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# Type error in recursion
Consider the following function:
def f(n) :
def retfunc(x) :
return 1 if n == 0 else x*add(f(n-k) for k in (1..n))
return retfunc
The test
w = f(9); print w, type(w)
gives
"function retfunc at 0x43d3de8" "type 'function'"
which looks fine to me. However an evaluation w(3) gives
TypeError: unsupported operand type(s) for +: 'int' and 'function'
How can I get around this error?
EDIT: One solution is, as indicated by DSM,
def f(n):
retfunc(x) = 1 if n == 0 else add(x*f(n-k) for k in (1..n))
return retfunc
What I am trying to do here? Let's see:
for i in [1..8] : [c[0] for c in expand(f(i)(x)).coeffs()]
[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 4, 6, 4, 1]
[1, 5, 10, 10, 5, 1]
[1, 6, 15, 20, 15, 6, 1]
[1, 7, 21, 35, 35, 21, 7, 1]
edit retag close merge delete
Sort by » oldest newest most voted
I think the confusion here involves the fact that your examples use f(1) as an expression, but Python functions and the values those functions take are two different things. There are several ways to get around this:
# method 1
def f(n):
if n == 0:
retfunc(x) = 1
else:
retfunc(x) = add(x*f(n-k) for k in (1..n))
return retfunc
sage: f(0)
x |--> 1
sage: f(1)
x |--> x
sage: f(2)
x |--> x^2 + x
sage: f(3)
x |--> (x^2 + x)*x + x^2 + x
sage: f(3)(4)
100
(2) Use expressions instead, and substitute manually when you want to call it later:
# method 2
def f(n):
if n == 0:
retexpr = 1
else:
retexpr = add(x*f(n-k) for k in (1..n))
return retexpr
sage: f(0)
1
sage: f(1)
x
sage: f(2)
x^2 + x
sage: f(3)
(x^2 + x)*x + x^2 + x
sage: f(3)(x=4)
100
(3) Evaluate the function at x before doing the addition in your code:
# method 3
def f(n) :
def retfunc(x) :
return 1 if n == 0 else x*add(f(n-k)(x) for k in (1..n))
return retfunc
sage: [f(i) for i in [0..3]]
[<function retfunc at 0x10eee66e0>, <function retfunc at 0x10eee6668>, <function retfunc at 0x10eee6758>, <function retfunc at 0x10eee67d0>]
sage: [f(i)(x) for i in [0..3]]
[1, x, (x + 1)*x, ((x + 1)*x + x + 1)*x]
sage: [f(i)(x) for i in [0..3]]
[1, x, (x + 1)*x, ((x + 1)*x + x + 1)*x]
sage: f(3)(x=4)
100
I'd prefer (1) or (2) because it's somewhat easier to work with Sage-level objects in a calculus-y way than Python functions, but YMMV.
more
Thank you! First time that I hear about Sage-level functions versus Python functions.
( 2012-03-03 02:22:16 -0500 )edit
Your error is not due to recursion. The following example (slightly simplified with respect to yours) produces the same error :
def f(n) :
def retfunc(x) :
if n==0:
return 1
return x*f(3)
return retfunc
w=f(9)
w(3)
The error is at x*f(3). f(3) is a function, not a number. I don't really understand what you are trying to do. Some kind of factorial ?
The following code does not produce the error :
def f(n) :
def retfunc(x) :
if n==0:
return 1
return x*f(n-1)(3)
return retfunc
w=f(9)
print w(3)
Note : f(n-1)(3)
Hope it helps
Laurent
more
Thank you Laurent for your help. "The following code does not produce the error." Unfortunately the code also does not produce what I need :) "I don't really understand what you are trying to do." Well, I hoped that this would happen: f(0) returns the function retfunc(x) = 1 f(1) returns the function retfunc(x) = x*add(f(1-k) for k in (1..1)) = x*f(0) = x f(2) returns the function retfunc(x) = x*add(f(2-k) for k in (1..2)) = x*(f(1)+f(0)) = x*(x+1) f(3) returns the function retfunc(x) = x*add(f(3-k) for k in (1..3)) = x*(f(2)+f(1)+f(0)) = x*(x*(x+1)+x+1)
( 2012-03-03 01:13:56 -0500 )edit |
# Which statistical test to use for comparing two groups?
I have data for 396 transcriptional factors (TF), where I have calculated the number of Transcriptional targets for each transcriptional factor present between the two groups:
Group-I(G-I): where co-exp genes have a correlation coefficient is more than 0.6 and
Group-II(G-II): where the co-exp genes have r values less than -0.6.
My question which statistical test I should use for evaluating the difference between the two groups?
TF G-I G-II
AATF 1 0
ABL1 5 1
AES 0 1
AHR 1 2
AIP 2 0
APBB1 0 1
APEX1 2 1
AR 0 2
ARNT 3 0
• What metric do you wish to compare? Is it the mean r value? What is your hypothesis? – Ram RS Jul 15 at 13:38
• Hi, I have co-expression map for all the human transcription factors. My hypothesis is that transcriptional targets of a transcriptional factor are among the highly co-expressed gene for that transcription factor. For that, I have divided my group into high and low co-exp genes – Riya Jul 15 at 13:44
• abacus.bates.edu/~ganderso/biology/resources/… – Alex Reynolds Jul 15 at 17:22
• Post a histogram or similar of the two groups, it would help to see if they have a particular distribution. – Devon Ryan Jul 16 at 7:18 |
# Physical interpretation of one of Hamilton's equations
1. Oct 11, 2012
### Syrus
1. The problem statement, all variables and given/known data
I've attached a picture from a passage of my book (Liboff, Quantum Mechanics) with which I am having difficulty. Specifically, equation 1.25 claims to possess a certripetal force factor (in the text underneith) and a moment arm factor. I see both of these terms present. However, shouldn't the centripetal force factor reduce to mv2/r? Also, i assume the moment arm is included because torque is formally (clasically) r x F. However, here θ denotes the polar angle, and so rcos(θ) is the z-component of r. As a result, I dont see how this expression amounts to the torque. Can someone clarify this for me?
2. Relevant equations
3. The attempt at a solution
#### Attached Files:
• ###### img003.jpg
File size:
63.4 KB
Views:
58
2. Oct 11, 2012
### Syrus
Okay, I may have made some progress already:
For everything below, phi = σ.
In this case, mv2/r is equal to m(rσ)2/r = mrσ2. Now, r x F = |r| |F| sin(ζ), where ζ is the angle between r and F. But this must mean here that sin(θ)cos(θ) = sin(ζ), and this is (perhaps?) what I don't see.
**Upon second thought, this centripetal force arises from motion in the phi direction at constant r and theta, so the cross product is simply the product of the magnitudes r and F (since the angle between these two vectors is always 90 degrees). Thus, disregard my comments above.
Last edited: Oct 11, 2012
3. Oct 13, 2012
### Syrus
Still looking for help! |
## PDE Seminar Construction of a Weyl Representation from a Weak Weyl Representation of the Canonical Commutation Relation
Date
2008-05-12 16:30 - 2008-05-12 17:30
Place
Faculty of Science Building #8 Room 309
Speaker/Organizer
Yasumichi MATSUZAWA (Hokkaido Univ.)
We consider weak Weyl representations of the canonical commutationrelation with one degree of freedom. We give a formula to constructa new weak Weyl representation from arbitrarily given weak Weylrepresentation. In the last, we discuss relation to the theory oftime operator in quantum mechanics.
http://coe.math.sci.hokudai.ac.jp/sympo/pde/index_en.html |
# VQE jobs do not appear in queue
I am trying to calculate the ground state energy of some simple molecules using 'MolecularGroundStateEnergy' tools in qiskit. When I run the calculations for very small molecules (e.g. H2, HeH+) the jobs appear in the queue very quickly and the calculations work independent of which backend I select. However, running the same code with slightly larger molecules (e.g. LiH, BeH2 and water) the jupyter notebook runs indefinitely and the jobs do not show in the IMBQ queue.
Note: I have run successful calculations for all these molecules using the state-vector simulator with no issues.
Here is the function i use to calculate energy via VQE given a molecule (created from PySCF driver):
def calc_vqe_energy(molecule):
driver = molecule
energy = MolecularGroundStateEnergy(driver = driver,
transformation=TransformationType('full'),
qubit_mapping=QubitMappingType('parity'),
two_qubit_reduction=True,
freeze_core=False,
z2symmetry_reduction='auto')
solver = energy.get_default_solver(quantum_instance)
#calculate energy using the above solver
calc = energy.compute_energy(solver)
return calc
this is set up to run using the quantum instance:
IBMQ.save_account(token)
provider = IBMQ.get_provider(hub='ibm-q')
backend = provider.get_backend("ibmq_qasm_simulator")
coupling_map = backend.configuration().coupling_map
quantum_instance = QuantumInstance(backend=backend, shots=1000,
coupling_map=coupling_map,
measurement_error_mitigation_cls=CompleteMeasFitter,
cals_matrix_refresh_period=30, skip_qobj_validation=False)
Thanks for the help!
• It takes more time for larger molecules. Also there is really no queue for the online simulator unless it is being heavily used. The cloud can spin up as many simulatorS as needed. Jul 20, 2020 at 12:56
• Thank you for your response! It makes sense that it should take longer, however, my notebook has been running for an entire day and nothing has appeared on my pending job list/queue. Surely on the simulator it should at least register the job? Do you also happen to know how long you can run jupyters for on the IBMQ notebooks? Thanks for your help. Jul 20, 2020 at 16:02
• Yes you should see the jobs in the results pane on the left. If not then there is a different issue popping up. Jul 20, 2020 at 17:18
If you want to see your job status or monitor it, you can try to use the method job.status() (see details here) or try the job monitor, you can see if it fails or if it runs with those.
• @Joshua you can do several things: you can access all the jobs from a backend using jobs = provider.backends.jobs(backend_name='ibmq_qasm_simulator', limit=0) but it will give you something quite heavy. A better way to access a job is through the IBM Quantum Experience site. You can access all the job_id of every job you launched, along with their status. Then you can access the job from the backend in qiskit by running jobtest=backend.retrieve_job('job_id'). Try this and tell me if it gives you what you need! :) |
## c++.command-line - compiler output directory
"chris elliott" <chris ampleforth.u-net.com> writes:
Hi
I am starting to try and improve the compilation of wxWindows
is it possible to provide a switch to make or dmc so that it puts the output
in a specified directory?
I'd like either to have a switch for dmc, like the bcc32 -noutputdir
or be able to parse the filename in make
make .... -oadir\$..obj where$. gives you the filename. I guess the first is more effieicnet
chris
currently I have make 5.02, dmc 8.29n
Jun 13 2003
mjs NOSPAM.hannover.sgh-net.de (Mark Junker) writes:
I am starting to try and improve the compilation of wxWindows
I'd like either to have a switch for dmc, like the bcc32 -noutputdir
or be able to parse the filename in make
make .... -oadir\$..obj where$. gives you the filename. I guess the first is more effieicnet
---------------- makefile -------------------
# makefile for test app.
all: $(OUTDIR)test.exe$(OUTDIR)test.exe : test.cpp
# your build command goes here
---------------------------------------------
-------------- commandline ------------------
make -DOUTDIR=test\
---------------------------------------------
Regards,
Mark Junker
Jun 14 2003
james westongold.com writes:
is it possible to provide a switch to make or dmc so that it puts the output
in a specified directory?
dmc does it. Unfortunately I just wasted some time (re)building ACE before
noticing that it doesn't do what you expect.
Unfortunately, unlike every other compiler you've used, '-o foo\bar.obj' DOES
NOT create file foo\bar.obj as the output. Its the optimise flag '-o' followed
by garbage.
Using '-ofoo\bar.obj' does what we know and love, but if your makefiles are
anything like the ACE build system its a pain.
I'm amazed, having started to try dmc I find a lot of the old Zortech stuff is
still there. I understand the history, I'm just surprised that the crufty old
codegen options etc etc didn't die when the driver got a new name.
I'll repoprt back on getting ACE (and maybe TAO) to build eventually. Takes a
while 'cos its a train-time project and -HX doesn't work with STLPort.
:-(
(To be fair, it is of course damn fast even with PCH turned off)
James
Jul 15 2003 |
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• ### $450 Tuition Credit & Official CAT Packs FREE February 15, 2019 February 15, 2019 10:00 PM EST 11:00 PM PST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) • ### Free GMAT practice February 15, 2019 February 15, 2019 10:00 PM EST 11:00 PM PST Instead of wasting 3 months solving 5,000+ random GMAT questions, focus on just the 1,500 you need. # What is the value of |x|? new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 52905 What is the value of |x|? [#permalink] ### Show Tags 10 Nov 2014, 06:38 1 10 00:00 Difficulty: 45% (medium) Question Stats: 65% (01:44) correct 35% (01:50) wrong based on 433 sessions ### HideShow timer Statistics Tough and Tricky questions: Absolute Values. What is the value of |x|? (1) |x^2 + 16| - 5 = 27 (2) x^2 = 8x - 16 Kudos for a correct solution. _________________ Intern Joined: 01 Nov 2014 Posts: 2 Re: What is the value of |x|? [#permalink] ### Show Tags 10 Nov 2014, 07:04 The answer should be option (2) - statement 2 alone is sufficient but statement (1) alone is not sufficient. Here's the solution: (1) Using Statement 1 alone, |x^2 + 16| - 5 = 27 It means that either (x^2 + 16) - 5 = 27 or (x^2 + 16) - 5 = - 27 Solving the first of these, x^2 + 16 = 32 => x^2 = 16 => x = -4 or 4 Solving the second, x^2 + 16 = -22 => x^2 = -6 => x = \sqrt{6}, -\sqrt{6} Using statement 1 alone, we don't get a unique answer. Using Statement 2 alone, x^2 = 8x - 16 => x^2 - 8x + 16 = 0 => x^2 - 4x - 4x + 16 = 0 => (x-4) (x-4) = 0 This gives x = 4 and |x| = 4. So, statement 2 alone is sufficient but statement 2 alone is not sufficient Manager Joined: 10 Sep 2014 Posts: 96 Re: What is the value of |x|? [#permalink] ### Show Tags 10 Nov 2014, 10:21 2 This was my rationale for picking choice D statement 1: sufficient After adding 5 to each side you can set x^2 + 16= -32 and x^2 + 16 = 32 since it has absolute value brackets around it. Thus, x = 4 or x = -4, and the absolute value of each of those values is 4. statement 2: sufficient - I subtracted 8x-16 from x^2 to get x^2 - 8x + 16 = 0 - Factored to get (x-4)(x-4) - x =4 which has absolute value of 4 Manager Joined: 27 May 2014 Posts: 81 Re: What is the value of |x|? [#permalink] ### Show Tags 10 Nov 2014, 11:25 2 D is correct, the absolute value portion of the equation will never be negative as X^2 is always positive Manager Joined: 22 Sep 2012 Posts: 129 Concentration: Strategy, Technology WE: Information Technology (Computer Software) Re: What is the value of |x|? [#permalink] ### Show Tags 10 Nov 2014, 18:12 3 Statement 1: |x^2 + 16| - 5 = 27 |X^2+16| = 32 Since X^2 will always be positive x^2 = 16 therefore |x| = 4. Sufficient Statement 2: x^2 = 8x - 16 (x-4)^2=0 x = 4 , therefore |x|=4. Sufficient D) should be the answer Senior Manager Joined: 12 Aug 2015 Posts: 283 Concentration: General Management, Operations GMAT 1: 640 Q40 V37 GMAT 2: 650 Q43 V36 GMAT 3: 600 Q47 V27 GPA: 3.3 WE: Management Consulting (Consulting) Re: What is the value of |x|? [#permalink] ### Show Tags 23 Aug 2015, 01:01 1 OA: Note that the question is asking for the absolute value of x rather than just the value of x. Keep this in mind when you analyze each statement. (1) SUFFICIENT: Since the value of x2 must be non-negative, the value of (x2 + 16) is always positive, therefore |x2 + 16| can be written x2 +16. Using this information, we can solve for x: |x2 + 16| – 5 = 27 x2 + 16 – 5 = 27 x2 + 11 = 27 x2 = 16 x = 4 or x = -4 Since |-4| = |4| = 4, we know that |x| = 4; this statement is sufficient. (2) SUFFICIENT: x2 = 8x – 16 x2 – 8x + 16 = 0 (x – 4)2 = 0 (x – 4)(x – 4) = 0 x = 4 Therefore, |x| = 4; this statement is sufficient. The correct answer is D. _________________ KUDO me plenty Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6949 GMAT 1: 760 Q51 V42 GPA: 3.82 What is the value of |x|? [#permalink] ### Show Tags 30 Dec 2015, 22:27 Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. What is the value of |x|? (1) |x^2 + 16| – 5 = 27 (2) x^2 = 8x – 16 There is 1 variable (x) in the original condition. In order to match the number of variables and the number of equations, we need 1 equation. Since the condition 1) and 2) both has 1 equation, there is high chance D is going to be the answer. In the case of the condition 1), we can obtain, |x^2+16|=5+27=32. Then, |x^2+16|=-32 or 32. However, since x^2+16=-32 is only possible when x^2=-48, this is not possible. In case of x^2+16=32, x=-4 or 4. However, since |x|=|-4|=|4|=4, the answer is unique and the condition becomes sufficient. In the case of the condition 2), we can obtain x^2-8x+16=0. Then, (x-4)^2=0. So, x=4. Since, |x| can be 4, the answer is unique, and the condition is sufficient. Therefore, the correct answer is D. For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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Re: What is the value of |x|? [#permalink]
### Show Tags
29 Mar 2016, 17:32
Bunuel wrote:
Tough and Tricky questions: Absolute Values.
What is the value of |x|?
(1) |x^2 + 16| - 5 = 27
(2) x^2 = 8x - 16
Kudos for a correct solution.
since we are asked for the absolute value of x, we need to find the value of either x or -x.
1. |x^2+16|=32
ok 2 options:
x^2 +16=32 => x^2 = 16 -> |x|=4
x^2 +16 = -32 -> x^2 = -48 - this is not possible
only one option: |x|=4. sufficient.
2. x^2 -8x+16 = 0
(x-4)(x-4)=0
x=4
|x|=4. sufficient.
D
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Re: What is the value of |x|? [#permalink]
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# Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line
Soonsik Kwon, Tadahiro Oh, Haewon Yoon
Research output: Contribution to journalArticlepeer-review
## Abstract
In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this simplified approach to an infinite iteration of normal form reductions in the context of the cubic nonlinear Schr\"odinger equation (NLS) and the modified KdV equation (mKdV) on the real line and prove unconditional well-posedness in $H^s(R)$ with (i) $s\geq 1/6$ for the cubic NLS and (ii) $s > 1/4$ for the mKdV. Our normal form approach also allows us to construct weak solutions to the cubic NLS in $H^s(R)$, $0 \leq s < 1/6$, and distributional solutions to the mKdV in $H^{1/4}(R)$ (with some uniqueness statements).
