be the Poisson kernel of the Dirichlet problem, and the area of the unit sphere.
Introduce the function
(22)
and consider the equation
(23)
The following statement holds.
Lemma 7Let, , , be continuous functions satisfying the condition (19). Then:
(1) If the condition (20) is realized, then problem (17)-(18) is uniquely solvable at any.
(2) If the condition (21) is realized, then problem (17)-(18) is solvable if the following condition is realized:
(24)
where the functionis a solution of equation (23), moreover the number of independent solutions of this equation under these conditions is equal to 1.
Proof Since is a harmonic function, a solution of problem (17)-(18) can be found in the form of the Poisson integral where is an unknown function. Substituting this function into the boundary condition (18), we obtain the integral equation with respect to the unknown function ,
(25)
Designate
Then equation (25) can be rewritten in the form of
(26)
To investigate the solvability of the integral equation (26), we study the properties of the kernel . We show that is a continuous function on .
In fact, since , we obtain for all , and therefore the function is continuous on . Further, the function has an integrable singularity, and that is why the function is continuous on . Then by virtue of the uniform convergence of the series , the kernel is also a continuous function on .
Hence, one can apply Fredholm theory to equation (26). Since in the case of and fulfillment of the condition (20), the solution of problem (17)-(18) can only be , for the integral equation (26) has only a trivial solution.
Hence, for any the solution of equation (26) exists, is unique, and belongs to the class . Using this solution, we construct the function which will satisfy all the conditions of problem (17)-(18).
If the condition (21) is valid, then satisfies the condition (18) at , i.e. the corresponding homogeneous equation (26) has the nonzero solution . Then the adjoint homogeneous equation has also a nonzero solution, and that is why in this case fulfillment of the condition (24) is necessary and sufficient for solvability of problem (17)-(18). The lemma is proved. □
### 6 Study of the basic problem
We now formulate the basic statement.
Theorem 1Let, , , , , , be continuous functions satisfying the condition (19). Then:
(1) If the condition (20) is fulfilled, then problem (8)-(9) is uniquely solvable at any.
(2) If the condition (21) is fulfilled, then the condition (24) is necessary and sufficient for solvability of problem (8)-(9) where the functionis a solution of equation (23). If a solution of the problem exists, then it is unique up to the constant summand.
(3) If a solution of problem (8)-(9) exists, then it is represented in the form of, whereis a solution of problem (17)-(18).
Proof (1) Let a solution of problem (8)-(9) exist. Apply to this function the operator and denote . Take the problem which the function satisfies. Since by Lemma 2, in the case of harmonicity of the function , the function is also harmonic in Ω, and the function is harmonic.
Further, since according to Lemma 5 the equality holds, the boundary condition of problem (8)-(9),
with respect to the function will be rewritten in the form of
In addition, since , we have . Thus, if is a solution of problem (8)-(9), then the function will be a solution of problem (17)-(18).
Now, let the conditions (19) and (20) be realized. Then by Lemmas 6 and 7, for any the solution of problem (17)-(18) exists, is unique, and designate , . Then we have by Lemma 5 in Ω, and therefore we get . Harmonicity of the function follows from Lemma 2, and fulfillment of the conditions (9) can be checked immediately:
The first statement of the theorem is proved.
(2) Let now the condition (21) be fulfilled, and let the solution of problem (8)-(9) exist. Consider the function . As in the first case, we show that the function satisfies the conditions of problem (17)-(18). Then according to Lemma 7, fulfillment of the condition (24) is necessary. Thus, we prove the necessity of the condition (24) at fulfillment of the equality (21).
We show that if the equality (21) is fulfilled, then the condition (24) is also sufficient for the existence of the solution of problem (8)-(9).
In fact, if the conditions (21) and (24) are realized, a solution of problem (17)-(18) exists, is unique up to constant summand, and . Then, similarly to the proof of the first statement of the theorem, the function satisfies all the conditions of problem (8)-(9). The theorem is proved. □
Remark 1 One can show that in the case of , the corresponding homogeneous problem (8)-(9) has nontrivial solutions.
Example 1 Let , , , and , . Further, let be a homogeneous harmonic polynomial of the power k. By virtue of the equality (11), we have .
Then
and
Hence, for the harmonic polynomial will be the solution of the homogeneous problem (8)-(9). If δ is a number close to zero, then we have .
If the dimension of the space , then the number of these polynomials is equal to [47].
### Competing interests
The authors declare that they have no competing interests.
### Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
### Acknowledgements
This work has been supported by the MON Republic of Kazakhstan under Research Grant No. 0713/GF.
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http://www.aimsciences.org/journal/1551-0018/2015/12/5 | # American Institute of Mathematical Sciences
ISSN:
1551-0018
eISSN:
1547-1063
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## Mathematical Biosciences & Engineering
2015 , Volume 12 , Issue 5
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2015, 12(5): 907-915 doi: 10.3934/mbe.2015.12.907 +[Abstract](369) +[PDF](1005.2KB)
Abstract:
This paper presents a mathematical model of heat transfer in a prevascular breast tumor. The model uses the steady state temperature of the breast at the skin surface to determine whether there is an underlying tumor and if so, verifies whether the tumor is growing or dormant. The model is governed by the Pennes equations and we present numerical simulations for versions of the model in two and three dimensions.
2015, 12(5): 917-936 doi: 10.3934/mbe.2015.12.917 +[Abstract](411) +[PDF](1013.6KB)
Abstract:
In this study, we consider a model of T cell homeostasis based on the Smith-Martin model. This nonlinear model is structured by age and CD44 expression. First, we establish the mathematical well-posedness of the model system. Next, we prove the theoretical identifiability regarding the up-regulation of CD44, the proliferation time phase and the rate of entry into division, by using the experimental data. Finally, we compare two versions of the Smith-Martin model and we identify which model fits the experimental data best.
2015, 12(5): 937-964 doi: 10.3934/mbe.2015.12.937 +[Abstract](442) +[PDF](639.5KB)
Abstract:
We consider an in-host model for HIV-1 infection dynamics developed and validated with patient data in earlier work [7]. We revisit the earlier model in light of progress over the last several years in understanding HIV-1 progression in humans. We then consider statistical models to describe the data and use these with residual plots in generalized least squares problems to develop accurate descriptions of the proper weights for the data. We use recent parameter subset selection techniques [5,6] to investigate the impact of estimated parameters on the corresponding selection scores. Bootstrapping and asymptotic theory are compared in the context of confidence intervals for the resulting parameter estimates.
2015, 12(5): 965-981 doi: 10.3934/mbe.2015.12.965 +[Abstract](442) +[PDF](721.6KB)
Abstract:
This work is the outcome of the partnership between the mathematical group of Department DISBEF and the biochemical group of Department DISB of the University of Urbino "Carlo Bo" in order to better understand some crucial aspects of brain cancer oncogenesis. Throughout our collaboration we discovered that biochemists are mainly attracted to the instantaneous behaviour of the whole cell, while mathematicians are mostly interested in the evolution along time of small and different parts of it. This collaboration has thus been very challenging. Starting from [23,24,25], we introduce a competitive stochastic model for post-transcriptional regulation of PTEN, including interactions with the miRNA and concurrent genes. Our model also covers protein formation and the backward mechanism going from the protein back to the miRNA. The numerical simulations show that the model reproduces the expected dynamics of normal glial cells. Moreover, the introduction of translational and transcriptional delays offers some interesting insights for the PTEN low expression as observed in brain tumor cells.
2015, 12(5): 983-1006 doi: 10.3934/mbe.2015.12.983 +[Abstract](380) +[PDF](653.5KB)
Abstract:
In this paper we formulate a dynamical model to study the transmission dynamics of schistosomiasis in humans and snails. We also incorporate bovines in the model to study their impact on transmission and controlling the spread of Schistosoma japonicum in humans in China. The dynamics of the model is rigorously analyzed by using the theory of dynamical systems. The theoretical results show that the disease free equilibrium is globally asymptotically stable if $\mathcal R_0<1$, and if $\mathcal R_0>1$ the system has only one positive equilibrium. The local stability of the unique positive equilibrium is investigated and sufficient conditions are also provided for the global stability of the positive equilibrium. The optimal control theory are further applied to the model to study the corresponding optimal control problem. Both analytical and numerical results suggest that: (a) the infected bovines play an important role in the spread of schistosomiasis among humans, and killing the infected bovines will be useful to prevent transmission of schistosomiasis among humans; (b) optimal control strategy performs better than the constant controls in reducing the prevalence of the infected human and the cost for implementing optimal control is much less than that for constant controls; and (c) improving the treatment rate of infected humans, the killing rate of the infected bovines and the fishing rate of snails in the early stage of spread of schistosomiasis are very helpful to contain the prevalence of infected human case as well as minimize the total cost.
2015, 12(5): 1007-1016 doi: 10.3934/mbe.2015.12.1007 +[Abstract](387) +[PDF](993.4KB)
Abstract:
A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.
2015, 12(5): 1017-1035 doi: 10.3934/mbe.2015.12.1017 +[Abstract](452) +[PDF](446.8KB)
Abstract:
A method for early diagnosis of parametric changes in intracellular protein synthesis models (e.g. the p53 protein - mdm2 inhibitor model) is developed with the use of a nonlinear Kalman Filtering approach (Derivative-free nonlinear Kalman Filter) and of statistical change detection methods. The intracellular protein synthesis dynamic model is described by a set of coupled nonlinear differential equations. It is shown that such a dynamical system satisfies differential flatness properties and this allows to transform it, through a change of variables (diffeomorphism), to the so-called linear canonical form. For the linearized equivalent of the dynamical system, state estimation can be performed using the Kalman Filter recursion. Moreover, by applying an inverse transformation based on the previous diffeomorphism it becomes also possible to obtain estimates of the state variables of the initial nonlinear model. By comparing the output of the Kalman Filter (which is assumed to correspond to the undistorted dynamical model) with measurements obtained from the monitored protein synthesis system, a sequence of differences (residuals) is obtained. The statistical processing of the residuals with the use of $\chi^2$ change detection tests, can provide indication within specific confidence intervals about parametric changes in the considered biological system and consequently indications about the appearance of specific diseases (e.g. malignancies)
2015, 12(5): 1037-1053 doi: 10.3934/mbe.2015.12.1037 +[Abstract](426) +[PDF](2010.9KB)
Abstract:
The goal of this paper is to analyze a model of cancer-immune system interactions from [16], and to show how the introduction of control in this model can dramatically improve the hypothetical patient response and in effect prevent the cancer from growing. We examine all the equilibrium points of the model and classify them according to their stability properties. We identify an equilibrium point corresponding to a survivable amount of cancer cells which are exactly balanced by the immune response. This situation corresponds to cancer dormancy. By using Lyapunov stability theory we estimate the region of attraction of this equilibrium and propose two control laws which are able to stabilize the system effectively, improving the results of [16]. Ultimately, the analysis presented in this paper reveals that a slower, continuous introduction of antibodies over a short time scale, as opposed to mere inoculation, may lead to more efficient and safer treatments.
2015, 12(5): 1055-1063 doi: 10.3934/mbe.2015.12.1055 +[Abstract](410) +[PDF](343.8KB)
Abstract:
Based on the reported data until 18 March 2015 and numerical fitting via a simple formula of cumulative case number, we provide real-time estimation on basic reproduction number, inflection point, peak time and final outbreak size of ongoing Ebola outbreak in West Africa. From our simulation, we conclude that the first wave has passed its inflection point and predict that a second epidemic wave may appear in the near future.
2015, 12(5): 1065-1081 doi: 10.3934/mbe.2015.12.1065 +[Abstract](391) +[PDF](447.0KB)
Abstract:
In this paper, we analyze a general predator-prey model with state feedback impulsive harvesting strategies in which the prey species displays a strong Allee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.
2015, 12(5): 1083-1106 doi: 10.3934/mbe.2015.12.1083 +[Abstract](458) +[PDF](617.3KB)
Abstract:
A multi-group epidemic model with distributed delay and vaccination age has been formulated and studied. Mathematical analysis shows that the global dynamics of the model is determined by the basic reproduction number $\mathcal{R}_0$: the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0\leq1$, and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. Lyapunov functionals are constructed by the non-negative matrix theory and a novel grouping technique to establish the global stability. The stochastic perturbation of the model is studied and it is proved that the endemic equilibrium of the stochastic model is stochastically asymptotically stable in the large under certain conditions.
2015, 12(5): 1107-1126 doi: 10.3934/mbe.2015.12.1107 +[Abstract](351) +[PDF](5989.7KB)
Abstract:
Mucociliary clearance is the first line of defense in our airway. The purpose of this study is to identify and study key factors in the cilia motion that influence the transport ability of the mucociliary system. Using a rod-propel-fluid model, we examine the effects of cilia density, beating frequency, metachronal wavelength, and the extending height of the beating cilia. We first verify that asymmetry in the cilia motion is key to developing transport in the mucus flow. Next, two types of asymmetries between the effective and recovery strokes of the cilia motion are considered, the cilium beating velocity difference and the cilium height difference. We show that the cilium height difference is more efficient in driving the transport, and the more bend the cilium during the recovery stroke is, the more effective the transport would be. It is found that the transport capacity of the mucociliary system increases with cilia density and cilia beating frequency, but saturates above by a threshold value in both density and frequency. The metachronal wave that results from the phase lag among cilia does not contribute much to the mucus transport, which is consistent with the experimental observation of Sleigh (1989). We also test the effect of mucus viscosity, whose value is found to be inversely proportional to the transport ability. While multiple parts have to interplay and coordinate to allow for most effective mucociliary clearance, our findings from a simple model move us closer to understanding the effects of the cilia motion on the efficiency of this clearance system.
2015, 12(5): 1127-1139 doi: 10.3934/mbe.2015.12.1127 +[Abstract](426) +[PDF](415.0KB)
Abstract:
The inflammatory response aims to restore homeostasis by means of removing a biological stress, such as an invading bacterial pathogen. In cases of acute systemic inflammation, the possibility of collateral tissue damage arises, which leads to a necessary down-regulation of the response. A reduced ordinary differential equations (ODE) model of acute inflammation was presented and investigated in [10]. That system contains multiple positive and negative feedback loops and is a highly coupled and nonlinear ODE. The implementation of nonlinear model predictive control (NMPC) as a methodology for determining proper therapeutic intervention for in silico patients displaying complex inflammatory states was initially explored in [5]. Since direct measurements of the bacterial population and the magnitude of tissue damage/dysfunction are not readily available or biologically feasible, the need for robust state estimation was evident. In this present work, we present results on the nonlinear reachability of the underlying model, and then focus our attention on improving the predictability of the underlying model by coupling the NMPC with a particle filter. The results, though comparable to the initial exploratory study, show that robust state estimation of this highly nonlinear model can provide an alternative to prior updating strategies used when only partial access to the unmeasurable states of the system are available.
2017 Impact Factor: 1.23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8034682869911194, "perplexity": 691.6997754043553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156252.62/warc/CC-MAIN-20180919161420-20180919181420-00340.warc.gz"} |
https://www.physicsforums.com/threads/poisson-and-binomial-distributions-corrupted-characters-in-a-file.370564/ | # Poisson and binomial distributions, corrupted characters in a file
1. Jan 18, 2010
### Kate2010
A text file contains 1000 characters. When the file is sent by email from one machine to another, each character (independent of other characters) has probability 0.001 of being corrupted. Use a poisson random variable to estimate the probability that the file is transferred with no errors. Compare this to the answer you get when modelling the number of errors as a binomial random variable.
Poisson:
Let x be the number of corrupted characters.
E(X) = 0.001 = a
P(X=n) = (0.001^n)(e^-0.001)/(n!)
P(X=0) = e^-0.001
Binomial:
np = 1000 x 0.001 = 1
When I approximate using poisson but with np instead I get e^-1 which is about 0.37.
This doesn't seem right?
2. Jan 18, 2010
### Staff: Mentor
Show us your calculation of P(X = 0) for the binomial case.
3. Jan 18, 2010
### Kate2010
I used poisson but approximated lambda as np, so ((np)^k)(e^-np)/k! where np = 1000x0.001 = 1 and k = 0, so we get (1^0)(e^-1)/0! = e^-1.
I don't know how I could use the proper binomial random variable.
4. Jan 18, 2010
### Staff: Mentor
I believe you are supposed to assume that X is a binomial r.v., where the probability for a given character being in error is .001. You want P(X=0). See this Wikipedia page for more information.
5. Jan 18, 2010
### Kate2010
Thanks, when I do it that way I get (1000 choose 0)(0.001^0)(1-0.001)^1000 = 0.3676...
So, is it when I have calculated my poisson RV that I've gone wrong and I should have done this as I was previously trying to approximate binomial?
6. Jan 18, 2010
### Kate2010
Actually, have just read up some more on Poisson distributions and I know where I've gone wrong. Thanks for all your help! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8378203511238098, "perplexity": 1110.1162309939662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647576.75/warc/CC-MAIN-20180321043531-20180321063531-00186.warc.gz"} |
http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts-Grade-8/r19/section/3.13/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 3.13: Solve Inequalities Using Addition or Subtraction
Difficulty Level: At Grade Created by: CK-12
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Practice Inequalities with Addition and Subtraction
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The students at Floyd Middle School have been working hard fundraising. They sold popcorn, had a bake sale and a car wash. Finally with the time for purchasing band uniforms rapidly approaching, Mrs. Kline gathered the whole band together one afternoon to discuss their profit.
“We did very well,” she started. “We raised a total of $12,000 and we had$1,000 in our account, so we have $13,000 to spend on our uniforms. I know that may seem like a lot of money, but uniforms are expensive. We are only going to be able to purchase new jackets for everyone. If we have any money left over, we’ll buy new fingerless gloves because some of the ones we’re using look awful.” “Mrs. Kline, did you already pick the design of the jacket?” Kayla asked from the second row. “Yes. It was one of the ones we voted on last year,” she said holding up a picture of the shiny, new navy jacket. “Now I need a few people to figure out the cost and if we have enough for the gloves too.” Kayla and Juan volunteered to work on the arithmetic. Here is the information Mrs. Kline gave them. The jackets each cost$99.95.
The total budget is $13,000. There are 144 students in the band. “We will need to spend$11,512.80 on the jackets,” Kayla said to Juan.
“Wow, that’s a lot of money. How much can we spend on the gloves?”
That is a great question. It is one that can be answered by writing an inequality. The students need their total to be equal to or less than 13,000. In this Concept, you will learn how to work with inequalities that have addition and/or subtraction in them. Then you will use what you have learned to help Kayla and Juan with the band uniforms. ### Guidance Sometimes, you will have an inequality that is not as straightforward as \begin{align*}x>4\end{align*}. With this example, we know that the variable will be equal to any number that is greater than four. This is quite easy to work with and we can write a set of numbers to make this inequality a true statement. What if it isn’t that simple? Sometimes, you will see an inequality like this one. \begin{align*}x+3>7\end{align*} Here we need to figure out the set of numbers that will make this a true statement. We are looking for a number that when added to three is greater than seven. To figure this out, we will need to solve this inequality. Solving an inequality is similar to solving an equation. Here are some number properties that can help you solve inequalities. The addition property of inequality states that if the same number is added to each side of an inequality, the sense of the inequality stays the same. In other words, the inequality symbol does not change. If \begin{align*}a>b\end{align*}, then \begin{align*}a+c>b+c\end{align*}. If \begin{align*}a \ge b\end{align*}, then \begin{align*}a+c \ge b+c\end{align*}. If \begin{align*}a, then \begin{align*}a+c. If \begin{align*}a \le b\end{align*}, then \begin{align*}a+c \le b+c\end{align*}. What about subtraction? Remember, subtracting a number, \begin{align*}c\end{align*}, is the same as adding its opposite, \begin{align*}-c\end{align*}. So, the addition property of inequality applies to subtraction as well. We can also state this as it’s own property. The subtraction property of inequality states that if the same number is subtracted from each side of an inequality, the sense of the inequality stays the same. In other words, the inequality symbol does not change. If \begin{align*}a>b\end{align*}, then \begin{align*}a-c > b-c\end{align*}. If \begin{align*}a \ge b\end{align*}, then \begin{align*}a-c \ge b-c\end{align*}. If \begin{align*}a, then \begin{align*}a-c < b-c\end{align*}. If \begin{align*}a \le b\end{align*}, then \begin{align*}a-c \le b-c\end{align*}. Applying these properties makes our work quite simple. You can think of solving inequalities in the same way that you thought of solving equations. The big difference is that your answer will be a set of numbers and not a single number. Just like with equations, you need to be sure that your answer makes the mathematical statement true. If it doesn’t, then you need to rethink your answer. Now let’s look at solving inequalities. Solve this inequality and graph its solution: \begin{align*}n-3<5\end{align*}. Solve the inequality as you would solve an equation, by using inverse operations. Since the 3 is subtracted from \begin{align*}n\end{align*}, add 3 to both sides of the inequality to solve it. Since you are adding the same number to both sides of the inequality, the addition property of inequality applies. According to that property, we know that the inequality symbol should stay the same when we add the same number, 3, to both sides of the inequality. Think back! To solve inequalities, you may need to remember how to add and subtract integers. Pay attention to the sign when you work with these values. \begin{align*}n-3 &< 5\\ n-3+3 &< 5+3\\ n+(-3+3) &< 8\\ n+0 &< 8\\ n &< 8\end{align*} Now, graph the solution. The inequality \begin{align*}n<8\end{align*} is read as “\begin{align*}n\end{align*} is less than 8.” So, the solution set for this inequality includes all numbers that are less than 8, but it does not include 8. Draw a number line from 0 to 10. Add an open circle at 8 to show that 7 is not a solution for this inequality. Then draw an arrow showing all numbers less than 8. Solve this inequality and graph its solution: \begin{align*}-2 \le x+4\end{align*} Use inverse operations to isolate the variable. Since the 4 is added to \begin{align*}x\end{align*}, subtract 4 from both sides of the inequality to solve it. Since you are subtracting the same number from both sides of the inequality, the subtraction property of inequality applies. According to that property, the inequality symbol should stay the same when we subtract the same number, 4, from both sides of the inequality. \begin{align*}-2 & \le x+4\\ -2-4 & \le x+4-4\\ -2+(-4) &\le x+0\\ -6 & \le x\end{align*} Now, we should graph the solution. However, before we can do that, we need to rewrite the inequality so the variable is listed first. The inequality \begin{align*}-6 \le x\end{align*} is read as “-6 is less than or equal to \begin{align*}x\end{align*}.” If we list the \begin{align*}x\end{align*} first, we must reverse the inequality symbol. That means changing the “less than or equal to” symbol \begin{align*}(\le)\end{align*} to a “less than or equal to symbol” \begin{align*}(\ge)\end{align*}. So, \begin{align*}-6 \le x\end{align*} equivalent to \begin{align*}x \ge -6\end{align*}. This makes sense. If -6 is less than or equal to \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} must be greater than or equal to -6. The inequality \begin{align*}x \ge -6\end{align*} is read as “\begin{align*}x\end{align*} is greater than or equal to -6.” So, the solutions of this inequality include -6 and all numbers that are greater than -6. Draw a number line from -10 to 0. Add a closed circle at -6 to show that -6 is a solution for this inequality. Then draw an arrow showing all numbers greater than -6. #### Example A \begin{align*}x+5<-12\end{align*} Solution: \begin{align*}x<-17\end{align*} #### Example B \begin{align*}y-8\le5\end{align*} Solution: \begin{align*}y\le13\end{align*} #### Example C \begin{align*}a-5\ge22\end{align*} Solution: \begin{align*}a\ge27\end{align*} Now let's go back to the dilemma at the beginning of the Concept. First, we write an inequality to represent the uniform cost, unknown cost of gloves, and the total budget for everything. The total of the uniforms and gloves must be less than or equal to the total budget. \begin{align*}\ 11,512 = \text{cost of uniforms}\end{align*} \begin{align*}x = \text{budget for gloves}\end{align*} \begin{align*}\le\end{align*}13,000
Here is the inequality.
\begin{align*}11,512 + x \le 13,000\end{align*}
We solve the inequality by using inverse operations.
\begin{align*}x &\le 13,000 - 11,512\\ x & \le \1487.20\end{align*}
The students will have a maximum budget of $1487.20 to spend on the gloves. ### Vocabulary Equation a mathematical statement using an equals sign where the quantity on one side of the equals is the same as the quantity on the other side. Inequality a mathematical statement where the value on one side of an inequality symbol can be less than, greater than and sometimes also equal to the quantity on the other side. The key is that the quantities are not necessarily equal. Addition Property of Inequality You can add a quantity to both sides of an inequality and it does not change the sense of the inequality. Subtraction Property of Inequality You can subtract a quantity from both sides of an inequality and it does not change the sense of the inequality. ### Guided Practice Here is one for you to try on your own. At the store, Talia bought one item—a$4.99 bottle of shampoo. Let \begin{align*}d\end{align*} represent the amount in dollars that she handed the clerk. She received more than 5 in change. a. Write an inequality to represent \begin{align*}d\end{align*}, the number of dollars Talia handed the clerk to pay for the shampoo. b. List three possible values of \begin{align*}d\end{align*}. Solution Consider part a first. Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. The fact that this problem involves “change” may help you see that you should write a subtraction expression to represent the first part of this problem. To represent how much change Talia received, you will need to subtract the cost of the shampoo from the amount she handed the clerk. The key words “more than”, in this case, indicate that you should use a \begin{align*} > \end{align*} symbol. Use this information to write an inequality for this problem. \begin{align*}& (\text{dollars handed to the clerk}) - (\text{cost of \4.99 bottle of shampoo}) > (\text{\5 in change})\\ & \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \quad \downarrow \qquad \quad \downarrow\\ & \qquad \qquad \qquad d \qquad \qquad \qquad - \qquad \qquad \qquad \quad 4.99 \qquad \qquad \qquad > \quad \quad \ \ 5\end{align*} So, this problem can be represented by the inequality \begin{align*}d-4.99 > 5\end{align*}. Next, consider part b. Solve the inequality to help you find three possible values for \begin{align*}d\end{align*}. To solve this inequality, add 4.99 to both sides. Do not change the inequality symbol. \begin{align*}d-4.99 & > 5\\ d - 4.99+4.99 &> 5+4.99\\ d+(-4.99+4.99) &> 5.00+4.99\\ d+0 &> 9.99\\ d &> 9.99\end{align*} According to the inequality above, the amount Talia handed the clerk was more than9.99.