Original language English 649-720 Annales de la Faculte des Sciences de Toulouse 29 3 https://doi.org/10.5802/afst.1643 Published - 16 Nov 2020
## Keywords
• nonlinear Schrödinger equation
• modified KdV equation
• normal form reduction
• unconditional well-posedness
• unconditional uniqueness
## Fingerprint
Dive into the research topics of 'Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line'. Together they form a unique fingerprint.
• ### ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
Oh, T.
EU government bodies
1/03/1529/02/20
Project: Research |
Placing Our Trust in Learners
Stage: 1, 2 and 3
In March 2010, the NRICH website had a slightly different feel, inspired by the way teachers at Kingsfield School in Bristol work with their students. Following an introduction to a potentially rich starting point, a considerable proportion of the lesson time at Kingsfield is dedicated to working on questions and ideas generated by children. So, rather than publishing a range of problems on NRICH that month, we offered starting points for 'projects'.
The aim was that these projects would give learners opportunities to work together, discuss ideas, develop conjectures of their own and suggest new lines of enquiry. This is how it is to be a mathematician, working alongside other mathematicians. Children could experience this within their own classrooms, but also in the wider world through the NRICH Projects Site, an online community we set up for this purpose. By placing ideas and thoughts onto the Projects Site, after some exploration, others could share in the excitement and investigate too. It was possible to see where others had gone from the same starting point and it meant that a user could choose to explore other people's ideas further. During the months of March and April, we added problems to the main NRICH website which were based on the questions and conjectures posted by learners.
It was the initial discussions with Alf Coles and Tracy Wylie, both teachers at Kingsfield at the time, which prompted the format of the March NRICH site. Alf, and Tracy, led us - the NRICH team - in some mathematics in the same way they would with their learners in school. Reflecting on what we had done together, we discussed our aim to enable children to experience what it means to work like a mathematician. Part of working in this way involves asking your own questions and pursuing them, rather than always investigating a question or conjecture suggested by someone else. Alf and Tracy's approach embraced this completely and I was struck in particular by Alf's assertion that we, as teachers, should perhaps trust pupils more. By this, I mean that we should trust them to 'play with' a mathematical starting point and be able to ask their own, appropriate, questions. This led me to wonder, do I do this when I work with children? Do I really trust them? Do I really 'let go' of the mathematics and allow children the freedom to pursue their own line of enquiry?
I was reminded of these discussions and contemplations one Saturday as I worked with a group of Year 5 and 6 children in Warwick. This was the first time I had met the majority of the children and we began by playing the Factors and Multiples Game.
To start with, I used a $1$ to $50$ grid on the interactive whiteboard. The group played in two teams against each other, taking it in turns to choose a number on the grid which was either a factor, or a multiple of, the previous number. The aim of the game is to prevent your opponent from being able to select a number. You may like to think about how you would win before reading on!
Quite often when I play this with a group of learners, the first game can go on for a while so I might suggest that we could be here 'for ever', and therefore I encourage children to play on their own grid against a partner instead. On this Saturday, I explained that I was looking for someone to develop a strategy so that they knew they would ALWAYS win the game. We gathered together as a whole group after a short time to discuss what they thought was important so far and this brought up the significance of prime numbers. After a little more playing time, there were many pairs of children who believed they could play against me, and beat me. Having proved this was the case, I encouraged them to articulate their strategy in as general a way as possible.
Here is an example of the beginning of a strategy involving prime numbers (circled in red). In this case though, the player does not always win:
The above idea led to a winning strategy, where both prime numbers chosen are large enough that the player must win:
Choosing the correct primes is the crux of the winning the game, and clearly depends on the size of the grid.
Indeed, one of the lovely things about this game is of course its potential for 'what if...' investigations. I have to admit that I stayed in control of these 'what ifs' myself and focused on the grid size: what if we played on a $1-100$ grid? What if we played on a $1-500$ grid etc. etc ...? This in itself led to some fantastic opportunities for generalising: Imagine that your friend is about to play this game against someone else, but you don't know what the grid is going to look like. How would you advise your friend to play, so that he or she is guaranteed to win the game?
What emerged was a very well-articulated strategy that worked for any size grid. Or so I thought ...
It was at this point that one child in the group raised his hand, "What if we had a $1$ to $10$ grid?" This is one of those moments in the classroom when an on-the-spot decision can make a huge difference to the direction of the lesson. I could have said, and in some ways I'm surprised I didn't, something along the lines of, "Well, yes, we'd use the same strategy, wouldn't we?" and I'd have left it there, with a question that I didn't really expect an answer to, or rather I simply expected a nod. Instead, I said something more along the lines of "Well, let's try it".
So, in pairs, children tried a $1-10$ grid. Try it yourself. Intriguing? We spent a good fifteen minutes exploring this simpler case, and on two occasions one pair thought they had developed a strategy, but when putting it to the test against another pair, it didn't work. I felt such a buzz. We were all, the other adults in the room - me included, were engaged in a challenge that was a genuine challenge for each and every one of us. None of us knew the answer. We didn't even know whether it would be possible to find a strategy. How exciting! We were working like mathematicians.
And the question that had drawn us in was posed by a child of ten, perhaps eleven, years old. I am glad that on that occasion I did trust the pupil, but I shudder when I think how close I was to dismissing his question. I hope it has given me the push I needed to 'let go' a bit more in the future. Do I trust the learners I work with and therefore give them that freedom to ask their own questions in a genuine, meaningful way? Do you? |
As per $IS \: 456:2000$ for $M20$ grade concrete and plain barsin tension the design bond stress $\tau_{bd}=1.2$ MPa. Further, $IS \: 456:2000$ permits this design bond stress value to be increased by $60 \%$ for HSD bars. The stress in the HSD reinforcing steel barsin tension, $\sigma_s=360$ MPa. Find the required development length, $L_d$, for HSD barsin terms of the bar diameter, $\phi$ ________ |
# Would there be any reason to bore a tunnel entirely through a world?
I am looking to have a Tunnel drilled through a world and deal with the physics of What happens in the centre of a planet, what happens with the heat at the core and whether it is faster to travel through a planet. However I'm interested in a reason for doing this?
Obviously with any form of surface transport on any kind of reasonable sized world, it would be impractical to drill a hole through a planet to get to the other side, as surface transport would be easier.
What kind of technological, environmental or chemical environment might force a society to decide to go through rather than around a planet?
• As it stands this is is very broad. Can you try and give as much detail as possible about the problem you're trying to solve? meta.worldbuilding.stackexchange.com/questions/14/… Sep 30 '14 at 10:27
• Yes, could you please add a little detail about the world? Technology, creatures, etc. I'm going to try and answer, but some of my answer may be invalid depending on what you say. Sep 30 '14 at 10:55
• "surface transport would be easier" -- Well, in the Total Recall remake, they decided to have a tunnel from Australia to England, so there's at least one person who things a tunnel would be worthwhile on Earth. Or maybe just "cool" :) Sep 30 '14 at 15:57
• About the why: I remember a sci-fi novel, where in the distant future the human population lived completely underground, because the surface was uninhabitable. They used tunnels almost entirely through the planet (not through the inner core, but through the lowest parts of the mantle and maybe even the outer core), because the conditions on the surface were not much better, and they had the technology to drill and maintain safe tunnels even very deep down, as practically the whole civilization was based on tunneling.
– vsz
Sep 30 '14 at 19:05
• As I'm sure you know, the inner part of a habitable world is highly dynamic. The inner core of Earth spins at a different rate to the rest of the planet. The mantle is turbulent and generates enough force to raise mountains and shift continents. A completely solid planet without an active core would likely lack a geomagnetic field and would have no protection from solar radiation. Oct 23 '14 at 10:33
How they would work
The classic design for the tunnels would be a tube with a near-vacuum inside. Objects are dropped into the tube and they fall through to emerge at the other side. The tunnel itself would need to curve so the object can fall matching the spin of the planet as it does so. You would also need something (perhaps magnetic systems built into the tunnel walls) to keep the speed of the object up as even in a near-vacuum it would lose some speed, the same systems could be used to keep the object away from the walls of the tunnel though as otherwise drift could bring it into contact.
The tunnel wouldn't need to go directly through the center of the planet and out at the same time, you can make the tunnel go to one side of the core and then bend around it to come out at any other point on the surface, like an orbit around the center of the planet although gravity gets weaker the deeper you go which makes the orbital math more interesting. For nearby destinations that would not be practicable but for anything a long way away it would be pretty fast and efficient once the tunnel is built.
The main problems are heat and pressure, particularly as you approach the core. Keeping out the high pressure high temperature magma would be a massive engineering challenge. Vessels passing through would actually not be too badly effected though as vacuum is a good insulator. This would be made much easier in a smaller world without a molten core.
Why they would be built
The main advantages of going through the planet are speed and security. For example a planet under siege or in a state of cold war might make such tunnels in order to transport resources rapidly to wherever they are needed.
The initial construction of the tunnels would be a huge effort but once they are built it would be very fast and efficient so ecologically sensitive civilizations may actually prefer it to air travel.
A common concept with warp drives is that they only function in a low enough gravitational gradient. There is no science that I'm aware of backing up that theory but it's a common literary device. Drives working that way normally means the ships need to be a long way out in space to jump. The center of the planet though also has no gravity gradient so the hypothetical warp drives would function there.
• You mention one of the challenges being magma; that assumes a planet with a fluid core (i.e. core of the moon is mostly solid). That said, at those pressures rock may still flow. Oct 3 '14 at 21:55
• One other advantage is that, barring the needed corrections to keep within the tunnel, and assuming equal heights on both ends; the energy cost of sending materials to the other side of the planet is basically 0. Oct 3 '14 at 21:58
I have to chime in here on a reason for the tunnel that no one has seemed to touch on. The most likely, and I think interesting reason to bore all the way through a planet is pure hubris. Why do we feel compelled to build the tallest building? Why do we care about record flights across bodies of water, or circumnavigating the world in a sail boat? The purest reason to make a tunnel through a planet is because you could.
This reason might be seen as a sort of cop-out, but the reality is fairly well documented and even better known in a metaphorical context. The story of icarus, the tower of babel, etc.
For world building this sort of idea could be really rich. Maybe an unknown ancient group made it, maybe it was made by "gods" or the like depending on your setting.
I think everyone has covered the how, but narrative-wise remember that it doesn't necessarily have to be stable, or well-engineered. If done well your tunnel could be believable without anyone even understanding its background or history, or the physics of the thing. It could be an imposing, mysterious character on its own.
On a small enough planet, where such a tunnel is feasible, dropping an object through to the other side (or via many possible gravity-assisted routes) has the same delivery time as a near-surface ballistic orbit, with less work required to accelerate and decelerate because the planet's mass does that for you. The tunnel would be need to be held at a near vacuum for this to work effectively.
If there was a constant requirement to deliver material one side to another, this might offset the rather large cost of building such a tunnel.
I'd like to propose a different cause for the tunnel. You mention being forced to go through it, but is there really a reason for them to be forced to make it before they're forced to use it?
A tunnel to the core, reaching the other side can be dug out for mining (impractical but it might be hitting two birds with one stone), exploitation of heat for energy generation, technological and scientific experimentation, cheap housing and storage (energy generation and heating are close by, transporting good is cheap).
It could also allow the easy manufacturing of various minerals and metals, by exploiting the extreme conditions through specialized scaffolding. Instead of creating artificial diamonds through explosions, you create pockets within the highly pressurized mantle and bake them there - just like how mom used to make them at home :P.
Magma taps could provide exotic materials or allow using the magma to alloy with common materials for specialized uses. The lower gravity, combined with controlled near-vacuum conditions (for the constructed route, as the rest of the others have proposed), provide tremendous opportunities for construction, experimentation and energy saving. You can build a huge construct, like a spacecraft or space station, out of heavy materials that are close by, then break it up modularly and transport it easily to the surface, or launch it through rails on the side of the tube (considering the depth, there would be enough time to slowly build up speed to avoid the massive acceleration otherwise required to catapult spacecraft into orbit).
And now, suppose everything goes wrong and the underground tunnel is mankind's only hope for survival. There it is, already furnished and ready for use, with housing available due to tourism and other market causes. Now they're forced to go underground and the tunnel is very efficient, necessary even - but they didn't build it out of necessity, which would make less sense no matter the overland threats and dangers. The planet is very, very, very deep - too deep to ever drill down that far just for cheap travel.
• The Part with the Magma taps remindet me on a documentytion where it is mentioned that beneath 100 km + there are dimonds about the sice of scyscrapers. That could be used FOR SCIENCE! Oct 1 '14 at 7:07
• @Fulli This could attract a lot of Dwarf Fortress players xD Oct 1 '14 at 9:56
Would there be any reason to bore a tunnel entirely through a world?
I can think of one reason and it's the same as why the chicken crossed the road - to get to the other side. ;)
So, why then would someone want to use a hole instead of surface transport like you suggest?
1) It's cheaper
2) It's more reliable
3) It's more 'eco-friendly'
4) It's faster
I could go on.
Before going further we need to clarify some apparent assumptions.
You seem to be comparing "a world" with our world. The primary reason I suggest this is because you suggest there is heat at the core - not all planets have hot cores.
...surface transport would be easier.
What kind of technological, environmental or chemical environment might force
a society to decide to go through rather than around a planet?
What if "a world" had a mountain range around the equator which was too high to get over easily (even with an airplane)?
What if "a world" was nothing but a mucky swamp, or a sandy desert, or an accordion of mountains that never ended? Again, assuming no airplanes. One does not need as much technology to dig a hole if a hot core is not an issue.
And those are just a few geological hazards. We could also talk about wind patterns that made the equator a 'hot zone' from natural or man made radiation. What about a group of people, animals, insects, ... that make it nearly impossible (or at least extremely hard) to traverse from one side of the globe to the other.
There are many scenarios one can imagine that makes it hard to get from one side of a world to another.
Lastly, as other have noted, gravity is the main force providing the 'work' to get from one side of the planet to the other so once the hole was 'built' transportation costs could be minimal. If friction-less the object would emerge to the same height above the ground as it was dropped. Since a friction-less system of any kind is very difficult one could easily use electromagnets, tethers or other simple means to bring the payload whatever extra distance to the surface was required due to the losses from friction.
There are always reasons.
• If a civilization how a highly effective creature to dig tunnels, but no good system of ground transportation, (No extra fast creatures or flying creatures) it could actually be an effective method of transportation. And a strong tunneling creature would make the cost reasonable. Of course, depending on the planets size, supporting the tunnel in the center may not be possible.
• Another option similar to the one above would be if the above ground world was unsafe. If going above ground for even a short time was highly dangerous, people might have to go through the earth to easily travel. If it's safe to go through the center of the planet, they might try to do so.
• The tunnel could be an elaborate secret passage. If someone country's worse enemy's capital was on the opposite side of the world, the best way to secretly get there would be to go through the world. Other paths would give the enemy too much time to prepare.
• Neil Slater's idea is also good. (Trying to be inclusive without copying)
Another reason to make such a tunnel: They are sending probes to nearby stars. The probes are boosted at extreme accelerations via a linear motor (or perhaps some technology we don't know) and a tunnel through the planet is the longest possible motor and thus the highest ejection velocity.
The tunnel could hold Ethernet cables to reduce latency for communication to the other side of the planet by pi/2.
That assumes the light speed couldn't be worked around despite massively advenced technology.
That shifts the problem to "why would low latency be so important"...
EDIT: @ Toby's quip about High-Frequency Trading actually points to a semi-plausible scenario: a rich Corporation building it to gain an edge in stock trading. See real-world analogy
Or there could be a permanent, highly automated war in which faster coordination can yield a big advantage.
Better yet: all inhabitants spend their days in Virtual Reality, some sort of MMORPG on steroids, and lags are inconvenient when interacting with someone half a world away. Resources are plentiful enough to make it worth it.
• HFT will even try to scam alien worlds! Oct 23 '14 at 21:29
• On an earth size planet, I would rather communicate with neutrino than make such a tunnel :D Feb 25 '16 at 19:30
A good example of the above ground world being unsafe (as suggested by DonyorM) would be to place it very close to a star. Lethal surface temperatures combined with extreme radiation and no atmosphere would make it necessary to have everything underground, at which point radial tunnels look quite attractive.
Another option would be a small planet with no atmosphere but a very high population, where the only remaining space for a new mass transit project is through the middle. There could be just a very high demand for sunlight and surface space (e.g. for farming)
The key bit to managing the core temperatures and pressures is not really the size of the planet, but the density.
It would also help if you were on a planet that rotated relatively slowly, since your radial transport vehicle will start the journey travelling at surface rotation speed and you will need a large part of your energy budget to slow that down as you descend and speed up on the other side (or risk hitting the wall of the tunnel).
• You should be able to curve the tunnel to allow for the effects of radial velocity and still drop the cargo in free-fall for the entire trip without needing anything more than basic stabilization to keep it centered in the tunnel.. Sep 30 '14 at 14:39
• @TimB That might work but I don't know if it'll really be necessary. How curved would it really have to be, considering conservation of angular momentum? Also, why wouldn't rails work much better than curving? Sep 30 '14 at 21:07
• With free-fall through vacuum you lose very little energy and can attain extremely high speed. If you place the capsule on rails then that will both slow down the maximum speed and require you to input more energy to keep it moving. Sep 30 '14 at 22:47
• @TimB - a subterranean orbital. Awesome! Oct 23 '14 at 10:36
Because, for whatever reason, this civilization is forbiden to reach orbit and going at the center is the only way to reach zero G
Well... parabolic flight is another way to get zero G but come on.... let's make a tunnel |
Browse Questions
# Three uniform spheres of mass M and radius R each are kept in such a way that each touches the other two. The magnitude of the gravitational force on any of the sphere due to the other two sphere is
$(a)\;\frac {\sqrt 3}{4}\frac{GM^2}{R^2} \quad (b)\;\frac{3}{2}\frac{GM^2}{R^2} \quad (c)\;\sqrt 3 \frac{GM^2}{R^2} \quad (d)\;\frac{\sqrt 3}{2} \frac{GM^2}{R^2}$
Force between two spheres will be
$F= \large\frac{GMM}{(2R)^2}=\frac{GM^2}{4R^2}$
Two forces of equal magnitude are acting at angle $60^{\circ}$ on any of the sphere
$F_{net}=\large\frac{\sqrt 3}{4} \frac{GM^2}{R^2}$
Hence a is the correct answer.
edited Feb 17, 2014 by meena.p |
## CCC '16 S4 - Combining Riceballs
View as PDF
Points: 15 (partial)
Time limit: 3.0s
Memory limit: 128M
Author:
Problem type
Alphonse has rice balls of various sizes in a row. He wants to form the largest rice ball possible for his friend to eat. Alphonse can perform the following operations:
• If two adjacent rice balls have the same size, Alphonse can combine them to make a new rice ball. The new rice ball's size is the sum of the two old rice balls' sizes. It occupies the position in the row previously occupied by the two old rice balls.