So, three possible values of \begin{align*}d\end{align*} are $10.00,$10.99, and $20.00. These are only 3 possible answers. You could choose any amount that is greater than$9.99.
### Practice
Directions: Solve the following inequalities.
1. \begin{align*}x+4>10\end{align*}
2. \begin{align*}y-11<20\end{align*}
3. \begin{align*}a+2<1\end{align*}
4. \begin{align*}b+3 \ge 5\end{align*}
5. \begin{align*}y-2 \le -4\end{align*}
6. \begin{align*}x+1 \ge -5\end{align*}
7. \begin{align*}x-3<11\end{align*}
8. \begin{align*}x-4>-3\end{align*}
9. \begin{align*}y+7>22\end{align*}
10. \begin{align*}a-6\ge-1\end{align*}
11. \begin{align*}b+14>20\end{align*}
12. \begin{align*}x-24>-11\end{align*}
13. \begin{align*}a+3\le-9\end{align*}
14. \begin{align*}x-12>1\end{align*}
15. \begin{align*}y+13>-33\end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
You can add a quantity to both sides of an inequality and it does not change the sense of the inequality. If $x > 3$, then $x+2 > 3+2$.
Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
inequality
An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
Subtraction Property of Inequality
The subtraction property of inequality states that the subtraction of equal amounts from both sides of an inequality will not change the sense of the inequality.
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http://quant.stackexchange.com/tags/option-pricing/hot?filter=year | # Tag Info
Q: What does the risk-neutral price represent if the option is not replicable? In an incomplete market, there is no unique martingale measure but instead a set $Q$ of equivalent martingale measures. Consequently, there is an interval of arbitrage-free prices: $\Big( inf_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX], sup_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX] ... 5 The following paper gives you really all of the missing steps in a very detailed form: Black-Scholes Option Pricing Formula by Michael Tomas and Ravi Shukla From the paper: "This presentation is purely for pedagogical purposes. In the course of doing work on option pricing, we found no complete solution for the Black-Scholes model. By complete, we mean ... 4 The dynamics of the underlying stock process are obviously crucial to the derivative's price. Thus if you don't necessarily assume$S_t$to be log normally distributed (B&S-Model) you won't get the same price even if the market is arbitrage free. Example: Assume$S_t=C \forall t \in \mathbb{R}^+$and$r=0$. Thus$S_t$is constant and the interest ... 4 Most of the time, when you have a simple SDE without a drift, it's a martingale because the Wiener process itself is a martingale. In your example, you have a constant with the Wiener process, therefore the whole process must also be a martingale because the expectation is clearly X(t). However, we can't conclude a driftless SDE is always a martingale. ... 4 I would use the following arguments: If the option were on the first throw of the dice, then we would price it using the expectation, which is$3.5$(=$(1+2+\cdots+6)/6$. Now we have a 2 stage game: First throw : if the player throws more than$3.5$points, i.e.$4,5,6$, then there is no sense in throwing again. If he throws$1-3$then it makes sense to ... 4 The condition $$ud=1\text{, or equivalently }u=1/d$$ is necessary to ensure convergence of the Binomial tree's mean$\mu$and standard deviation$\sigma$to nonfinite values when$n$(number of steps) goes to infinity. Cox-Rubinstein-Ross showed in their famous paper, that to achieve this, we must have: $$u=e^{\sigma\sqrt{t/n}}\text{, ... 4 Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these have been relaxed.) In particular, there are two schools of criticism, one is that time homogeneity makes little sense in the real market. In fact, the Recovery ... 4 In three bullet points: Efficiency: the obtained prices maximize assumed utilities of different agents. In their paper "The Valuation of Option Contracts and a Test of Market Efficiency", Cohen, Black and Scholes compare the theoretical value of options to their market price. The efficiency is in this sense: can agents obtain more or less in practice than ... 4 it doesn;t imply \ln S_T=\ln S_0+rT+σW^Q_T it implies \ln S_T=\ln S_0+(r-0.5\sigma^2)T+σW^Q_T look up Ito's lemma. This is covered in just about any book on financial maths including my own Concepts etc 4 Your characterisation is correct but incomplete. 1) The most important part of Black-Scholes is not the model but the more general framework of dynamic hedging: you can replicate your payoff by continuously trading the underlying and the amount (delta) you should hold is the derivative of the current premium with respect to the current spot. This is a much ... 3 The Black-Scholes price of this option is approximately 14.8. When I run a Monte Carlo simulation with 10000 paths and "exact" time stepping, I get results very close to this value. You are simulating the terminal asset price with the first-order Euler approximation over multiple time steps:$$S(t+\Delta t)= S(t) + rS(t)\Delta t + \sigma ... 3 Benoit Mandelbrot applied fractals and self-similarity to financial markets and the hurst exponent has its roots in chaos theory. Look at this article from Wilmott magazine. Just a personal note: I have not worked that much with this kind of theory so far but I also have not seen any of my peers being exceptionally sucessfull with these methods. 3 Asian options: strike is average of underlying over tenor. Underlying is stochastic. Options with kock-ins/knock-outs: Underlying is stochastic and may cross the kock threshold as it evolves. Option value depends on this cross or lack thereof (boolean). Options on Options, too. Motivations for Asian options you can google. Kock-ins and knock-outs ... 3 The error is, you are not storing the random numbers for the same path at the end: xbefore = x + c*tau + sigma*sqrt(tau)*randn() A = muA + sigmaA*randn(); xafter = xbefore + A; But then at end you set a different path here by creating a new random number: xT = log(S0)+(c+muA*lambda)*T+sqrt((sigma^2+(muA^2+sigmaA^2)*lambda)*T)*randn(); randn() ... 3 These options can be priced by adding an early exercise premium value to the intrinsic value: http://www.statistics.nus.edu.sg/~stalimtw/PDF/lb-float.pdf 3 you don't need$ud=1.$In fact, there are now about 30 binomial trees which converge to Black--Scholes in the large step limit. Most of them do not have$ud=1.$All you need is $$d < e^{r \Delta t} < u$$ The tree recombines provided$u$and$d$don't change from step to step. See my book More Mathematical Finance for a comprehensive review and ... 3 There are lots of papers online and here are a few I would suggest math.umn riskworx G. Dimitroff, J. de Kock Nowak, Sibetz I you have matlab there is an step step example to calibrate SABR model. Since it uses the financial toolbox of matlab for a few functions I dont think you can replicate it in any other language. There must be C++ code available ... 3 You can view the price of an option as the cost to dynamically replicate it. The more volatility, the more costs you will have trading the underlying to keep your delta equal to 0 (I'm assuming you sold the option, hence a negative gamma position). So, if at any spot, any date your local vol is above 0.194, rebalancing the portfolio will be constantly more ... 3 Think of moving volatility in the other direction. As volatility approaches zero, any call strike strictly smaller than the ATM strike,$K<K_{ATM}$, will have zero probability of ending in the money, and the corresponding option value will be zero. An infinitesimally small change in stock price will not move$K$past$K_{ATM}$, so the option value ... 3 I found and answer to my own question. So, I post it here for people who maybe have the same problem. The answer, however, is quite intuitive. The last observation used for the estimation of the physical density is also the time point where the investors know the most about the physical density because at this point the most possible historical observations ... 3 Simply put, no. Vega depends on a variety of factors (including the level/price of the underlying asset). However, vomma/volga/vega convexity (whatever you want to call dVega/dIV) is always positive. So as IV increases, the vega of an option increases - I think this might have been what you were getting at. It's important to understand that IV is an input ... 2 LSM is very fiddly. The most important things in my view are 1) don't believe anyone who says that the choice of basis functions doesn't matter. 2) implement an upper bounder, eg Andersen--Broadie (2003) or Joshi-Tang (2014) so you can tell if your prices are good 3) do two passes, one to build the strategy, one to price, if they give very different ... 2 American options pricing (swaption is just a kind of option) is a bit tricky due to the early exercise. Here is a page listing possible approaches, including some numeric methods, and some close form approximation formula. As I understand, lattice methods (tree, PDE discretization such as forward shooting) are fine to price American options. There're ... 2 As I see it the question does not enforce that the market is free of arbitrage. This is why you can get to contradicting prices. Thus you can't actually apply a risk-neutral argument here without making additional assumptions. You yourself provide the example of such an arbitrage. If the underlying process had a B&S dynamics you could just borrow money ... 2 Assume the price follows a lognormal process. We can convert it into a problem of finding the probability of a standard Brownian motion particle starting from$0$and hitting$x$before time$t$, or its first passage time$\tau_x$being less than$t$. This can be derived through the reflection principle. The paths crossing$x$are exactly paired up by the ... 2 There is a whole family of GARCH option pricing models; ones with complex distributions, leverage effects, skewness parameters etc. For an example see Christoffersen and Jacobs (2004). Some example GARCH models: NGARCH EGARCH TGARCH Some distributions that can be used: Hyperbolic Normal Inverse Gaussian Variance Gamma and the generalized form ... 2 Generally speaking, if you have two or three sources of noise, you are still going to be much better off pricing American options on a lattice than via LSMC. Too often, LSMC becomes the refuge of academics lacking patience to learn proper lattice techniques. Now, you can frequently reduce the difficulty of pricing American options by considering the ... 2 The problem with your formula is the equation sign$=$. The second order finite difference is only an approximation to the true gamma: $$f^{\prime \prime}(x) \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}.$$$h$can not be a result. Ideally, it should be small (whatever that means), so your original choice of$1\text{bp}$seems appropriate for this ... 2 Historical returns are not to be used 'untreated' for the calculation of option prices. The expectation that you will be using in Monte Carlo will take the form $$C(K,T) = E^Q\{D(T)\ \max[0, S_T-K, 0]\}$$ where$T$is the maturity,$K$is the strike price,$S$is the stock price and$D\$ is the discount factor. But the expectation is taken under the 'risk ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8352516293525696, "perplexity": 1022.0228491058127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131297195.79/warc/CC-MAIN-20150323172137-00124-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/125281/coutour-integral-of-gamma-functions | # Coutour Integral of Gamma Functions
How do I solve the Integral $$\frac{1}{2\pi j} \oint \frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{ (2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$
This integral is an inverse Mellin transform. Therefore, the contour extends from $l+j\infty$ to $l-j\infty$, where $l\in\mathbb{R}$.
$j=\sqrt{-1}$
$b \in\mathbb{R},\quad b>0$
$i\in\mathbb{R}, \quad i\ge 0$
$\Gamma(.)$ is the gamma function. Does it make any difference when $i$ becomes an integer?
-
you mean without telling us how a depends on s? – Carlo Beenakker Mar 22 '13 at 15:08
sorry...just fixed it – Remy Mar 22 '13 at 15:11
What does $ii$ mean? And is $j^2=-1$? – Daniel Loughran Mar 22 '13 at 17:00
sorry ..fixed it again ... it is hard to type using a tablet ... ii is an integer, I updated the equation – Remy Mar 22 '13 at 19:11
Why not use $\Gamma[3+i-s]=(2+i-s)\Gamma[2+i-s]$ to cancel a Gamma function in the numerator and denominator? – Stopple Mar 22 '13 at 22:45
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https://hal.in2p3.fr/in2p3-00087373 | # Gamma and Cosmic-Ray Tests of Special Relativity
Abstract : Lorentz symmetry violation (LSV) at Planck scale can be tested (see e.g. physics/0003080) through ultra-high energy cosmic rays (UHECR). In a deformed Lorentz symmetry (DLS) pattern where the effective LSV parameter varies like the square of the momentum scale (quadratically deformed relativistic kinematics, QDRK), a 10E-6 LSV at Planck scale would be enough to produce observable effects on the properties of cosmic rays at the 10E20 eV scale: absence of GZK cutoff, stability of unstable particles, lower interaction rates, kinematical failure of any parton model and of standard formulae for Lorentz contraction and time dilation... Its phenomenological implications are compatible with existing data. If the effective LSV parameter is taken to vary linearly with the momentum scale (linearly deformed relativistic kinematics, LDRK), a LSV at Planck scale larger than 10E-7 eV seems to lead to contradictions with data above TeV energies. Consequences are important for high-energy gamma-ray experiments, as well as for high-energy cosmic rays and gravitational waves.
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Conference papers
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http://hal.in2p3.fr/in2p3-00087373
Contributor : Dominique Girod Connect in order to contact the contributor
Submitted on : Monday, July 24, 2006 - 10:11:54 AM
Last modification on : Friday, November 6, 2020 - 3:26:28 AM
### Citation
L. Gonzalez-Mestres. Gamma and Cosmic-Ray Tests of Special Relativity. International Symposium on High Energy Gamma-Ray Astronomy, Jun 2000, Heidelberg, Germany. pp.878-881. ⟨in2p3-00087373⟩
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https://www.ann-geophys.net/36/381/2018/angeo-36-381-2018.html | Journal topic
Ann. Geophys., 36, 381–404, 2018
https://doi.org/10.5194/angeo-36-381-2018
Special issue: Space weather connections to near-Earth space and the...
Ann. Geophys., 36, 381–404, 2018
https://doi.org/10.5194/angeo-36-381-2018
Regular paper 16 Mar 2018
Regular paper | 16 Mar 2018
# Variability and trend in ozone over the southern tropics and subtropics
Variability and trend in ozone over the southern tropics and subtropics
Abdoulwahab Mohamed Toihir1, Thierry Portafaix1, Venkataraman Sivakumar2, Hassan Bencherif1,2, Andréa Pazmiño3, and Nelson Bègue1 Abdoulwahab Mohamed Toihir et al.
• 1Laboratoire de l'Atmosphère et des Cyclones, Université de La Réunion, St-Denis, Réunion Island, France
• 2Discipline of Physics, School of Chemistry and Physics, University of KwaZulu-Natal, Durban, South Africa
• 3Laboratoire Atmosphères, Milieux, Observations Spatiales, CNRS, Université Versailles Saint Quentin, Guyancourt, France
Correspondence: Abdoulwahab Mohamed Toihir (fahardinetoihr@gmail.com)
Abstract
Long-term variability in ozone trends was assessed over eight Southern Hemisphere tropical and subtropical sites (Natal, Nairobi, Ascension Island, Java, Samoa, Fiji, Reunion and Irene), using total column ozone data (TCO) and vertical ozone profiles (altitude range 15–30 km) recorded during the period January 1998–December 2012. The TCO datasets were constructed by combination of satellite data (OMI and TOMS) and ground-based observations recorded using Dobson and SAOZ spectrometers. Vertical ozone profiles were obtained from balloon-sonde experiments which were operated within the framework of the SHADOZ network. The analysis in this study was performed using the Trend-Run model. This is a multivariate regression model based on the principle of separating the variations of ozone time series into a sum of several forcings (annual and semi-annual oscillations, QBO (Quasi-Biennial Oscillation), ENSO, 11-year solar cycle) that account for most of its variability.
The trend value is calculated based on the slope of a normalized linear function which is one of the forcing parameters included in the model. Three regions were defined as follows: equatorial (0–10 S), tropical (10–20 S) and subtropical (20–30 S). Results obtained indicate that ozone variability is dominated by seasonal and quasi-biennial oscillations. The ENSO contribution is observed to be significant in the tropical lower stratosphere and especially over the Pacific sites (Samoa and Java). The annual cycle of ozone is observed to be the most dominant mode of variability for all the sites and presents a meridional signature with a maximum over the subtropics, while semi-annual and quasi-biannual ozone modes are more apparent over the equatorial region, and their magnitude decreases southward. The ozone variation mode linked to the QBO signal is observed between altitudes of 20 and 28 km. Over the equatorial zone there is a strong signal at ∼26km, where 58 % ±2 % of total ozone variability is explained by the effect of QBO. Annual ozone oscillations are more apparent at two different altitude ranges (below 24 km and in the 27–30 km altitude band) over the tropical and subtropical regions, while the semi-annual oscillations are more significant over the 27–30 km altitude range in the tropical and equatorial regions. The estimated trend in TCO is positive and not significant and corresponds to a variation of $\sim \mathrm{1.34}±\mathrm{0.50}$ % decade−1 (averaged over the three regions). The trend estimated within the equatorial region (0–15 S) is less than 1 % per decade, while it is assessed at more than 1.5 % decade−1 for all the sites located southward of 17 S. With regard to the vertical distribution of trend estimates, a positive trend in ozone concentration is obtained in the 22–30 km altitude range, while a delay in ozone improvement is apparent in the UT–LS (upper troposphere–lower stratosphere) below 22 km. This is especially noticeable at approximately 19 km, where a negative value is observed in the tropical regions.
1 Introduction
Atmospheric ozone protects all life on earth from damaging solar ultraviolet radiation. About 90 % of ozone is observed in the stratosphere where it is strongly influenced by various photochemical and dynamical processes. Three processes determine stratospheric ozone distribution and concentration, namely ozone production, destruction and transport. While ozone concentration in the upper stratosphere (35–50 km) is primary determined by the processes of photochemical creation and destruction, ozone concentration in the lower and middle stratosphere (below 30 km) is strongly affected by transport (Portafaix et al., 2003; Bencherif et al., 2007, 2011; El-Amraoui et al., 2010). Ozone is formed primarily in tropical regions as a result of a reaction between atmospheric oxygen and the ultraviolet component of incident solar radiation; it is then subsequently transported and distributed to higher latitudes following the Brewer–Dobson circulation (Weber et al., 2011). The combination of these processes contributes to the overall spatiotemporal distribution and variability of atmospheric ozone.
The variability in ozone levels also depends on several dynamic proxies and the chemical evolution of ozone-depleting substances (ODS). Since 1980, anthropogenic ODS emissions reaching the stratosphere have led to a decline in global ozone concentration (Sinnhuber et al., 2009). However, recent observations have indicated that the decreasing trend in stratospheric ozone has been halted from the mid-90s (1995–1996), and an upward trend has been observed at 60 N–60 S since 1997 (Jones et al., 2009; Nair et al., 2013; Kyrölä et al., 2013; Chehade et al., 2014; Eckert et al., 2014). This increasing trend in stratospheric ozone is associated with a decrease in atmospheric chlorine and bromine compounds as a direct result of the Montreal Protocol which regulates ODS emission (United Nation Environment Program, UNEP 2009). From the present and going forward, this increase in stratospheric ozone is expected to begin to be more widely observable on a global scale (Steinbrecht et al., 2009; WMO (World Meteorological Organization, 2010, 2014). However, Butler et al. (2016) have shown that tropical ozone will not recover to typical levels of the 1960s by the end of the 21st century. Depending on the specific processes involved, climate change can induce modifications in the stratosphere that can delay or accelerate ozone recovery. Greenhouse gas (GHG) induced stratospheric cooling should lead to increased photochemical ozone production and slower destruction rates via the Chapman cycle (Jonsson et al., 2009). Furthermore, and with direct relevance to the region under study, acceleration of the Brewer–Dobson circulation due to tropospheric warming is thought to lead to decreases/increases in tropical/middle latitude stratospheric ozone.
It is important to create a consistent and reliable dataset in order to quantify ozone variability, to estimate trends and to validate models used for predicting future evolution of ozone levels (Randel and Thompson, 2011). The work presented here investigates the period 1998–2012 where an increase in stratospheric ozone is expected to be measurable. The aim of this study is to investigate the current behaviour of seasonal and inter-annual ozone variability over the southern tropics and subtropics and to analyse whether the expected ozone recovery is effective in the selected study region and at which altitude level ozone increase is significant. The WMO (2014) reported a divergence in ozone profile trends computed from individual instruments in the lower stratospheric tropical region. Sioris et al. (2014) showed a declining ozone trend for OSIRIS data, while Gebhardt et al. (2014) reported an increasing ozone trend using measurements from the SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY) and Aura Microwave Limb Sounder (MLS) instruments. Therefore, a study of ozone levels at tropical latitudes and present in the lower and middle stratosphere is of paramount importance. The trend analysis of two types of ozone data is relevant with respect to this study, namely, total column ozone (TCO) and vertical ozone profile measurements. While the TCO analysis offers the best way to provide information on ozone variability for the complete ozone layer, profile analysis allows for the investigation of ozone behaviour at different altitude levels (Nair et al., 2013).
Ozone profile measurement is usually achieved by ground-based lidar instruments, satellite observation and ozonesondes. However, observation by ozonesonde is often preferred due to its high vertical resolution. SHADOZ (Southern Hemisphere Additional Ozonesonde) is the only network which provides ozonesonde profiles over the southern tropics and subtropics. It has a precision of approximately 5 % with a vertical resolution between 50 and 100 m (Thompson et al., 2003a). Previously radiosonde observations from SHADOZ networks have been used to validate tropospheric ozone observations obtained from satellites (Ziemke et al., 2006, 2011), to study the stratospheric altitude profile of ozone and water vapour in the Southern Hemisphere (Sivakumar et al., 2010), to investigate the dynamic characteristics of the Southern Hemisphere tropopause (Sivakumar et al., 2006, 2011; Thompson et al., 2012), to study seasonal and inter-annual variation of ozone and temperature between the atmospheric boundary layer and the middle stratosphere (Thompson et al., 2003b; Diab et al., 2004; Sivakumar et al., 2007; Bègue et al., 2010; Mze et al., 2010) and to analyse the variability and trends in ozone and temperature in the tropics (Lee et al., 2010; Randel and Thompson, 2011).
In this paper, SHADOZ profiles recorded at eight Southern Hemisphere stations were used to investigate seasonal and inter-annual variability of ozone and to estimate the trends at different stratospheric pressure levels. The TCO measurements were performed based on Dobson and SAOZ (Système d'Analyse par Observation Zénithale) ground-based instruments (Pommerau and Goutail, 1988; Dobson, 1931). However, ground-based data were only available for three of the eight stations considered in this work. Satellite observations were therefore used in order to complete ground-based observations, particularly at tropical sites (Nairobi, Java, Samoa, Fiji and Ascension Island) where ground-based datasets were incomplete or non-existent. The chosen satellite datasets are OMI (Ozone Monitoring Instrument) and TOMS (Total Ozone Mapping Spectrometer) (McPeters et al., 1998; Bhartia, 2002) because of their high-quality ozone data. The TOMS instrument has facilitated the recording of long-term ozone measurements on both global and regional scales. TOMS instruments were successfully flown aboard several satellites from 1978 to 2005, with the latest instrument aboard the Earth Probe (EP) satellite operational from July 1996 to December 2005. In the framework of the EOS (Earth Observing System) programme, TOMS was replaced by the OMI instrument launched onboard the Aura satellite (July 2004 and operational to the present). The combined TCO data recorded from ground-based instrumentation and satellites were used to investigate ozone variability and trends over the selected study region. Data analysis was conducted using a multivariate regression model known as Trend-Run (Bencherif et al., 2006; Bègue et al., 2010).
Multivariate regression models are considered powerful tools to study stratospheric ozone variability and trends (Randel and Thompson, 2011; Kyrölä et al., 2013; Nair et al., 2013; Bourassa et al., 2014; Gebhardt et al., 2014; Eckert et al., 2014). In these models, the choice of proxies is important as the indexes of these proxies are often chosen based on atmospheric forcings that have been historically accepted as having an influence on ozone variability (Damadeo et al., 2014). Butchart et al. (2003) analysed equatorial ozone variability based on a coupled stratospheric chemistry and transport model. They observed a strong QBO (Quasi-Biennial Oscillation) signal in total column ozone variability with an observed correlation between the stratospheric ozone anomaly and the westerly wind distribution higher than 50 %. Brunner et al. (2006) demonstrated the existence of a strong QBO signal in ozone variability throughout most of the lower stratosphere, with a peak amplitude in the tropics of the order of 10–20 %. The ozone–QBO relationship has been discussed in several papers, as well as ENSO (El Niño–Southern Oscillation) influence on ozone variability. ENSO events are linked to coherent variations of the ozone zonal mean in the tropical lower stratosphere, tied to fluctuations in tropical upwelling (Randel et al., 2009). Randel and Thompson (2011) observed a negative signal in ozone variability exhibited by the ENSO index with a magnitude of around 6 % in the lower stratosphere.