• If two rice balls have the same size, and there is exactly one rice ball between them, Alphonse can combine all three rice balls to make a new rice ball. (The middle rice ball does not need to have the same size as the other two.) The new rice ball's size is the sum of the three old rice balls' sizes. It occupies the position in the row previously occupied by the three old rice balls.
Alphonse can perform each operation as many times as he wants.
Determine the size of the largest rice ball in the row after performing 0 or more operations.
#### Input Specification
The first line will contain the integer, ().
The next line will contain space separated integers representing the sizes of the riceballs, in order from left to right. Each integer is at least and at most .
• For 1 of the 15 available marks, .
• For an additional 2 of the 15 available marks, .
• For an additional 2 of the 15 available marks, .
#### Output Specification
Output the size of the largest riceball Alphonse can form.
#### Sample Input 1
7
47 12 12 3 9 9 3
#### Output for Sample Input 1
48
#### Explanation for Sample Input 1
One possible set of moves to create a riceball of size 48 is to combine 12 and 12, forming a riceball of size 24. Then, combine 9 and 9 to form a riceball of size 18. Then, combine 3, 18 and 3 to form a riceball of size 24. Finally, combine the two riceballs of size 24 to form a riceball of size 48.
#### Sample Input 2
4
1 2 3 1
#### Output for Sample Input 2
3
#### Explanation for Output for Sample Input 2
There are no moves to make, thus the largest riceball in the row is size 3.
• Dordor1218 commented on Feb. 3, 2019, 7:18 p.m.
Could anyone check what i did wrong? I can't seem to find anything wrong with it. thx
• aurpine commented on Feb. 20, 2016, 10:54 p.m.
Why? What am I doing wrong? Is anyone else getting this?
• awaykened commented on Feb. 20, 2016, 11:32 p.m.
your solution has a complexity of , which is theoretically too slow...
that being said, there are other AC solutions, you just need to optimize a bit more
• Zander commented on Feb. 20, 2016, 7:25 p.m.
Could someone give me a hint on how to store states?
• kobortor commented on Feb. 20, 2016, 7:36 p.m.
Use a 2d array of integers
• Zander commented on Feb. 20, 2016, 9:05 p.m.
thank you :) |
College Math Teaching
April 5, 2018
A talk at University of South Alabama
Filed under: advanced mathematics, knot theory, topology — Tags: — collegemathteaching @ 3:27 pm
My slides (in order, more or less), can be found here.
August 11, 2016
Post Promotion Summer
Filed under: editorial, topology — Tags: — collegemathteaching @ 12:02 am
This is my first “terminal promotion” summer. And while I have something that I have “sort of” written up…I just don’t like the result; it basically fills in some gaps in a survey article. But I think that my thinking about this article has lead me to something that I can add to the paper so that I’ll actually LIKE what I submit.
Then again, my quandary can be summed up in this tweet:
If I wait until I am absolutely in love with my work before I send it out, it will never get sent out.
Hopefully, I’ll have more material to add to this blog this semester.
What I am working on: equivalence classes of simple closed curves; these are one to one, continuous images of the unit circle in 3-space. The objects that I am studying are so pathological that these curves fail to have a tangent at ANY point. One of these beasts can be constructed by taking the intersection of these nested, solid tori.
June 7, 2016
Pop-math: getting it wrong but being close enough to give the public a feel for it
Space filling curves: for now, we’ll just work on continuous functions $f: [0,1] \rightarrow [0,1] \times [0,1] \subset R^2$.
A curve is typically defined as a continuous function $f: [0,1] \rightarrow M$ where $M$ is, say, a manifold (a 2’nd countable metric space which has neighborhoods either locally homeomorphic to $R^k$ or $R^{k-1})$. Note: though we often think of smooth or piecewise linear curves, we don’t have to do so. Also, we can allow for self-intersections.
However, if we don’t put restrictions such as these, weird things can happen. It can be shown (and the video suggests a construction, which is correct) that there exists a continuous, ONTO function $f: [0,1] \rightarrow [0,1] \times [0,1]$; such a gadget is called a space filling curve.
It follows from elementary topology that such an $f$ cannot be one to one, because if it were, because the domain is compact, $f$ would have to be a homeomorphism. But the respective spaces are not homeomorphic. For example: the closed interval is disconnected by the removal of any non-end point, whereas the closed square has no such separating point.
Therefore, if $f$ is a space filling curve, the inverse image of a points is actually an infinite number of points; the inverse (as a function) cannot be defined.
And THAT is where this article and video goes off of the rails, though, practically speaking, one can approximate the space filling curve as close as one pleases by an embedded curve (one that IS one to one) and therefore snake the curve through any desired number of points (pixels?).
So, enjoy the video which I got from here (and yes, the text of this post has the aforementioned error)
May 20, 2016
Student integral tricks…
Ok, classes ended last week and my brain is way out of math shape. Right now I am contemplating how to show that the complements of this object
and of the complement of the object depicted in figure 3, are NOT homeomorphic.
I can do this in this very specific case; I am interested in seeing what happens if the “tangle pattern” is changed. Are the complements of these two related objects *always* topologically different? I am reasonably sure yes, but my brain is rebelling at doing the hard work to nail it down.
Anyhow, finals are graded and I am usually treated to one unusual student trick. Here is one for the semester:
$\int x^2 \sqrt{x+1} dx =$
Now I was hoping that they would say $u = x +1 \rightarrow u-1 = x \rightarrow x^2 = u^2-2u+1$ at which case the integral is translated to: $\int u^{\frac{5}{2}} - 2u^{\frac{3}{2}} + u^{\frac{1}{2}} du$ which is easy to do.
Now those wanting to do it a more difficult (but still sort of standard) way could do two repetitions of integration by parts with the first set up being $x^2 = u, \sqrt{x+1}dx =dv \rightarrow du = 2xdx, v = \frac{2}{3} (x+1)^{\frac{3}{2}}$ and that works just fine.
But I did see this: $x =tan^2(u), dx = 2tan(u)sec^2(u)du, x+1 = tan^2(x)+1 = sec^2(u)$ (ok, there are some domain issues here but never mind that) and we end up with the transformed integral: $2\int tan^5(u)sec^3(u) du$ which can be transformed to $2\int (sec^6(u) - 2 sec^4(u) + sec^2(u)) tan(u)sec(u) du$ by elementary trig identities.
And yes, that leads to an answer of $\frac{2}{7}sec^7(u) +\frac{4}{5}sec^5(u) + \frac{2}{3}sec^3(u) + C$ which, upon using the triangle
Gives you an answer that is exactly in the same form as the desired “rationalization substitution” answer. Yeah, I gave full credit despite the “domain issues” (in the original integral, it is possible for $x \in (-1,0]$ ).
What can I say?
January 27, 2016
A popular video and covering spaces…
Filed under: media, popular mathematics, topology — Tags: , , , , — collegemathteaching @ 11:16 pm
Think back to how you introduced the sine and cosine functions on the real line. Ok, you didn’t do it quite this way, but what you did, in effect, is to define $sin(u) = Im(e^{iu})$ and $cos(u) = Re(e^{iu})$ and then use “elementary trigonometry” to relate the “angle” $u$ to the arc length subtended on the circle $|z| = 1$. One notes that the map $\rho: R^1 \rightarrow C^1$ defined by $\rho(u) = e^{iu}$ has period $2\pi$
Note: the direction “to the right” on the real line is taken to be “counterclockwise” on the circle (red arrows).
Skip if you haven’t had a topology class
The top line is known as the “universal covering space” for the circle. The reason for the terminology has to do with topology. Depending on how long ago you had your topology course, you might remember that the fundamental group of the real line is trivial and the associated group of deck transformations is infinite cyclic (generated by the map $d(u) = u + 2\pi$ ). One then shows that the fundamental group of the circle is the quotient of the group of deck transformations with the fundamental group of the real line; hence the fundamental group of the circle is infinite cylic.
Resume if you haven’t had a topology class
Notice the following: if one, say, “takes a walk” along the line in the direction of the red arrow, the action of the “covering mapping” is to take the same walk in the counter clockwise direction of the circle. That is, the covering action does the following: a walk on the line in the direction of points $A_1, B_1, C_1, A_2, B_2, C_2....$ corresponds to a walk on the circle $A, B, C, A, B, C....$. That is, walking from $A_1$ to $A_2$ corresponds to a complete lap of the circle.
(that is, on the real line, $A_{n+1} = A_{n} + 2\pi$)
Now note the following: for BOTH the line and the circle, the direction is well defined. “To the right” on the real line” is “counter clockwise” on the circle.
However: on the real line, it makes perfect sense to say that $A_1$ is “before” $B_1$ which is “before” $C_1$ which is “before” $A_2$ and so on; this is merely:
$A_1 < B_1 < C_1 < A_2 < B_2 ...$. This is order is valid no matter where one starts on the line.
However, this “universal ordering” makes no sense on the circle, UNLESS one specifies a start point. True, one moves from $A$ to $B$ to $C$ and back to $A$ again..but if one started at $B$ and started to walk, it would appear that $A$ came AFTER $B$ and not before.
So what?
This quirky animation from CraveFX starts off innocently enough, a janitorial worker mops up a leaky refrigerator and then picks up a coin on the ground. It’s not until you see what causes the refrigerator to leak and why the coin is on the ground that you realize that you’re watching an intricate moving puzzle piece before your eyes. The characters are stuck in an infinite loop caused by another character in their own infinite loop. It’s chaotic and great and hard to keep up with.
The video is below. Now the question: “what action occurred before what other action”? and the answer is “it depends on when you started watching”. The direction of time corresponds to the red arrows in the above diagrams; THAT is well defined. Why? The reason is the Second Law of Thermodynamics; spills do NOT reverse themselves, hence the direction is set in stone, so to speak. But as far as order, it depends ON WHEN THE VIEWER STARTED WATCHING.
Anyway, this video reminded me of covering spaces.
October 29, 2015
A quick break from the routine…
Filed under: editorial, topology — Tags: — collegemathteaching @ 9:23 pm
But this semester, I’ve been up to my eyeballs in this new (to me) course. If I never see actuarial mathematics again, it will be too soon. 🙂
July 1, 2015
Embarrassing gaps in my mathematical knowledge
Filed under: mathematician, topology — Tags: , — collegemathteaching @ 1:56 pm
Yes, mathematics is a huge, huge subject and no one knows everything. And, when I was a graduate student, I could only focus on 1-2 advanced courses at a time, and when I was working on my thesis, I almost had a “blinders on” approach to finishing that thing up. I think that I had to do that, given my intellectual limitations.
So, even in “my area”, my knowledge outside of a very narrow area was weak at best.
Add to this: 20+ years of teaching 3 courses per semester; I’ve even forgotten some of what I once knew well, though in return, I’ve picked up elementary knowledge in disciplines that I didn’t know before.
But, I have many gaps in my own “area”. One of these is in the area of hyperbolic geometry and the geometry of knot complements (think of this way: take a smooth simple closed curve in $R^3$, add a point at infinity to get $S^3$ (a compact space), now take a solid torus product neighborhood of the knot (“thicken” the knot up into a sort of “rope”) then remove this “rope” from $S^3$. What you have left over is a “knot complement” manifold.
Now these knot complements fall into one of 3 different types: they are torus knot complements (the knot can live on the “skin” of a torus),
satellite knot complements (the knot can live inside the solid torus that is the product neighborhood of a different, mathematically inequivalent knot,
or the knot complement is “hyperbolic”; it can be given a hyperbolic structure. At least for “most” knots of small “crossing number” (roughly: how many crossings the knot diagram has), are hyperbolic knots.
So it turns out that the complement of such knots can be filled with “horoballs”; roughly speaking, these are the interior of spheres which are “tangent to infinity”; infinity is the “missing stuff” that was removed when the knot was removed from $S^3$. And, I really never understood what was going on at all.
I suppose that one can view the boundary of these balls (called “horospheres”) as one would view, say, the level planes $z = k$ in $R^3$; those planes become spheres when the point at infinity is added. This is a horoball packing of the complement of the figure 8 knot; missing is the horosphere at $z = 1$ which can be thought of as a plane.
But the internet is a wonderful thing, and I found a lecture based on the work of Anastasiia Tsvietkova and Morwen Thistlethwaite (who generated the horoball packing photo above) and I’ll be trying to wrap my head around this.
June 25, 2015
Workshop in Geometric Topology: TCU 2015 morning session 1
Filed under: advanced mathematics, conference, editorial, topology — Tags: — collegemathteaching @ 3:59 pm
I’ll be blunt: I’ve been teaching at a 11-12 hour load (mostly 11; one time I had a 9 hour load; 3 courses) since fall, 1991. Though I’ve published, most of what I’ve done has been extremely “bare handed”; it is tough to learn the most advanced techniques (which is a full time job in and of itself)
So, at math conferences, I get to see how much further behind I’ve fallen.
But these things help in the following way:
1. They are an excellent change of pace from the usual routine of teaching calculus.
2. I do learn things, even if it is “looking up” a definition or two; for example I looked up the definition of “pure braid group” in between the 20 minute talks.
3. I have to review my own stuff to see if I am indeed making progress; I don’t want to say something idiotic in front of some very smart, informed people.
But yes, the talks have been given by smart, (mostly) young, energetic people who have been studying the topic that they are talking about very intensely for a long time; frequently it is tough to hang in to the second half of the 20 minute talks. But I can see WHAT is being studied, what tools are being used and, as I said before, find stuff to look up.
The final talk: didn’t understand much beyond the general gist but it was well organized, well presented..exactly what you get when you have a brilliant energetic young researcher working full time in mathematical research.
On one hand, I envy his talent. On the other hand, I am glad that we have some smart humans among us; they benefit all of us.
The trip here The plane was about 2.5 hours late getting in, then there was a long ride to the car rental place and a 35 minute drive to campus, then finding my way around in the dark. So no morning run; I might do a gentle “after the talks” focused walk (5K-ish?).
I talk at 9 am tomorrow and I want to make it worth their while.
May 31, 2015
And a Fields Medalist makes me feel better
Filed under: calculus, editorial, elementary mathematics, popular mathematics, topology — Tags: — collegemathteaching @ 10:30 pm
I have subscribed to Terence Tao’s blog.
$\frac{d^{k+1}}{dx^{k+1}}(1+x^2)^{\frac{k}{2}}$ for $k \in \{1, 2, 3, ... \}$. Try this yourself and surf to his post to see the “slick, 3 line proof”.
But that really isn’t the point of this post.
This is the point: I often delight in finding something “fun” and “new to me” about an established area. I thought “well, that is because I am too dumb to do the really hard stuff.” (Yes, I’ve published, but my results are not Annals of Mathematics caliber stuff. 🙂 )
But I see that even the smartest, most accomplished among us can delight in the fun, simple things.
That makes me feel better.
Side note: I haven’t published much on this blog lately, mostly because I’ve been busy updating this one. It is a blog giving notes for my undergraduate topology class. That class was time consuming, but I had the teaching time of my life. I hope that my students enjoyed it too.
March 17, 2015
Compact Spaces and the Tychonoff Theorem IV: conclusion
Filed under: topology — Tags: , , — collegemathteaching @ 9:14 pm
We are finishing up a discussion of the Tychonoff Theorem: an arbitrary product of compact spaces is compact (in the product topology, of course). The genesis of this discussion comes from this David Wright article.
In the first post in this series, we gave an introduction to “compactness”.
In the second post, we gave a proof that the finite product of compact spaces is compact.
In the third post, we gave come equivalent definitions of compactness
In particular, we showed that:
1. A space is compact if and only if the space has the following property: if $A \subset X$ is an infinite union of open sets with no finite subcover, then $A$ is a proper subset of $X$; that is, $X -A \neq \emptyset$ and
2. A space is compact if and only if the space has the following property: every infinite subset $E$ has a perfect limit point. Note: a perfect limit point for a set $E$ is a point $x \in X$ such that, for every open $U, x \in U$, $|U \cap E| = |E|$ (the intersection of every open neighborhood of a perfect limit point with $E$ has the same cardinality as $E$.
Note the following about these two facts: each of these facts promises the existence of a specific point rather than the existence/non-existence of a cover of a particular type. Fact 1 promises the existence of an excluded point, and fact 2 promises the existence of a perfect limit point.
When it comes to a point in an infinite of topological spaces, constructing a point is really like constructing a sequence of points (in the case of countable products) or a net of points (in the case of uncountable products). That is, if one wants to construct a point in an infinite product of spaces, one can assume some well ordering of the index used in the product, then construct the first coordinate of the point from the first factor space, the second coordinate from the second factor space, and so on.
We’ll use fact one: the excluded point property to prove Tychonoff’s Theorem.
Proof. Assume that $X = \Pi_{\alpha \in I} X_{\alpha}$ and that $I$ is well ordered. We start out by showing that the product of two compact spaces is compact, and use recursion to get the general result.
Let $\mathscr{O}$ be an infinite union of open sets in $X_1 \times X_2$ with no finite subcover. First, we show that there is some $a \in X_1$ such that for each open set $U, a \in U$, no finite subcollection of $\mathscr{O}$ covers $U \times X_2$. Now if there is some open $U \subset X_1$ where $U$ is disjoint from every $\pi_1 (O), O \in \mathscr{O}$ we are done with this step. So assume not; assume that every $U$ is a subset of the first factor of some $O \in \mathscr{O}$. If it isn’t the case that there is $x \in X_1$ where $U_x \times X_2$ has no finite subcover of elements of $\mathscr{O}$, for each such $U_x \subset X_1, x \in X_1$ there is a finite number of elements of $\mathscr{O}$ that covers $U_x \times X_2$. Now since $X_1$ is compact, a finite number of $U_x$ covers $X_1$, hence a finite subcover of $\mathscr{O}$ covers ALL of $X_1 \times X_2$. Hence some point $a \in X_1$ exists such that no finite subcover of $\mathscr{O}$ covers $U \times X_2$ for any open $U \subset X_1, a \in U$.
Similarly, we can find $b \in X_2$ so that for all open $V \subset X_2, b \in V$, no finite subcollection of $\mathscr{O}$ covers $U \times V$ where $U$ is a basic open set in $X_1$ that contains $a$. This shows that $(a,b) \notin \cup_{O \in \mathscr{O}}$ because, if it were, this single point would lie in some basic open set $U \times V$ which, by definition, is a finite subcover.
Now given an arbitrary product with a well ordered index set $I$ we can now assume that there is some collection of open sets that lacks a finite subcover and inductively define $a_{\gamma} \in X_{\gamma}$ so that, if $U$ is any basic open set containing $\Pi_{\alpha \leq \gamma} \{a_{\alpha} \} \times \Pi_{\alpha > \gamma} X_{\alpha}$ then no finite subcollection of $\mathscr{O}$ covers $U$. The point $(a_{\gamma})$ thus constructed lies in no $\mathscr{O}$.
Note: if you are wondering why this “works”, note that we assumed NOTHING about the compactness of the remaining product space factors $\Pi_{\alpha > \gamma} X_{\alpha}$.
And remember that we are using the product topology: an open set in this topology has the entire space as factors for all but a finite number of indices. So we only exploit the compactness of the leading factors.
Older Posts » |
# Multiplication Made Easy
Created by
barang
Includes
8 Strategies for Multiplication Powerpoints = for multiplying by x2/x3/x4/x5/x8/x9/x0/&x1
Using Place Value to Multiply with Arrays PowerPoint
Doubling and Halving Strategy to Multiply PowerPoint
Also added a cool multiplication wheels powerpoint
and some editable multiplication certificates.
Some of my best powerpoints. I have found them extremely useful and clear to use for grades 3 and 4.
\$10.36
\$10.90);
(5% off)
Save for later |
# Problem in trying to solve a differential equation [closed]
I'm trying to solve a differential equation but not getting a solution.
Some references I read gave me hits about using a runge kutta method. I'm new to the software, any hints regarding this error or how to solve?
• The error message is telling you didn't give any initial conditions in your NDSolve expression. – m_goldberg Jan 24 '16 at 9:03
The $\theta(t-\Delta)$ makes it a Delay Differential Equation, which the documentation here shows how to solve. You have to specify a history function like this:
u[t /; t <= 0] == f[t]
NDSolve[{u''[t] + u[t - 1] == 0, |
# Where Should My Writing Ideas Go?
edited June 2020
Where Should My Writing Ideas Go?
A Zettelkasten is a personal tool for thinking and writing that creates an interconnected web of thought. Its emphasis is on connection and not mere collection of ideas.
• edited June 2020
they are files in the same Zettel note archive without themselves having an ID
Why do the outline notes not get IDs?
• edited June 2020
@pat They used not to, so they stay afloat in a directory listing. This post predates me using structure notes for smaller hierarchies: my line of reasoning was that these outlines are work-in-progress project files, and they have nothing to do with the """real""" Zettelkasten work. For example, I will hardly have a reason to ever link to a book project outline.
Nowadays, I think that distinction doesn't hold up. I can make an overview of book project outlines, and then use this overview of book projects to navigate around. So there's definitely benefit in assigning these things Ids as well.
Some old - O notes are still there, but I'm transforming one after another to a structure note with an ID as I encounter them. (The boy scout principle of leaving your ZK in a tidier state than you found it -- bulk renaming hardly ever payed off in my past ) I updated the post accordingly.
Author at Zettelkasten.de • https://christiantietze.de/
• edited June 2020
@ctietze Thank you for the clarification. I was very surprised, because I find the ID principle so useful that I didn’t understand why you would choose to forego it.
I do understand your reasoning at the time, and appreciate you sharing how your thinking has evolved. |
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# By examining the chest X ray,the probability that TB is detected when a person is actually suffering is 0.99.The probability of an healthy person diagnosed to have TB Is 0.001.In a certain city,1 in 1000 people suffers from TB.A person is selected at random and is diagnosed to have TB.What is the probability that he actually has TB?
\$\begin{array}{1 1}(A)\;\large\frac{110}{221}\$$B)\;\large\frac{120}{221}\\(C)\;\large\frac{130}{221}\\(D)\;\large\frac{112}{221}\end{array} ## 1 Answer Comment A) Need homework help? Click here. Toolbox: • \(E_1=A\;person\;actully\;has\;T\:B$$
• $$E_2=A\;person\;actully\;has\;T\:B$$
• $$A\; person\;actuley\;has\;T\;B\;/he\;is\;diagonised\;as\;having\;T\;B$$
• P(E$$_2$$/A)=$$\Large\frac{{P(E_2)}{P(A/E_2)}}{{P(E_1)}{P(A/E_1)}+{P(E_2)}{P(A/E_2)}}$$
•
•
•
$$P(A/E_2$$)=$$P(diagonised\;as\;having\;T\;B\;/actully\;has\;T\:B$$)
=$$0.99$$
$$P(A/E_2$$)=$$P(diagonised\;as\;having\;T\;B\;/has\;not\;have\;T\:B$$)
=$$0.001$$
$$P(E_1$$)=$$P(one\; in\; thousand \;having\; T\;B$$)
=$$\frac{1}{1000}$$
$$p(E_2)=1-\Large\frac{1}{1000}=\frac{999}{1000}$$
P(E$$_1$$/A)
$$\Large\frac{\frac{1}{1000}\times0.99}{\frac{1}{1000}\times0,99+\frac{999}{1000}\times0.001}$$
=$$\Large\frac{.999}{1.989}$$
=$$\Large\frac{110}{221}$$ |
# The Communicative Approach
## The Communicative Approach
The communicative approach is the recent and latest approach of teaching English. It enables the students to communicate his ideas in a better way. The socio-linguists Dell Hymes propagates this approach.
Trim, David and Henny has developed this approach as National Functionalism and the communicative approach.
### Characteristics of Communicative Approach of English teaching
1. The communicative approach is based upon need analysis and planning to prepare communicative curriculum and syllabus.
2. It is based upon the concept of how language is used and what is functional utility of language.
3. It lays less stress on grammar.
4. It is based upon the concept of how language is used and what is functional utility of language.
5. It lays emphasis on language in use rather than language as structure.
6. It gives emphasis on the semantic objective of the language which means the meaning of language in real life situations and contexts.
7. The skills of speaking and writing are included in communicative approach.
8. It provides the communicative opportunities where the students may be able to communicate their ideas through dialogue, discussions,and debate literary and cultural activities of the school.
### Merits of communicative Approach-
• It develops the speech ability among the students.
• It teaches different ways of expression.
• This approach is based on the practical utility.
• It lays more stress on the functional value of the language.
• It enables the students to communicate their ideas both inside and outside the class room
### Demerits of communicative Approach
• This approach ignores grammar and structures.
• It is not properly and scientifically developed .
• It is a new approach and it is to be used and tested in our schools for language teaching.
• Practical utility of this approach is yet to be confirmed.
• Trained teachers are not available in this approach to teach English language.
### Suggestions-
• A systematic theory of this approach should be developed.
• Re-orientation program for the English teachers should be organized by the concerning agencies time and again.
• Teaching learning material should be developed
Note: Did you liked the post? Please tell us by your comment below..!! |
Exercise: Creating Circles of Evaluation from Arithmetic Expressions
Directions: For each math expression on the left, draw its Circle of Evaluation on the right.
• $(2 + 17) * (12 - 8)$
• $(23 * 14) * (3 + 20)$
• $(5 - 17) + (14 * 5)$ |
# AC Currents and Voltages Solver and Calculator
A calculator to calculate the voltages and currencts of an AC circuit with a load $Z_L$ given $Z_1$, $Z_2$, $Z_3$ and $Z_L$. The calculator computes all currents and voltages in polar form.
## Formulae for Currents and Voltages Used in the Calculator
Use Kirchhoff's law of current to write
$I_1 = I_2 + I_3$
and Kirchhoff's law of voltage
$V_i - V_{Z_1} - V_{Z_2} = 0$
$V_{Z_2} - V_{Z_3} - V_{Z_L} = 0$
Use Ohm's law to rewrite the above equations as
$I_1 = I_2 + I_3$
$V_i - Z_1 I_1 - Z_2 I_2 = 0$
$Z_2 I_2 - Z_3 I_3 - Z_L I_3 = 0$
Rewrite the above system in standard form
$I_1 - I_2 - I_3 = 0$
$Z_1 I_1 + Z_2 I_2 = V_i$
$Z_2 I_2 - (Z_3 + Z_L) I_3 = 0$
Solve the above system to obtain currents
$I_3 = \dfrac{Z_2 V_i}{(Z_1+Z_2)(Z_3+Z_L)+Z_1Z_2}$
$I_2 = \dfrac{(Z_3+Z_L) V_i}{(Z_1+Z_2)(Z_3+Z_L)+Z_1Z_2}$
$I_1 = I_2 + I_3$
Use Ohm's law to claculate voltages as follows
$V_{Z_1} = Z_1 I_1$
$V_{Z_2} = Z_2 I_2$
$V_{Z_3} = Z_3 I_3$
$V_o = Z_L I_3$
>br>
## Example Using the Calculator
In the AC circuit below, we are given $v_i = 10 \angle 0^{\circ}$ , $R_1 = 100 \; \Omega$, $C = 0.47 \; \mu F$, $R_2 = 120 \; \Omega$, $R_3 = 200 \; \Omega$, $R_4 = 400 \; \Omega$, $L = 20 \; mH$ , frequency $f = 2$ kHz.
Find the currents $I_1$, $I_2$ , $I_3$ and the voltages across each resistor.
Let
$z_1 = R_1 = 100 \; \Omega \angle 0$
$\dfrac{1}{Z_2} = \dfrac{1}{R_2} + j 2 \pi f C$ , resisitor $R_2$ and capacitor $C$ are in parallel
Use Parallel RC circuit Impedance Calculator to calculate $Z_2$ and obtain
$Z_2 = 97.9040 \; \Omega \angle -35.3269^{\circ}$
$Z_3 = R_3 = 200 \; \Omega \angle 0$
$\dfrac{1}{Z_L} = \dfrac{1}{R_4} + \dfrac{1}{j 2 \pi f L }$ , resisitor $R_4$ and inductor $L$ are in parallel
Use Parallel RL circuit Impedance Calculator to calculate $Z_L$ and obtain
$Z_L = 212.8072 \; \Omega \angle 57.8581^{\circ}$
The above values for $Z_1$, $Z_2$, $Z_3$ and $Z_L$ are the default values for the calculator but of course you may change these values.
## Use of the calculator
Enter the impedances $Z_1$, $Z_2$, $Z_3$ and $Z_L$ as complex numbers in polar form (modulus and argument in degress) then press "calculate".
The calculator presented may be used to calculate the ac currens and voltages in any circuit that may be reduced to the basic circuit shown above.
The currents and voltages are in polar form.
Peak Source Voltage V = 10 V Impedance $Z_1$ = 100 $\angle$ 0 $^{\circ}$ Impedance $Z_2$ = 97.9040 $\angle$ -35.3269 $^{\circ}$ Impedance $Z_3$ = 200 $\angle$ 0 $^{\circ}$ Load Impedance $Z_L$ = 212.8072 $\angle$ 57.8581 $^{\circ}$ |
2016-11-06
Database transaction handling in C++ systems
Concurrency (either by multithreading or asynchonicity) and database usage are present in every software system. Transactions are the mechanism given by relational databases to provide ACID properties to the execution of multiple statements.
In order to use them, most database systems expect transactions to take place in a single database connection by using explicit boundaries. This means that in order to perform X database actions as a single logical operation the system needs to: get a connection to the database, start a transaction on the connection, execute the actions over that connection and finish the connection’s ongoing transaction. The application can tell whether to finish the transaction by applying the changes or rollback all the performed changes and leave the DB in the exact same form as it was before starting the transaction. Also, the DBMS might forcefully finish abort the transaction due to conflicts generated by other transactions.
In most C++ systems I have worked on I have found that transactions are not handled in a nice way from the architectural point of view. I can basically classify the transaction management and handling in one of two options:
• Database layer abstracted but not allowing transactions. This situation is represented by the architecture where there are multiple classes acting as repositories which interface with the database. But the abstraction handles the acquisition and release of the DB connection, thus not allowing to have a transaction that can cover multiple methods across one or more repositories. To give an example, there is a repository for the Account object with two methods: save and getById. Each of the methods acquires a connection performs the DB operation and then releases the connection. As each method acquires its own connection, there is no way to have a transaction across the two methods.
• Database implementation polluting business logic. In this case, we have the business logic acquiring a connection and starting a transaction and passing the DB connection to other methods so that they can pass it to the DB layer. Now, you are able to use transactions in the system but the business logic needs to know the underlying layer in order to start the transaction and also all the functions that might be called need to take the DB connection object as an argument.
None of the aforementioned solutions is a good one. We would like to have the abstraction of the first scenario but with the possibility that the business logic can declare that a certain set of actions should be done within a transaction.
Solution Approach
Architecture
Let’s see how we can tackle this issue in a nice way. The following UML diagram shows an architectural approach to the transaction handling program that decouples all the components as much as possible. The decoupling is important because we want to be able to mock most of the things in order to successfully test our code.
If the business logic requires actions to be run under the context of a single transaction, then it will need to know about the transaction manager.
As the management of transactions is very specific to each implementation, what the business logic actually knows and interacts with is an object that implements the TransactionManager interface. This interface will provide an easy way of executing actions within a single transaction context.
In order to interact with the database layer, the business logic will still need to have knowledge and interact with the model’s repositories. Of course, this is done through an interface and not a concrete type.
Entity repositories
The repositories don’t change at all. They use the same interface to retrieve the connections they need to execute the queries. The difference will be in the setup, instead of using the DbConnectionManagerdirectly, they will use the ConcreteTransactionManager which also implements the DbConnectionManager interface.
This means that the repositories don’t actually care whether a transaction is happening or not.
Transaction Manager
The transaction manager is the key piece in this design. Its purpose is to provide a clear way for the business logic to perform a sequence of actions in the same transaction, while hiding the details of how that transaction is handled.
By using the transaction manager, the business logic doesn’t need to know the underlying implementation of how the persistence layer initiates or closes the transaction.
It also removes the responsibility from the business layer to keep moving the transaction state from call to call. Thus, allowing a component of the business logic that requires transactional behavior to call another who doesn’t but still get transactional results if the latter one fails.
In the diagram above, I assumed the scenario where the transaction is handled through the DB connection, which might not be the case. The way in which the transaction manager and repositories interact will depend entirely on the technicality of how the transaction mechanism needs to be implemented.
Behavior and usage
From the business logic’s point of view what we want to achieve is that the usage is somewhat like this:
void BusinessLogic::foo()
{
transactionManager.performInTransaction([&]() {
Entity a = entityRepo.getById(123);
if (a.x > 20) {
a.x -= 20;
}
entityRepo.save(a);
});
}
Everything that is executed inside the lambda given to the transaction manager, should be done in the context of a single transaction. When the lambda returns, the transaction gets committed.
That way would be fine if transactions cannot fail, but that is not the case. The easiest way for the TransactionManager to communicate the failure of a transaction is through an exception.
In that case we should enclose the call to the transaction manager in a try-catch clause like this:
void BusinessLogic::foo()
{
try {
transactionManager.performInTransaction([&]() {
Entity a = entityRepo.getById(123);
if (a.x > 20) {
a.x -= 20;
}
entityRepo.save(a);
});
}
catch (TransactionAborted&) {}
}
There is one more thing we need to add to make it more complete at a basic level. The business logic needs to be able to trigger a transaction rollback on its own. For that we can also use C++ exception mechanism, the lambda can throw an AbortTransaction which will make the transaction to be rolled back silently. As it was the user who asked for the rollback, the performInTransaction call should finish normally and not through an exception as was the case for the failed transaction.
Nested transactions
The purpose of this blog-post is not to write a full-fledged transaction manager, but to show an architectural solution that is decoupled, versatile and easily expandable.
For the sake of simplicity I am going to assume that if transaction nesting occurs the nested transaction is irrelevant. That is, everything will be done in the context of the outer transaction.
PoC Implementation: a transaction manager for SQLite
As a proof of concept of of this architecture, I am going to write a simple transaction manager for SQLite and write a couple of test cases to show the behavior.
The code is in the following gist: TransactionManagerArchitecturePoC.cpp.
Let’s disassemble the code into the different components.
ScopedDbConnection and ConcreteConnectionManager
In order to make things simple, the ScopedDbConnection is just a std::unique_ptr that takes a std::function<void(sqlite3*)> as its deleter. Doing this, allows us to return a ScopedDbConnection from the TransactionManager that when it gets destructed, it either actually closes the connection or does nothing if the connection belongs to an on-going transaction.
The ConcreteTransactionManager::getConnection() method, simply creates and returns a ScopedDbConnection to the DB we are using. When this scoped connection gets destructed, the underlying connection gets closed.
Transaction manager
There are two important components in the transaction manager: the protocol for transactions and its internal state to give support to the transaction protocol.
The state is composed by one internal data type (TransactionInfo) and a std::map from threads to TransactionInfo instances. A std::mutex also forms part of the transaction manager’s state and is used to synchronize access to the map.
The TransactionInfo is a structure that holds a ScopedDbConnection and a counter. This counter is used to keep track of transaction nesting.
The transaction handling protocol is the meat of the class, this is what actually handles when a transaction gets initialized and when it gets committed or aborted.
Let’s take a look at that code:
void ConcreteTransactionManager::performInTransaction(const std::function<void()>& f)
{
try {
f();
}
catch (AbortTransaction&) {
if (--transactionInfo.count > 0) {
throw;
}
abortTransaction(transactionInfo);
return;
}
catch (...) {
if (--transactionInfo.count > 0) {
throw;
}
abortTransaction(transactionInfo);
throw;
}
if (--transactionInfo.count > 0) {
return;
}
commitTransaction(transactionInfo);
}
The protocol is really simple: when starting into a performInTransaction block, we setup the transaction which results in a TransactionInfo object reference. Then we execute the given function and when that function exits, we reduce the count on the transaction information object. When the count reaches 0, depending on how we reached the count, the transactions gets committed or aborted. In case the transaction finished by an exception different than AbortTransaction, the exception is re-thrown.
commitTransaction and abortTransaction are really simple functions that just execute a statement using the connection from the transaction. They also remove the transaction from the on-going transaction map.
The setupTransaction is also really simple: it checks whether there is an on-going transaction for the given threadId, if there is then it just increments its count and returns. If there’s not it initializes a new TransactionInfo, places it in the map and executes the statement to start a transaction.
The other important function in this class is getConnection. This needs to handle two cases: (a) if something is executed outside the context of a transaction, the returned connection needs to destroy itself when going out of scope; and (b) if a connection is requested within the context of a transaction, the ScopedDbConnection returned must not close the inner connection when going out of scope.
The way I decided to handle the latter is to return a new ScopedDbConnection containing the sqlite3 connection of the transaction but with an innocuous deleter. This way for the client is totally transparent whether the connection comes from a transaction context or not.