It is known that solar flux has also contributed to the timescale variability of ozone (Zerefos et al., 1997; Randel and Wu, 2007). Several studies have shown that when the solar activity is high, larger amounts of ozone are formed in the upper stratosphere. Maximum ozone levels should therefore occur during periods of high solar activity (Haigh, 1994; Labitzke et al., 2002; Gray et al., 2010). However, Efstathiou and Varotsos (2013) demonstrated that the effect of the solar cycle response on total column ozone was caused by dynamical changes which were related to solar activity. It has been reported by Austin et al. (2008) that the solar flux response to long-term ozone variation is around 2.5 %, with a maximum in the tropics. Soukharev and Hood (2006) have reported that the solar cycle response to ozone variability is positive and statistically significant in the lower and upper stratosphere but insignificant over the middle stratosphere. In this investigation, QBO, ENSO and solar flux proxies were used as inputs in the Trend-Run model to simulate inter-annual ozone variation over the selected study region. Annual and semi-annual cycles were taken into account in order to quantify their impact on seasonal and annual variability.
This paper is organized as follows: Sect. 2 describes data and instrumentation used in this investigation and Sect. 3 describes the Trend-Run model and presents the method adopted for data analysis. Results detailing ozone trends and variability are presented in Sect. 4. Section 5 concludes this investigation with a summary of important results.
2 Ozone data source
## 2.1 TCO from ground-based observations
The ground-based instruments employed in this investigation are Dobson and SAOZ spectrometers. TCO data obtained from these instruments are available from the World Ozone Ultraviolet radiation Data Centre (WOUDC, http://www.woudc.org/) and the Network for the Detection of Atmospheric Composition Change (NDACC, http://www.ndsc.ncep.noaa.gov/) websites. The Dobson spectrophotometer was the first instrument developed to measure TCO (Dobson, 1931) and the first TCO measurement was made in 1926 at Arosa, Switzerland. Dobson spectrophotometers were initially distributed at northern middle latitudes and then later moved to include observations in the Southern Hemisphere. At present, ground-based ozone measurements using Dobson spectrophotometers are widespread globally. The Dobson network currently consists of more than 80 stations, with most instruments calibrated following the standard reference D 83 which is recognized by the ESRL (Earth System Research Laboratory) and the WMO (Komhyr et al., 1993). The relative uncertainty associated with the Dobson instrument has been assessed at approximately 2 % (Basher, 1985). In this work, TCO recorded from Dobson instruments located at Natal (equatorial site: 35.38 W, 5.42 S) and Irene (subtropical site: 28.22 E, 25.90 S) was used. The measurement principle of the Dobson spectrometer is based on a UV radiation differential absorption technique in a wavelength range where ozone is strongly absorbed compared to one where ozone is weakly absorbed. Total column ozone observations are performed by measuring the relative intensities at selected pairs of ultraviolet wavelengths. The most used wavelengths are the double pair (305.5/325.5 nm and 317.6/339.8 nm) and (311.45/332.4 nm and 316.6/339.8 nm) emanating from the Sun, moon or zenith sky (WMO, 2003, 2008). For further information regarding the Dobson spectrometer functioning, the reader may refer to the studies outlined by Komhyr et al. (1989, 1993).
The SAOZ spectrometer was developed by the CNRS (Centre National de la Recherche Scientifique). In 1988, it was used for the first time at Dumont d'Urville (Antarctica) to measure stratospheric ozone during the polar winter. SAOZ is a passive remote sensing instrument operating in the visible and ultraviolet ranges. It measures the sunlight scattered from the zenith sky in the wavelength range between 300 and 600 nm. The SAOZ instrument is able to retrieve the total column of ozone and NO2 in the visible band with an average spectral resolution of around 1 nm using the differential optical absorption spectroscopy (DOAS) technique. At Reunion, a SAOZ instrument has been operational since 1993 within the framework of NDACC. SAOZ ozone measurements have been performed during sunrise and sunset with a precision of <5 % and an accuracy of <6 % (Hendrick et al., 2011). In the present work, the daily average of TCO was taken as the mean value of sunrise and sunset observations (86–91 SZA) recorded at Reunion. SAOZ TCO measurements have been shown to be in good agreement with OMI-TOMS (Pastel et al., 2014; Toihir et al., 2013) and Infrared Atmospheric Sounding Interferometer (IASI; Toihir et al., 2015a) measurements. Further details can be found in Pazmiño (2010).
## 2.2 TCO from satellite observations
Satellite data employed in this investigation were obtained from the EP-TOMS and OMI-TOMS L2 data products. The EP-TOMS data product is the most recent (1996–2005) and the duration of other important satellite observations include the TOMS instrument onboard the Nimbus-7 satellite (1978–1993), Meteor-3 (1991–1994) and ADEOS (1996–1997). On 2 July 1996, EP-TOMS was the only instrument launched onboard the Earth Probe satellite. It was launched into a polar orbit with an initial altitude of 500 km and an inclination angle of 98. However, after the failure of the ADEOS satellite, the EP-TOMS instrument was raised to an altitude of 739 km with an inclination angle of 98.4 This was done in order to provide complete global coverage of ozone and other species such as sulfur dioxide. The TOMS instrument is a downward nadir viewing spectrometer that measures both incoming solar energy and backscatter ultraviolet radiance at six different wavelengths (379.95, 359.88, 339.66, 331.06, 317.35 and 312.34 nm) with a spatial resolution of 50 km×50 km. The instrument uses a single monochromator and a scanning mirror to sample the backscattered solar ultraviolet radiation at 35 sample points at 3 intervals along a line perpendicular to the orbital plane (Bramstedt et al., 2003). TOMS data used in this work are an overpass product and are available online from the website link http://acdisc.gsfc.nasa.gov/opendap/EarthProbe_TOMS_Level3/contents.html. For more details on the TOMS instrument and the associated data product, the reader may refer to the EP-TOMS data product user guide document (McPeters et al., 1998).
As previously mentioned, the EP-TOMS instrument was operational from July 1996 to December 2005. TCO from TOMS was therefore combined with that from OMI recorded overpass for the eight selected investigation sites in order to produce a complete ozone dataset running to December 2012. Two OMI L2 data products are available: OMI-DOAS and OMI-TOMS. However, the OMI-TOMS L2 product was chosen for this investigation due to its good agreement with the EP-TOMS data product (Toihir et al., 2014). OMI is a compact nadir viewing instrument launched aboard the Aura satellite in July 2004 into a near-polar helio-synchronous orbit at approximately 705 km in altitude. OMI operates at a spectral resolution within 0.5 nm in three spectral regions referred to as UV-1, UV-2 and VIS. In terms of spatial coverage, its viewing angle is 57 under a swath width of 2600 km. The ground pixel size of each scan is 13×24km2 in the UV-2 (310–365 nm) and visible (350–500 nm) channels, and 13 km×48 km for the UV-1 (270–310 nm) channel. OMI data used in this work are overpass L2 products and are accessible from http://avdc.gsfc.nasa.gov/pub/data/satellite/Aura/OMI/V03/L2OVP/. OMI-TOMS ozone data are retrieved based on two wavelengths (317.5 and 331.2 nm are applicable under most conditions, while 331.2 and 360 nm are used for conditions of high ozone concentration and high solar zenith angle). The L2 product has a precision of ∼3 % and has shown good agreement with Dobson and SAOZ measurements over the southern tropics and subtropics (Toihir et al., 2013, 2015a). Further details relating to the OMI instrument and mode of operation can be found in the OMI Algorithm Theoretical Basis Document Volume II (Bhartia, 2002).
## 2.3 Ozone profiles from the SHADOZ network
Balloon-borne electrochemical concentration cell (ECC) ozonesonde devices were used to provide profile ozone measurements over the eight stations (Nairobi, Ascension Island, Java, Samoa, Fiji, Natal, Reunion and Irene) investigated in this work. These stations form part of the SHADOZ network. The geographical location of each site is illustrated in Fig. 1, while geographical coordinates and altitude a.s.l. are given in Table 1. Equipped with radiosondes for temperature, humidity and pressure measurements, the ECC ozonesonde provides vertical profiles with a resolution between 50 and 100 m from ground to the altitude where balloon burst occurs (∼26–32 km). The precisions of SHADOZ measurements have been evaluated to be ∼5 % (Thompson et al., 2003b). Further details on ECC ozonesonde instrument validation, operating mode and algorithm used for ozone partial pressure (in mPa) retrieval in each of the SHADOZ stations can be obtained from Thompson et al. (2003a, 2007) and Smit et al. (2007). In this work, analysis of vertical ozone variability was performed using SHADOZ profiles recorded over 15 years (1998–2012). These data are available from http://croc.gsfc.nasa.gov/shadoz/. The frequency of SHADOZ observations is between one and six balloon launches per month. The monthly mean profiles for a given station were taken as the average of recorded profiles during the month for that station.
Figure 1Geolocation of the SHADOZ sites selected for this investigation.
Table 1Geolocation and mean sea level (MSL) of stations investigated in this study.
3 Data analysis
## 3.1 Methodology
Prior to analysis of TCO variability and trends, preliminary work was completed in order to create a reliable ozone dataset for each station by merging different available measurements. The first combination exercise was performed on satellite data (TCO form OMI and TOMS) by using the monthly values of ozone recorded during the overlap observation period of the two satellites (October 2004–December 2005). The relative difference (RD) between TOMS and OMI with respect to OMI observation was calculated for individual sites as follows:
$\begin{array}{}\text{(1)}& {\text{RD}}_{m}=\mathrm{100}×\frac{{\text{TOMS}}_{m}-{\text{OMI}}_{m}}{{\text{OMI}}_{m}},\end{array}$
where “m” is the month in which both OMI and TOMS observations were performed. After assessing the relative difference, the bias and root mean square (rms) associated with the difference were calculated. For further details outlining the method used to calculate bias and rms, see Toihir et al. (2015a), Toihir et al. (2015b) and Anton et al. (2011). The results obtained for each individual site are presented in Table 2. The recorded biases between TOMS and OMI are positive, thereby indicating an overestimation of TOMS total column ozone with respect to OMI. However, the bias is less than 2.5 % and correlation coefficients (C) between OMI and TOMS are higher than 0.9 (see column 4 of Table 2); it is therefore reasonable to combine the two observations. As OMI measurements recorded during the study period for the chosen sites show better agreement with ground-based (Dobson and SAOZ) measurements than TOMS (see Toihir, 2016, chapter 2), the combination of OMI and TOMS was performed using the OMI observations as a reference. The data combination follows two steps: the first step consists of adjusting the TOMS dataset (January 1998 to December 2005) with respect to OMI by using the obtained rms. The absolute rms is considered the mean systematic error in Dobson units between TOMS and OMI. As TOMS ozone values are always higher than OMI values, TOMS time series (1996–2005) can be adjusted to OMI measurements as follows:
$\begin{array}{}\text{(2)}& {\text{TOMS}}_{\text{adjusted}}\left(m\right)=\text{TOMS}\left(m\right)+\text{rms}.\end{array}$
The second step is to average the measurements from OMI with the adjusted TOMS measurements recorded during the overlap observation period of the two satellites.
Table 2The computed bias, rms (root mean square) and correlation coefficient C observed between OMI and TOMS for individual stations during the period from August 2004 to December 2006.
Figure 2 presents the time evolution of monthly mean TCO measurements over Reunion. Blue and black curves represent TOMS and OMI data respectively. Figure 2 highlights the existing good agreement between both TOMS and OMI observations, with similar results being reported in Toihir et al. (2014). The merged satellite time-evolution data are shown as a dotted red line in Fig. 2. The combination of the above satellite data is performed for each site under study by using the rms obtained for the site (see Table 2).
Figure 2Monthly mean of TCO as measured by TOMS (blue) and OMI (black) over Reunion. The combined measurements of OMI and TOMS data are indicated by the dashed red line.
A second method to validate this data combination consisted of a comparison of satellite data with available ground-based measurements. TCO from ground-based instruments were available at Natal, Reunion and Irene stations and ground-based and TOMS adjusted-OMI time evolutions of the TCO monthly mean over these stations are shown in Fig. 3. Figure 3 illustrates the fact that satellite data show good agreement with ground-based observation before and after the simultaneous period of TOMS and OMI observations, thereby indicating a good agreement between the merged satellite and ground-based data. Correlation coefficients C between the merged satellite measurements and ground-based data are 0.88, 0.90 and 0.97 over Natal, Irene and Reunion respectively. The obtained relative bias between satellite and ground-based observation with respect to the ground-based observation is assessed to be less than 2.5 % for individual sites. Due to this consistency existing between merged satellite data and ground-based measurements, the study of TCO variability and trends was performed using the merged satellite data for sites where no ground-based measurements existed. In the case of three stations where ground-based measurements were available, satellite and ground-based measurements were merged by adjusting the satellite dataset with respect to ground-based data.
Figure 3Temporal evolution of TCO from ground-based spectrometers (blue) compared with that obtained by a combination of TOMS and OMI TCO overpass over Natal (a), Reunion (b) and Irene (c) for the period January 1998 to December 2012.
The study of vertical ozone variability and associated trends was performed based on SHADOZ ozone profile data available for the eight selected sites. In this work, 3431 profiles were examined. The number of examined profiles and the temporal coverage of data recorded for individual stations are presented in Table 3. Although the eight SHADOZ stations started measurements in 1998, the temporal coverage of the data recorded and the number of observations at each station differ. For example, observations at Ascension Island were discontinued in 2010, while no data were available in Natal during 2012. Irene SHADOZ observations were discontinued in 2006 and restarted in November 2012. Considering the discontinuity in observations and the limited number of monthly profiles, monthly data for stations with the same climatological behaviour were averaged (Mzé et al., 2010). Stations were classified into three groups based on low variability in zonal ozone distribution, the proximity of stations and the dynamical structure of the stratosphere (Ziemke et al., 2010). These groupings were near-equatorial (Nairobi, Natal, Java and Ascension Island), tropical (Samoa and Fiji) and subtropical (Reunion and Irene) and corresponded to stations located between latitude bands 0–10 S, 10–20 S and 20–30 S respectively. Geographical coordinates of stations are given in Table 1. Monthly profiles recorded from stations located in the same altitude band were averaged to represent the monthly mean for the selected altitude range. Figure 4 shows the time and height evolution of the constructed monthly mean ozone profile. The top (a), middle (b) and bottom (c) panels represent the monthly vertical distribution of ozone concentration between 15 and 30 km over the equatorial, tropical and subtropical regions respectively. In order to follow a more uniform data processing approach, monthly TCO values for stations from the same latitude range were averaged for specific cases. The constructed TCO time series from the procedure defined above are shown by the blue line in Fig. 5.
Figure 4Time–height section of ozone concentration (mol cm−3) obtained by mean monthly profiles recorded over stations located in the equatorial (a), tropical (b) and subtropical (c) regions from January 1998 to December 2012.
Table 3Time period and total number of height profiles used in this analysis for selected stations.
Figure 5Time evolution of monthly mean total ozone values (blue) over the equatorial (a), tropical (b) and (c) subtropical regions. The black lines represent TCO values as calculated by the Trend-Run model.
## 3.2 Description of the Trend-Run model
Trend-Run is a multi-regression model adapted by Reunion University and dedicated to the study of ozone and temperature variability and associated trends (Bencherif et al., 2006; Bègue et al., 2010; Toihir et al., 2014). Input parameters include monthly mean ozone values (in DU for TCO or concentration at a given pressure for high-altitude profiles) and variables that represent a significant contribution to stratospheric ozone variability. Considering the selected region (0–30 S) and the temporal coverage defined in this work (1998–2012), and in order to simplify analysis and interpretation of results, the parameters included in this analysis were QBO, ENSO and solar flux. Note that aerosols constitute a source of uncertainty that may affect TCO variability, notably following a major event such as a volcanic eruption. However, by using the Trend-Run model, Bencherif et al. (2006) and Bègue et al. (2010) showed that volcanic aerosol forcing from the Pinatubo eruption was weak and could be assumed negligible beyond 1996. Aerosol index is therefore not included in the framework of this present study as the ozone time series starts in 1998, 2–3 years after the post-Pinatubo eruption period. In trend calculations, a long-term linear function is generated by the model to characterize the trend index which is used among the model parameters. The trend value is calculated based on the slope of the normalized linear function. As ozone variability is also affected by annual and semi-annual oscillations (AO and SAO), these two parameters were taken into account in terms of model inputs. They may be described by a sinusoidal function as shown below:
$\begin{array}{}\text{(3)}& \left\{\begin{array}{l}\text{AO}\left(m\right)=\mathrm{cos}\left(\frac{\mathrm{2}\mathit{\pi }m}{\mathrm{12}}\right)+\mathit{\phi },\\ \text{SAO}\left(m\right)=\mathrm{sin}\left(\frac{\mathrm{2}\mathit{\pi }m}{\mathrm{6}}\right)+\mathit{\phi },\end{array}\right\\end{array}$
where “m” is the monthly temporal parameter and $\mathit{\phi }=\frac{\mathrm{2}\mathit{\pi }I}{\mathrm{180}}$.
I is a phasing coefficient between the temporal signal of ozone and the sinusoidal annual or semi-annual function. The ENSO parameter is based on the Multivariate ENSO Index (MEI), while the solar cycle is the solar 10.7 cm radio flux. These two parameters were obtained from the NCEP/NCAR website: http://www.esrl.noaa.gov/psd/data/climateindices/list/. The MEI is defined by positive values during El Niño and negative values for the duration of La Niña. The chosen QBO index is the time series of zonal wind at 30 hPa over Singapore and is available from the following link: http://www.cpc.ncep.noaa.gov/data/indices/qbo.u30.index. ENSO and QBO parameters are phased with respect to ozone time series based on phasing time which define the temporal point where the variable response to ozone is maximum. The index time series of the considered parameters are first normalized and filtered to remove a possible 3-month response. The output geophysical signal Y(z,m) of ozone for a given altitude “z” is modelled as follows:
$\begin{array}{}\text{(4)}& \begin{array}{l}Y\left(z,m\right)=C\left(z{\right)}_{\mathrm{1}}+C\left(z{\right)}_{\mathrm{2}}\text{AO}\left(m\right)+C\left(z{\right)}_{\mathrm{3}}\text{SAO}\left(m\right)\\ +C\left(z{\right)}_{\mathrm{4}}\text{SC}\left(m\right)+C\left(z{\right)}_{\mathrm{5}}\text{QBO}\left(m\right)+C\left(z{\right)}_{\mathrm{6}}\text{MEI}\left(m\right)\\ +C\left(z{\right)}_{\mathrm{7}}\text{TREND}\left(m\right)+\mathit{\epsilon }\left(z,m\right),\end{array}\end{array}$
where C(z)(1–7) are regression coefficients representing the weighting of parameterized variables on geophysical signal Y and ε(z,m) is the residual term representing a noise and/or contribution of parameters not included in the model. The regression coefficients are determined based on the least-squares method in order to minimize the sum of the residual squares. It is worth noting that the degree of data independency is assessed through the autocorrelation coefficient φ of the residual term (Bencherif et al., 2006). More details on how φ is calculated for ozone data are found in Portafaix (2003). The uncertainties in coefficient C(z)(1–7) are assessed by taking into account the autocorrelation coefficient and are formulated as follows:
$\begin{array}{}\text{(5)}& {\mathit{\sigma }}_{a}^{\mathrm{2}}=v\left(k\right)\cdot {\mathit{\sigma }}_{s}^{\mathrm{2}}\cdot \frac{\mathrm{1}+\mathit{\varphi }}{\mathrm{1}-\mathit{\varphi }},\end{array}$
where ${\mathit{\sigma }}_{s}^{\mathrm{2}}$ and v(k) represent the variance of the residual term and the covariance matrix of the different proxies input in the linear multi-regression model respectively. Equation (5) is used to estimate error associated with trend estimation.
The temporal function $C\left(z{\right)}_{p+\mathrm{1}}\cdot {Y}_{p}\left({m}_{\mathrm{1}\to n}\right)$ is defined as the factorized signal of input proxy “P”, and $\left|C\left(z{\right)}_{p+\mathrm{1}}\right|\cdot \mathit{\sigma }\left({Y}_{p}\left({m}_{\mathrm{1}\to n}\right)\right)$ is the corresponding SD (Brunner et al., 2006). The contribution of a given parameter to the total variation of inter-annual data is the ratio of a factorized signal sum of squares to the total sum of squares of the inter-annual ozone data. The amplitude of the response of the parameterized parameter (in percentage by unit of the parameter) with respect to the total variance of ozone time series is assessed based on the corresponding regression coefficient. It may be expressed as follows:
$\begin{array}{}\text{(6)}& A\left(z{\right)}_{P}=\mathrm{100}×\frac{C\left(z{\right)}_{P+\mathrm{1}}}{C\left(z{\right)}_{\mathrm{1}}}.\end{array}$
A is positive when the temporal signal of the parameter is in phase with the ozone time series, and it is negative in the opposite case.
4 Results and discussion
In this investigation, ozone inter-annual variability and trends were studied using the Trend-Run model as described above. The contribution and response of dynamical forcings to TCO variability are presented and discussed in Sect. 4.2. The aim of this study was to analyse the behaviour of each individual forcing contribution and to quantify the effect of its contribution on ozone variability in terms of TCO and mixing ratio profiles for the 15–30 km altitude range. The observed solar flux response to ozone over the selected altitude range is very weak (less than 1 %); the solar flux contribution to ozone vertical distribution is therefore not discussed. Ozone trends (for the observation period) as estimated by the model are presented (per site) in the final subsection.
## 4.1 Model assessment
In order to quantify the fit of the regression model with original ozone data, a statistical coefficient R2 was used. The coefficient R2 is defined as the ratio of a regression sum of squares to the total sum of squares (Bègue et al., 2010) and it measures the proportion of the total variation in ozone as described by the model. When the regression model accounts for most of the observed variation in ozone time series, the value of R2 is close to unity. For the cases where the model shows little agreement with measured data, R2 decreases toward zero (Bègue et al., 2010; Toihir et al., 2014). Figure 5 shows the time evolution of total column ozone as simulated by the multi-regression model (black). The blue lines correspond to TCO measurements performed over the (a) equatorial, (b) tropical and (c) subtropical latitude bands. A good agreement between model and observations is apparent over the three regions and the best fit is observed for the extratropics. This is because the variability is usually influenced by the seasonal cycle which is well accounted for by the regression model (Eq. 3). The values obtained for the statistical coefficient R2 were 0.75, 0.71 and 0.91 for the equatorial, tropical and subtropical bands respectively. These results indicate that the model describes approximately 75, 71 and 91 % of TCO variability. This implies that the sum of the contributions of the annual cycle, semi-annual cycle, QBO, ENSO, solar flux and trend to TCO variability reached 75, 71 and 91 % for the equatorial, tropical and subtropical bands respectively. That represents how the model is able to reproduce most of the variability of the studied ozone time series. Statistical coefficients (R2) higher than 0.82 were highlighted by Toihir et al. (2014) over sites located at 30–40 S, while Bencherif et al. (2006) and Bègue et al. (2010) demonstrated that the Trend-Run model is able to describe approximately 80 % of the temperature variability observed over Durban (South Africa) and Reunion.
Regarding vertical ozone distribution, profiles plotted in Fig. 4 were interpolated at a vertical resolution of 0.1 km which resulted in 150 altitude levels for the range between 15 and 30 km. The multi-regression model calculation was performed for each altitude level and profiles of the coefficient of determination R2 obtained at (a) equatorial, (b) tropical and (c) subtropical bands are shown in Fig. 6. The calculated R2 values were higher than 0.6, and this indicates that the input parameters account for more than 60 % of ozone variability in the 15–30 km altitude range.
Figure 6Vertical profiles of R2 (determination coefficient) calculated by the Trend-Run model for the selected latitude range: (a) equatorial, (b) tropical and (c) subtropical regions.
A study by Brunner et al. (2006) demonstrated that high values of R2 characterized a regime dominated by seasonal variability, while R2 values between 0.3 and 0.7 described cases where ozone variability was dominated by QBO. This is in agreement with results from the present study and confirms the capacity of the Trend-Run model to retrieve consistent ozone time series. As seen from Fig. 6, values of R2 greater than 0.8 were recorded at different altitude layers depending on the region, namely, in the 16.3–26.0 km range for the equatorial region, in the 17.3–22.2 km layer for the tropics and between 18.9 and 24.5 km for the extratropics. Furthermore, this analysis demonstrates that the noted altitude ranges are dominated by a seasonal variability which is expressed by an annual cycle function in the model.
## 4.2 TCO variability
### 4.2.1 Seasonal variability
Figure 7 illustrates ozone climatology obtained by compilation of 15 years of data recorded between January 1998 and December 2012. Three regions are explored: (a) equatorial, (b) tropical and (c) subtropical. Over these selected regions, total ozone exhibited high values from July to November. A positive gradient was observed from May to the maximum annual peak in September (equatorial region) but occurred a month later in October for tropical and near-subtropical regions. The recorded positive gradient over the subtropics was the largest and this indicated a high seasonal variability in this region compared to tropical and equatorial zones. Low ozone levels were recorded during the austral summer/autumn period over the three regions. Ozone recorded in the equatorial region was high in March, April and May compared to the tropics. Two peaks were clearly observed for the equatorial region, and these highlight a semi-annual cycle with a maximum in September–October and a second maximum in March–May.
Figure 7Seasonal variations of total ozone over three different regions: equatorial (a), tropical (b) and subtropical (c) regions. Error bars represent ±1σ.