ScopedDbConnection ConcreteTransactionManager::getConnection()
{
std::lock_guard<std::mutex> _(_currentTransactionsMutex);
if (it == _currentTransactions.end()) {
return _connectionManager.getConnection();
}
return ScopedDbConnection(it->second.dbConnection.get(), [](sqlite3* c) {});
}
The developed transaction manager has the following properties:
• Transactions are per thread. This means that the transaction is associated to the thread who started it. Only the actions performed by that thread included in the transaction.
• Transactions cannot be shared. This derives from the former item and means that one thread cannot give its current transaction to other threads. So, if thread A opens a transaction and launches thread B, the actions that thread B performs are not covered by the transaction initiated by thread A and there is no way to make that happen.
• There are no nested transactions. If while in the context of a transaction a new call to executeInTransaction is made, this second call doesn’t have any practical effect. A call to abort aborts the already on-going transaction and a successful exit from the inner transaction doesn’t trigger a commit of the outer transaction.
Conclusion
One of the most important advantages this design has is that it is so decoupled that mocking it is really easy to do. This helps testing the code that uses it with minimal effort and no dependencies.
It is also really easy to use and transparent that it is really non-intrusive to the code that uses it. And only the code that wants to use transaction capabilities requires knowledge of it. The rest of the code that needs to interact with it in order to provide the transaction functionalities (i.e the repos or whoever uses the DB) don’t know about transactions. This gets masked by the TransactionManager providing the ConnectionManager interface.
Also all the transactional functionality, is encapsulated by the TransactionManager. This makes it easier to test implementations in isolation without requiring other components to know any logic about transactions.
What can be improved?
The PoC is really simple and there are many edge cases that it doesn’t take into account.
When implementing a production solution, one has to be aware that the repositories can fail due to transaction problems and probably mask those as a TransactionAborted.
For my PoC, I decided to use abstract classes and virtual functions to provide polymorphism, but this can be implemented using templates.
It is also arguable that the ScopedDbConnection returned carries no guarantees that it is not going to be retained by the callee. A different approach needs to be taken to guarantee that: maybe the connection manager can have a method executeOnConnection(std::function<void(const ScopedDbConnection&)>).
2016-05-08
Syncing external hard-drive with dropbox for backup
This little project started because Bitcasa is dropping their Personal Drive product which I used to use. This forced me to change to another cloud storage provider and I decided to use Dropbox. (During this process I found out how broken Bitcasa is/was and got really furious. But it will be a topic for another blogpost)
One of the things I liked about Bitcasa is that they provided a FUSE that I could just mount anywhere. There was no “syncing” of the files in the sense that the files only existed in the cloud provider. It would download the chunks of the requested files on demand and keep them in a cache. This allowed me to not have to worry about disk-space in my physical hard-drive.
Dropbox, on the other hand, doesn’t work like this. When you setup the daemon, you select a folder to be mirrored to the cloud. The daemon monitors any changes in the folder or cloud and keeps both copies synced. The problem with this is that it requires to have as much space in the device where the dropbox folder is as the contents stored in Dropbox. For my immediate situation, that would work but it is definitely not going to scale. I have a 256Gb disk and around 100Gb of data to store in Dropbox.
One possibility is to restrict the content to be mirrored. With this you get a partial syncing of your Dropbox account in your local folder. But after what happened to me with Bitcasa (I lost files, MANY files), I want to have a physical backup copy in an external HD to be on the safe side in any event.
Approach
After doing some research I decided to take the following approach in order to tackle the problem.
I run an instance of Dropbox solely for the purpose of syncing my external hard-drive. In this way it doesn’t interfere with the files that I actually want to have always synced in my desktop.
I run the external hd Dropbox instances manually and I haven’t automated this process. The reason behind this decision is that if I accidentally delete something from Dropbox, the backup will still have it and it won’t sync until I tell it to do so.
Running a second instance of Dropbox
Dropbox installs the folders .dropbox and .dropbox-dist under the home directory.
The first one has all the configuration for the Dropbox instance, while the latter has the binary dropboxd and the files required by it.
If you try executing dropboxd, it will complain saying that Dropbox is already running (for syncing the folder in the home directory).
The key to be able to run more than one Dropbox instance is to know how Dropbox determines the location of the .dropbox configuration folder. As it is in this folder where all the configuration for an instance is stored, where all the cached elements are kept and also where the pid file is kept what prevents multiple instances using the same config.
The location used by Dropbox for the configuration directory is $HOME/.dropbox. Thus by changing the value of the HOME environmental variable when we execute dropboxd, we can change the configuration folder and have as many instances as we want. I mount my external hard-drive on /mnt/external-hd/, so I just execute HOME=/mnt/external-hd/ /home/santiago/.dropbox-dist/dropboxd. The first time it will ask for the instance setup information: account, password, location of the mirrored folder, etc. After the first time, it will run silently. One caveat is that if the mount directory of your external hard-drive changes, then you should be careful when starting the external-hd’s Dropbox service. If dropbox thinks you have deleted the data, it will sync that upstream and you will lose the data. To prevent this, before running it, create a symlink from the old location to the new and then move the location to the new one using Dropbox’s configuration setup. 2016-04-10 Setup NAT Network for QEMU in Mac OSX I am working on a really cool project where we have to manage virtual machines that are launched through QEMU. The project is meant to run in Linux, but as I (and all my colleagues) have a Mac to develop, we wanted to be able to run the project under Mac. Yes, I know QEMU’s performance on Mac sucks! We couldn’t use QEMU’s user mode network stack due to its limitations. We needed to use TAP interfaces, the machines should be able to acquire the network configuration through DHCP and should be NATted. An schema of how I wanted things to be is as follows: Enable TAP interfaces in Mac The first issue I stumbled upon is the fact that Mac does not have native support for TAP interfaces. In order to have TAP interfaces in Mac we need to download and install TunTap. Once it is installed, we are going to see a list of TAP nodes installed with the scheme /dev/tapX. TunTap determines the max number of possible TAP interfaces and sets them up. Creating the bridge interface Once we are able to use TAP interfaces, we need to create the bridge where we can attach them. This is really straight-forward in Mac. For a temporal bridge we just need to issue the following command with elevated privileges: $ sudo ifconfig bridge1 create
The next step is to configure the address for that newly created bridge. Which IP we give it, will depend on the network we want to use for our VM network. As an example, I will use the 192.168.100.0/24 network, so I will assign 192.168.100.1 to the bridge. It will act as the default gateway for all the virtual machines, that is why we need to assign that IP statically.
$sudo ifconfig bridge1 192.168.100.1/24 Packet forwarding and NAT The next step to reach our goal is to configure our Mac so that the packets that arrive from the bridge1 interface are routed correctly. We also need to NAT these packets as otherwise they won’t find their way back. Enabling packet forwarding is really easy, we just need to execute: $ sudo sysctl -w net.inet.ip.forwarding=1
For the NAT we need to create a pfctl configuration file where we state the rule that will do the NAT. In a file we write the following:
nat on en0 from bridge1:network to any -> (en0)
This rule tells pfctl to NAT packets that:
1. Go through en0 (you should replace this interface for the one connected to the internet) and
2. Have source IP from the network range associated to bridge1 (here goes the bridge name of our VM network)
The address to use for the NAT is indicated after the ->. We need to put the interface connected to the internet between parenthesis. The parenthesis are important because it forces the evaluation of the address associated with the interface each time the rule is applied. If we don’t add it, the address to use will be resolved at load time and if it changes, the used address will be incorrect.
Now we need to enable the pfctl with the given rule.
$sudo pfctl -F all # This flushes all active rules$ sudo pfctl -f /path/to/pfctl-nat-config -e # Enable pf with the config
Setting up the DHCP server
Before setting up the virtual machine, we need to set up the DHCP server so the VM will be able to acquire the network configuration easily.
Fortunately, Mac OS comes with a DHCP server installed by default and the only thing we need is to set it up and start it. The server is bootpd and is located under the /usr/libexec/ directory.
The DHCP server reads the configuration from the file /etc/bootpd.list and we need to edit it.
The bootpd.plist file has 3 main sections (detailed explanation in the official documentation):
• Root dictionary keys: this properties are used to control the behavior of the daemon as a whole.
• Subnets property: An array of the subnetworks that the DHCP server has associated.
• NetBoot property: This is used to configure the netboot options.
We are going to be interested in the first two sections as they are the ones that are needed to have the DHCP service up and running.
Here’s the file as we need it:
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
<key>bootp_enabled</key>
<false/>
<key>dhcp_enabled</key>
<array>
<string>bridge1</string>
</array>
<key>netboot_enabled</key>
<false/>
<key>relay_enabled</key>
<false/>
<key>Subnets</key>
<array>
<dict>
<key>name</key>
<string>VM NAT Network (192.168.100.0/24)</string>
<string>255.255.255.0</string>
<string>192.168.100.0</string>
<key>net_range</key>
<array>
<string>192.168.100.2</string>
<string>192.168.100.254</string>
</array>
<key>allocate</key>
<true/>
<key>dhcp_router</key>
<string>192.168.100.1</string>
<key>dhcp_domain_name_server</key>
<array>
<string>8.8.8.8</string>
</array>
</dict>
</array>
</dict>
</plist>
Let’s drill down to the important elements in the xml file.
The dhcp_enabled key in the root dictionary is used to state which interfaces we want to associate the DHCP with. We must here add the bridge interface name, otherwise the DHCP service won’t listen to the DHCP requests on that interface.
The other required thing we need to do is add an entry in the array associated with the Subnets key. Each entry is a dictionary that will describe a subnetwork to be used by the DHCP service. The following is a description of the main keys used above (again, for the complete list see the documentation):
• name: A string just to give the subnetwork a human readable aspect.
• net_address: this is the subnetwork base address. In our case 192.168.100.0.
• net_range: which range in this subnetwork is managed by the DHCP server. The value of this property is an array that contains two strings: the lower and upper bound of the addresses to manage. In our case, we want the DHCP to manage all the hosts but the one assigned to the host, then our range is: 192.168.100.2-192.168.100.254.
• alocate: this boolean property tells the DHCP server whether to assign or not IP addresses from the range. We must set it to true.
The other two keys are used to push configuration to the DHCP clients. We want to push the default gateway as well as the DNS, for that we use the dhcp_router and dhcp_domain_name_server option.
Now that the configuration has been set up, we need to start the DHCP server. To do that, we just execute $sudo /usr/libexec/bootpd -D. This will launch the server in the background with DHCP capabilities on. If we want to have it in the foreground and see how it is working, it can be launched using the -d flag. QEMU and interface setup The last thing to do is to launch the virtual machines and correctly set up the attached interface so that it is correctly attached to the bridge. We are going to be using a TAP interface setup in QEMU using a virtio NIC. We cannot use the bridge setup in Mac due to the inexistent qemu-bridge-helper for the platform. To configure the virtio device we need to use the following command line arguments: -net nic,model=virtio. Here is where we would also specify the MAC address for the interface if we want to. The command line argument specification to setup the interface as TAP is like this: -net tap[,vlan=n][,name=name][,fd=h][,ifname=name][,script=file][,downscript=dfile][,helper=helper] From those arguments we are interested in 2 of them in particular, script and downscript. The files given in those arguments are executed right after the TAP interface is created and right before the TAP interface is destroyed, respectively. We need to use those scripts to attach and detach the interface from the bridge. The scripts receive one command line argument with the name of the interface involved. We need to create two scripts: • qemu-ifup.sh will be used as the start script and will attach the interface to the bridge: #!/bin/bash ifconfig bridge1 addm$1
• qemu-ifdown.sh will be used in the downscript to detach the interface from the bridge before it is destroyed:
#!/bin/bash
ifconfig bridge1 deletem \$1
All that’s left is start the VMs and enjoy the newly created NAT network. |
# 2.12 Machine learning lecture 12 course notes (Page 7/8)
Page 7 / 8
For example, one may choose to learn a linear model of the form
${s}_{t+1}=A{s}_{t}+B{a}_{t},$
using an algorithm similar to linear regression. Here, the parameters of the model are the matrices $A$ and $B$ , and we can estimate them using the data collected from our $m$ trials, by picking
$arg\underset{A,B}{min}\sum _{i=1}^{m}\sum _{t=0}^{T-1}{∥{s}_{t+1}^{\left(i\right)},-,\left(A,{s}_{t}^{\left(i\right)},+,B,{a}_{t}^{\left(i\right)}\right)∥}^{2}.$
(This corresponds to the maximum likelihood estimate of the parameters.)
Having learned $A$ and $B$ , one option is to build a deterministic model, in which given an input ${s}_{t}$ and ${a}_{t}$ , the output ${s}_{t+1}$ is exactly determined. Specifically, we always compute ${s}_{t+1}$ according to Equation [link] . Alternatively, we may also build a stochastic model, in which ${s}_{t+1}$ is a random function of the inputs, by modelling it as
${s}_{t+1}=A{s}_{t}+B{a}_{t}+{ϵ}_{t},$
where here ${ϵ}_{t}$ is a noise term, usually modeled as ${ϵ}_{t}\sim \mathcal{N}\left(0,\Sigma \right)$ . (The covariance matrix $\Sigma$ can also be estimated from data in a straightforward way.)
Here, we've written the next-state ${s}_{t+1}$ as a linear function of the current state and action; but of course, non-linear functions are also possible.Specifically, one can learn a model ${s}_{t+1}=A{\phi }_{s}\left({s}_{t}\right)+B{\phi }_{a}\left({a}_{t}\right)$ , where ${\phi }_{s}$ and ${\phi }_{a}$ are some non-linear feature mappings of the states and actions. Alternatively, one can also use non-linear learning algorithms, such as locally weighted linearregression, to learn to estimate ${s}_{t+1}$ as a function of ${s}_{t}$ and ${a}_{t}$ . These approaches can also be used to build either deterministic or stochastic simulatorsof an MDP.
## Fitted value iteration
We now describe the fitted value iteration algorithm for approximating the value function of a continuous state MDP. In the sequel, we will assumethat the problem has a continuous state space $S={\mathbb{R}}^{n}$ , but that the action space $A$ is small and discrete. In practice, most MDPs have much smaller action spaces than state spaces. E.g., a car has a 6d state space, and a2d action space (steering and velocity controls); the inverted pendulum has a 4d state space, and a 1d action space; a helicopter has a 12d state space, and a4d action space. So, discretizing ths set of actions is usually less of a problem than discretizing the state space would have been.
Recall that in value iteration, we would like to perform the update
$\begin{array}{ccc}\hfill V\left(s\right)& :=& R\left(s\right)+\gamma \underset{a}{max}{\int }_{{s}^{\text{'}}}{P}_{sa}\left({s}^{\text{'}}\right)V\left({s}^{\text{'}}\right)d{s}^{\text{'}}\hfill \\ & =& R\left(s\right)+\gamma \underset{a}{max}{\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{sa}}\left[V\left({s}^{\text{'}}\right)\right]\hfill \end{array}$
(In "Value iteration and policy iteration" , we had written the value iteration update with a summation $V\left(s\right):=R\left(s\right)+\gamma {max}_{a}{\sum }_{{s}^{\text{'}}}{P}_{sa}\left({s}^{\text{'}}\right)V\left({s}^{\text{'}}\right)$ rather than an integral over states; the new notation reflects that we are now working in continuous states rather than discrete states.)
The main idea of fitted value iteration is that we are going to approximately carry out this step, over a finite sampleof states ${s}^{\left(1\right)},...,{s}^{\left(m\right)}$ . Specifically, we will use a supervised learning algorithm—linear regression in our description below—to approximate the value functionas a linear or non-linear function of the states:
$V\left(s\right)={\theta }^{T}\phi \left(s\right).$
Here, $\phi$ is some appropriate feature mapping of the states.
For each state $s$ in our finite sample of $m$ states, fitted value iteration will first compute a quantity ${y}^{\left(i\right)}$ , which will be our approximation to $R\left(s\right)+\gamma {max}_{a}{\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{sa}}\left[V\left({s}^{\text{'}}\right)\right]$ (the right hand side of Equation [link] ). Then, it will apply a supervised learning algorithm to try to get $V\left(s\right)$ close to $R\left(s\right)+\gamma {max}_{a}{\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{sa}}\left[V\left({s}^{\text{'}}\right)\right]$ (or, in other words, to try to get $V\left(s\right)$ close to ${y}^{\left(i\right)}$ ).
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
Got questions? Join the online conversation and get instant answers! |
# inverse and the domain...
• October 26th 2009, 02:45 AM
pop_91
inverse and the domain...
i cant seem to get the inverse of this function. (oh and sorry if this is in the wrong place i didnt know where to put it)
f(x)=1-2^x x E R
ive got so far but i cant seem to finish it .... and how do you find the domain of this invserse (is it x E R)
thanks :)
• October 26th 2009, 03:18 AM
Raoh
hi.
$f(x)=1-2^{x}\Leftrightarrow y=1-2^{x}\Leftrightarrow x=1-2^{y}$...then solve for y,and change it to $f^{-1}(x)$.
the domain of the inverse is equal to the range of f.
http://www1.wolframalpha.com/Calcula...image/gif&s=36 |
How to create an algorithm when given the invariant
I am given the following invariant:
Invariant The greatest $i$ keys of an Array are always in sorted order on the last $i$ indices of the Array.
I am supposed to create a sorting Algortihm using this invariant. But I seem to have understood the concept of an Invariant differently, because with the information I find online, I can't find an idea for starting this problem. Do I have to create an algorithm that sorts the first $n-1$ elements in the array? Thanks for your help
• How do you understand the concept of “invariant”? An invariant is a property that is supposed to hold for some thing. – Guildenstern Nov 7 '17 at 13:08
• Yes, this is why I don't understand how I'm supposed to create an algorithm where something like that holds true at all times.. – chala Nov 7 '17 at 13:10
• Don’t you understand the concepts of invariants in general, or this invariant in particular? I’ll invite you to ask about it in chat so that you can hash this out. ☺ – Guildenstern Nov 7 '17 at 13:12
• At every step of your algorithm, the invariant should hold. I think you can use selection sort starting from the end. – klaus Nov 7 '17 at 16:56
1. For $i=0$, you don't know anything about the array.
2. For $i=n$, the array is sorted.
So the idea is to go from $i=0$ to $n$ while maintaining the invariant. This is similar to an inductive proof; in fact, you'll get an inductive algorithm.
• Start with the input array and $i=0$; clearly, the invariant is maintained.
• For some $i \geq 0$, assume that the invariant holds. What do you do to ensure the invariant holds for $i+1$? |
All Questions
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Can you help me understand what a cryptographic “salt” is?
I'm a beginner to cryptography and looking to understand in very simple terms what a cryptographic "salt" is, when I might need to use it, and why I should or should not use it. Can anyone offer me a ...
3k views
Can one generalize the Diffie-Hellman key exchange to three or more parties?
Does anyone know how to do a Diffie-Hellman or ECDH key exchange with more than two parties? I know how to do a key exchange between 2 parties, but I need to be able to have a key agreement between 3 ...
3k views
What is the relation between RSA & Fermat's little theorem?
I came across this while refreshing my cryptography brain cells. From the RSA algorithm I understand that it somehow depends on the fact that, given a large number (A) it is computationally ...
655 views
Why are bitwise rotations used in cryptography?
Any understanding I have of cryptography stops right around the cipher level. As such, I'm just curious as to why bit shifts and moreover circular bit shift are so prevalent in cryptography.
355 views
In what way is XXTEA really vulnerable?
I'm looking at using the XXTEA algorithm to encrypt a small amount of data (say, less than 32KB) in the context of a software licensing algorithm. That is, we wish to make it difficult (not ...
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# Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.
#### crypt50
##### New member
Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.
Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I don't really understand what I am supposed to do. Is $G$ a subset of $V \oplus V$ or a subset of the ordered pairs $(x, T(x))$. Please help.
#### Opalg
##### MHB Oldtimer
Staff member
Re: Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.
Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I don't really understand what I am supposed to do. Is $G$ a subset of $V \oplus V$ or a subset of the ordered pairs $(x, T(x))$. Please help.
By definition, $V \oplus V$ consists of all ordered pairs $(x,y)$ for $x$ and $y$ in $V$.
$G$ consists of those elements $(x,y)$ in $V \oplus V$ for which $y = Tx$. So $G$ is a subset of $V \oplus V$, and your first task is to show that it is a subspace.
#### Chris L T521
##### Well-known member
Staff member
Re: Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V.
Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I don't really understand what I am supposed to do. Is $G$ a subset of $V \oplus V$ or a subset of the ordered pairs $(x, T(x))$. Please help.
Well you're told that elements of $G$ consist of all ordered pairs $(x,T(x))$ for $x\in V$; i.e.
$G=\{(x,T(x)):x\in V\}\subset V\oplus V$
To show that $G$ is a subspace of $v$, you want to take $u=(x_1,T(x_1))$ and $v=(x_2,T(x_2))$ and show that for any $a,b\in \Bbb{F}$, you have that $au+bv \in G$ (this is rather easy to show because addition is done component wise; furthermore you also know that $T$ is linear).
This is me going off on a limb (kind of; so I'd appreciate it if someone could point out if I'm wrong here), but I believe that in order to show that $(V\oplus V) / G\cong V$, you need to construct a morphism $\varphi : V\oplus V \rightarrow V$ where $\mathrm{Im}(\varphi)=V$ and $\ker\varphi = G$ (and thus you obtain the result by the First Isomorphism Theorem). |
# Maximise the Absolute
Let $m$ and $n$ be distinct positive integers.
Find the maximum value of $|x^m - x^n|$, where $x$ is a real number in the interval $(0, 1)$.
Note by Sharky Kesa
5 years, 4 months ago
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Simplest thing I can think of $|1^1-0^1|$.... It has to be more complicated than this...
- 4 years, 10 months ago
That doesn't work at all. I think you typoed.
- 4 years, 10 months ago
He put (0,1), which means 0 & 1 not included, so it must be something else!!
- 4 years, 10 months ago
0
- 4 years, 10 months ago
I dont think it will have a numeric value as answer as "x" is variable and there are infinite nos. between 0 and 1...
- 4 years, 10 months ago
Note that it asks for a maximum value.
- 4 years, 10 months ago
- 4 years, 10 months ago
In terms of m and n?
- 4 years, 10 months ago
The answer should be "x" !
- 4 years, 10 months ago
Note that $x^r$ is a strictly decreasing function when $x\in(0,1)$, and the maximum value would be achieved when $m$ is the least and $n$ is the most, in this case, $m=1$ and $n\to\infty$, where $|x^m-x^n|=|x^1-0|=x$.
Note: @Sparsh Goyal posted the numerical answer first.
- 4 years, 8 months ago
× |
# American Institute of Mathematical Sciences
December 2017, 22(10): 3663-3669. doi: 10.3934/dcdsb.2017147
## Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
* Corresponding author
Received December 2016 Revised December 2016 Published April 2017
Fund Project: Supported by the National Natural Science Foundation of China (11101060,11671066) and by the Fundamental Research Funds for the Central Universities (DUT16LK24)
In this paper we study the global boundedness of solutions to the fully parabolic chemotaxis system with singular sensitivity:$u_t=\Delta u-\chi\nabla·(\frac{u}{v}\nabla v)$, $v_t=k\Delta v-v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}$ ($n\ge 2$), where $\chi, \, k>0$. It is shown that the solution is globally bounded provided $0<\chi<\frac{-(k-1)+\sqrt{(k-1)^2+\frac{8k}{n}}}{2}$. This result removes the additional restriction of $n \le 8$ in Zhao, Zheng [15] for the global boundedness of solutions.
Citation: Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
##### References:
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##### References:
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2018 Impact Factor: 1.008 |
# Abstract
Public Health Observatories (PHOs) were created in England in 2000 as an important adjunct to the public health system in the country. The observatories were networked together, which allowed pooling of expertise and rapid dissemination of methods and results. The network grew to include the whole of the UK and Ireland and was a very successful force for change until PHOs were subsumed into the new Government Agency Public Health Organization, Public Health England. This paper describes the lessons learnt from their existence in the public health system in England for fourteen years.
Health Systems; Knowledge Management for Health Research; Urband Health
# Resumo
Os Observatórios de Saúde Pública foram criados na Inglaterra em 2000 como instâncias auxiliares importantes para o sistema de saúde pública no país. Os observatórios constituíram uma rede, que permitiu o compartilhamento de experiências e a disseminação rápida de métodos e resultados. Essa rede cresceu até cobrir todo o Reino Unido e Irlanda, servindo como força altamente bem-sucedida em favor de mudanças, até que os observatórios foram subsumidos pela nova agência governamental de saúde pública, chamada Public Health England. O artigo descreve as lições aprendidas ao longo dos 14 anos de existência dos observatórios junto ao sistema de saúde pública da Inglaterra.
Sistemas de Saúde; Gestão do Conhecimento para a Pesquisa em Saúde; Saúde Urbana
# Resumen
Los Observatorios de Salud Pública se crearon en Inglaterra en 2000 como importantes órganos auxiliares para el sistema de salud pública en el país. Los observatorios establecieron una red, permitiendo el intercambio de experiencias y la rápida difusión de los métodos y resultados. Esta red creció hasta cubrir todo el Reino Unido e Irlanda, sirviendo como fuerza de gran éxito para el cambio, hasta que los observatorios fueron subsumidos por la nueva salud pública agencia gubernamental, la Public Health England. El artículo describe las lecciones aprendidas durante los 14 años de existencia de los observatorios en el sistema de salud pública en Inglaterra.
Sistemas de Salud; Gestión del Conocimiento para la Investigación en Salud; Salud Urbana
# Introduction
Public Health Observatories (PHOs) in England were established in 2000 following the publication of the UK Government’s white paper (a statement of proposed Government policy), Saving Lives: Our Healthier Nation11. Department of Health. Saving lives: our healthier nation. London: The Stationery Office; 1999.. “Saving Lives” was an ambitious document, claiming to herald a new approach to public health. It focused action in four areas with targets (Figure 1). To attain these ambitious aims, “Saving Lives” called for the creation of a regional network of public health observatories. They operated very successfully until 2013 when reorganization of the health service subsumed the observatories into a new Government agency called Public Health England. This article describes the establishment and evolution of public health observatories in England and offers some reflections on their lasting legacy. The English PHOs had a distinctly “regional focus”, serving populations of between 2.6 and 8.7 million 22. Office for National Statistics. Reference table: region and country profiles - key statistics tables, October 2013. http://www.ons.gov.uk/ons/regional-statistics/index.html#tab-data-tables (accessed on 25/Oct/2014).
http://www.ons.gov.uk/ons/regional-stati...
. Although having a slightly different role, the Office for National Statistics, some would argue already served some of the functions of a national PHO, especially if some of the activities of the Department of Health were also included.
Figure 1
Targets set out in Saving Lives: Our Healthier Nation 1.
“Saving Lives” proposed an integrated approach to health improvement across Government, and the document itself was significant in that it was signed by 12 Government Ministers and the Prime Minister. The section relevant to information was as described in Figure 2.
Figure 2
Section on information from Saving Lives: Our Healthier Nation 1.
# Establishment and evolution of the observatories
## Context
“Saving Lives” was a wide-ranging set of proposals on public health. As well as creating PHO, which were referred to in a very short paragraph, this proposed legislation also established a number of other new public health initiatives. This included the Health Development Agency (HDA – now abolished), which was charged with translating policy into action. A regional structure was established to facilitate this, backed up by a strong central function. It had a strong remit to encourage evidence-based practice. But the HDA was abolished in 2005 with a number of its functions being subsumed into the National Institute for Clinical Evidence (NICE). NICE then underwent a name change to the National Institute for Health and Care Excellence, though it retained the same acronym.
It was stated that the PHOs would be linked together to form a national network of knowledge, information and surveillance in public health, and would be a major new resource for local bodies working in public health. The observatories were said to be modeled on the Liverpool Health Observatory, which had been created in 1990.
One of the important developments in the creation of PHOs in England was the creation of the Association of Public Health Observatories (APHO). Collaboration between regions of England had always been strong, and so a national structure linking these regional bodies was a very natural part of the evolution.
APHO was set up at the same time as PHOs were established. Its aim was to coordinate work between the PHOs, to avoid duplication, disseminate good practice, and to allow the principle of “doing once for all” to be applied. We had many successes as APHO, where expertise was pooled. Health profiles 33. Bradford C, Hill A, Wilkinson J. English health profiles – did they do what was expected? An evaluation of health profiles 2006. Public Health 2009; 123:311-5. for all local authorities and the production of single topic reports for the Chief Medical Officer were examples of outputs from this collaboration. APHO was modeled on the cancer registries association in the UK (the UKACR), and also a similar organization in France (Fédération National Observatoires Régionaux de Santé – FNORS; http://www.fnors.org) which coordinates the work of the French Health Observatories.
Initially established as an England only body, the English Observatories quickly extended membership to other parts of the UK. Health profiles were borne out of similar work in Scotland, and much was learnt from our Scottish colleagues. Wales was establishing its own PHO at the same time as the English PHOs, and was very keen to do this in collaboration with the rest of the UK. The Institute of Public Health in Ireland was established in 1999 (under the 1999 “Good Friday” agreement), and the chair of APHO at the time was an advisor to establishing the all-Ireland observatory. That observatory, INIsPHO, was formed and became a member of APHO. Over the years all the “Celtic countries” were full members and active participants in APHO. A number of pieces of joint work were undertaken. The coverage of PHOs in UK and Ireland also proved to be a great advantage in joining in European Union-funded projects.
## PHO functions
“Saving Lives” had proposed some very specific functions for public health observatories; these are shown in Figure 3.
Figure 3
Proposed roles of Public Health Observatories from Saving Lives: Our Healthier Nation 1.
Public Health Observatories were set up initially in the eight National Health Service (NHS) regions of England. Subsequently in 2004, an additional ninth observatory was created and the boundaries of the PHOs moved to those of the Government Office Regions. The decision to create regional or local observatories was made in order that they would be able to serve a defined geographical population and to relate to the people living and working in that geographical area. These local relationships were critical to their success. National organizations are often seen as “distant and remote”, particularly the further that one gets from London.
PHOs were based in a number of different organizations, such as universities or government offices of the regions which existed at that time. The London Health Observatory had a variety of host organizations, and some were based in local NHS organizations. For those outside the NHS, the piecemeal nature of the funding was a problem in that host organizations were not happy for PHOs to commit resources and appoint staff on permanent contracts. This had a detrimental effect on recruitment. Subsequently an enlightened official in the Department of Health provided assurances that should PHOs be wound up, any costs of cessation would be met by the Department of Health. This was a major step forward and enabled PHOs to appoint permanent staff.
Shortly after their inception, six out of nine of the observatories took on the National Drug Treatment and Monitoring Service (a system to monitor and report on the level of drug misuse). This added a welcome new income stream to these observatories.
Observatories very quickly became organizations which were seen to be able to get things done, and were increasingly commissioned to take on additional short term funded pieces of work. In the North East PHO, in the last two years of its operation, the organization had a turnover of around £2million ($3.39 million), with core income amounting to around £500,000 ($850,000).
In some parts of the country the synergy with cancer registries and their functions was recognised. In the South West for example, the PHO and the registry had worked as a combined operation from the beginning; in other parts of the country, increasingly when cancer registry directors left, the cancer registry function was transferred to the responsibility of the PHO.
All this was made possible by a very supportive Chief Medical Officer in England, and a supportive Department of Health. Indeed in 2010, after the first ten years of PHOs, they were described as the jewel in the crown of public health 44. Wilkinson JR, Ferguson B. The first ten years of public health observatories in England – and the next. Public Health 2010; 124:245-7.. The Chief Medical Officer, in his meetings with PHOs and the regional directors of public health, made it very clear that he wanted PHOs to be mildly irritating and to raise questions and issues which might be too uncomfortable to be raised by those closer to government. The independent role was emphasized and supported. However, this did not last, with the change of government to Conservative (more right wing) and the change of Chief Medical Officer, PHOs became to be seen as “too independent”.
As well as “core functions” all PHOs had a number of specialist areas (three to four per PHO). This enabled PHOs to develop expertise in particular areas such as mental health and inequalities. It was a move that was very popular with national organizations, as it allowed these to talk to a single PHO and also one which had developed a degree of expertise in that particular field. One of the specialist areas was focused on international collaboration. Set out below are some activities which were developed in this area, which helped to establish PHOs international credentials.
## International collaboration
The North East PHO led the international work of the APHO and worked closely with other PHOs who had interests in an international dimension. The London Health Observatory was particularly involved as most visitors from abroad find it easy to visit the capital city.
Observatories were very successful over the years in participating in international projects and some of these are detailed below.
### • ISARE – Indicators of Health in the Regions of Europe 5,6,7
This was a series of projects funded by the European Commission (DG Sanco) starting in 2001. The project dealt with the development of health indicators at a sub-national level in Europe. It was a series of projects widely welcomed by the Commission and culminated in the publication of 265 health profiles for the regions of Europe in 2010. The project had to deal with the definition of “regions” and difficulties in using comparative data across the different countries in Europe. Additional work had to be carried out when the EU expanded from 15 to 27 countries. It is expected that this work will be repeated in 2-3 years time. It was coordinated by FNORS, the French association of health observatories, with which PHOs in the UK and Ireland had very close links over the years.
### • Public Health Genomics European Network (PHGEN)
This was an international project which examined the implications of the genetic revolution on public health in Europe. It was coordinated by the University of Maastricht in the Netherlands, and the UK was a partner in the work. PHOs contributed to thinking what the future intelligence needs to support this revolution would be. In the future, we concluded, it will be necessary not only to collect information about health and disease, but also to assemble data on the genetic composition (and therefore the risks) of the population. Some initial thoughts were published elsewhere 88. Wilkinson JR, Ells LJ, Pencheon D, Flowers J, Burton H. Public health genomics: the interface with public health intelligence and the role of public health observatories. Public Health Genomics 2011; 14:35-42.. A number of reports have been published and there are extremely important implications for health systems around the world.
# Spreading the PHO model globally and helping build capacity
## • Example 1: building the global observatory movement
PHOs in the UK and Ireland were at the forefront of the development of the global observatory movement, in the making and receiving of international visits, supporting relevant WHO work, and spreading the equity messages.
In 2005 a delegation from the UK and Ireland visited Canada at their request to advise on the development of “Health Canada” and in particular its information role. The contacts developed have been maintained and a follow-up meeting was held in Gateshead in 2009. Subsequently one senior member of staff spent a six-month secondment in Canada, contributing to the creation of a health observatory in Saskatoon, based on the English moel.
## • Example 2: Injury Observatory for Britain and Ireland (IOBI)
The Injury Observatory for Britain and Ireland (IOBI) is a collaborative effort, with no dedicated funding, between a number of Public Health Observatories and some key academic institutions. Its work still continues to this date (see: http://www.injuryobservatory.net, accessed on 01/Aug/2014).
The purpose is to support those working on the prevention of injuries caused by accidents, violence or self harm, by making important and relevant information and tools available in one site, including: (i) Analyses of trend in injury deaths, hospital admissions and injury occurrence across countries and regions; and (ii) Policy support for prevention.
# Funding
All the developments in “Saving Lives” were backed up with an important level of funding, named the public health development fund, which amounted to at least £96 million (\$162 million) over three years.
However, from the outset, it was made clear that the core funding made available for PHOs was to be seen as “seed” funding and that PHOs were expected to look for alternative sources of funding. Although funding was never withdrawn (until the latter years), PHOs continued to receive a level of core funding from the Department of Health. Initially this was received on an annual basis, which caused some problems for longer-term planning and resource allocation.
This additional level of funding was quickly followed up by observatories being asked to take on a regional function for hospital inpatient data (the hospital episode statistics service, HES). Again this came with an additional level of funding, which added to the security and permanence of the PHOs.
# Some examples of the broader work of the Observatories
## • Legacy and future of the PHOs in England
The newly elected government in England in 2010 quickly passed legislation that was to reform the public health system and bring into being a new organization called Public Health England (PHE). This subsumed much of the public health infrastructure in England, including the PHOs, but also included the eight regional cancer registries, the quality assurance reference centres from the cancer screening programmes, the services responsible for monitoring the levels of drug and alcohol misuse in the population, the Health Protection Agency, and some of the remaining regional public health infrastructure. A total of 40 different organisations eventually went into PHE, creating massive organizational change and considerable inertia while transition was taking place. After much debate the government decided that PHE would be part of the civil service. At the same time there was radical change to public health in the NHS. Public health directors and their staff were transferred from the NHS where they had been since 1974, into local authorities where it was felt they would be better placed to influence the wider determinants of health.
One of the problems was the way in which the reorganization was protracted. Because of the concerns about the NHS reorganization that was taking place at the same time (this had not been signaled in any manifesto) and was highly unpopular, this was all put on hold for a year – “the pause”. Unfortunately this led to a further attrition of staff who were not prepared to live with the uncertainty involved.
The functions of PHOs were transferred into PHE and continue there to this date. However, PHOs were effectively dismantled, as was the highly successful APHO. The remaining staff were transferred into PHE to become the Knowledge and Intelligence function, and the word “region” was dropped. Why this was the case still remains open to speculation, as at the same time, some of the other organizations going into PHE were largely unscathed and still continue to exist in PHE with their own identity.
PHOs were remarkably successful, in an NHS which is constantly being reformed in England, simply lasting 13 years was a major achievement. It is regrettable, that although much good work was undertaken that a formal evaluation was never undertaken. This is an important lesson for the future. Anticipating scrutiny, the PHOs themselves established peer review scrutiny and audit, which itself was very helpful in communicating good practice across PHOs.