These results can be explained as follows: stratospheric ozone is formed through a photochemical reaction (Chapman, 1930) that requires a sufficient quantity of solar radiation. The equatorial region is therefore the primary site of ozone production (Coe and Webb, 2003). During the period close to equinox, solar radiation leads to an increase in ozone production over the equatorial region. During summer, ozone transport consists primarily of vertical upwelling movements which are confined to the low-latitude region. This results in less ozone being distributed to the subtropics. This effect is the main contributing factor to why a total ozone peak is observed in autumn over the equatorial region (Mzé et al., 2010). However, the semi-annual equinox process may not be the sole explanation for the maximum value recorded in the equatorial region during September spring equinox, because it is generalized over the three regions. High ozone levels recorded in winter/spring are due to transport and accumulation of ozone in the Southern Hemisphere as a result of Brewer–Dobson circulation on a regional (air mass transport from the tropics to subtropics) and global scale (from summer hemisphere to winter hemisphere) (Holton et al., 1995; Fioletov, 2008). Transport of ozone from the tropical to extratropical regions is the most dominant process during the winter period (Portafaix, 2003) and constitutes the principal reason for the observed annual maximum over the region 20–30 S. The decline of ozone levels during late spring and summer is explained by a decrease in ozone transport coupled with ozone photochemical loss (Fioletov, 2008). However, the observation of maximum ozone in September over the equatorial region but 1 month later over the subtropics may be explained by a delay of ozone transport between the two regions. As shown in Fig. 7, the curve of the seasonal distribution of TCO attributed to (b) the tropical region is below that of (a) the equatorial zone from January to September. While considering the process of formation and transport of ozone as described above, TCO annual records in tropical regions should be higher than that recorded over equatorial regions. However, it is important to note that both sites (Samoa and Fiji, in the 10–20 latitude range) are located in the Pacific Ocean. The low TCO quantity observed in the Pacific is due to a low tropospheric ozone levels over the region as a result of a zonal Wave-One observed on tropical tropospheric ozone (Thompson et al., 2003a and 2003b). Similar results were reported by Thompson et al. ( 2003a and 2003b) where low ozone levels over SHADOZ Pacific sites were observed with respect to African and Atlantic sites due to the zonal Wave-One.
The contribution of seasonal cycles (annual and semi-annual oscillations) to ozone variability obtained from the regression analysis is presented in the first and second columns of Table 4. The calculated values show that except for Nairobi, the annual cycle constitutes the predominant mode of ozone variability over the studied sites and its influence is strongly observed over the subtropics. The contribution of annual oscillations to ozone variability exhibits a latitudinal signature, with its minimum and maximum at equatorial and subtropical zones respectively. The percentage contribution of annual oscillations to TCO variability decreases equatorward from Irene (65.96±3.93 %) to Nairobi (20.33±1.89 %). As explained in Sect. 3, the annual cycle is modeled by a sinusoidal function with maximum and minimum in winter/spring and autumn/summer. This kind of annual variation mode characterizes the ozone climatology of southern tropics and subtropics (Toihir et al., 2013, 2015a). The response of annual oscillations to ozone variability obtained for individual sites and reported in Table 5 is positive. This indicates that the modeled annual cycle function is in phase with the original ozone data. However, the amplitude of the annual cycle on total variance of ozone over the subtropics is the highest, and this indicates that ozone variance is sensitive to annual oscillation in the subtropics compared to tropics. The amplitude was evaluated at approximately 5.6 % by unit of the standardized annual function in subtropical sites (Reunion and Irene) and varied between 1 and 3 % from Fiji towards the Equator.
Table 4Percentage of the contribution and corresponding SD of the annual cycle, semi-annual cycle, 11-year solar cycle, QBO and ENSO to total ozone variability for individual sites, as obtained by the Trend-Run regression model. The last column shows the corresponding value for the coefficient of determination R2.
Table 5Response values of the chosen proxies (annual cycle, semi-annual cycle, solar flux, QBO and ENSO) to total ozone variability for individual sites as obtained by the Trend-Run model. The response is given in percent by unit of the normalized proxy.
Results presented in Table 4 show that on average, 13.86±1.26 % of ozone variability can be explained by semi-annual oscillations over stations located at 5–10 S. However, the semi-annual pattern was not apparent over Nairobi compared to the rest of the equatorial sites. The predominant mode of ozone variability in this equatorial site was the QBO (20.48±1.92 %), followed by the annual oscillations (20.33±1.89 %). The semi-annual oscillation pattern of TCO was observed to be significant at low latitude, with its maximum contribution over Ascension Island. The response values associated with semi-annual oscillation to ozone variability are presented in column 2 of Table 5 and show that a maximum in amplitude over Ascension Island was observed. Both quantities (contribution and response) decreased gradually while moving away from the Ascension Island site towards the Equator or southward. This indicates that the semi-annual oscillations of ozone are weighted at approximately 8 of latitude and coincide with the location where the influence of equinox processes on ozone variability is the greatest.
### 4.2.2 QBO contribution
The QBO constitutes the second most important parameter influencing ozone variability after seasonal oscillations. The QBO behaviour on ozone time series can be observed by removing the monthly climatological value from its monthly mean. In this investigation, the deseasonalized ozone signal was further subjected to 3-month smoothing in order to filter out smaller perturbations from intra-seasonal oscillations and to address the component due to ozone biennial oscillations.
Figure 8 illustrates the deseasonalized ozone (blue dashed curve) time series (1998–2012) for equatorial and subtropical regions. These data are superimposed with the monthly mean zonal wind data at 30 hPa over Singapore (black curve). From Fig. 8 it can be seen that the predominant variability of deseasonalized monthly ozone data is an approximately 2-year cycle linked to the QBO and represented by the zonal wind data at 30 hPa. However, the equatorial and subtropical ozone signals seem to be opposite in phase. This indicates that if an excess of ozone is recorded over the equatorial region, the subtropical zone shows a trend illustrating a deficit with respect to the monthly climatological value and vice versa. The QBO change from the Equator to the subtropics is due to a secondary circulation having an upwelling branch in the subtropics and a subsidence branch in the tropics during the westerly phase of the QBO (Chehade et al., 2014). This secondary circulation is characterized by a deceleration or an acceleration of the Brewer–Dobson circulation (BDC) during the westerly or easterly phase of the QBO respectively. The deceleration of the BDC leads to an increase in ozone in the tropics, while the acceleration leads to a decrease in ozone in the tropics. Due to this circulation, the deseasonalized TCO data over the equatorial region are in phase with the QBO index. Positive ozone anomalies correspond to the QBO westerly phase and negative anomalies are linked to the QBO easterly phase. Similar results have been determined in previous studies (Randel and Wu, 2007; Butchart et al., 2003; Zou et al., 2000; Bourassa et al., 2014; Peres et al., 2017).
Figure 8Deseasonalized time series of total ozone (blue line) for equatorial (0–10 S) and subtropical (20–30 S) regions calculated by subtracting the ozone monthly mean from the corresponding monthly climatological value. The black line is the time series in months of zonal wind recorded at 30 hPa over Singapore.
The above process was explained by Peres et al. (2017) and may be outlined as follows: the descending westerly phases of the QBO are associated with a vertical circulation characterized by downward motion in the tropics and upward motion in the subtropics. This leads to a weakening of the normal speed of the Brewer–Dobson circulation. In this way the upward motion of air mass is slowed down and the tropopause height decreases. As the ozone mixing ratio increases with the increase in altitude in the lower stratosphere, ozone production can occur for a longer period than normal (Cordero et al., 2012). This mechanism leads to a positive column ozone anomaly at low latitudes and a negative anomaly in the subtropics. During the QBO descending easterly phase, ozone formed spends less time at low latitude due to the enhanced Brewer–Dobson circulation (Cordero et al., 2012). The newly created ozone is rapidly transported to the subtropics, resulting in a negative ozone anomaly at lower latitudes and a positive ozone anomaly in the extratropics.
Butchart et al. (2003) explained that the equatorial ozone anomalies due to QBO forcing extend to 15–20 of latitude. In order to investigate the border between equatorial and subtropical QBO signals on ozone, the response of the QBO signal to TCO variability for individual sites was calculated based on the regression coefficient associated with the QBO on the Trend-Run model (Eq. 4). The response values are given in column 5 of Table 5. It is seen that the response of QBO to TCO variability by unit of normalized QBO signal is positive when the QBO is in phase with the ozone signal and negative for the reverse case. In this study a positive response (of decreasing magnitude) is obtained with the decrease in latitude moving from Samoa (14.13 S) to Nairobi (1.27 S). Conversely the responses obtained are negative at Fiji (18.13 S), Reunion (21.06 S) and Irene (25.90 S). These results indicate that the border between the equatorial and subtropical QBO-ozone signature is found at around 15 in latitude. Similar results were reported by Chehade et al. (2014) by calculating the coefficients of regression associated with the influence of the QBO on ozone variability. The Chehade et al. (2014) study reported positive and negative regression coefficients between 0 and 15 S and from 15 S towards the pole respectively.
In this work, positive ozone anomalies with amplitude varying between 7.5 and 13.9 DU were recorded over the equatorial sites during the zonal wind westerly phase, while the negative anomalies observed during the easterly phase oscillated between 13.3 and 2.80 DU. These results indicate that inter-annual oscillations of TCO over the equatorial regions have maximum amplitude varying within ±14 DU during the study period (1998–2012). This is in agreement with the results of Butchart et al. (2003). For 1980 to 2000 they obtained a deseasonalized ozone signal which varied within ±14 DU over the Equator.
It is worth noting that the maximum amplitude of ozone anomalies observed in the subtropics is lower (within ±11 DU) than the equatorial region (±14 DU). These amplitude values indicate that the QBO modulation on inter-annual variability of ozone is higher over the equatorial region in comparison to the extratropics. Values reported in Table 4 show a clearly decreasing QBO contribution poleward, i.e. from Nairobi (20.48±1.92 %) in the equatorial zone to Irene (3.60±1.07 %) in the subtropics.
### 4.2.3 ENSO contribution
Column 6 of Table 4 presents the percentage of ENSO contributions to inter-annual ozone variability for individual sites. It is clear that the ENSO parameter does not exert much influence on ozone variability over the subtropical sites (Reunion and Irene), where its contribution to TCO variability is assessed to be less than 1 %. However, the ENSO contribution is more pronounced over tropical and low-latitude sites located between 19 S and the Equator. In this latitude band, the ENSO contributions observed at Fiji, Samoa, Ascension Island and Java are high compared to Natal and Nairobi. An average ENSO contribution of 9.40±0.65 % to total ozone variability over Fiji, Samoa, Ascension Island and Java is observed, while a corresponding value of 5.73±0.75 % for Natal and Nairobi is seen (see Table 4). These results indicate that the ENSO influence on total ozone variability is high over the tropics compared to that in the equatorial zone and subtropics. It has been reported by Randel et al. (2009) that ENSO originates in the tropics and is linked to coupled atmosphere–ocean dynamics. This factor may be the main reason for the observed high ENSO contribution over sites located in the tropics compared to low-latitude and subtropical sites. Zerefos et al. (1992) removed the QBO, seasonal and solar cycles from the ozone signal at Samoa and obtained a good correlation between ENSO and ozone signal. They mentioned a possible ENSO–ozone relationship beyond the tropical region only during very large ENSO events (e.g. 1982–1983, 1997). In this study, the low estimated ENSO contribution (less than 1 %) for the subtropical region may also be explained by the fact that there were few large ENSO events recorded during the investigated period (1998–2012).
The tropical sites located in the Pacific Ocean (Samoa, Fiji and Java) were more influenced by ENSO when compared to tropical and equatorial sites (Ascension Island, Natal and Nairobi) due to the high ENSO activity usually observed in the Pacific Ocean (Randel and Thompson, 2011; Chandra et al., 1998, 2007; Logan et al., 2008; Ziemke and Chandra, 2003; Ziemke et al., 2010). In addition, the Ascension Island ENSO contribution was higher compared to those observed at Natal and Nairobi. As ENSO is a coupling ocean–atmosphere event generated in the Pacific Ocean (Ziemke and Chandra, 2003; Rieder et al., 2013), the low ENSO contribution recorded over Natal and Nairobi compared to Ascension Island can be explained by their continental location. As Ascension Island is located in the Atlantic Ocean, the high ENSO contribution observed at this site is probably due to a Pacific–Atlantic Ocean connection.
The ENSO response to ozone variance is assessed and shown in the last column of Table 5. The aim of this work is to define the mean behaviour of ozone variance for individual sites during ENSO events. It is worth noting that the mean ENSO influence is observed on tropospheric ozone (Randel and Thompson, 2011; Ziemke and Chandra, 2003). However, due to weak variability in stratospheric ozone over the tropics (Ziemke et al., 1998) during periods of normal conditions, tropospheric ozone changes were shown to affect the stratospheric ozone and total column ozone variability over the tropics during ENSO events (Rieder et al., 2013). As the MEI is characterized by positive values during El Niño, a positive ENSO response indicates an increase in total column ozone (TCO) and a negative response indicates a decrease in TCO. The ENSO response to ozone variability over subtropical sites (Reunion and Irene) is positive, while it is generally negative over the tropics (except over Java and Nairobi). These results are in good agreement with those reported by Chehade et al. (2014) which are obtained using a regression analysis model. Through their study they obtained negative (positive) ENSO regression coefficients indicating a negative (positive) ENSO response to ozone variability over the tropics (extratropics). It has been mentioned in several papers (Rieder et al., 2013; Randel et al. 2009; Randel and Thompson, 2011; Frossard et al., 2013; Chehade et al., 2014) that a negative ENSO response to ozone variability in the tropics is linked to enhanced transport of an ozone-rich air mass from the tropics to the extratropics due to strengthening of the Brewer–Dobson circulation in the stratosphere during ENSO warm events. As reported by Rieder et al. (2013), ENSO events are associated with more frequent stratospheric warming, an increase in tropopause height and a decrease in stratospheric ozone in the tropics. By comparison, in the subtropics, the tropopause height decreases and stratospheric ozone increases. This effect could be the reason for the observed ENSO positive response over Irene and Reunion (subtropics). The ENSO warm events produce suppressed convective movements over the western Pacific, leading to a positive anomaly of ozone in this region, while the air mass convection is enhanced over the eastern Pacific that leads to a reduction of ozone (Ziemke and Chandra, 2003). For the period 1970–2001, Ziemke and Chandra (2003) obtained a positive El Niño response corresponding to an average peak of positive ozone anomaly evaluated at 5 DU over the Indonesia area. Furthermore, they found a negative ozone anomaly in the eastern Pacific, the location of Fiji and Samoa. According to Ziemke and Chandra, (2003), this could be the main reason for the observed positive ENSO response at Java in the western Pacific and negative response over Samoa and Fiji in the eastern Pacific. The connection between the western Pacific and the Indian Ocean could be among the reasons for the observed positive ENSO response to ozone variability at Nairobi in the eastern African and Reunion in the western Indian Ocean region.
### 4.2.4 Solar flux contribution
Variation in total ozone concentration also depends on solar flux intensity over the tropics and subtropics. Figure 9 illustrates the variation in annual total ozone anomaly calculated between 1998 and 2012 over the three regions (bottom panel) and the annual 10.7 cm solar radio flux variation during the same period (top panel). Inspection of Fig. 9 illustrates the effect of the 11-year solar cycle on inter-annual total ozone variations. Negative ozone anomalies are observed during periods of low sunspot intensity, thereby confirming the dependence of ozone production on incident solar radiation flux. The converse is also true; that is, high annual total ozone levels are associated with periods of high sunspot intensity. The results presented here are in good agreement with Soukharev and Hood (2006). The latter obtained negative (positive) monthly mean ozone anomalies during periods with low (high) Mg II solar UV intensity. The dependence of annual ozone levels on annual solar radiation is also highlighted by Efstathiou and Varotsos (2013). In this investigation, the high levels of TCO were observed over the tropics when solar activity was at a maximum (2001) and compared to values recorded during a minimum in solar activity in 2008. This illustrated the impact of the solar cycle on global ozone concentrations. In the work presented here, considering the year with solar maximum (2001) to the year with minimum solar intensity, a variation in ozone levels was observed corresponding to approximately 1.05, 1.52, and 1.60 % for the (a) equatorial, (b) tropical and (c) extratropical regions respectively. It is expected that the observed percentage change in ozone levels is directly related to solar flux activity. A study by Zerefos et al. (1997) confirmed that the solar flux component in TCO between 1964 and 1994 was approximately 1–2 % over decadal timescales. It can therefore be concluded from the results presented here, together with those of Zerefos et al. (1997), that the solar flux contribution to ozone variability did not vary significantly during the past 6 decades.
Figure 9(a) Annual mean of 11-year solar flux recorded from 1998 to 2012 and (b) the annual TCO anomaly recorded for the same period in equatorial, tropical and subtropical regions.
As the solar flux index is one of the input parameters in the regression model, the contribution and response of the 11-year solar cycle to total ozone variability were assessed. The effect of changes in solar flux on ozone levels is presented in the fourth column of Table 5. The obtained responses are always positive, indicating that the solar flux index is in phase with total ozone time evolution as reported in the WMO (2014). The increase in ozone occurs when the sunspot intensity increases and vice versa. The average magnitude of the solar flux response to ozone variability is about 1.57 % by unit of solar flux index. The contribution of solar flux to total ozone variability over the study region varies by between approximately 4.38 and 7.81 %. The contribution of the solar cycle is high compared to the 3 % reported by the WMO (2010, 2014) on the global scale. This is probably due to (1) the length of the data records (less than two solar cycles), (2) the time period under investigation (1998–2012) and (3) the studied region (from equatorial to around 40 S). However, the contribution values obtained exhibit a minimum at Reunion in the subtropics and a maximum at Ascension Island in the lower-latitude region. These results indicate a high solar flux influence over the low-latitude region in comparison with the subtropics. The contribution of solar flux to total ozone variability decreases gradually by moving away from Ascension Island equatorward or southward. This solar flux contribution pattern is similar to that observed in semi-annual oscillations (see Sect. 4.2.1). From the results presented in this study it can be inferred that semi-annual variations are modulated by solar activity over the tropics and subtropics and that both semi-annual and 11-year solar forcings control ozone distributions.
## 4.3 Height profile ozone variability
The main objective of this section is to provide additional details regarding the contribution of proxies to ozone variability for different altitude levels from 15 to 30 km. This exercise is based on vertical distribution of ozone concentration as presented in Fig. 4. Here the ozone profiles are interpolated with 0.1 km vertical resolution giving 150 altitude levels in the 15–30 km altitude range. The Trend-Run model is applied to each altitude level.
### 4.3.1 Seasonal contribution to ozone profile variability
The contributions of annual and semi-annual oscillations to ozone variability (15 to 30 km) are shown in Fig. 10. Figure 10 reveals that seasonal cycles are the most dominant forcings to ozone variability in the lower stratosphere. The contribution of annual oscillation (AO) to ozone variability is highlighted over the three studied regions and accounts for more than 40 % in the UT–LS (upper troposphere–lower stratosphere). AO maximum amplitudes in equatorial and tropical regions are found around the tropical tropopause layer (TTL) below 20 km and around 22.5 km in the subtropics. These results are in good agreement with recent findings. Gebhardt et al. (2014) showed that in the lower stratosphere (21 km), ozone variability is dominated by AO in the tropics (20 N–20 S). Furthermore, Eckert et al. (2014) explained that vertical motion in the TTL had a significant impact on ozone variance and that seasonal variations contributed more than 50 % to ozone variability in this layer. A strong annual cycle signature in the UT–LS has also been reported by Bègue et al. (2010). The important contribution of the annual cycle in the UT–LS may be due to the STE (stratosphere–troposphere exchange) processes which occur seasonally in the tropics. These exchanges are linked to the upwelling air mass through the tropopause in summer and the submersion of stratospheric air mass to the troposphere during winter. These air mass exchanges affect the seasonal budget of ozone and temperature from equatorial to middle latitude. As seen in Fig. 10, annual cycle contributions higher than 40 % are found from 15 to 20 km over the equatorial region. However, the annual cycle contributions higher than 40 % observed over the tropical region are found from 17.5 to 22 km. In addition, the maximum annual cycle contribution for the equatorial region is observed at 18.5 km, whilst in the tropics it occurs at 19.5 km. These observations support the existence of upward propagation of the AOs of ozone below 25 km and the amplitude of these AOs increases with the increase in latitude poleward. This may explain the high annual cycle contribution observed in the 15–24.7 km altitude range in the subtropics. The presence of an annual signature in ozone temporal evolution in this wide altitude band (15–24.7 km) in the subtropics is inconsistent with the strong annual variability observed in TCO, as discussed in the previous subsection.
Figure 10Height profile in % of the contribution of the annual cycle (a1–c1) and the semi-annual cycle (a2–c2) to ozone variability as calculated by the Trend-Run model over the equatorial (a1, a2), tropical (b1, b2) and subtropical (c1, c2) regions. SDs associated with contributions are presented in grey dotted lines.
It is worth noting that the seasonal cycles of ozone observed at altitudes below 25 km are linked to dynamical processes, while above an altitude of 27 km, seasonal cycles are modulated by ozone photochemical processes. Contributions of the annual forcing of greater than 10 % are observed at altitudes above 27 km. Here the contribution part of annual oscillation increases with altitude, especially over the tropical and subtropical regions. These results support the existence of an annual cycle signature on ozone temporal evolution over altitudes above 30 km in the tropical and subtropical regions. Eckert et al. (2014) found that ozone variation due to AO was greater than 10 % above 27 km in the 10–30 S latitude band. However, this amplitude decreases above 35 km, where ozone variability is strongly controlled by semi-annual oscillation (SAO). Furthermore, the maximum amplitude of the ozone SAO signal observed by Eckert et al. (2014) was centered slightly above 30 km over the tropics. Mze et al. (2010) observed a strong semi-annual cycle in equatorial ozone climatology between 27 and 36 km, with a maximum peak at 31 km, and Gebhardt et al. (2014) observed ozone variability dominated by SAO in the middle (35 km) and upper tropical stratosphere (44 km) in the tropics (20 N–20 S).
The (SAO) contribution to ozone variability is highlighted in the present work for altitudes higher than 27 km for the equatorial and tropical regions. Above 27 km, the SAO contribution accounts for more than 10 % of total ozone variability over the equatorial region, while this effect is reduced to less than 10 % for the tropics. This confirms the high contribution of SAO to ozone variability at low latitudes with respect to other regions as discussed in Sect. 4.1.1. As supported by Eckert et al. (2014), beyond the tropics the SAO amplitude decreases gradually to near zero. As reported in previous studies, the SAO is linked to change in the zonal wind regime at the equatorial stratopause, where the maximum component appears during solstice (when easterly) and equinox (when westerly) (Belmont, 1975; Hirota, 1978; Delisi and Dunkerton, 1988). This mechanism leads to temperature anomalies corresponding to 2–4 K in the upper stratosphere (Nastrom and Belmont, 1975), thereby driving a change in the rate of ozone production and loss (Maeda, 1984) during the solstice and equinox periods. As demonstrated in Sect. 4.2.1, the equinox period is characterized by maximum peaks of TCO production over the equatorial region, while the solstice corresponds to minimum ozone levels (or ozone loss) over equatorial regions.
### 4.3.2 QBO contribution to ozone profile variability
QBO contributions to ozone profile variability between 15 and 30 km are presented in Fig. 11 for the three regions, namely (a1) equatorial, (b1) tropical and (c1) subtropical. A strong contribution of the QBO to ozone variability is observed over the 20–28 km altitude range for the equatorial region. Here the QBO is the most dominant mode and accounts for more than 30 % of ozone variability. These results are in agreement with previous work in which a strong signature of the QBO on inter-annual variability of ozone with large amplitudes over the tropics at altitudes of approximately 20–27 km has been reported (Randel and Thompson, 2011; Gebhardt et al., 2014; Eckert et al., 2014). Figure 11a1 shows two strong peaks at 60 % corresponding to altitudes of 21.5 and 25 km. The QBO contribution is important and does not change much over the altitude range 24 to 26.5 km, indicating that the maximum amplitude of the QBO signature occurs in this altitude range, where the degree of QBO influence on ozone variability is approximately invariant. Fadnavis and Beig (2009) found the QBO maximum over tropics at approximately 26 km, while Eckert et al. (2014) found this maximum at 25 km. The QBO signal on ozone variability is linked to the downward propagation of zonal wind throughout the time. This process leads to a positive (negative) ozone anomaly when in the westerly (easterly) mode with a cycle of around 2 years (Randel and Thompson, 2011; Lee et al., 2010; Butchart et al., 2003). Note that the maximum contributions of the QBO over the tropical and subtropical regions (Fig. 11b1 and c1) are less than 12 %. These results confirm the latitudinal signature of the QBO which is expressed by a decrease in its effect on ozone variability poleward. The inter-annual ozone anomaly linked to the QBO time evolution can be positive or negative depending on the altitude range for a given period due to the downward propagation pattern of the QBO with time (Randel and Thompson, 2011). The responses of the QBO index used are presented in Fig. 11 as (a2) equatorial, (b2) tropical and (c2) subtropical regions. Over the equatorial region, the QBO index profile exhibits positive values below 23 km and negative values above. These results indicate a strong contribution of the QBO westerly regime to ozone variability for equatorial regions below 23 km during the studied period (1998–2012). In contrast, the easterly regime had more influence on variability of ozone above altitudes of 23 km. The opposite case is observed in the subtropics, where the QBO index profile exhibits positive values for altitudes between 18 and 23 km and negative values between altitudes of 23 to 27.5 km. Similar results have been reported by Bourassa et al. (2014). The response amplitude obtained in this work over the subtropics varies within ±5 %, whilst it is at ±7 % (by unit of the normalized QBO index) over the equatorial region. These results reaffirm the key role of the QBO in ozone variability highlighted in equatorial regions in contrast to the subtropics.