There is now a strategic review being carried out into PHE. There is even some talk among senior colleagues that the loss of the observatories should be reversed as it was a mistake. The way in which observatories have developed in other parts of the world has been something of a wake up call to those in the public health system in England.
# Main functions and policy impact of the PHOs
Given the proliferation of organizations calling themselves health-related observatories, we decided to attempt to clarify the roles and functions of PHOs to protect the brand. We published our views on what constituted a health observatory 99. Hemmings J, Wilkinson JR. What is a public health observatory? J Epidemiol Community Health 2003; 57:324-6.. We concluded that the main tests of an observatory were if it carried out the following functions: (i) The primary characteristic of these observatories is that they produce and disseminate intelligence for their host area, such as on regional health development; (ii) PHOs reflect the increasing importance of cross-agency work, health inequalities and evidence-based policy making.
It of course is always difficult to quantify the benefits of an organization such as a PHO. However we set out some of the achievements of the first ten years of PHOs in a publication at the time 44. Wilkinson JR, Ferguson B. The first ten years of public health observatories in England – and the next. Public Health 2010; 124:245-7..
We also did operate in a market and competed against private sector, other parts of the NHS, and academia. We were very successful at this, which could be measured by our income.
The international profile for PHOs in England was well recognized. Our expertise was requested from all over the world.
# Conclusions and discussion
So what were the main challenges we faced on transition to a new civil service organization, and the challenges and opportunities for new observatories which are developing worldwide?
## Challenges
### • Retaining the talent developed over 10 years
The observatories in the UK grew exponentially over their first ten years of existence and were a major force for change in public health in the UK and Ireland. Once reorganization was a possibility, many staff left and retention became a serious problem. The problem was made worse by the protracted nature of the NHS reorganization in England, which was “paused” for a year because of political concerns. Many staff were able to find alternative employment in other research organizations and health charities.
### • Maintaining an independent public health voice: a real challenge for PHE
A former Chief Medical Officer in England had welcomed PHOs in being able to say things, which he was not. Many of the PHO directors had had experience of being Directors of Public Health, and were therefore very familiar with the tension between speaking out on behalf of the public’s health and the need for organizational corporacy.
PHE, being part of the civil service, has the additional tension of being primarily there to support the government and ministers. Its communications divisions became much more powerful and were able to prevent the successors to PHOs from publishing any material which might be seen as contentious.
### • Civil service culture
The civil service culture was a very difficult pill for many former NHS employees to swallow. The emphasis of PHE in serving ministers was difficult. Indeed a select committee report on PHE suggested that it needed to be more independent.
# Opportunities
Some of the enthusiasm and commitment which we saw in PHOs in England in the early days was inevitably lost. Giving a number of experienced and committed people a small amount of relatively unconstrained resource had reaped huge rewards. It is rewarding to see that some of the principles of the PHOs are now being replicated around the world. The opportunity which PHOs have to work between academic and service public health is an unparalleled opportunity and is a very exciting area to work in. PHE is now undertaking a strategic review which may even herald the return of PHOs.
Looking further afield, it is very encouraging to see how observatories are developing worldwide, enabling information and intelligence to be translated into public health policy. The close relationship between those working in information and those in policy areas within observatories offers a great advantage, and encourages the adoption of evidence-based decision making. The developing observatories worldwide also have the opportunity to advance in the following areas:
## • Further developing capacity
One of our prime objectives in the UK was to develop skills and capacity, especially for information specialists. In the numerous reorganizations of the English NHS, although public health consultants had been relatively well protected, many of the other specialist staff such as analysts were not. Many of these working in the NHS quickly found alternative employment in other sectors, and in particular in universities and the drug industry. So it was a prime function for PHOs to develop capacity.
We developed training programs which led to accreditation of specialist status. PHOs were instrumental in bringing about consultant status for senior information specialists. Some PHOs were able to jointly organize and fund masters’ programs and a number of PHOs funded PhD students. In addition, PHOs were often co-applicants for research grant funding which often included the funding of additional PhD students. Much of the training infrastructure remains in place and has been built on since its inception.
## • Developing a community of specialists in public health intelligence with a more direct relationship with policy makers
Being in a public health observatory can encourage staff to be much more outward facing than is sometimes possible if working in an information unit as part of a larger organization. Having an effective network, has a huge multiplier effect on impact. Work being carried out in one part of the country (or in one country) is more often than not directly relevant to another part of the country (or other country). Working together reduces duplication and produces much greater clarity of the message. Information specialists can become familiar with work being carried out in other places, other countries and so the opportunities for learning and disseminating good practice become greatly enhanced.
## • Being relevant to local areas and to local public health
Observatories were well placed to be able to forge close relationships with their local communities and to become known to them. In England, some were more successful than others. Our experience shows that they can be at the centre of regional public health policy, and can be heavily involved in public health training in their local areas.
Health intelligence and its usage can be a major driver of change in public health, as the PHOs in England demonstrated in their thirteen years of existence. Having a degree of independence to state uncomfortable messages is a critical role and a role that a mature democratic system should be able to cope with. It is debatable whether there has been as much profile given to public health in the media since the demise of PHOs in England. The new PHE struggles to establish its place as both authoritative and independent, albeit as part of the government civil service. PHOs are a great concept and it is hoped that the spark that has been kindled in the UK will continue to be a major contributing factor in improving public health globally.
# Acknowledgments
All former colleagues in the UK and Ireland Public Health Observatories and colleagues and friends from around the world.
# References
• 1
Department of Health. Saving lives: our healthier nation. London: The Stationery Office; 1999.
• 2
Office for National Statistics. Reference table: region and country profiles - key statistics tables, October 2013. http://www.ons.gov.uk/ons/regional-statistics/index.html#tab-data-tables (accessed on 25/Oct/2014).
» http://www.ons.gov.uk/ons/regional-statistics/index.html#tab-data-tables
• 3
Bradford C, Hill A, Wilkinson J. English health profiles – did they do what was expected? An evaluation of health profiles 2006. Public Health 2009; 123:311-5.
• 4
Wilkinson JR, Ferguson B. The first ten years of public health observatories in England – and the next. Public Health 2010; 124:245-7.
• 5
Ochoa A, Ledésert B. Health indicators in the European regions. ISARE Project. http://www.isare.org/fichiers_isare/pdf/ISAREEn.pdf. (accessed on 28/Feb/2015).
» http://www.isare.org/fichiers_isare/pdf/ISAREEn.pdf
• 6
Wilkinson J, Berghmans L, Imbert F, Ledésert B, Ochoa A; ISARE II Project Team. Health Indicators in the European regions - ISAREII. Eur J Public Health 2008; 18:178-83.
• 7
Wilkinson JR, Berghmans L, Imbert F, Ledésert B, Ochoa A; ISARE III Project Team. Health indicators in the European regions: expanding regional comparisons to the new countries of the European Union - ISARE III. Public Health 2009; 123:490-5.
• 8
Wilkinson JR, Ells LJ, Pencheon D, Flowers J, Burton H. Public health genomics: the interface with public health intelligence and the role of public health observatories. Public Health Genomics 2011; 14:35-42.
• 9
Hemmings J, Wilkinson JR. What is a public health observatory? J Epidemiol Community Health 2003; 57:324-6.
# Publication Dates
• Publication in this collection
Nov 2015 |
## [pcductape] Re: Internet Explorer will not open
• From: Bob Vandersteen <bvan@xxxxxxxxxxx>
• To: pcductape@xxxxxxxxxxxxx
• Date: Tue, 25 Feb 2003 19:31:42 -0700
Try this: copy iexplore.exe c:\progra~1\intern~1
Cheers,
Bob...
At 07:51 PM 2/25/03 -0600, Carl wrote:
Hi All,
I found my computer with a headache this morning. I tried to open Internet Explorer and it will not. The error message is that iexplore.exe is corrupt. I tried to open Windows Explorer and get the same error message. My other programs seem to be working ok, for now.
I tried run > sfc /scannow and the system asked me for my Windows XP cdrom and I cannot find it. I still have my receipt but no cdrom.
So, I copied iexplore.exe off a computer at work and then I tried using dos to copy to c:\program files\Internet explorer and I keep getting the error message shown below:
Of course, I don't know if that would even work, copying a good iexplore.exe over the bad one. I do have a good iexplore.exe on c:\ drive; what if I deleted the other one, would Windows pick up on the good one.
Carl
ctm007@xxxxxxxxxxxxxx
• Follow-Ups: |
# Batch normalization vs batch size
I have noticed that my performance of VGG 16 network gets better if I increase the batch size from $$64$$ to $$256$$. I have also observed that, using batch size $$64$$, the with and without batch normalization results have lot of difference. With batch norm results being poorer. As I increase the batch size the performance of with and without batch normalization gets closer. Something funky going on here.
So I would like to ask is the increase in batch size have effect on batch normalization?
• Adding batchnorm right after the relu layers? – Aditya Nov 30 '18 at 6:57
• tried both after relu and before relu, both results are poorer than without BN. – Haramoz Nov 30 '18 at 15:39 |
# Coin Errors
04 september 2021
# Coin Errors
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Jun 11
# More LaTeX Tips for Your Thesis – Aliases and the xspace Package
This is a follow-up post of this one
Consistency and speed are two major issues in writing your thesis. First, you want important and particularly technical terms to appear typeset in a consistent manner throughout the whole thesis. Everything else would just look unprofessional, and given that you chose LaTeX you seem to be serious about that point. Secondly, these kinds of names tend to be quite long and also appear frequently. So instead of typing your fingers bloody, you should make use of Latex being essentially a programming language and define variables, a.k.a. aliases or new commands. This is usually done using \newcommand. You can do that anywhere in the document, but I would recommend grouping all aliases together in the preamble. Let’s look at an example (sorry for the logic domain, but this is where all my current LaTeX texts are set in …):
%preamble
\usepackage{xspace}
\newcommand{\fol}{First-Order Logic\xspace}
\newcommand{\sig}{\Sigma}
\newcommand{\powerset}[1]{\mathcal{P}(#1)
}
%\newcommand{\powerset}[1]{2^{#1}}
\newcommand{\oversim}[1]{\,{\buildrel #1 \over \sim }\,}
\newcommand{\lEdownarrow}{ML$\,+\,E\,+\downarrow$\xspace
}
%document
\begin{itemize}
% usage in normal text mode
\item It is good that when writing about \fol, I don't have to write \fol twice!
% works in math mode as well
\item Greek letter: $\sig$
% useful for consistency, if you want to change the notation later (see above)
\item Let $N=\{1,2,3\}$ and regard the elements of $\powerset{N}$ \dots
% hides complicated TeX stuff behind a simple command
\item They agree on everything but $x$: $g \oversim{x} g'$
% custom fine-grained spacing, consistent across your whole thesis
\item That's a hybrid language: \lEdownarrow
\end{itemize}
The output looks like this:
So what is going on? The first two examples do not have any parameters, but show that aliasing works both in text and math mode. Normally, these aliases bring up problems with spaces directly after them. You always need to have a space to delimit the command use from the normal text following it, so usually Latex does not print this space. This would lead to the first example ending like this:
The \xspace command provided by equally named package “just does the right thing”, so no trailing space in the first case, but in the second one. The next two examples show how to use parameters. The number in brackets gives the number of parameters, and you can subsequently access them with #index. This is useful e.g. when you don’t want to fix the notation yet (as in the power set example), or when building shortcuts for complex operators.
This is pretty much all that is important about aliases in LaTeX, xspace mostly works out of the box, so for once no need to RTFM.
More LaTeX tips for your thesis are coming up soon. If you have any wishes of packages I should write about (or even a guest article), please tell me about it. So long, happy writing… |
# Centralizer of a cyclic sylow $p$-subgroup
Let $G$ be a group where $|G|=p^nm$ and $p$ is the smallest prime dividing $|G|$. Suppose that $P \in \operatorname{Syl}_p(G)$ is cylic. Then $C_G(P)=N_G(P)$
Now I know $|N_G(P)/C_G(P)|$ must divide $|\operatorname{Aut}_G(P)| = p^{n-1}(p-1)$ but I can't seem to show why $|N_G(P)/C_G(P)|$ divides $m$?
• Since $p$ is the smallest prime dividing $|G|$ and the argument you suggest we get $N_G(P)$ is a $p$-group since $([N_G(P):C_G(P),p-1)=1$. Then $P=N_G(P)$ and since $P$ is cyclic, $C_G(P)=P$. – Levent Jul 4 '16 at 2:35
• Thanks. I still I don't see the part where $N_G(P)$ is a $p$-group. Could you please expand on that? – R Maharaj Jul 4 '16 at 2:46
Since $p$ is the smallest prime dividing $|G|$, we get $([N_G(P):C_G(P)],p-1)=1$, so $[N_G(P):C_G(P)]$ divides $p^{n-1}$. Clearly, $P\subseteq C_G(P)$ since $P$ is abelian, then $[N_G(P):C_G(P)]$ cannot be divisible by $p$. These two facts together implies $C_G(P)=N_G(P)$.
• I am still having a problem with seeing that $([N_G(P) : C_G(P)] , p-1) =1$ – R Maharaj Jul 4 '16 at 3:17
• Well, $p$ is the smallest prime dividing $|G|$, so if you take a prime $q$ dividing $[N_G(P):C_G(P)]$, necessarily $q>p$. Therefore the index is coprime to $p$. – Levent Jul 4 '16 at 3:20
• aha! I see. On that note, how does $|G|$ relate to $|\operatorname{Aut}(G)|$? – R Maharaj Jul 4 '16 at 3:26
• All you can say is $[G:Z(G)]$ divides $|Aut(G)|$, nothing more as I know in the general case. They are not that related. – Levent Jul 4 '16 at 3:28 |
## data structures – Algorithm for Estimating Number of Unique Monthly Visitors
Is there a way to estimate the number of unique monthly visitors to a site based on a sample of, say, one week of data? This isn’t as simple as just multiplying the number of unique visitors the first week by 4, due to the hotel problem. If 10 people visit your site the first week and the same people are the only visitors to your site the second, third, and fourth week, the total number of monthly unique visitors to your site is only 10.
I know you can use HLL to estimate the number of unique visitors to a site in O(1) space. I’m wondering if there’s a similar approach to estimate how many unique visitors there will be after a month.
## optics – Estimating the optical resolution of a lens
In Tony & Chelsea Northrup’s recent video titled 5 Lies Camera Companies Tell You, they mention that the optical resolution of a kit lens is often lower than the resolution of the sensor at around 5 minutes in. Wikipedia has a section dedicated to measuring the optical resolution.
However, this is not comprehensive. Is there a relatively simple method of estimating the resolution of a lens? I don’t have an actual use for this information, so if it’s significantly easier to estimate rather than calculate, that is satisfactory too.
Some lenses – such as the Raspberry Pi lenses for the High Quality Camera – do quote a number for resolution, but none of the lenses on my system (Micro-Four Thirds) as far as I was able to find.
## algorithms – Estimating users standard deviation given avg, min, max for various tests
Given a series of tests, where we are given one users score, the overall minimum, the overall maximum, and the overall average, how would I estimate the user’s standard deviation on total score (ie. sum of all their tests)?
We cannot assume that the lowest scoring person from one test was the lowest scoring in the next test, but I think it is fair to assume that people generally stay within some score bands (although if this can be done without that assumption, that would be better).
My intuition tells me that this seems to be some sort of application of Monte Carlo, but I can’t seem to figure out how to actually do this.
## What process should follow when estimating the SharePoint project
I have new customer needs. I wanted to establish for each component. What standard process should I follow.
## Estimating the use of data for Google Voice App
I use the Google Voice app for iPhone. With a roaming cost of $0.2 per MB, I would like to understand what to expect if the iPhone GV application: • cost to simply keep it on: no message sent or received • cost per message The goal is to avoid an unreasonable bill ($ 50 +) for the simple sending of text messages.
## statistics – Find the method of estimating moments of θ, MLE and MSE for the Bernoulli distribution
Have I made mistakes? I feel like my MLE is a little messy and I have not used the fact that $$0 theta the 1/2$$.
With $$X_1, …, X_n$$ iid $$f (x | theta) = theta ^ {x} (1- theta) ^ {1-x}, x = 0$$ or $$1,0 the theta the 1/2$$
First I try:
Suppose a sample estimator: $$bar {X} = frac { sum_ {i = 1} ^ {N} x_i} {N}$$ and $$E (x) = sum_ {x = 1.0} ^ {} x theta ^ {x} (1- theta) ^ {1-x} = 0 + theta$$
So $$hat { theta} = bar {X} = frac { sum_ {i = 1} ^ {N} x_i} {N}$$ is my estimation moments
With MSE:
$$E ([frac{sum_{i=1}^{N}x_i}{N}-theta]^ 2) = var ( frac { sum_ {i = 1} ^ {N} x_i} {N}) +[E(frac{sum_{i=1}^{N}x_i}{N})-theta]^ 2$$
$$= var ( frac { sum_ {i = 1} ^ {N} x_i} {N}) +[theta-theta]^ 2 = var ( frac { sum_ {i = 1} ^ {N} x_i} {N}$$
For MLE
$$prod_ {i = 1} ^ {N} theta ^ {x_i} (1- theta) ^ {1-x_i} = theta ^ { sum _ {}} {_ x}} prod_ {i = 1} ^ {N} (1- theta) ^ {1-x_i}$$
take ln
$$log (1- theta) sum _ {} ^ {} (1-x_i) + ln ( theta) sum _ {} ^ {} x_i$$
will find $$frac {d} {d theta} = 0$$ for $$theta$$with $$frac {d} {d theta} = frac { sum _ {} ^ {} (x_i)} { theta} + frac { sum _ {}} {{} (1-x_i)} {1- theta}$$
So $$theta = frac { sum _ {} ^ {} (x_i)} { sum _ {} ^ {} (x_i) – sum _ {} ^ {} (1-x_i)}$$ is my daughter
With MSE:
$$E ([frac{sum_{}^{}(x_i)}{sum_{}^{}(x_i)-sum_{}^{}(1-x_i)}-theta]^ 2) = var ( frac { sum _ {} ^ {} (x_i)} { sum_ {} ^ {} (x_i) – sum _ {} ^ {} (1-x_i)}) +[E(frac{sum_{}^{}(x_i)}{sum_{}^{}(x_i)-sum_{}^{}(1-x_i)})-theta]^ 2$$
and
$$sum _ {} ^ {} (x_i) – sum _ {} ^ {} (1-x_i) = N,$$
$$= var ( frac { sum_ {i = 1} ^ {N} x_i} {N}$$
And so they have the same ESM, but my method estimator of moments can have negative values while the MLE is always positive if $$x_i$$ is all positive or negative values. The MLE is therefore a better estimator, is not it?