Figure 11Height profile in % of the QBO contribution of (a1–c1) and response by unit of the QBO index (a2–c2) as calculated by the Trend-Run model over the equatorial (a1, a2), tropical (b1, b2) and subtropical (c1, c2) regions. SDs associated with contributions are presented in grey dotted lines.
### 4.3.3 ENSO contribution to ozone profile variability
ENSO contributions to ozone profile variability over the 15–30 km altitude range are presented in Fig. 12 for the three regions (a1) equatorial, (b1) tropical and (c1) subtropical. These calculations show that ENSO events weakly affect subtropical ozone levels. In contrast, the ENSO contribution and its associated response to ozone variability appear to be important below an altitude of 25 km at tropical latitudes. These results are in good agreement with the study by Sioris et al. (2014) in which a statistically significant ENSO contribution was reported at an altitude of 18.5–24.5 km in the tropics. Such results indicate that ENSO influence on ozone variability is essentially focused on the troposphere and lower stratosphere. ENSO contribution to the lower stratospheric ozone is highlighted by the 20–25 km altitude band with peaks toward 15 and 13 % over the tropical (at 23 km) and equatorial (at 21 km) regions respectively. The response linked to this contribution over the equatorial region oscillates with amplitude, varying between 8 and 4 % (Fig. 12, curve a2). Here the ENSO response values are negative below an altitude of 23 km and positive in the 23–26 km altitude range, indicating that the multivariate ENSO index is in phase with ozone time evolution above 23 km and in opposite phase below 23 km. As reported in previous studies (Ziemke and Chandra, 2003; Rieder et al., 2013), ENSO events contribute to the variation in tropopause height associated with decrease/increase in TCO due to the enhancing/suppressing of tropical convection. The results obtained in the equatorial region indicate that the ENSO effect, associated with enhanced convection, generally affects the lower part of the stratosphere. This leads to a decrease in ozone levels below 23 km, while the suppressed convection present generally results in a positive response to ozone variance above 23 km.
Figure 12Height profile in % of the contribution of ENSO (a1–c1) and response by unit of the ENSO index (a2–c2) as calculated by the Trend-Run model over the equatorial (a1, a2), tropical (b1, b2) and subtropical (c1, c2) regions. SDs associated with contributions are presented in grey dotted lines.
However, the amplitude obtained for the negative ENSO response is the largest. These results suggest that ozone variability linked to ENSO events is more sensitive to the enhanced equatorial convection in the UT–LS. In a recent study, similar results were reported by Randel and Thompson (2011), in which they observed a large negative change in ozone variance due to enhanced Brewer–Dobson circulation over the tropics associated with ENSO variability. In addition, the obtained ENSO response over the tropical region (Fig. 12b2) is negative, indicating a decrease in ozone due to ENSO events occurring during the study period. From the troposphere to approximately 20 km, the ENSO signal exhibits an important contribution and response to ozone variability over the tropics as reported in previous published works (Bourassa et al., 2014; Sioris et al., 2014; Randel and Thompson, 2011). However, the response obtained has a larger amplitude (maximum of 38 % at 16.5 km) than those reported by the above-mentioned papers due to the location of the Fiji and Samoa stations (the two sites that represent the tropical region in this work). As mentioned above, Fiji and Samoa are located in the eastern Pacific, a region where ENSO events are significant. Here the ENSO contribution to total ozone variability is observed to be greater than 10 % below 18.7 km. The negative ENSO response associated with this contribution is explained by an enhanced tropical upwelling during ENSO events which led to a decrease in ozone levels in the lower stratosphere of the eastern Pacific Ocean region as reported by previous studies (Randel and Thompson, 2011; Randel et al., 2009; Calvo et al., 2010).
## 4.4 The trend estimates
### 4.4.1 Total column ozone trends
The trend analysis of TCO was performed for each site and results presented in Fig. 13. The TCO trends are assessed based on the Trend-Run model and expressed in percentage per decade. The obtained trend values are positive, suggesting an increase in TCO during the study period. However, the mean values of TCO trends obtained from the Samoa site (14.13 S) equatorward are lower than 1 %, while they are higher than 1.5 % for Fiji (18.13 S), Reunion (21.06 S) and Irene (25.90 S). These results illustrate that the increase in TCO in the subtropics is greater than in the tropics. Figure 14 summarizes the trend evolution over the three latitudinal bands. An increase in the trend with latitude southward is observed. The average trends (in percentage per decade) are $+\mathrm{0.74}±\mathrm{0.6}$, $+\mathrm{1.55}±\mathrm{0.4}$ and $+\mathrm{1.74}±\mathrm{0.40}$ (±1σ) over the equatorial, tropical and subtropical regions respectively. These TCO trend results are consistent with the WMO (2014), where a positive trend of approximately 1±1.7 (±2σ) was reported at 60 N–60 S for the period 2000–2013. These positive trends are probably linked to the decline of effective equivalent stratospheric chlorine (EESC) over the globe as supported by previous studies (Yang et al., 2006; Anderson et al., 2000; Waugh et al., 2001; WMO 2010). However, compared with the tropics and subtropics, the delay in ozone improvement observed in the equatorial region is probably due to the tropical strengthening of the Brewer–Dobson circulation (WMO 2014; Randel and Thompson, 2011).
Figure 13Time evolution of monthly total ozone values (blue) observed at each site. The black lines represent the time evolution of the TCO obtained by the Trend-Run model, the blue lines show observation data, and the straight black line illustrates the obtained decadal trend.
Figure 14The same as Fig. 13 but per latitudinal region: equatorial (a), tropical (b) and subtropical (c).
It should be noted that approximately 10 % of ozone is tropospheric and is produced as a result of chemical reactions between species such as nitrogen oxide (NOx), carbon monoxide (CO) or volatile organic compounds (VOCs). In the context of climate change due to anthropogenic pollutant emissions and in spite of the efforts by the international community to reduce GHG emissions, pollution has continued to increase in the Southern Hemisphere, leading to a systematic increase in tropospheric ozone. Thompson et al. (2014) assumed that growth in ozone precursors like NOx and VOC may account for the characteristic free tropospheric ozone increase, especially in wintertime. The observed positive trend in TCO time series is therefore probably due also to an increase in and long-range transport of pollutants (mainly from industrial and biomass burning activity) in the free troposphere of the southern tropics and subtropics, as suggested by previous studies (Diab et al., 2004; Thompson et al., 2014; Clain et al., 2009). Thompson et al. (2014) highlighted significant increases throughout the free troposphere, 20–30 % decade−1 over Irene and up to 50 % decade−1 over Reunion in the southern subtropical region.
Furthermore, Heue et al. (2016) showed that the tropical tropospheric ozone trend is 0.7±0.12 DU (approximately 3.5 %) decade−1 between 1995 and 2015. There results show that there is an improvement in ozone from the ground to the middle atmosphere in the studied regions. Tropospheric ozone affects total column ozone trends; however, the contribution is not sufficient to modify the TCO trend compared to stratospheric ozone, which represents 90 % of the total ozone present.
### 4.4.2 Vertical distribution of the trends
In this investigation, TCO analysis indicated a positive trend over the study region. However, a trend investigation in terms of vertical ozone distribution is necessary in order to understand trend variation through different altitude levels. Equatorial, tropical and subtropical ozone trend profiles estimated by the Trend-Run model are presented in Fig. 15, panels (a), (b) and (c) respectively. Considering the temporal coverage of Irene ozonesonde data (1998–2008 and some profiles recorded in 2012), the Irene and Reunion ozone profiles were separated for the sake of trend analysis. Figure 15c1 shows the ozone trend profile for Reunion, while that for Irene (January 1998 to December 2008) is shown in panel c2. Overall, the ozone trends obtained are negative in the upper troposphere and positive for altitudes higher than 22 km. This indicates an improvement in stratospheric ozone. For the equatorial region, the trend profile shows a positive gradient from 15 km to the tropopause (approximately 18 km), followed by a maximum positive ozone trend in the 18–22 km layer (about +2 %). A negative trend is seen between 22 and 25 km and a zero trend for heights above 25 km. The tropical trend evolution below 22 km is similar to the trend observed over Reunion (Fig. 15c1), which is marked by a negative peak around 19 km. Such a negative peak is probably a characteristic of the tropical trend and highlighted at Reunion due to the proximity of this station to the tropical zone.
Figure 15Vertical profiles of ozone decadal trends derived by the Trend-Run model from 15 to 30 km at the Equator (a), tropics (b), Reunion (c1) and Irene (c2). Irene and Reunion are separated for this present case due to the lack of ozone profiles at the Irene station from January 2008 to October 2012. The Irene trend profile is derived based on ozone profiles recorded from January 1998 to December 2007 (10 years).
The results in trend estimates over the equatorial region are similar to previous studies based on a combination of GOMOS and SAGE II data recorded between 1984 and 2011 performed by Kyrölä et al. (2013). This work showed an apparent improvement in ozone trends onwards from 1997. The recorded trend (1997–2011) in the 20–30 km altitude range is consistent with results obtained in the present study and can be summarized as follows: a positive trend of approximately 2 % in the altitude range 19–21 km and approximately 0 % between altitudes 21 and 30 km. A trend varying between 0 and 1 % was also observed by Randel and Thompson (2011) in an altitude range of 23 to 35 km by averaging SHADOZ and SAGE II ozone profiles recorded between 1984 and 2009 for latitudes 20 N–20 S. Furthermore, Randel and Thompson (2011) observed a negative peak at 19 km, similar to what was observed in this study over Reunion and the tropical region. This negative peak is explained as resulting from a systematic increased tropical lower stratospheric upwelling. As reported by the WMO (2014), the increase in tropical upwelling is associated with a strengthening of the Brewer–Dobson circulation caused by GHG-induced climate change. Furthermore, Lamarque and Solomon (2010) associated the declining trend in ozone levels in the lower tropical stratosphere with an increase in carbon dioxide and sea surface temperature. They also demonstrated that a decrease in ozone in the upper stratosphere could be strongly linked to an increase in halogen compounds at high altitudes. The results obtained in this work indicate that the positive trends observed over Reunion and Irene from TCO analyses are probably due to an increase in ozone levels in the stratosphere, which is associated with a decrease in halogen compounds and EESC at high altitudes. Declining EESC levels in the subtropical region constitutes the main reason for ozone increase as a result of the Montreal Protocol and its amendments (Yang et al., 2006; Anderson et al., 2000; Reinsel, 2002; WMO, 2010, 2014).
The positive trend of 2 % in ozone levels recorded over the equatorial region in the 18–22 km altitude range cannot be explained by the decline of EESC compounds in the stratosphere. Instead, this trend could rather be linked to an increase in NOx (NO+NO2) in this altitude range (Gebhardt et al., 2014; Nevison et al., 1999). Furthermore, Gebhardt et al. (2014) obtained a positive ozone trend at altitudes higher than 18 km and the variation pattern observed is similar to that obtained in our study up to an altitude of approximately 23 km. The investigation of Gebhardt et al. (2014) was based on SCIAMACHY satellite observations (2002–2012). They associated the observed positive trend at 18–22 km with an increase in NOx concentration in the lower stratosphere. The idea that NOx levels may have a direct effect on stratospheric ozone variability is not new and was first suggested by Nevison et al. (1999). In this study they used a model in order to identify the NOx contribution to ozone variability at different altitudes in the stratosphere. Through these calculations it was found that the increase in NOx compounds in the lower stratosphere leads to an increase in ozone levels at this altitude. It is important to note that NOx can have a buffering effect on the Clx and HOx ozone-destruction cycles in the tropical lower stratosphere as explained by Nevison et al. (1999).
5 Conclusions
This work describes and discusses the variability and associated trends in ozone measurements recorded between January 1998 and December 2012 over eight southern tropical and subtropical sites. Total ozone products from different instruments were presented and a very good agreement was found in products recorded by different instruments at the same site. This confirmed the good quality of products and the validity of combining these products in order to create a long-term and homogenous TCO dataset for the study of ozone variability and long-term trends over the southern tropics and subtropics. SHADOZ data, used for the study of vertical ozone distribution, were found to be good quality according to Thompson et al. (2003a, b). The analysis of the contribution and response of some parameters to total ozone variance was achieved through the use of a multi-regression model called Trend-Run. This model is able to describe 71–91 % of ozone variability influenced by seasonal variation of climate and by dynamic parameters such as QBO, ENSO and solar flux. The obtained results were found to be in good agreement with recent findings and can be summarized as follows.
• Ozone variability over the study region is dominated by annual oscillation (AO) which strongly affects ozone variance in the UT–LS as a result of seasonal troposphere–stratosphere exchanges. The amplitude of ozone annual oscillations is high in the subtropics and decreases with latitude equatorward. In contrast, the ozone SAO is more apparent and contributes significantly above 27 km, with its maximum influence at the lower latitudes (around 8 S), where its amplitude decreases with movement toward the Equator or subtropics.
• The QBO signal constitutes the second most important parameter in terms of effect on ozone variability in the tropics after the seasonal oscillation. QBO heavily affects ozone variance in 20–28 km of the altitude range, with its maximum amplitude in the equatorial region. The pattern of the QBO response to ozone variability in the equatorial region is out of phase with that of the subtropics. The border between equatorial and subtropical patterns is found at around 15 in latitude.
• The ENSO contribution over the subtropics is less than 1 %. In contrast, an important contribution of ENSO to ozone variance is highlighted in the tropics, below 25 km with its maximum amplitude in the Pacific area. The tropical ENSO response is basically negative and was explained in previous work (Randel and Thompson, 2011; Sioris et al., 2014) as resulting from an enhancement of tropical convection leading to ozone decrease during warm ENSO events.
• Solar flux and total ozone are in phase. Increase in ozone levels occurs when sunspot intensity increases and conversely periods of low sunspot activity correspond to periods of low ozone production. The maximum solar flux contribution is at the tropical latitudes. The mean contribution of solar flux to total ozone variability over the studied region is high (6.03±0.84 %) compared to the 3 % reported globally by the WMO (2010, 2014). This is probably due to the region selected for this study (tropics and subtropics), the length of the data records (less than two solar cycles) and the time period under investigation in this work (1998–2012).
The regression model is also used to quantify ozone trends over the selected regions. Results obtained exhibit an upward trend of TCO during the period of study. The trend values were in good agreement with the WMO (2014) report (1±1.7 (2σ) at 60 N–60 S for the period 2000–2013) and exhibited an increase in ozone with increase in latitude from the Equator to the subtropics. The results of vertical profile analysis illustrated a positive ozone trend from 22 km upward and negative values in the upper troposphere, indicating that the ozone recovery observed in tropical and subtropical regions occurs in the stratosphere. It has been explained in recent studies (Sioris et al., 2014; Bourassa et al., 2014; Randel and Thompson et al., 2011) that the delay in ozone recovery at upper tropospheric and lower stratospheric levels was partly associated with a tropical enhancement of the Brewer–Dobson circulation over the tropics, while ozone recovery observed from an altitude of 22 km and upward was probably linked to declining levels of stratospheric chlorine and bromine compounds in the atmosphere. A further study continuing from this work would involve the inclusion of Brewer–Dobson and EESC indexes in the model not only to quantify the sources of observed trends, but also to improve the model quality.
Author contributions
Author contributions.
AMT was the project leader; TP was the supervisor of the project; VS and HB participated in interpretation of the results and the review of the manuscript; AP and NB contributed to data analysis and the review of the manuscript.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Data availability
Data availability.
The data used in the present work are available and publicly accessible from different sites: Dobson and SAOZ TCO data, available at https://woudc.org/data/explore.php; TOMS TCO data, available at https://acdisc.gesdisc.eosdis.nasa.gov/opendap/ EarthProbe_TOMS_Level3/TOMSEPL3.008/contents.html; OMI data, available at https://gs614-avdc1-pz.gsfc.nasa.gov/pub/data/satellite/Aura/OMI/V03/L2OVP/OMTO3/; SHADOZ ozonesonde data, available at https://tropo.gsfc.nasa.gov/shadoz/Archive.html; ENSO, solar, and QBO flux indexes, available at https://www.esrl.noaa.gov/psd/data/climateindices/list/.
Special issue statement
Special issue statement.
Acknowledgements
Acknowledgements.
The LACy (Laboratoire de l'Atmosphère et des Cyclones) is supported by the INSU (Institut National des Sciences de l'Univers), a CNRS institute, and by the Regional Council of Reunion (Conseil Regional de La Réunion). This present work is supported by the RAMI project, a regional doctoral school in the Indian Ocean zone, under the “Horizons Francophones” programme of the AUF (Agence Universitaire de la Francophonie) and the GDRI-ARSAIO (Atmospheric Research in Southern Africa and the Indian Ocean), a French–South African cooperative programme network supported by the French Centre National de la Recherche Scientifique (CNRS) and the South African National Research Foundation (NRF). The first author (Abdoulwahab Mohamed Toihir) acknowledges the Indian Ocean Bureau of AUF and the University of KwaZulu Natal for travel and hosting under invited student exchange in the above NRF-CNRS bilateral research project (UID 68668). The authors also thank Barbara Lynn Duigan (NASSP-UKZN) for improving the readability of the manuscript. The authors are thankful to agencies supporting NDACC activities: INSU/CNRS, CNES as well as the Canada WOUDC team, which is under the auspices of the WMO, the SHADOZ network, and the OMI and TOMS programmes, for providing access to data via their websites.
Edited by: Petr Pisoft
Reviewed by: Lucas Vaz Peres and two anonymous referees
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Zou, H., Chongping, J., and Libo, Z.: QBO signal in total ozone over Tibet, Adv. Atmos. Sci., 17, 562–568, 2000. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 14, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.870343804359436, "perplexity": 4499.0834286403715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00077.warc.gz"} |
http://umr-math.univ-mlv.fr/evenements/exposes/seminaire_cristolien_d_analyse_multifractale.1338536700 | ## Sparse stochastic processes, fractals and wavelet analysis
Type:
Site:
Date:
01/06/2012 - 09:45 - 10:45
Salle:
Amphithéatre Vert
Orateur:
UNSER Michael
Localisation:
EPF Lausanne
Localisation:
Suisse
Résumé:
The ”self-similar” processes described in this talk are generalized versions of the classical Lévy processes. They are generalized stochastic processes (in the sense Gelfand and Vilenkin) that are solutions of (unstable) fractional stochastic differential equations (fSDE). They are described by a general innovation model that is specified by: 1) a whitening operator (fractional derivative or Laplacian), which shapes their second-order moments, and 2) a Lévy exponent $f$, which controls the sparsity of the (non-Gaussian) innovations (white Lévy noise). We give a complete characterization these processes in terms of their characteristic form (the infinite-dimensional counterpart of the characteristic function). This allows us to prove that they admit a sparse representation in a wavelet bases. We also provide evidence that wavelets allow for a better $N$ term approximation than the classical Karhunen-Loève transform (KLT), except in the Gaussian case where the processes are equivalent to Mandelbrot’s fractional Brownian motion. We also highlight a fundamental connection with spline mathematics and the construction of maximally localized basis functions (B-splines). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9415873885154724, "perplexity": 924.1001524229092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812758.43/warc/CC-MAIN-20180219171550-20180219191550-00625.warc.gz"} |
http://philpapers.org/s/Field%20Theory | ## Search results for 'Field Theory' (try it on Scholar)
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Bibliography: Quantum Field Theory in Philosophy of Physical Science
1. Hartry Field (1974). Quine and the Correspondence Theory. Philosophical Review 83 (2):200-228.score: 280.0
A correspondence theory of truth explains truth in terms of various correspondence relations (e.G., Reference) between words and the extralinguistic world. What are the consequences of quine's doctrine of indeterminacy for correspondence theories? in "ontological relativity" quine implicitly claims that correspondence theories are impossible; that is what the doctrine of 'relative reference' amounts to. But quine's doctrine of relative reference is incoherent. Those who think the indeterminacy thesis valid should not try to relativize reference, They should abandon the relation (...)
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2. Harty Field (2004). The Consistency of the Naïve Theory of Properties. Philosophical Quarterly 54 (214):78 - 104.score: 280.0
If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering (...)
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3. Hartry Field (1972). Tarski's Theory of Truth. Journal of Philosophy 64 (13):347-375.score: 240.0
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4. Hartry Field (1973). Theory Change and the Indeterminacy of Reference. Journal of Philosophy 70 (14):462-481.score: 240.0
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5. Sunny Y. Auyang (1995). How is Quantum Field Theory Possible? Oxford University Press.score: 240.0
Quantum field theory (QFT) combines quantum mechanics with Einstein's special theory of relativity and underlies elementary particle physics. This book presents a philosophical analysis of QFT. It is the first treatise in which the philosophies of space-time, quantum phenomena, and particle interactions are encompassed in a unified framework. Describing the physics in nontechnical terms, and schematically illustrating complex ideas, the book also serves as an introduction to fundamental physical theories. The philosophical interpretation both upholds the reality of (...)
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6. Harvey R. Brown & Rom Harré (eds.) (1988). Philosophical Foundations of Quantum Field Theory. Oxford University Press.score: 240.0
Quantum field theory, one of the most rapidly developing areas of contemporary physics, is full of problems of great theoretical and philosophical interest. This collection of essays is the first systematic exploration of the nature and implications of quantum field theory. The contributors discuss quantum field theory from a wide variety of standpoints, exploring in detail its mathematical structure and metaphysical and methodological implications.
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7. R. G. Beil (2003). Finsler Geometry and Relativistic Field Theory. Foundations of Physics 33 (7):1107-1127.score: 240.0
Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of “gauge” transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.
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8. Hartry Field (1992). A Nominalistic Proof of the Conservativeness of Set Theory. Journal of Philosophical Logic 21 (2):111 - 123.score: 240.0
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9. Mark A. Rubin (2002). Locality in the Everett Interpretation of Quantum Field Theory. Foundations of Physics 32 (10):1495-1523.score: 240.0
Recently it has been shown that transformations of Heisenberg-picture operators are the causal mechanism which allows Bell-theorem-violating correlations at a distance to coexist with locality in the Everett interpretation of quantum mechanics. A calculation to first order in perturbation theory of the generation of EPRB entanglement in nonrelativistic fermionic field theory in the Heisenberg picture illustrates that the same mechanism leads to correlations without nonlocality in quantum field theory as well. An explicit transformation is given (...)
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10. Gerard ’T. Hooft (2013). Duality Between a Deterministic Cellular Automaton and a Bosonic Quantum Field Theory in 1+1 Dimensions. Foundations of Physics 43 (5):597-614.score: 240.0
Methods developed in a previous paper are employed to define an exact correspondence between the states of a deterministic cellular automaton in 1+1 dimensions and those of a bosonic quantum field theory. The result may be used to argue that quantum field theories may be much closer related to deterministic automata than what is usually thought possible.
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11. Friedrich W. Hehl & Yuri N. Obukhov (2008). An Assessment of Evans' Unified Field Theory II. Foundations of Physics 38 (1):38-46.score: 240.0
Evans attempted to develop a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for a theory that implements some of the ideas that have been (imprecisely) indicated by Evans and show that it yields two field equations. The second field equation is algebraic in (...)
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12. Friedrich W. Hehl (2008). An Assessment of Evans' Unified Field Theory I. Foundations of Physics 38 (1):7-37.score: 240.0
Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. This geometry can be characterized by an orthonormal coframe ϑ α and a (metric compatible) Lorentz connection Γ α β . These two potentials yield the field strengths torsion T α and curvature R α β . Evans tried to infuse electromagnetic properties into this geometrical framework by putting the coframe ϑ α to be proportional to (...)
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13. score: 240.0
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14. score: 240.0
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15. score: 240.0
In the paper it will be shown that Reichenbach’s Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ , respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ and commuting with (...)
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16. Meinard Kuhlmann (2010). Why Conceptual Rigour Matters to Philosophy: On the Ontological Significance of Algebraic Quantum Field Theory. [REVIEW] Foundations of Physics 40 (9):1625-1637.score: 240.0
I argue that algebraic quantum field theory (AQFT) permits an undisturbed view of the right ontology for fundamental physics, whereas standard (or Lagrangian) QFT offers different mutually incompatible ontologies.My claim does not depend on the mathematical inconsistency of standard QFT but on the fact that AQFT has the same concerns as ontology, namely categorical parsimony and a clearly structured hierarchy of entities.
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17. A. Zee (2010). Quantum Field Theory in a Nutshell. Princeton University Press.score: 240.0
Since it was first published, Quantum Field Theory in a Nutshell has quickly established itself as the most accessible and comprehensive introduction to this profound and deeply fascinating area of theoretical physics. Now in this fully revised and expanded edition, A. Zee covers the latest advances while providing a solid conceptual foundation for students to build on, making this the most up-to-date and modern textbook on quantum field theory available. -/- This expanded edition features several additional (...)
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18. score: 240.0
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19. Yuichiro Kitajima (2013). EPR States and Bell Correlated States in Algebraic Quantum Field Theory. Foundations of Physics 43 (10):1182-1192.score: 240.0
A mathematical rigorous definition of EPR states has been introduced by Arens and Varadarajan for finite dimensional systems, and extended by Werner to general systems. In the present paper we follow a definition of EPR states due to Werner. Then we show that an EPR state for incommensurable pairs is Bell correlated, and that the set of EPR states for incommensurable pairs is norm dense between two strictly space-like separated regions in algebraic quantum field theory.
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20. Miklos Redei & Stephen J. Summers (2002). Local Primitive Causality and the Common Cause Principle in Quantum Field Theory. Foundations of Physics 32 (3):335-355.score: 240.0
If $\mathcal{A}$ (V) is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V 1 and V 2 are spacelike separated spacetime regions, then the system ( $\mathcal{A}$ (V 1 ), $\mathcal{A}$ (V 2 ), φ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A∈ $\mathcal{A}$ (V 1 ), B∈ $\mathcal{A}$ (V 2 ) correlated in the normal state φ there exists a (...)
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21. Paul Teller (1995). An Interpretive Introduction to Quantum Field Theory. Princeton University Press.score: 240.0
Quantum mechanics is a subject that has captured the imagination of a surprisingly broad range of thinkers, including many philosophers of science. Quantum field theory, however, is a subject that has been discussed mostly by physicists. This is the first book to present quantum field theory in a manner that makes it accessible to philosophers. Because it presents a lucid view of the theory and debates that surround the theory, An Interpretive Introduction to Quantum (...)
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22. score: 240.0
This paper will examine the implications of an extended “field theory of information,” suggested by Wolfhart Pannenberg, specifically in the Christian understanding of creation. The paper argues that the Holy Spirit created the world as field, a concept from physics, and the creation is directed by the logos utilizing information. Taking into account more recent developments of information theory, the essay further suggests that present creation has a causal impact upon the information utilized in creation. In (...)
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23. P. A. Marchetti (2010). Spin-Statistics Transmutation in Quantum Field Theory. Foundations of Physics 40 (7):746-764.score: 240.0
Spin-statistics transmutation is the phenomenon occurring when a “dressing” transformation introduced for physical reasons (e.g. gauge invariance) modifies the “bare” spin and statistics of particles or fields. Historically, it first appeared in Quantum Mechanics and in semiclassical approximation to Quantum Field Theory. After a brief historical introduction, we sketch how to describe such phenomenon in Quantum Field Theory beyond the semiclassical approximation, using a path-integral formulation of euclidean correlation functions, exemplifying with anyons, dyons and skyrmions.
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24. score: 240.0
If the general arguments concerning theinvolvement of variation and selection inexplanations of fit'' are valid, then variationand selection explanations should beappropriate, or at least potentiallyappropriate, outside the paradigm historisticdomains of biology and knowledge. In thisdiscussion, I wish to indicate some potentialroles for variation and selection infoundational physics â specifically inquantum field theory. I will not be attemptingany full coherent ontology for quantum fieldtheory â none currently exists, and none islikely for at least the short term future. Instead, I (...)
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25. Mark A. Rubin (2011). Observers and Locality in Everett Quantum Field Theory. Foundations of Physics 41 (7):1236-1262.score: 240.0
A model for measurement in collapse-free nonrelativistic fermionic quantum field theory is presented. In addition to local propagation and effectively-local interactions, the model incorporates explicit representations of localized observers, thus extending an earlier model of entanglement generation in Everett quantum field theory (Rubin in Found. Phys. 32:1495–1523, 2002). Transformations of the field operators from the Heisenberg picture to the Deutsch-Hayden picture, involving fictitious auxiliary fields, establish the locality of the model. The model is applied to (...)
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26. G. C. Field (1953). What Is Political Theory? Proceedings of the Aristotelian Society 54:145 - 166.score: 240.0
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27. G. C. Field (1923). Aristotle's Account of the Historical Origin of the Theory of Ideas. Classical Quarterly 17 (3-4):113-.score: 240.0
Whatthe influences were which led to the development and formulation of the so-called Theory of Ideas, usually associated with the name of Plato, is a question of perennial interest. And the interest has been increased by the vigorous controversy that, during the last ten years, has been conducted round the question of the exact part played by Socrates in the development of this theory. All the available evidence on the question is accessible and familiar to students of Greek (...)
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28. score: 240.0
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29. Francesco Giacosa (2012). Non-Exponential Decay in Quantum Field Theory and in Quantum Mechanics: The Case of Two (or More) Decay Channels. Foundations of Physics 42 (10):1262-1299.score: 240.0
We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between t and t+dt. An important quantity is the ratio of the probability (...)
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30. G. C. Field (1932). Moral Theory. London, Methuen & Co., Ltd..score: 240.0
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31. G. C. Field (1966). Moral Theory: An Introduction to Ethics. London, Methuen.score: 240.0
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32. G. C. Field (1928). The Inaugural Address: The Origin and Development of Plato's Theory of Ideas. Aristotelian Society Supplementary Volume 8:1 - 30.score: 240.0
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33. score: 224.0
An examination is made of the way in which particles emerge from linear, bosonic, massive quantum field theories. Two different constructions of the one-particle subspace of such theories are given, both illustrating the importance of the interplay between the quantum-mechanical linear structure and the classical one. Some comments are made on the Newton-Wigner representation of one-particle states, and on the relationship between the approach of this paper and those of Segal, and of Haag and Ruelle.
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34. Bert Schroer (2010). Localization and the Interface Between Quantum Mechanics, Quantum Field Theory and Quantum Gravity I. Studies in History and Philosophy of Science Part B 41 (2):104-127.score: 216.0
It is shown that there are significant conceptual differences between QM and QFT which make it difficult to view the latter as just a relativistic extension of the principles of QM. At the root of this is a fundamental distiction between Born-localization in QM (which in the relativistic context changes its name to Newton–Wigner localization) and modular localization which is the localization underlying QFT, after one separates it from its standard presentation in terms of field coordinates. The first comes (...)
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35. Bert Schroer (2010). Localization and the Interface Between Quantum Mechanics, Quantum Field Theory and Quantum Gravity II. Studies in History and Philosophy of Science Part B 41 (4):293-308.score: 216.0
The main topics of this second part of a two-part essay are some consequences of the phenomenon of vacuum polarization as the most important physical manifestation of modular localization. Besides philosophically unexpected consequences, it has led to a new constructive “outside-inwards approach” in which the pointlike fields and the compactly localized operator algebras which they generate only appear from intersecting much simpler algebras localized in noncompact wedge regions whose generators have extremely mild almost free field behavior. -/- Another consequence (...)
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36. H. Kleinert (2014). Quantum Field Theory of Black-Swan Events. Foundations of Physics 44 (5):546-556.score: 216.0
Free and weakly interacting particles are described by a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. They describe Gaussian random walks with collisions. By contrast, the fields of strongly interacting particles are governed by effective actions, whose extremum yields fractional field equations. Their particle orbits perform universal Lévy walks with heavy tails, in which rare events are much more frequent than in Gaussian random walks. Such rare events are observed in exceptionally strong windgusts, monster or rogue waves, (...)
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37. Thomas Fuß (2002). SU(3) Local Gauge Field Theory as Effective Dynamics of Composite Gluons. Foundations of Physics 32 (11):1737-1755.score: 216.0
The effective dynamics of quarks is described by a nonperturbatively regularized NJL model equation with canonical quantization and probability interpretation. The quantum theory of this model is formulated in functional space and the gluons are considered as relativistic bound states of colored quark-antiquark pairs. Their wave functions are calculated as eigenstates of hardcore equations, and their effective dynamics is derived by weak mapping in functional space. This leads to the phenomenological SU(3) gauge invariant gluon equations in functional formulation, i.e., (...)
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38. Olivier Driessens (2013). Celebrity Capital: Redefining Celebrity Using Field Theory. [REVIEW] Theory and Society 42 (5):543-560.score: 216.0
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39. Jan Rzewuski (1967). Field Theory. London, Iliffe.score: 216.0
v. 1. Classical theory.--v. 2. Functional formulation of S-matrix theory.
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40. J. McFadden (2002). The Conscious Electromagnetic Information (Cemi) Field Theory: The Hard Problem Made Easy? Journal of Consciousness Studies 9 (8):45-60.score: 210.0
In the April 2002 edition of JCS I outlined the conscious electromagnetic information field theory, claiming that consciousness is that component of the brain's electromagnetic field that is downloaded to motor neurons and is thereby capable of communicating its informational content to the outside world. In this paper I demonstrate that the theory is robust to criticisms. I further explore implications of the theory particularly as regards the relationship between electromagnetic fields, information, the phenomenology of (...)
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41. Susan Pockett (2002). Difficulties with the Electromagnetic Field Theory of Consciousness. Journal of Consciousness Studies 9 (4):51-56.score: 210.0
The author's version of the electromagnetic field theory of consciousness is stated briefly and then three difficulties with the theory are discussed. The first is a purely technical problem: how to measure accurately enough the spatial properties of the fields which are proposed to be conscious and then how to generate these artificially, so that the theory can be tested. The second difficulty might also be merely technical, or it might be substantive and fatal to the (...)
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42. Heinz Werner & Seymour Wapner (1952). Experiments on Sensory-Tonic Field Theory of Perception: IV. Effect of Initial Position of a Rod on Apparent Verticality. Journal of Experimental Psychology 43 (1):68.score: 210.0
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43. score: 210.0
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44. Seymour Wapner & Heinz Werner (1952). Experiments on Sensory-Tonic Field Theory of Perception: V. Effect of Body Status on the Kinesthetic Perception of Verticality. Journal of Experimental Psychology 44 (2):126.score: 210.0
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45. score: 210.0
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46. score: 210.0
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47. score: 210.0
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48. Kullervo Rainio (1986). Stochastic Field Theory of Behavior. Academic Bookstore [Distributor].score: 210.0
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49. score: 210.0 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8428409695625305, "perplexity": 2317.790334102624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131295084.53/warc/CC-MAIN-20150323172135-00160-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.dummies.com/programming/electronics/diy-projects/how-rectifier-circuits-work-in-electronics/ | How Rectifier Circuits Work in Electronics - dummies
# How Rectifier Circuits Work in Electronics
One of the most common uses for rectifier diodes in electronics is to convert household alternating current into direct current that can be used as an alternative to batteries. The rectifier circuit, which is typically made from a set of cleverly interlocked diodes, converts alternating current to direct current.
In household current, the voltage swings from positive to negative in cycles that repeat 60 times per second. If you place a diode in series with an alternating current voltage, you eliminate the negative side of the voltage cycle, so you end up with just positive voltage.
If you look at the waveform of the voltage coming out of this rectifier diode, you’ll see that it consists of intervals that alternate between a short increase of voltage and periods of no voltage at all. This is a form of direct current because it consists entirely of positive voltage. However, it pulsates: first it’s on, then it’s off, then it’s on again, and so on.
Overall, voltage rectified by a single diode is off half of the time. So although the positive voltage reaches the same peak level as the input voltage, the average level of the rectified voltage is only half the level of the input voltage. This type of rectifier circuit is sometimes called a half-wave rectifier because it passes along only half of the incoming alternating current waveform.
A better type of rectifier circuit uses four rectifier diodes, in a special circuit called a bridge rectifier.
Look at how this rectifier works on both sides of the alternating current input signal:
• In the first half of the AC cycle, D2 and D4 conduct because they’re forward biased. Positive voltage is on the anode of D2 and negative voltage is on the cathode of D4. Thus, these two diodes work together to pass the first half of the signal through.
• In the second half of the AC cycle, D1 and D3 conduct because they’re forward biased: Positive voltage is on the anode of D1, and negative voltage is on the cathode of D3.
The net effect of the bridge rectifier is that both halves of the AC sine wave are allowed to pass through, but the negative half of the wave is inverted so that it becomes positive. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8922244310379028, "perplexity": 671.2039715201803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526888.75/warc/CC-MAIN-20190721040545-20190721062545-00049.warc.gz"} |
https://space.stackexchange.com/tags/ulysses/hot | Tag Info
25
The performance of Shuttle-Centaur would have greatly exceeded that of either the Atlas-Centaur or Titan-Centaur combination. Neither the Atlas nor Titan were able to put a fully fueled Centaur into Earth orbit using only their lower stages. The Centaur would have burned part of its propellant completing the orbital insertion. In contrast, the shuttle ...
5
Yes. Ulysses was quite a light weight, only 370 kg. For comparison, the Galileo probe, which was put in to Jupiter's atmosphere, weighted almost that much, and that was a small part of Galileo, which in total weighed about 2562 kg. Much of that mass was to orbit the planet, also the communications was supposed to be better, as the prime part of Galileo was ...
4
Several terse online references state that the nutation anomaly was managed by the use of "Conscan" or CONSCAN, for example The only other problem of any significance has been a nutation-like motion which built up following deployment of the 7.5-meter (25-foot) axial radio wave experiment antenna shortly after launch. This disturbance was apparently ...
4
In simple terms, gravity pulls an object directly towards another object. As an analogy, if you are running down the street and grab hold of a lamp post with your left hand you will swing around it to the left, if you grab it with your right you will swing to the right. You can use this to do a u-turn, letting go whenever you or going the right direction, or ...
3
I don't know did anything change since 2017, but here is complete and bulky pdf https://www.researchgate.net/publication/313365623_Laser_Interferometer_Space_Antenna. What exactly is eLISA How large will eLISA be? After several iterations current mission design (LISA, not eLISA anymore) is planned to be: -Space gravitational detector consisting of 3 ...
2
A polar LEO orbit is still a equatorial heliocentric orbit (since that describes the Earth's orbit around the sun). To change from a equatorial heliocentric to polar heliocentric requires a lot of Delta v. The final orbit wasn't circular, but we can get an order-of-magnitude figure for the energy required by using the circular plane change approximation. ...
2
As stated in the comments above, the trajectory adjustment took place on the eighth of July 1991, before Ulysses swung by Jupiter. For more information on its trajectory, please consult: http://adsbit.harvard.edu//full/1992A%26AS...92..207W/0000209.000.html By changing the start date of the simulation to the first of August 1991, half a year before the ...
1
I'll elaborate on some points because I find the Ulysses mission really compelling. I'll quote further from the references cited in @OrganicMarble's award-winning answer The ULYSSES Mission, Wenzel, K. P., Marsden, R. G., Page, D. E., & Smith, E. J., Astronomy and Astrophysics Supplement, Vol.92, NO. 2/JAN, P. 207, 1992 Spacecraft description 5.1 ...
Only top voted, non community-wiki answers of a minimum length are eligible | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8345178365707397, "perplexity": 2374.0761019342444}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738573.99/warc/CC-MAIN-20200809192123-20200809222123-00305.warc.gz"} |
https://infoscience.epfl.ch/record/188522 | ## Measurement of the dynamics in ski jumping using a wearable inertial sensor-based system
Dynamics is a central aspect of ski jumping, particularly during take-off and stable flight. Currently, measurement systems able to measure ski jumping dynamics (e.g., 3D cameras, force plates) are complex and only available in few research centres worldwide. This study proposes a method to determine dynamics using a wearable inertial sensor-based system which can be used routinely on any ski jumping hill. The system automatically calculates characteristic dynamic parameters during take-off (position and velocity of the center of mass perpendicular to the table, force acting on the center of mass perpendicular to the table, and somersault angular velocity) and stable flight (total aerodynamic force). Furthermore, the acceleration of the ski perpendicular to the table was quantified to characterize the skis lift at take-off. The system was tested with two groups of 11 athletes with different jump distances. The force acting on the center of mass, acceleration of the ski perpendicular to the table, somersault angular velocity, and total aerodynamic force were different between groups and correlated with the jump distances. Furthermore, all dynamic parameters were within the range of prior studies based on stationary measurement systems, except for the center of mass mean force which was slightly lower.
Published in:
Journal of Sports Sciences, 32, 6, 591-600
Year:
2014
Publisher:
Abingdon, Taylor & Francis
ISSN:
0264-0414
Keywords:
Laboratories: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8570501804351807, "perplexity": 1472.0708209802042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823618.14/warc/CC-MAIN-20181211104429-20181211125929-00461.warc.gz"} |
http://math.stackexchange.com/questions/279771/analytic-function-in-open-connected-set-that-is-bounded-by-another-analytic-func | # Analytic function in open connected set that is bounded by another analytic function
Let $G$ be an open connected set and $f, g$ analytic functions on $G$. If $|f|\le |g|$ then there exists an analytic function $h$ such that $f(z)=h(z)g(z)$.
We know $|f/g|\le 1$ everywhere in $G$, and so $f/g$ is bounded near any singularities. (The singularities must be isolated, for otherwise they have a limit and the function is identically zero.) So we can extend $f/g$ to be analytic in all of $G$.
But where do I go from here? Presumably the connectedness of $G$ is important, but I don't see how it helps in manipulating $f$ and $g$ to make an $h$ appear
EDIT: As pointed out in the comments, I am already done. Take $h= f/g$.
-
What's the problem? You already extended $f/g$ to be analytic in $G$. That's your $h$. – Robert Israel Jan 16 '13 at 3:24
Excuse my while I facepalm myself to death. Thank you – Bey Jan 16 '13 at 3:26
Bey: You might as well post an answer. – Jonas Meyer Jan 16 '13 at 3:31
## 1 Answer
As has been pointed out, there's actually nothing left for me to do. For completeness, here's the solution:
Since $|f(z)|\le |g(z)|$ for all $z\in G$, we have $|f(z)/g(z)|\le 1$ everywhere in $G$. So $h(z):=f(z)/g(z)$ is bounded in a neighborhood of any possible singularities, meaning any singularity is removable. Hence, we can define $h$ to be analytic on all of $G$. Now we simply notice that $f(z)=h(z)g(z)$ and we're done.
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http://crypto.stackexchange.com/questions/16083/computational-diffie-hellman-problem-over-the-group-of-quadratic-residues | Computational Diffie-Hellman problem over the group of quadratic residues
Suppose that $N=pq$ where $p$ and $q$ are safe primes. $\mathbb{QR}_N$ is the group of quadratic residues which is a cyclic group with order $\frac{\phi(N)}{4}$. Let $g$ be the generator of $\mathbb{QR}_N$.
The computational Diffie-Hellman problem is defined as : given $U=g^u\in\mathbb{QR}_N$ and $V=g^v\in\mathbb{QR}_N$ where $u,v$ are chosen uniformly at random from $\mathbb{Z}_{\frac{\phi(N)}{4}}$, compute $CDH(U,V)=g^{uv}$.
Now, if $N$ can be efficiently factored, then computing $CDH(U,V)=g^{uv}$ is still hard ?
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That depends entirely on the size of $p$ and $q$.
Given a factorization of $N = pq$, an attacker can compute $g^u \bmod p$ and $g^v \bmod p$, and then attempt to solve the CDH problem modulo $p$, giving him $g^{uv} \bmod p$.
Then, he can then compute $g^u \bmod q$ and $g^v \bmod q$, and then attempt to solve the CDH problem modulo $q$, giving him $g^{uv} \bmod q$.
Then, he can combine them to form $g^{uv} \bmod pq$.
The only parts that might not be straightforward is the CDH problem modulo $p$ and $q$ -- if one if the two primes is large enough to make this infeasible, then he cannot do that (and conversely, if he cannot solve the CDH problem modulo $p$, he obviously cannot solve it modulo $pq$.
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how large of $p$ or $q$ at least to make sure the hardness of $CDH(U,V) \pmod{p}$ or $CDH(U,V) \pmod{q}$ ? – T.B May 9 at 1:58
@Alex: Well, with the Number Field Sieve, a large and determined attacker is known to be able to compute discrete logs of 768 bits (they've factored numbers that large, and NFS can be used to compute discrete logs of the same size without too much more work), and (depending on the amount of resources available), possibly a bit more. 1024 bits may be safe for now. – poncho May 9 at 2:02
so if the CDH assumption over $\mathbb{QR}_N$ holds, then CDH assumption $\pmod{p}$ or CDH assumption $\pmod{q}$ may not hold; from other direction if CDH assumption $\pmod{p}$ or CDH assumption $\pmod{q}$ holds ,then certainly CDH assumption over $\mathbb{QR}_N$ holds. If I want to use this correctly, I'd better supposing CDH assumption $\pmod{p}$ or CDH assumption $\pmod{q}$ holds, is that right? – T.B May 9 at 2:22
@Alex: as for CDH over $N$ does not imply CDH over $p$, well, that's obvious; consider $N = 7p$ where $p$ is a large safe-prime; the CDH problem over 7 is known to be easy. Now, it's possible that CDH over $N$ might be difficult even if CDH over $p$ and $q$ is feasible; that would require that $N$ be hard to factor. Also: is there a specific reason why you're asking for CDH over a composite? CDH over a prime of the same size is much safer. – poncho May 9 at 2:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9422410130500793, "perplexity": 534.9072494643261}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400379083.43/warc/CC-MAIN-20141119123259-00219-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/153902/un-countable-union-of-open-sets | # (Un-)Countable union of open sets
Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well.
Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets.
My question is, does thereby follow that $\bigcap_{i \in \mathbb{N}} A_i$ and $\bigcup_{i \in \mathbb{N}} A_i$ are open sets as well?
And what about $\bigcap_{i \in I} A_i$ and $\bigcup_{i \in I} A_i$ for uncountabe $I$?
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Ok, it is an axiom for a topologies. But then there should be a proof for metrik spaces, say $\mathbb{R}$ with the canonical metrik? So to say, that every point $x \in \bigcap_{i \in I} A_i$ is an inner point. – Haatschii Jun 4 '12 at 20:16
A remark: regardless of whether it is true that an infinite union or intersection of open sets is open, when you have a property that holds for every finite collection of sets (in this case, the union or intersection of any finite collection of open sets is open) the validity of the property for an infinite collection doesn't follow from that. In other words, induction helps you prove a proposition for any natural number, but not for any transfinite cardinal. You'll have to use different techniques to prove or disprove the statement in your questions. – talmid Jun 4 '12 at 20:43
The union of any collection of open sets is open. Let $x \in \bigcup_{i \in I} A_i$, with $\{A_i\}_{i\in I}$ a collection of open sets. Then, $x$ is an interior point of some $A_k$ and there is an open ball with center $x$ contained in $A_k$, therefore contained in $\bigcup_{i \in I} A_i$, so this union is open. Others have given a counterexample for the infinite intersection of open sets, which isn't necessarily open. By de Morgan's laws, the intersection of any collection of closed sets is closed (try to prove this), but consider the union of $\{x\}_{x\in (0,1)}$, which is $(0,1)$, not closed. The union of an infinite collection of closed sets isn't necessarily closed.
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Any union of a set of open sets is again open. However, infinite intersections of open sets need not be open. For example, the intersection of intervals $(-1/n,1/n)$ on the real line (for positive integers $n$) is precisely the singleton $\{0\}$, which is not open.
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An arbitrary union (coutable or not) of open sets is open, but even for a countable intersection it's not true in general. For example, when $\Omega$ is the real line endowed with the usual topology, and $A_i:=\left(-\frac 1i,\frac 1i\right)$, $A_i$ is open but $\bigcap_{i\in \Bbb N}A_i=\{0\}$ which is not open.
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Moreover (exercise for the reader), any subset of the ambient space (e.g. the reals) can be written as the intersection of some collection of open sets. – Dave L. Renfro Jun 4 '12 at 21:51
@DaveL.Renfro: can you give some hints how to prove this? – Thomas E. Jun 5 '12 at 11:28
Actually, Dave's assertion fails for the indiscrete topology... – GEdgar Jun 5 '12 at 12:16
@Thomas E.: Assume singleton sets are closed sets (to take care of GEdgar's observation). Then, given any subset $E,$ we can write $E$ as the union of a collection of singleton sets (use the points belonging to $E$), and hence $E$ can be written as the union of a collection of closed sets. By applying De Morgan's Law, we can now write the complement of $E$ as the intersection of a collection of open sets (the open sets will be co-singleton sets). Finally, note that as $E$ varies over all subsets of the ambient space, the complement of $E$ will vary over all subsets of the ambient space. – Dave L. Renfro Jun 5 '12 at 16:25
@DaveL.Renfro. True, I didn't somehow even think about that :-) So $T_{1}$ would be the least requirement for this construction. – Thomas E. Jun 5 '12 at 19:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9280970096588135, "perplexity": 152.50981383271494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990112.92/warc/CC-MAIN-20150728002310-00068-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://linearalgebras.com/understanding-analysis-exercise-4-2.html | If you find any mistakes, please make a comment! Thank you.
## Solution to Understanding Analysis Exercise 4.2
##### Exercise 4.2.1
See Understanding Analysis Instructors’ Solution Manual Exercise 4.2.5
##### Exercise 4.2.2
(a) We would like $|(5x-6)-9|<1$, that is $|5(x-3)|<1$. Hence we need$|x-3|<\frac{1}{5}.$Therefore, the largest possible $\delta$ is $\dfrac{1}{5}$.
(b) We would like $|\sqrt{x}-2|<1$, that is$-1<\sqrt{x}-2<1\Longrightarrow 1<\sqrt{x}<3.$Hence we need $1<x < 9$, which also implies$-3< x-4<5.$Therefore, we must have$|x-4|<\min\{|-3|,|5|\}=3.$The largest possible $\delta$ is $3$.
(c) We would like $|[[x]]-3|<1$. Since $[[x]]$ is an integer, this may happen only if $[[x]]=3$, therefore $3\leqslant x<4$. We need$3-\pi \leqslant x-\pi < 4-\pi.$Hence$|x-\pi|<\min \{|3-\pi|,|4-\pi|\}=\pi-3.$The largest possible $\delta$ is $\pi-3$.
(d) We would like $|[[x]]-3|<0.01$. Since $[[x]]$ is an integer, this may happen only if $[[x]]=3$, therefore $3\leqslant x<4$. We need$3-\pi \leqslant x-\pi < 4-\pi.$Hence$|x-\pi|<\min \{|3-\pi|,|4-\pi|\}=\pi-3.$The largest possible $\delta$ is $3$.
##### Exercise 4.2.3
See Understanding Analysis Instructors’ Solution Manual Exercise 4.2.4
##### Exercise 4.2.4
(a) We would like$\left|\frac{1}{[[x]]}-\frac{1}{10}\right|<\frac{1}{2},$namely$-\frac12<\frac{1}{[[x]]}-\frac{1}{10}<\frac{1}{2}.$Therefore$-\frac{2}{5}<\frac{1}{[[x]]}<\dfrac{3}{5}.$Since $[[x]]$ cannot be zero, we have $x\geqslant 1$. Then we have $[[x]]>\frac{3}{5}$. Because $[[x]]$ is an integer, $[[x]]\geqslant 2$. Thus $x\geqslant 2$ and $x-10\geqslant -8$. We have $|x-10|\leqslant 8$. Therefore the largest possible $\delta$ is $8$.
(b) We would like$\left|\frac{1}{[[x]]}-\frac{1}{10}\right|<\frac{1}{50},$namely$-\frac{1}{50}<\frac{1}{[[x]]}-\frac{1}{10}<\frac{1}{50}.$Therefore$\frac{2}{25}<\frac{1}{[[x]]}<\frac{3}{25}.$Hence $\frac{25}{3}<[[x]]<\frac{25}{2}.$Because $[[x]]$ is an integer, $9\leqslant [[x]]\leqslant 12$. Thus $9\leqslant x < 13$ and $-1\leqslant x-10\leqslant 3$. We have $|x-10|\leqslant 1$. Therefore the largest possible $\delta$ is $1$.
(c) The largest $\varepsilon$ satisfying the property is $\dfrac{1}{90}$. For any $V_{\varepsilon}(10)$, there is a number $x$ in it such that $9<x<10$ . Hence $[[x]]=9$, we have$\frac{1}{[[x]]}-\frac{1}{10}=\frac{1}{90}.$Hence there is no suitable $\delta$ response possible.
##### Exercise 4.2.5
(a) Let $\varepsilon>0$. Definition 4.2.1 requires that we produce a $\delta>0$ so that $0<|x-2|<\delta$ leads to the conlcusion $|(3x+4)-10|<\varepsilon$. Note that$|(3x+4)-10|=|3x-6|=3|x-2|.$Thus if we choose $\delta=\varepsilon/3$, then $0<|x-2|<\delta$ implies $|(3x+4)-10|<\varepsilon$.
(b) Let $\varepsilon>0$. Definition 4.2.1 requires that we produce a $\delta>0$ so that $0<|x|<\delta$ leads to the conlcusion $|x^3|<\varepsilon$. Note that$|x^3|=|3x-6|=|x|^3.$Thus if we choose $\delta=\sqrt[3]{\varepsilon}$, then $0<|x|<\delta$ implies $$|x^3-0|=|x|^3<(\sqrt[3]{\varepsilon})^3=\varepsilon.$$(c) Let $\varepsilon>0$. Definition 4.2.1 requires that we produce a $\delta>0$ so that $0<|x-2|<\delta$ leads to the conlcusion $|(x^2+x-1)-5|<\varepsilon$. Note that$|(x^2+x-1)-5|=|x-2|\cdot |x+3|.$We can choose $\delta\leqslant 1$, then $|x+3|\leqslant 6$. Thus if we choose $\delta=\min \{1, \varepsilon/6\}$, then $0<|x|<\delta$ implies $$|(x^2+x-1)-5|=|x-2|\cdot |x+3|<\frac{\varepsilon}{6}\cdot 6=\varepsilon.$$(d) Let $\varepsilon>0$. Definition 4.2.1 requires that we produce a $\delta>0$ so that $0<|x-3|<\delta$ leads to the conlcusion $|1/x-1/3|<\varepsilon$. Note that$\left|\frac{1}{x}-\frac{1}{3}\right|=\frac{|x-3|}{3|x|}.$We can choose $\delta\leqslant 1$, then $2\leqslant |x|\leqslant 4$. Thus if we choose $\delta=\min \{1, 6\varepsilon\}$, then $0<|x-3|<\delta$ implies $$\left|\frac{1}{x}-\frac{1}{3}\right|=\frac{|x-3|}{3|x|}<\frac{6\varepsilon}{3\cdot 2}=\varepsilon.$$
##### Exercise 4.2.6
(a) True. A property is true for some set, then it is also true for a subset of this set.
(b) False. In the Definition 4.2.1, the value of $f(a)$ is not involved. In general, it can be any number.
(c) True by Corollary 4.2.4.
(d) False. Take the example $f(x)=x-a$ and $g(x)=1/(x-a)$ with domain $\mb R\setminus \{a\}$. Then $\lim_{x\to a}f(x)g(x)=1$.
##### Exercise 4.2.7
See Understanding Analysis Instructors’ Solution Manual Exercise 4.2.6
##### Exercise 4.2.8
(a) Does not exist. Note that $\lim(2-1/n)=2$ and $\lim (2+1/n)=2$, however$\lim f(2-1/n)=-1,\quad \lim f(2+1/n)=1.$By Corollary 4.2.5, limit $\lim_{x\to 2}f(x)$ does not exist.
(b) The limit is 1. For $0<\delta<1/4$ and any $x\in V_{\delta}(7/4)$, we have $x<2$. Hence $f(x)=-1$, for all $x\in V_{\delta}(7/4)$. Hence the limit is 1.
(c) Does not exist. Note that $\lim 1/(2n+1)=0$ and $\lim 1/(2n)=0$, however$\lim f(1/(2n+1))=\lim (-1)^{2n+1}=-1,$$\lim f(1/(2n))=\lim (-1)^{2n}=1.$By Corollary 4.2.5, limit $\lim_{x\to 0}f(x)$ does not exist.
(d) The limit is zero. For any $\varepsilon>0$, let $0<|x|<\varepsilon^3$, then$|\sqrt[3]{x}(-1)^{[[1/x]]}|=|\sqrt[3]{x}|<\varepsilon.$Hence the limit is zero.
##### Exercise 4.2.9
See Understanding Analysis Instructors’ Solution Manual Exercise 4.2.7
##### Exercise 4.2.10
(a) We say that $\lim_{x\to a^+}f(x)=L$ provided that, for all $\varepsilon >0$, there exists a $\delta>0$ such that whenever $0< x- a<\delta$ it follows that $|f(x)-L|<\varepsilon$.
We say that $\lim_{x\to a^-}f(x)=M$ provided that, for all $\varepsilon >0$, there exists a $\delta>0$ such that whenever $0< a-x<\delta$ it follows that $|f(x)-M|<\varepsilon$.
(b) By definition, it is clear that if $\lim_{x\to a}f(x)=L$ then both the right and left-hand limits equal $L$.
Conversely, if both the right and left-hand limits equal $L$. Since the right limit is $L$, for all $\varepsilon >0$, there exists a $\delta_1>0$ such that whenever $0< x- a<\delta_1$ it follows that $|f(x)-L|<\varepsilon$. Since the left limit is $L$, for all $\varepsilon >0$, there exists a $\delta_2>0$ such that whenever $0< a- x<\delta_2$ it follows that $|f(x)-L|<\varepsilon$.
Let $\delta=\min\{\delta_1,\delta_2\}$, then whenever $0<|x-a|< \delta$, we have $|f(x)-L|<\varepsilon$. Hence $\lim_{x\to a}f(x)=L$.
##### Exercise 4.2.11
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https://physics.stackexchange.com/questions/306515/mean-field-equations-in-the-bcs-theory-of-superconductivity/327425 | # Mean field equations in the BCS theory of superconductivity
In BCS theory, one takes the model Hamiltonian
$$\sum_{k\sigma} (E_k-\mu)c_{k\sigma}^\dagger c_{k\sigma} +\sum_{kk'}V_{kk'}c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger c_{-k'\downarrow} c_{k'\uparrow}$$
This Hamiltonian clearly conserves particle number. Thus, we expect the ground state to have a definite particle number. It's possible the ground state is degenerate, but that could be lifted by perturbing $\mu$.
Then, one makes a mean field approximation. One replaces $$c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger c_{-k'\downarrow} c_{k'\uparrow}$$
with $$\langle c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger\rangle c_{-k'\downarrow} c_{k'\uparrow}+c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger \langle c_{-k'\downarrow} c_{k'\uparrow}\rangle-\langle c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger\rangle \langle c_{-k'\downarrow} c_{k'\uparrow}\rangle$$
This doesn't make any sense to me. This seems to be saying that we know the terms $c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger$ and $c_{-k'\downarrow} c_{k'\uparrow}$ don't fluctuate much around their mean values. But we also know that in the actual ground state, the mean values are given by $\langle c_{-k'\downarrow} c_{k'\uparrow}\rangle=\langle c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger\rangle=0$ since the ground state will have definite particle number. Thus the fluctuations about the mean value aren't small compared to the mean value.
How can this mean field treatment be justified?
• Maybe this helps : link.springer.com/article/10.1007/BF02731446 – jjcale Jan 23 '17 at 22:12
• @JahanClaes I think I start understanding your problem. Is it because in the superconducting construction, you feel that one continues to suppose the ground state $\left|N\right\rangle$ is a Fermi sea, for which one should clearly get $\left\langle N\right|cc\left|N\right\rangle =0$, whereas one supposes in contrary that $\left\langle N-2\right|cc\left|N\right\rangle \propto\Delta\neq0$ when constructing the BCS order parameter $\Delta$ ? Is that your problem ? – FraSchelle Jan 27 '17 at 5:23
• @JahanClaes Otherwise, an other way to justify the mean-field approximation is given in this answer : physics.stackexchange.com/a/257639/16689. It is the same thing as below [physics.stackexchange.com/a/257639/16689]. Questions about the symmetry breaking get partial answer here physics.stackexchange.com/questions/133780/… and question about particle number conservation get partial answer here : physics.stackexchange.com/questions/44565/… – FraSchelle Jan 27 '17 at 5:31
• But if you want to understand why one must enlarge the class of accessible ground state(s) when discussing second order symmetry breaking, and in particular for the case of superconductivity, then you first have to restate your question, because it is not clear at all (it took me 3 days to understand it ...), then I may give an answer if I have time to write it. This question is indeed interesting, and I never tried to put words on it :-) – FraSchelle Jan 27 '17 at 5:35
• A lot of the confusion here comes from people taking the mean field approach too literally. No matter how you count, the number of electrons in a superconductor is fixed. Pairing, etc. cannot change that. However, dealing with particle conservation is very unwieldy (think about dealingwith particle conservation in the microcanonical ensemble of a Fermi gas without a chemical potential!). I think the best reference on this issue of particle conservation is Tony Leggett's textbook "Quantum Liquids". He goes through the entire BCS calculation without forgoing electron number conservation. – KF Gauss Jan 31 '17 at 15:02
First, why do we choose those terms to put the brakets around? Do we somehow know that these are the terms that won't fluctuate much?
The heuristic motivation is that the superconductivity phenomenon can be understood as the non-relativistic analog of the Higgs mechanism for the EM $U_{\text{EM}}(1)$ (this idea explains the main properties of the superconductor, such as absence of resistivity, magnetic field expulsion and other). For the Higgs mechanism one need a condensate of some particles. In the solid body, the only candidates are electrons. The simplest choice of the condensate is the scalar condensate (very-very heuristically, we require the Galilei invariance into the superconductor). Therefore the only allowed simplest scalar bounded state $|\psi\rangle$ is constructed from the electron pair with zero total spin, which can be imagine as two fermions moving in opposite directions: $$|\psi\rangle = c^{\dagger}_{\mathbf k, \uparrow}c^{\dagger}_{-\mathbf k, \downarrow}|0\rangle$$ This is indeed realized when we calculate the four-fermion vertex in the microscopic theory of electron-phonon interactions. Indeed, we start from the interaction Hamiltonian $$H_{\text{int}} = g\psi^{\dagger}\psi \varphi,$$ where $\psi$ is electron field, while $\varphi$ is phonon field, and assume the second order in $g$ diagrams of electron-electron scattering; let's denote their momenta as $\mathbf p_{i}$. After tedious calculations of the corresponding diagram one finds that the vertex $\Gamma^{(4)}(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3},\mathbf{p}_{4})$ (with $\mathbf{q} = \mathbf{p}_{1} + \mathbf{p}_{2}$) has the pole at zero momentum $\mathbf q$ and zero total spin of fermions $1,2$ and $3,4$ or ($1,4$ and $2,3$). The pole quickly disappears once one enlarges $|\mathbf q|$.
Finally, since 4-fermion vertex is expressed through two-particles Green $G^{(2)}$ function, then the presence of the pole in $\Gamma^{(4)}$ means the presence of the pole in $G^{(2)}$. This means that bounded states of electrons appear.
Second, why can't we just immediately say $\langle c_{-k'\downarrow}c_{k'\uparrow}\rangle=\langle c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger\rangle=0$ since we know the ground state will have definite particle number, and $c_{-k'\downarrow} c_{k'\uparrow}$ and $c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger$ don't conserve particle number?
(Added) Let's again use the idea of Higgs mechanism. It is clearly that the number of particles isn't strictly conserved in superconducting state. Really, because of the SSB of $U_{\text{EM}}(1)$ group the system loses the definition of the conserved particle number (being associated to the global phase invariance current) in the ground state.
Microscopically the formation of the condensate is argued in a following way: the presence of the two-fermions bounded states leads to instability of the electron gas inside the superconductor, and fermions are converted in Cooper pairs. Really, for bosons the lowest energy state can be filled by infinitely many number of particles, while for fermions this is not true due to Pauli principle. This (under some assumptions, see below) leads to formation of new vacuum state on which the operator $\hat{c}^{\dagger}_{\mathbf k,\uparrow}\hat{c}^{\dagger}_{-\mathbf k, \downarrow}$ has non-zero VEV: $$\langle \text{vac} | \hat{c}^{\dagger}_{\mathbf p,\uparrow}\hat{c}^{\dagger}_{-\mathbf p, \downarrow}|\text{vac}\rangle = \Psi \neq 0$$ It's not hard to construct this state: $$|\text{vac}\rangle = \prod_{\mathbf k = \mathbf k_{1}...k_{\frac{N}{2}}}(u_{\mathbf k} + v_{\mathbf k}\hat{c}^{\dagger}_{\mathbf k,\uparrow}\hat{c}^{\dagger}_{-\mathbf k, \downarrow})|0\rangle, \quad |v_{\mathbf k}|^{2} + |u_{\mathbf k}|^{2} = 1$$ This state called the coherent state really doesn't have definite number of particles.
For the particles number operator $\hat{N} = \sum_{\mathbf k}\hat{c}^{\dagger}_{\mathbf k,\uparrow}\hat{c}^{\dagger}_{-\mathbf k, \downarrow}$ the VEV $\langle \text{vac}|\hat{N}|\text{vac}\rangle$ is $$\tag 1 N = \langle \text{vac}|\hat{N}|\text{vac}\rangle \sim \sum_{\mathbf k}|v_{\mathbf k}|^{2},$$ Why the number of particles is not so indefinite
However, the root $\sqrt{\Delta N^{2}}$ of quadratic deviation from while the quadratic deviation $\langle \text{vac}|(\hat{N} - N)^{2}|\text{vac}\rangle$ is $$\tag 2 \Delta N^{2} = \langle \text{vac}|(\hat{N} - N)^{2}|\text{vac}\rangle = \sum_{\mathbf k}|u_{\mathbf k}|^{2}|v_{\mathbf k}|^{2}$$ is negligible in the limit of large particle number (and volume). Really, both of $(1)$ and $(2)$ are proportional to the volume $V$ (note only one summation over wave-numbers $\mathbf k$ in $(2)$), and hence $\sqrt{\Delta N^{2}} \sim \sqrt{V}$ which is negligible in the large volume limit. This argument is seemed to be first published in the BCS article.
(Added) This means that the state of superconductor with good accuracy can be projected on the state with definite particle number $N$. Precisely, define the phase dependent ground state $$|\text{vac}(\theta)\rangle = |\text{vac}[c^{\dagger}_{\mathbf k, \uparrow/\downarrow} \to c^{\dagger}_{\mathbf k, \uparrow/\downarrow}e^{-i\theta}]\rangle$$ The state with definite particle number $N$ is defined as $$|N\rangle = \int \limits_{0}^{2\pi} d\theta e^{i\hat{N}\theta}|\text{vac}(\theta)\rangle$$
• I understand why once you solve the mean field equations you get a state with non-definite particle number. But why doesn't imply that we've made a very bad mistake, since we know the ground state does have a definite particle number? – Jahan Claes Jan 21 '17 at 3:17
• @FraSchelle I don't think spontaneous symmetry breaking can be the cause of this, since even if the ground state is degenerate, that degeneracy must take place in a definite particle number sector. Or if it doesn't, it will upon perturbing $\mu$. – Jahan Claes Jan 23 '17 at 17:38
• @FraSchelle I'm not saying I'm smarter than Bardeen. I'm saying I don't understand why something happens, and I'd like a simple explanation why. – Jahan Claes Jan 25 '17 at 19:59
• Hi Jahan, while the above answer is correct you are justified in not being completely satisfied with the normal treatment of number conservation in BECs/superconductors and the relation to SSB. I strongly suggest you take a look at Leggett's book Quantum Liquids, chapter 2, which has a very interesting discussion about this. – Rococo Jan 26 '17 at 20:49
• @FraSchelle, the number of electrons in a material above and below the superconducting transition is fixed. Proof: charge conservation. The number of cooper pairs however is not fixed. There is plenty of discussion on number conservation and why it holds, but still can be put aside for calculations dating back even to Robert Schrieffer's thesis. – KF Gauss Jan 31 '17 at 15:10
After some thoughts, I think an other (and hopefully better) answer might go along the following lines, perhaps more rigorous. Nevertheless, I will not give the full mathematical details, since they require too much writing.
At the heart of perturbative treatment of many-body problems lies the Wick theorem. For two body interaction of fermionic nature, it states that
$$\left\langle c_{1}^{\dagger}c_{2}^{\dagger}c_{3}c_{4}\right\rangle =\left\langle c_{1}^{\dagger}c_{2}^{\dagger}\right\rangle \left\langle c_{3}c_{4}\right\rangle -\left\langle c_{1}^{\dagger}c_{3}\right\rangle \left\langle c_{2}^{\dagger}c_{4}\right\rangle +\left\langle c_{1}^{\dagger}c_{4}\right\rangle \left\langle c_{2}^{\dagger}c_{3}\right\rangle$$
As any theorem it can be rigorously proven. In fact, the proof is not that cumbersome, and I refer to
where the demonstration is the same in both papers. The key point is that one needs a Gaussian state to demonstrate the theorem, namely one needs a statistical operator in the form $\rho=e^{-\sum\epsilon_{n}c_{n}^{\dagger}c_{n}}$. In any other case the Wick's decomposition does not work (or at least I'm not aware of such a decomposition). Importantly, the demonstration in the above references uses only the anti-commutation properties between the $c$'s operators. So any time one has a statistical average made over Gaussian states of some anti-commuting (i.e. fermionic) operators, the Wick's theorem applies.
Now, the strategy of the mean-field treatment is the following. Write the Hamiltonian (I use the symbol $\sim$ to withdraw integrals and/or sums over all degrees of freedom)
$$H\sim H_{0}+c_{1}^{\dagger}c_{2}^{\dagger}c_{3}c_{4} \sim H_{0}+H_{\text{m.f.}}+\delta H$$
with $H_{0}$ the free particle Hamiltonian (which can be written in the form $H_{0}\sim\epsilon_{n}c_{n}^{\dagger}c_{n}$ in a convenient basis), $H_{\text{m.f.}}=\Delta c_{1}^{\dagger}c_{2}^{\dagger}+\Delta^{\ast}c_{3}c_{4}$ the mean-field Hamiltonian, and $\delta H=c_{1}^{\dagger}c_{2}^{\dagger}c_{3}c_{4}-H_{\text{m.f.}}$ the correction to the mean-field Hamiltonian.
The mean-field Hamiltonian can be diagonalised using a Bogliubov transformation, namely, one can write
$$H_{0}+H_{\text{m.f.}}=\sum_{n}\epsilon_{n}\gamma_{n}^{\dagger}\gamma_{n}$$
where the $\gamma$'s are some fermionic operators verifying
$$c_{i}=u_{ik}\gamma_{k}+v_{ik}\gamma_{k}^{\dagger}\\c_{i}^{\dagger}=u_{ik}^{\ast}\gamma_{k}^{\dagger}+v_{ik}^{\ast}\gamma_{k}$$
note: there are some constraints imposed on the $u$'s and $v$'s in order for the above Bogoliubov transformation to preserve the (anti-)commutation relation ; in that case one talks about canonical transformation, see
• Fetter, A. L., & Walecka, J. D. (1971). Quantum theory of many-particle systems. MacGraw-Hill.
for more details about canonical transformations. Once one knows a few properties of the mean-field ground state, one can show that $\lim_{N\rightarrow\infty}\left\langle \delta H\right\rangle \rightarrow0$ when $N$ represents the number of fermionic degrees of freedom, and the statistical average $\left\langle \cdots\right\rangle$ is performed over the mean-field ground state, namely $\left\langle \cdots\right\rangle =\text{Tr}\left\{ e^{-\sum\epsilon_{n}\gamma_{n}^{\dagger}\gamma_{n}}\cdots\right\}$. See
• De Gennes, P.-G. (1999). Superconductivity of metals and alloys. Advanced Book Classics, Westview Press
for a clear derivation of this last result. In fact deGennes shows how to choose the $u$ and $v$ in order for the Bogoliubov transformation to diagonalise the mean-field Hamiltonian, and then he shows that this choice leads to the best approximation of the zero-temperature ground state of the interacting (i.e. full) Hamiltonian. The idea to keep in mind is that the mean-field treatment works. Up to now, we nevertheless restrict ourself to operator formalism. When dealing with quantum field theory and its methods, one would prefer to use the Wick's theorem.
So, now we come back to the Wick's decomposition. One can manipulate the statistical average and show that
$$\left\langle c_{1}^{\dagger}c_{2}^{\dagger}\right\rangle =\text{Tr}\left\{ \rho c_{1}^{\dagger}c_{2}^{\dagger}\right\} \\=\text{Tr}\left\{ e^{-\varepsilon_{n}c_{n}^{\dagger}c_{n}}c_{1}^{\dagger}c_{2}^{\dagger}\right\} =\text{Tr}\left\{ c_{1}^{\dagger}c_{2}^{\dagger}e^{-\varepsilon_{n}c_{n}^{\dagger}c_{n}}\right\} =e^{-2\varepsilon_{n}}\text{Tr}\left\{ e^{-\varepsilon_{n}c_{n}^{\dagger}c_{n}}c_{1}^{\dagger}c_{2}^{\dagger}\right\} \\=e^{-2\varepsilon_{n}}\left\langle c_{1}^{\dagger}c_{2}^{\dagger}\right\rangle =0$$
using the cyclic property of the trace and the Bakker-Campbell-Hausdorf formula to pass the exponential from right to left of the two creation operators. The quantity should be zero because it is equal to itself multiplied by a positive quantity $e^{-2\varepsilon_{n}}$. The manipulation is the same as one does for the Wick's theorem, so I do not write it fully, check the above references.
The conclusion is that clearly, the quantity $\left\langle c_{1}^{\dagger}c_{2}^{\dagger}\right\rangle$ is zero in the Wick's decomposition. But due to the Cooper problem (or what we learned from the mean-field treatment), one knows that we should not take the statistical average over the free fermions $e^{-\sum\varepsilon_{n}c_{n}^{\dagger}c_{n}}$, but over the Bogoliubov quasi-particles $e^{-\sum\epsilon_{n}\gamma_{n}^{\dagger}\gamma_{n}}$, since they represent the approximately true (in the limit of large $N$ it is exact) ground state. Say differently, the Fermi surface is not the ground state of the problem, and the statistical average should not be taken over this irrelevant ground state. We should choose instead the statistical average over the Bogoliubov's quasiparticles $\gamma$. Namely, one has
$$\left\langle c_{1}^{\dagger}c_{2}^{\dagger}\right\rangle =\text{Tr}\left\{ e^{-\varepsilon_{n}\gamma_{n}^{\dagger}\gamma_{n}}c_{1}^{\dagger}c_{2}^{\dagger}\right\} \neq0$$
and this quantity is not zero, as you can check when writing the $c$'s in term of the $\gamma$'s. Importantly, note that the Wick's theorem can be prove rigorously for this statistical operator (see above).
This complete the proof that the mean-field approximation can be made rigorous, and that one can take the quantity $\left\langle c_{1}^{\dagger}c_{2}^{\dagger}\right\rangle$ as the order parameter of a phase whose ground state is not filled with free electrons, but with Bogoliubov quasi-particles generated by the $\gamma$ operators. One usually refers to this ground state as the Cooper sea. If you expand the exponential with the $\gamma$'s over the zero-electron state noted $\left|0\right\rangle$, you will see you end up with something like $e^{-\sum\epsilon_{n}\gamma_{n}^{\dagger}\gamma_{n}}\left|0\right\rangle \sim\prod\left(u+vc_{1}^{\dagger}c_{2}^{\dagger}\right)\left|0\right\rangle$ which is the BCS Ansatz. Using rigorous notations you can make a rigorous proof of this last statement.
Now if you want to make the derivation completely rigorous, it requires to demonstrate the Wick's theorem, to manipulate the mean-field Hamiltonian in order to show the Bogoliubov mean-field transformation works quite well, and to calculate the statistical average over the Bogoliubov ground state. This goes straightforwardly ... over many many pages which I'm too lazy to write here.
I will try to answer the question : Why do we have to enlarge the number of possibilities for the ground state ? I guess this is at the heart of your problem. Before doing so, let us quickly discuss the notion of ground state itself, which are called Fermi sea in the case of fermion.
# At the beginning: the Fermi surface or quasiparticles
So at the beginning is a collection of fermions, and the notion of Fermi surface. An important element of the forthcoming discussion is the notion of quasiparticle. In condensed matter systems, the Fermi surface is not built from free electrons, or bare electrons, or genuine electrons. To understand how this notion comes in, let's take a free electron, i.e. a particle following the Dirac equation with charge $e$ and spin $1/2$, and put it into any material (say a semi-conductor or a metal for instance). By electrons any condensed matter physicist means that in a complex system (s)he studies, it is impossible to take into account all the possible interactions acting on the bare electron. Most of the interaction will be of bosonic nature (think especially about phonons). So one hopes that taking all the complicated bosonic interactions on the fermionic bare electron results in a composite particle which still has a fermionic statistics, and hopefully behaves as a Schrödinger equation, possibly with kinetic energy $E_{c}=\dfrac{p^{2}}{2m_{2}^{*}}+\dfrac{p^{4}}{4m_{4}^{*}}+\cdots$ with effective masses $m_{2}^{*}$ that one can suppose $p$-independent and possibly $m_{2}^{*}\ll m_{4}^{*}$, such that one has the usual Schrödinger equation.
Note that
• the Schrödinger equation is anyways invariant with respect to the statistics of the particle it describes
• the above construction can be made a bit more rigorous using the tools of effective field theory and renormalisation
The important thing is that these electrons are still fermions, and so they pile up to form a Fermi sea. From now on we write electrons or quasiparticle without making distinction, since in condensed matter there is only quasiparticle. A quasiparticle of charge $e$ and spin $1/2$ will be called an electron. A Fermi surface is a stable object, as has been discussed in an other question. Stable ? Well, not with respect to the Cooper mechanism, which allows bound states of electrons to be generated on top of a Fermi sea. These bound states are of charge $2e$ and their total spin makes them some kind of bosons. They are as well quasiparticles, but we will call them Cooper pairs, instead of quasiparticle of charge $2e$ and spin $1$ or $0$.
Now we identified the ground state of a metal as a Fermi sea of fermionic quasiparticles called electrons, we can try to understand how this ground state becomes unstable and why we should then take into account several ground states, of possibly different statistical nature, as the bosonic versus fermionic ground state in a superconductor. The reason why we need to enlarge the available ground states is clearly due to the symmetry breaking, as we review now.
# Symmetry breaking and space of ground states
First, think about the para-ferromagnetic transition. Before the transition (paramagnetic phase) you can choose the orientation of the spin the way you want: they are random in the electron gas. A nice picture about that is to say: the Fermi surface is just the same for all the electrons.
Now comes the ferromagnetic phase: the system chooses either to align all the spins in the up or down direction (of course the direction is not fixed and the system is still rotationally invariant unless a magnetic field is applied, but clearly the electron spins are polarised). What about the Fermi surface ? Well, it becomes two-fold ... There is now one Fermi surface for the spin-up electrons and one Fermi surface for the spin-down electron. So the number of available ground states increases.
The link to the symmetry breaking is clear: the more you want of symmetries, the less possible states you allow. Say in the other way: breaking the symmetry allows for more states to exist. This is also quite straightforward from the following argument: once you allow an interaction responsible for the parra-to-ferro transition, you must first answer the question: is the unpolarised Fermi surface or one of the polarised Fermi surfaces the true ground state ? So you need a way to compare the spin-unpolarised and the spin-polarised ground states. So clearly the number of accessible ground states must be greater once a phase transition and a symmetry breaking is under the scope.
Now, about superconductivity and the relation to symmetry breaking, I refer to this (about particle number conservation) and this (about the $\text{U }\left(1\right)\rightarrow\mathbb{Z}_{2}$ symmetry breaking in superconductors) answers. The important thing is that the Cooper mechanism makes the Fermi surface instable. What results ? A kind of Bose-Einstein condensate of charged particles (the Cooper pairs of charge $2e$) and some electrons still forming a Fermi-Dirac condensate and so a Fermi sea, with less fermions than before the transition (hence the number of electrons is not conserved). So now the available ground states are i) the genuine Fermi sea made of electrons, ii) the charged Bose-Einstein condensate made with all the electrons paired up via Cooper mechanism and iii) a mixture of the two Fermi-Dirac and Bose-Einstein condensates (be careful, the terminology is misleading, a Bose-Einstein condensate and a Cooper pair condensate are not really the same thing).
Unfortunately, the real ground state is a kind of mixture, but at zero-temperature, one might suppose that all the conduction electrons have been transformed in Cooper pairs (in particular, this can not be true if one has an odd number of electrons to start with, but let forget about that). Let us call this complicated mixture the Cooper condensate, for simplicity.
In any case, we have to compare the Fermi sea with the Cooper pairs condensate. That's precisely what we do by supposing a term like $\left\langle cc\right\rangle \neq0$.
# A bit of mathematics
We define a correlation as $\left\langle c_{1}c_{2}\right\rangle$ with creation or annihilation operators. In a paramagnetic phase, we have $$\left\langle N\right|c^{\dagger}c\left|N\right\rangle =\left\langle n_{\uparrow}\right|c_{\uparrow}^{\dagger}c_{\uparrow}\left|n_{\uparrow}\right\rangle +\left\langle n_{\downarrow}\right|c_{\downarrow}^{\dagger}c_{\downarrow}\left|n_{\downarrow}\right\rangle$$ as the only non-vanishing correlation, with $\left|N\right\rangle$ a Fermi sea filled with $N$ electrons. Note in that case the polarised Fermi seas $\left|n_{\uparrow,\downarrow}\right\rangle$ make no sense, since there is no need for an internal degree of freedom associated to the electrons ; or these two seas are the two shores of the unpolarised Fermi ocean... Now, in the ferromagnetic phase, the correlations $\left\langle n_{\uparrow}\right|c_{\uparrow}^{\dagger}c_{\uparrow}\left|n_{\uparrow}\right\rangle$ and $\left\langle n_{\downarrow}\right|c_{\downarrow}^{\dagger}c_{\downarrow}\left|n_{ \downarrow}\right\rangle$ starts to make sense individually, and the ground states with polarised electrons as well. In addition, we must compare all of these inequivalent ground states. One way to compare all the possible ground states is to construct the matrix $$\left\langle \begin{array}{cc} c_{\uparrow}c_{\downarrow}^{\dagger} & c_{\uparrow}c_{\uparrow}^{\dagger}\\ c_{\downarrow}c_{\downarrow}^{\dagger} & c_{\downarrow}c_{\uparrow}^{\dagger} \end{array}\right\rangle =\left\langle \left(\begin{array}{c} c_{\uparrow}\\ c_{\downarrow} \end{array}\right)\otimes\left(\begin{array}{cc} c_{\downarrow}^{\dagger} & c_{\uparrow}^{\dagger}\end{array}\right)\right\rangle$$ where the ground state is a bit sloppily defined (i.e. I did not refer to $\left|n_{\uparrow,\downarrow}\right\rangle$, and I simply put the global $\left\langle \cdots\right\rangle$ for simplicity). The fact that the construction is a tensor product (the symbol $\otimes$ in the right-hand-side just does what appears in the left-hand-side) clearly shows that you can restaure the different ground states as you wish. In a sense, the problem of defining the different ground states is now put under the carpet, and you just have to deal with the above matrix. Clearly the diagonal elements exist only in the paramagnetic phase and the off-diagonal elements exists only in the ferromagnetic case, but this is no more a trouble, since we defined a tensorial product of several ground states and we are asking: which one is the good one?
Now, for the superconductor, one does not polarise the spins of the electrons, one creates some bosonic correlations on top of two fermionic excitations. So the natural choice for the matrix is $$\left\langle \begin{array}{cc} cc^{\dagger} & cc\\ c^{\dagger}c^{\dagger} & c^{\dagger}c \end{array}\right\rangle =\left\langle \left(\begin{array}{c} c\\ c^{\dagger} \end{array}\right)\otimes\left(\begin{array}{cc} c^{\dagger} & c\end{array}\right)\right\rangle$$ where still, the diagonal part exists already in a simple metal, and the off-diagonal shows up once the system transits to the superconducting phase. Clearly, if you call $\left|N\right\rangle$ the Fermi-Dirac condensate with $N$ electron of charge $e$, and $c$ the operator destroying an electron in this Fermi sea, then you must define $\left\langle cc\right\rangle \equiv\left\langle N-2\right|cc\left|N\right\rangle \propto\left\langle N\right|cc\left|N+2\right\rangle \propto\left\langle N-1\right|cc\left|N+1\right\rangle$ (note you can do the way you want, and this has profound implications for Josephson physics, but this is not the story today) for the correlation to exist. What are the states $\left|N-2\right\rangle$ then ? Well, it is clear from the context: a Fermi sea with two electrons removed. Without the Cooper mechanism, we would have no idea what is this beast, but thanks to him, we know this is just the instable Fermi sea with one Cooper pair removed due to the action of a virtual phonon.
Whereas the parra-ferromagnetic transition was seen in the competition between unpolarised versus polarised spins, the normal-superconducting transition can be seen in the competition between particle and hole versus particle-hole mixtures.
Now, how do we construct the mean-field Hamiltonian ? We simply use the two body interaction term, and we apply the Wick's theorem. That is, one does $$\left\langle c_{1}c_{2}c_{3}^{\dagger}c_{4}^{\dagger}\right\rangle =\left\langle c_{1}c_{2}\right\rangle \left\langle c_{3}^{\dagger}c_{4}^{\dagger}\right\rangle -\left\langle c_{1}c_{3}^{\dagger}\right\rangle \left\langle c_{2}c_{4}^{\dagger}\right\rangle +\left\langle c_{1}c_{4}^{\dagger}\right\rangle \left\langle c_{2}c_{3}^{\dagger}\right\rangle$$ valid for any average taken over Gaussian states. Clearly, one has (replace the numbers by spin vectors eventually) : the Cooper pairing terms, the Heisenberg-ferromagnetic coupling and some anti-ferromagnetic coupling (not discussed here). Usually, since a system realises one ground state, we do not need to try all the different channels. For superconductivity we keep the first term on the right-hand-side.
• This is something I'm going to have to go over a few times to fully understand, I think. But I still don't understand why we're suddenly taking the expectation values as $\langle N-2 |cc|N\rangle$, when in general expectation values and mean fields are always defined relative to a single state. I also don't understand why we consider a state with a cooper pair to have two fewer electrons than a state with no cooper pair--surely if you count the electrons using the number operator $\sum c^\dagger c$ you'll get the same answer regardless of whether the electrons have formed a pair? – Jahan Claes Jan 28 '17 at 2:51
• expectation values and mean fields are always defined relative to a single state : well, superconductivity theory tells you that it is not always true for mean field. So, either you change the nomenclature, and no more call the BCS treatment a mean-field treatment. After all, you can name things the way you want, and BCS method could be used indeed. What I've tried to explain is that both $\left|N\right\rangle$ and $\left|N+2\right\rangle$ are in the same class of solutions/ground-states: they are both Fermi seas, but the second one has 2 electrons more. This ground-state is reachable. – FraSchelle Jan 29 '17 at 8:08
• Somehow both quantum distributions appear in normal-superconductor phase transition, and so let us suppose they do. When you loose two fermions, you may get one boson. This mechanism is called Cooper instability. Not all materials present a Cooper instability of course. The important thing is: $c^{\dagger}c$ does not count the number of electrons, it counts the number of fermion modes, and $cc$ counts the Cooper-pairing generated bosons, so he number of boson modes. So in fact by Cooper pairing you loose two fermions and get one Cooper-pair/one boson. – FraSchelle Jan 29 '17 at 8:16
• The total number of charge is the same in both the normal and superconducting states. The crucial question is: how may we count them, because clearly $c^{\dagger}c$ no more does the job. This is a hard question in quantum field theory, since the U(1) symmetry breaking also destroys the conventional way of defining current through Noether theorem. In the mean-field/BCS treatment of the problem, you may try to define current in other way, and you will get something like $c^{\dagger}c+\Delta cc$ (picturesquely speaking, there is a post on SE where someone derives the current in the BdG frame) – FraSchelle Jan 29 '17 at 8:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9169586300849915, "perplexity": 326.37716609641063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487616657.20/warc/CC-MAIN-20210615022806-20210615052806-00487.warc.gz"} |
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# Density
## A measure of how tightly the matter in a substance is packed together.
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Density
The man in this cartoon is filling balloons with helium gas. What will happen if he lets go of the filled balloons? They will rise up into the air until they reach the ceiling. Do you know why? It’s because helium has less density than air.
### Defining Density
Density is an important physical property of matter. It reflects how closely packed the particles of matter are. When particles are packed together more tightly, matter has greater density.
Activity .-See the pìcture and explain which cube has more density.
Density depends how closely packet the particles of matter are. [Figure1]
Differences in density of matter explain many phenomena, not just why helium balloons rise. For example, differences in density of cool and warm ocean water explain why currents such as the Gulf Stream flow through the oceans. You can see a colorful demonstration of substances with different densities at this URL:
To better understand density, think about a bowling ball and volleyball, pictured in the Figure below . Imagine lifting each ball. The two balls are about the same size, but the bowling ball feels much heavier than the volleyball. That’s because the bowling ball is made of solid plastic, which contains a lot of tightly packed particles of matter. The volleyball, in contrast, is full of air, which contains fewer, more widely spaced particles of matter. In other words, the matter inside the bowling ball is denser than the matter inside the volleyball.
Balls with same size but difference material, so they have different density.
Q: If you ever went bowling, you may have noticed that some bowling balls feel heavier than others even though they are the same size. How can this be? Draw a picture with particles to explain it
A: Bowling balls that feel lighter are made of matter that is less dense.
### Calculating Density
The density of matter is actually the amount of matter in a given space. The amount of matter is measured by its mass, and the space matter takes up is measured by its volume. Therefore, the density of matter can be calculated with this formula:
$\text{Density} = \frac{\text{mass}}{\text{volume}}$
Assume, for example, that a book has a mass of 500 g and a volume of 1000 cm 3 . Then the density of the book is:
$\text{Density} = \frac{500 \ \text{g}}{1000 \ \text{cm}^3} = 0.5 \ \text{g/cm}^3$
Q: What is the density of a liquid that has a volume of 30 mL and a mass of 300 g?
A: The density of the liquid is:
$\text{Density} = \frac{300 \ \text{g}}{30 \ \text{mL}} = 10 \ \text{g/mL}$
### Summary
• Density is an important physical property of matter. It reflects how closely packed the particles of matter are.
• The density of matter can be calculated by dividing its mass by its volume.
### Vocabulary
• density : Amount of mass in a given volume of matter; calculated as mass divided by volume.
### Practice
Go to this URL and take the calculating-density quiz. Be sure to check your answers!
### Review
1. What is density?
2. Find the density of an object that has a mass of 5 kg and a volume of 50 cm 3 .
3. Create a sketch that shows the particles of matter in two substances that differ in density. Label the sketch to show which substance has greater density.
4. The the table below gives the density for some common materials.Can order from less density to bigger dendity
[Figure2]
6. Anna has calculated the mass and the volum of the unknow block, to be Mass = 79.4g Volum = 29.8 cm3 ; a second block has the same volum but diferrent mass, Mass= 25.4g. Using the table below identify what is each material block material | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 3, "texerror": 0, "math_score": 0.9011743664741516, "perplexity": 865.4014834847893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736673081.9/warc/CC-MAIN-20151001215753-00126-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://www.quantumuniverse.nl/and-action | And… Action!
Actie! De kreet heeft niet alleen een betekenis op de filmset; ook natuurkundigen hoor je vaak praten over de ‘actie’ van een systeem. In dit Engelstalige artikel legt Christian Ventura Meinersen uit wat een actie is, en hoe het begrip de volledige evolutie van een systeem bepaalt.
F=ma. It’s the one equation most people seem to remember from their high school physics classes. That is no coincidence: it is a beautifully small equation that captures most of our everyday life.
At every point in time we can look at the position and velocity of an object and, with the knowledge of the forces acting on that object at that instance, compute its future trajectory via the small equation $$F=ma$$! Just like a movie where the motion of objects comes to life by having many frames per second, Newton’s second law manages to trace out trajectories. It computes the forces at each point in time and updates the motion of the particle accordingly.
We come to one obvious problem with the equation $$F = ma$$: what happens for objects like light that have $$m=0$$? Does the vanishing mass mean that forces can never pull or push light? This cannot be the case as we know that the gravitational force can deflect light. This is an underlying issue with Newton’s second law; it is not a general law that can be applied to any object.
As so often in science, we therefore try to find patterns between phenomena and try to understand them, hence, under the umbrella of a more fundamental set of laws. The mathematician Joseph-Louis Lagrange was thinking about the patterns underlying each path that any object would take. The way he accomplished to say something useful about those patterns was by defining a new mathematical tool now famously known as the Lagrangian $$L$$ of a system. It is computed from the kinetic energy $$T$$ and the potential energy $$V$$ of an object, in the following way
$$L(x(t),\dot{x}(t))=T-V$$,
where
• $$x(t), \dot{x}(t)$$ are the position and velocity respectively and
• $$t$$ is the time coordinate.
Physically, the Lagrangian describes the transfer of energy between the two types of energy1: potential and kinetic. For example, you can think of an object falling from some height: the initial potential energy will be transformed into kinetic energy as the object is falling. So we exchange the energy corresponding to location for one corresponding to velocity and vice versa.
The next insight lies in the fact that, roughly speaking, the universe is lazy. It wants to do as little as possible over time. ‘Doing something’ physically means that there is a transfer of energy, as this results in dynamical processes. Let’s try to formulate this mathematically. We want to define something that encapsulates the transfer of energy over time. For that we use the Lagrangian and integrate it over all possible times $$t$$. This motivates us to define a mathematical tool, called the action $$S$$:
$$S[x(t)]=\int_{t_1}^{t_2} \; L(x(t),\dot{x}(t)) dt$$.
Using the ‘principle of universal laziness’2 we will find the path $$x(t)$$ that minimizes the action
$$\delta S[x(t)]=0$$.
Here the symbol $$\delta$$ means that we look at changes in the action, so we look for the minimum of the action as a function of the path. This will constitute the most ‘lazy’ path. Therefore the action, compared to Newton’s second law, is not a cause-effect law whereby one computes every frame of the trajectory and sews them together. It probes all possible paths and picks out the one with least action.
Indeed one can find such a path from the action through the Euler-Lagrange equation1:
$$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}} \right) – \left(\frac{\partial L}{\partial x} \right) =0$$.
This equation may look rather scary and not easy to solve, but we won’t ask you to – it’s just important to know that such an equation exists, and that therefore problems defined by a Lagrangian have a perfectly well-defined solution.
Summarizing: the Lagrangian formalism allows us to define the motion of an object independent of what type of object it is – even massless light. It relies on the fact that all paths that objects actually take share the same pattern: they minimize the action. Much later, this key insight even allowed physicists to describe situations where quantum mechanics and relativity are present, making the Lagrangian and the action very powerful tools!
[1] Of course you might have heard of many other ‘types’ of energy like thermal energy, chemical energy, gravitational energy, et cetera. But all of them rely on the fundamental ingredients of the position and velocity any object may have, hence we only require potential and kinetic energy. For instance, thermal energy can be thought of as the average kinetic energy.
[2] Of course this has a proper name in physics and goes under the name of principle of least action.
[3] One can actually derive the Euler-Lagrange equation via the principle of least action, but this would take a full course in calculus to derive. Feel free to still look it up! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8960434794425964, "perplexity": 413.7996901493483}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103941562.52/warc/CC-MAIN-20220701125452-20220701155452-00360.warc.gz"} |
https://www.cuemath.com/ncert-solutions/q-3-exercise-11-1-constructions-class-9-maths/ | # Ex.11.1 Q3 Constructions Solution - NCERT Maths Class 9
Go back to 'Ex.11.1'
## Question
Construct the angles of the following measurements:
(i) $$30^{\circ}$$
(ii) $$22 \frac{1}{2}^{\circ}$$
(iii) $$15^{\circ}$$
Video Solution
Constructions
Ex 11.1 | Question 3
## Text Solution
$${\rm{(i)}}\;\;30^{\circ}$$
Reasoning:
We need to construct an angle of $$60$$ degrees and then bisect it to get an angle measuring $$30^\circ$$
Steps of Construction:
(i) Draw ray $$PQ$$.
(ii) To construct an angle of $$60^{\circ}$$ .
With $$P$$ as centre and any radius, draw a wide arc to intersect $$PQ$$ at $$R$$. With $$R$$ as centre and same radius draw an arc to intersect the initial arc at $$S$$. $$\angle {SPR}=60^{\circ}$$
(iii) To bisect $$\angle {SPR}$$
With $$R$$ and $$S$$ as centres and same radius draw two arcs to intersect at $$T$$. Join $$P$$ and $$T$$ i.e. $$PT$$ is the angle bisector. Hence,
\begin{align}\angle {TPR}=\frac{1}{2} \angle {SPR}=30^{\circ}\end{align}
$${\rm{(ii)}}\;\;22 \frac{1}{2}^{\circ}$$
Reasoning:
We need to construct two adjacent angles of and bisect the second one to get a angle. This has to be bisected again to get a $$45^\circ$$ angle. The $$45^\circ$$ angle has to be further bisected to get \begin{align}22 \frac{1}{2}^{\circ}\end{align} angle.
\begin{align} 22 \frac{1}{2}^{\circ} &=\frac{45^{\circ}}{2} \\ 45^{\circ} &=\frac{90^{\circ}}{2}=\frac{30^{\circ}+60^{\circ}}{2} \end{align}
Steps of Construction:
(i) Draw ray $$PQ$$
(ii) To construct an angle of $$60^{\circ}$$
With $$P$$ as center and any radius draw a wide arc to intersect $$PQ$$ at $$R$$. With $$R$$ as center and same radius draw an arc to intersect the initial arc at $$S$$. $$\angle {SPR}=60^{\circ}$$
(iii)To construct adjacent angle of $$60^{\circ}$$ .
With $$S$$ as the center and same radius as before, draw an arc to intersect the initial arc at $$T$$ $$\angle {TPS}=60^{\circ}$$ .
(iv)To bisect $$\angle {TPS}$$
With $$T$$ and $$S$$ as centers and same radius as before, draw arcs to intersect each other at $$Z$$ Join $$P$$ and $$Z$$ $$\angle {ZPQ}=90^{\circ}$$
(v) To bisect $$\angle {ZPQ}$$
With $$R$$ and $$U$$ as centers and radius than half of $$RU$$, draw arcs to intersect each other at $$V$$. Join $$P$$ and $$V$$. $$\angle {VPQ}=45^{0}$$
(vi) To bisect $$\angle {VPQ}=45^{0}$$
With $$W$$ and $$R$$ as centers and radius greater than half of $$WR$$, draw arcs to intersect each other at $$X$$. Join $$P$$ and $$X$$. $$PX$$ bisects $$\angle {VPQ}$$
Hence,
\begin{align} \angle {XPQ} &=\frac{1}{2} \angle {WPQ} \\ &=\frac{1}{2} \times 45^{0} \\ &=22 \frac{1}{2} \end{align}
$${\rm{(iii)}}\;\;15^{\circ}$$
Reasoning:
We need to construct an angle of 60 degrees and then bisect it to get an angle measuring $$30^\circ$$. This has to be bisected again to get a $$15^\circ$$ angle.
\begin{align}15^{0}=\frac{30^{\circ}}{2}=\frac{\frac{60^{0}}{2}}{2}\end{align}
Steps of Construction:
(i) Draw ray $$PQ$$.
(ii) To construct an angle of $$60^{\circ}$$ .
With $$P$$ as center and any radius draw a wide arc to intersect $$PQ$$ at $$R$$. With $$R$$ as center and same radius draw an arc to intersect the initial arc at $$S$$. $$\angle {SPR}=60^{\circ}$$
(iii) Bisect $$\angle {SPR}$$ . With $$R$$ and $$S$$ as centers and radius greater than half of $$RS$$ draw arcs to intersect each other at $$T$$. Join $$P$$ and $$T$$ i.e. $$PT$$ is the angle bisector of $$\angle {SPR}$$ .
\begin{align} \angle {TPQ} &=\frac{1}{2} \angle{SPR} \\ &=\frac{1}{2} \times 60^{\circ} \\ &=30^{\circ} \end{align}
(iv)To bisect $$\angle {TPQ}$$
With $$R$$ and $$W$$ as centers and radius greater than half of $$RT$$, draw arcs to intersect each other at $$U$$ Join $$P$$ and $$U$$. $$PU$$ is the angle bisector of $$\angle {TPQ}$$ .
\begin{align}\angle {UPQ}=\frac{1}{2} \angle {TPQ}=15^{\circ}\end{align}
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