## Computer Vision – When estimating the motion of the camera from the fundamental matrix, why can not the translation be restored to scale using epipolar geometry?
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## pr.probability – Estimating the size of the remainder in a random partition
Choose a sequence of real numbers $$x_i$$ as following. To put $$x_0 = 1$$. Yes $$x_i$$ is chosen, then choose $$x_ {i + 1} in[0, x_i]$$ according to the uniform distribution. Obviously we have $$x_i rightarrow 0$$ with probability 1. Put $$I_i =[x_i, x_{i-1}]$$.
Then we choose $$n$$ random numbers $$y_1, ldots, y_n$$ in $$[0, 1]$$ independently of the uniform distribution. Let $$J$$ to be the union of all $$I_i$$, for which there are $$y_j$$ with $$y_j in I_i$$. Again, it is obvious that $$| J | rightarrow 1$$ as $$n rightarrow infty$$ with probability 1. My question is how fast $$1- | J |$$ rot? I would expect that the distribution of $$1- | J |$$ is not very focused, so I am interested in both an estimate of the expected value, median values and extreme values.
The above problem occurs quite naturally in the analysis of algorithms, so I would expect that this problem has been dealt with by someone. Several random structures have size parts according to the distribution of $$x_i$$and picking $$y_i$$ corresponds to the selection of random points in a random structure and to the study of the component containing this point. The question then is how much of the entire structure has been left unexplored.
## co.combinatorics – Estimating Maximum Clique Size via Matrix Ranking
let $$mathrm {M} in lbrace0,1 rbrace ^ n$$ to be the matrix of adjacency of a graph $$mathrm {G} left (V, E subskeq lbrace lbrace u, v rbrace | u, v in V rbrace right)$$ control $$n$$.
Question:
is it true that $$mathrm {rank} ( mathrm {M}) = max_ {n} mathrm {K} _ {n n subseteq mathrm {G}$$
resp. how is the rank of the adjacency matrix related to the size of the maximum clique? |
# Barratry
Re-inventing anarcho-capitalism in a French context.
topics: politics
created: 02 Nov 2009; modified: 27 Mar 2019; status: finished; confidence: remote;
Alexis de Tocqueville remarks (Democracy in America) that a size comparison of budgets between the early United States of America and the post-French Revolution Republic would be misleading. The Americans had dispersed their governments, while the French government had grown centralized.
The American budgets included allowance for waste caused by decentralization, but more importantly, he says, Americans expected that problems would be dealt with by local or state governments, or by charities and civil associations. They would handle even problems that the French public expected to be handled by the government. (Charity for the poor or orphaned being a case in point.) This observation may still be true; American foreign aid as a percentage of GDP are less than European aid, but competitive when private charity is included.
This difference in culture is one of the themes in de Tocqueville’s work. Cultural differences are his major explanations for why the French Revolution was so much more horrifying and unstable compared to the American Revolution. Tocqueville remarks, in a chapter comparing the morals of democratic & aristocratic societies, that in the past 50 years, no man had died for his politics in North America. This hyperbole is not strictly true - consider Alexander Hamilton or David C. Broderick, and Presidential assassinations - but sobering if you compare the respective magnitudes.
But then, one of the remarkable aspects of post-Revolution France was the growth of its territory under Napoleon and the growth of its government. That both occurred simultaneously is a little odd. The two growths needn’t be linked, after all - the British Raj was administered with 500 Englishmen1, and the American empire (Philippines etc.) did not bring with it the growth in government seen later with the New Deal & WWII. The military historians have spent centuries studying exactly how the Republic & Napoleon could control most of Europe, so the growth of government is a more interesting question.
# Meet the new boss
Ironically, neither Revolution much changed the actual administration of government. One change, thanks to the Enlightenment’s interest in Confucianism, was the attempt to shift the army & bureaucracy wholesale into the ‘civil service’ model: educated middle-class employees, with a fixed salary, employed for long periods. In particular, the judicature was made a lifetime appointment on good behavior, admitting only highly-qualified and experienced lawyers to the judicial black.
Previously, many positions and roles were filled by sale, or ‘barratry’. (The ecclesiastic analogue, ‘simony’, was common but not important for us.) For example, a tax collector for a district would be the highest bidder; the ‘salary’ was what the bidder could collect beyond what he had paid the monarch.
No doubt your good little democratic soul recoils at this practice - why, it’s even worse than outright nepotism or bribery! The system is utterly perverse: it doesn’t merely allow, but forces the officials to be corrupt, and the whole system is ripe for abuse. Tocqueville hated it:
"Louis XI had limited municipal freedoms because their democratic character frightened him; Louis XIV destroyed them without being afraid of them. What proves this is that he returned their liberties to all the towns that could buy them back. In fact, he wanted less to abolish their rights than to buy and sell them, and if in fact he did abolish them, it was without intending to, purely because it was financially expedient; and oddly enough the same game went on for eighty years.
Seven times, in that period, the right to elect their magistrates was sold to the towns, and when they had once again become accustomed to the pleasure, it was taken away from them in order to be sold once again. The motive behind the action was always the same, and often openly admitted. “Our financial needs”, it states in the preamble to the edict of 1722, “require us to look for the most secure means of meeting them.”
The means were sure but disastrous for those who fell under this strange tax. “I am struck by the huge sums which have always been paid to redeem municipal offices”, writes an intendant to the controller-general in 1764. “The sum of all that money, expended in useful works, would have been to the profit of the town, which, however, has experienced only the weight of authority and the privileges of the officeholders.” I cannot find any more shameful aspect of the old regime anywhere."2
# When is a tax not a tax?
But let’s take a contrarian view, perhaps an anarcho-capitalist one. What is bad about it?
Consider the case of the towns. They had a choice about whether to ‘redeem’ their offices or not. That alone ameliorates the offense; an optional tax is much less offensive than a mandatory one. And suppose they had chosen to not redeem the election of the magistracies? Then presumably the right would have reverted to the King, who would then have appointed his own.
Is this a bad thing? Let’s consider.
Either the King can be defied or he cannot be defied; he is either king or not king, able to enforce his decisions or not able to. If he can be defied, then the town is no worse off for the demand as it can both not pay & simply elect the magistrates it wants if the King’s choices are suboptimal. If he cannot be defied, then he can appoint whomever he wants. But, you say, what if he appoints a bad’un, or just an indifferent one, when the wise locals familiar with their peculiar area & issues could select an eminent town father instead?
Well, if the King is bad at picking magistrates or he is malicious, then the town is screwed. Since he cannot be defied, he could do other things, such as simply impose a massive tax or order the town razed. He could appoint magistrates regardless of whether or not the town pays; if one must trust the King in great matters, then it’s little worse trusting him in small matters. If he maliciously appoints magistrates upon nonpayment, the town has larger issues to worry about.
I’ve not heard of Louis XIV frequently besieging and leveling his towns, and France did reasonably well under him; we can probably assume that he wasn’t too incompetent or malicious. So the town might not be worse off letting the king pick the magistrates. (It’s worth noting that local rule is not always better; the Romans had a long-standing rule that none could be governor of their birth province3, and the British Raj provided better government and law to the patchwork of maharajahs it conquered.)
Selling offices might not be such a bad thing. Revenue has to be raised somehow, and selling offices is more easily controlled than sending the soldiers out to shake down the peasants and less economically destructive than debasing the coinage. Indeed, if we squint, selling offices could have all the advantages of capitalism.
# Defending markets in everything
## Pros: Supply & demand
Suppose we have a district which needs 1 judge. The king sells the judgeship for a year to some distant noble, who gives it to his capable (yet disinherited) 7th son.
The son knows that the judgeship cost, say, 1000 French livres. If he doesn’t have 1000 livres ready-money at the end of the year, he heads home in shame. And say there are 100 cases a year - these farmers are a bumptious lot. So the judge will want an average of at least 10 livres in bribe money per case, or 5 livres per side of a case. (We’ll use ‘case’ for each major judicial task, and use ‘bribe’ to refer to all compensation, fee, bribe, or otherwise.)
But the judge in the neighboring district only paid 500 livres, and is charging each side 2.5 livres! So now all the infatuated yet penurious youngsters are walking a few miles to the next town and paying that other judge to marry them. What’s the original going to do - refuse to recognize the marriages? His authority from the king doesn’t cover undoing the actions of another judge (and for good reason - the King knows that it would be tantamount to anarchy).
So here is competition of a sort, preventing the judge from capturing most of the value flowing through the district’s transactions.
But there’s competition within a district as well as between districts.
Perhaps the case must be settled by the first judge and not the neighboring. Let’s say the case is over ownership of local property (worth 100 livres). What the neighboring judge says is irrelevant - this is something the first judge has near-exclusive jurisdiction over. Villainously steepling his hands, he demands that the 2 sides engage in a bribery war. But if the plaintiff offers 1 livre, then the defendant will offer 2 livres (because 100 > 2), and then the plaintiff will offer 3, and so on right up to 99 livres, leaving one side with 0 livres, and the other with just 1. (This is similar but not identical to a dollar auction.)
The next such case that comes up, the 2 sides mutually agree to cut out the judge entirely & to just flip a coin, since an expected value of 50 livres beats the 1 livre left after a bribery war. The judge, not being a fool, realizes that he cannot charge more than 50 livres; and if there should happen to be some cheaper arbitrator like a village elder? Then he might have to charge even less. If he’s not careful, he’ll cause the formation of a entirely independent private legal & commercial system from which he cannot extract bribes (such systems are not uncommon; eg. halawa or the Lex mercatoria4).
But the judge can’t even charge up to the costs of the private legal system! Suppose the district has 100 transactions a year (as before), the alternative system costs 15 livres per transaction, and that the judge costs 14 livres a transaction (giving him 1400 livres: 1000 to buy the position next year, and 400 livres to live nobly), but there is a village elder willing to be a judge for 50 livres (regardless of how many transactions). Now the village can’t resort to the alternative system, but now it has an incentive to go collect 10 livres from the parties of the 100 transactions, scrape together another 50 for the elder (who has agreed to do transactions for 0 livres, aside from his salary of 50), and throw in another livre - paying the king 1001 livres. The judge is now out of a job, and the villagers have now paid just , or 10.5 livres per transaction, or a district-wide savings of 350 livres ().
So the judge can’t charge more than the neighboring judges, more than the expected value, more than the cost of alternative legal systems, or more than would make it profitable for the villagers to collectively buy his position.
That’s quite a few limitations.
## Cons
### Sovereign abuses
One might point out that there is a different incentive for the sovereign - to constantly permute government offices & dispossess those who hold office at its pleasure (such as the town example). The Ottoman Empire engaged in this sort of ‘planned obsolescence’, if you will:
“[Turkey in Europe, by ‘Odysseus’] says (p. 86) that so far back as the eleventh century a Vizier (Nizami-‘l-Mulk) wrote a work called ’The Science of Government’, in which he ‘recommended that provincial governors and agents should be often moved, and not allowed to become too powerful’. Speaking of the period of Turkish history when the Phanariots had risen to positions of importance, he says (p. 309): ‘Hospodars, dragomans, and patriarchs alike bought their offices for enormous sums… The Porte changed them all as often as possible, in order to increase the number of sales, but left them a free hand in the matter of filling their own pockets.’”5
The counter-measure on the part of office holders is to prefer offices with fixed terms, or to lower their bids to reflect their expected tenure. From the judge example, if the son expects to be a judge for only 6 months, then he will only bid half the yearly sum. If he loses the position to someone else, so be it: the winner will suffer from winner’s remorse and not he. He can find better investments for his 500 livres. The king may be able to exploit a few suckers, but he will run out sooner or later and be forced to accept a lower & fairer bid. (Or if there really are that many gullible wealthy office-buyers, then the king has hit upon a highly progressive taxation scheme!)
Or the king could create swarms of offices. Selling many offices a few times is as good as selling few offices many times.
But if the offices are empty, who will purchase them? In our day, few people seek to purchase or marry into honorable yet worthless titles, and the only titles respected are the ones either extremely high (Americans scoff at anything less than royalty), or are well-merited (England’s knighthoods).
And if the offices are not empty, if they come with remunerative opportunities or power, then where could those powers or opportunities come from, but another office? It is a zero-sum game.
If the sovereign is regularly expanding his domain and has new positions to fill, some office-holder is losing out on opportunity to gain by selling or filling the office himself. Suppose a district grows and now requires 2 judges to handle all cases; the King may sell a new judgeship, but by doing so, hasn’t he deprived the attorney-general of the opportunity? As the economists will tell you, a loss of a gain is as bad as a gain of a loss.
Thus, traffickers in barratry can be expected to oppose such abuses by the King, or indeed any change from the status quo:
“…[France and Spain] both had close personal and political ties to the Curia. Thus, it seems very likely that they were following the lead of the papacy in selling offices as a financial expedient to pay for their military ventures. Alas, venality was a Rubicon of sorts. Once created, venal offices were virtually impossible to destroy; for venal officeholders became powerful veto holders, who would oppose reform tooth and claw, something that the great French ministers of state from Richelieu on would learn to their chagrin.”6
### Official abuse
Perhaps the sovereign has no incentive to monkey around if there is an efficient market for offices. But the sovereign is just one piece of the puzzle. What about the officials themselves? What stops them from being the sponsored creature of some private interest, such as corporations? Regulatory capture is an issue with salaried civil services despite many laws and regulations; doesn’t a system of barratry make regulatory capture universal? There are many businesses which would like to legally buy an official.
But what one business buys today, another can tomorrow. If there are 2 companies competing over land usage - a railroad & an airline - and 1 judgeship would decide the issue, then the resolution is simple: whichever is more profitable will be able to outbid the other, leading to a more efficient outcome (just an application of the general capitalistic principle that the high bidder is more worthy of resources than the unprofitable losers). As with this instance, so with them all. In the end, all offices will end up being controlled by whomever they are most valuable to; this is the Coase theorem in action.
## Who watches the salarymen?
On the gripping hand, salaried bureaucracy has its own abuses:
• when large enough, the bureaucracy can literally vote in candidates who promise it money or increased funding7
• it inherently seeks to aggrandize itself; see Parkinson’s Law
• disconnected from the market-place, it has all the vices of capitalism (corruption) and none of its virtues (an inherent pressure towards competence) The low salaries which at first seem to be more economically efficient also mean that officials can be corrupted for trivial sums - federal senators and representatives will scurry to do the bidding of donors who gave a few thousand dollars.
• Institutional inertia can mean the administrative tail wags the elected dog. (Here we have the reverse of offices churning faster than kings!)
• The civil service exams do not assure competence
• In the case of China, the stress on ossified academic qualifications has been cited as having been a malign influence on the wider culture
Does selling offices remedy all of those ills? As we have seen, there is competition from official peers, so tenure is never assured & greed is punished by customer counter-measures. Individual officials are like a salesman working on commission, have an incentive to handle as many duties as possible: useless subordinates are just soaking up bribe-money the official could pocket. And as a whole, office-holders seek not an expansion of government bureaucracy, but a diminishment. Certainly they would not vote for a Napoleon III who will add millions to the civil service & dilute their dearly-bought prerogatives.
Does it fall prey to abuses of its own? Might it be a cure worse than the disease? Well, it is somewhat notable that most Western industrialized democracies do not sell any but the highest offices, and they generally are at the very top in various measures of governmental & economic efficiency. But it’s also worth noting that some governments like Singapore pay officials astronomical salaries to elicit performance, sums likely surpassing what they could earn via bribes. Further, many countries (particularly in Africa) with civil services and bureaucracies modeled after the Western exemplars wind up with administration worse than they had before, as rife with corruption & incompetence, though focused at simple nepotism or minimal personal exertion - suggesting that the superior Western performance may simply be a reflection of other factors such as being wealthy industrialized Western nations or culture, and not due to superior bureaucracy. I’ve not heard that the US federal government signally improved when patronage was officially ended in the late 1800s, for example, and Britain during the height of empire continued to sell off military and foreign positions.
1. “How ridiculous it is that a few hundred Englishmen should dominate India, a continent as big as Europe!” –Josef Stalin, to Joachim von Ribbentrop (quoted pg 2 of Looking back on India by Hubert Evans, and pg 176 of India’s Bismarck, Sardar Vallabhbhai Patel by B. Krishna; cf. the review of David Gilmour’s The Ruling Caste: Imperial Lives in the Victorian Raj)
2. Alexis de Tocqueville, Book 2, chapter 3 of The Old Regime and the Revolution (translated by Alan S. Kahn; ISBN0-226-80529-8)
3. Edward Gibbon, Decline and Fall, c. xvii
“Once a sufficiently high percentage of voters are unionized public employees, there is essentially no limit to the obligations imposed on the state. Because it would cause too much backlash from non-union non-government employed voters, most of the money extracted from taxpayers will be taken in the form of long-term health care and pension promises. A voter working at Wal-mart gets upset hearing that a bus driver is earning $130,000 per year. If instead the bus driver is paid$70,000 per year and able to retire at age 41 (MBTA here in Boston), it is tougher for a voter to figure out how much is being spent..” |
## Preference relationsEdit
The preference relation provides a foundation upon which classical microeconomics erects a theory of rational choice. This section describes preference relations and their properties. The rational preferences approach to studying human decision making treats preferences as given, imposing axiomatic assumptions intended to represent rational choice. We begin by envisioning a set of mutually exclusive choices facing a decision maker, X. The binary relation ${\displaystyle \succeq }$ represents a preference over the elements of X. In the natural language ${\displaystyle x\succsim y}$ is read as "x is at least as good as y" or "x is weakly preferred to y".
Two useful derivative relations are
Strict Preference: ${\displaystyle x\succ y\iff x\succsim y}$ and not ${\displaystyle y\succsim x}$ .
${\displaystyle x\succ y}$ is read as "x is preferred to y", or "x is strictly preferred to y".
Indifference: ${\displaystyle x\sim y\iff x\succsim y{\text{ and }}y\succsim x}$ .
${\displaystyle x\sim y}$ is read as "x is indifferent to y".
### Rational PreferencesEdit
The preference relation ${\displaystyle \succsim }$ on X is called rational if it is both complete and transitive. Completeness requires all pairs ${\displaystyle (x,y)}$ can be compared.
Completeness: ${\displaystyle \forall (x,y)\in \mathbf {X} }$ , either ${\displaystyle x\succeq y,\quad y\succeq x}$ , or both.
Transitivity imposes a 'consistency' requirement, enabling a ranking or ordinal mapping onto the elements of X. For example, if a shopper strictly prefers apples to oranges and prefers oranges to bananas, the transitivity assumption requires him/her to also prefer apples to bananas.
Transitivity: For all choices x, y, and z, if ${\displaystyle x\succeq y\land y\succeq z}$ then ${\displaystyle x\succeq z}$
properties of rational preferences:
1. ${\displaystyle \succeq }$ is reflexive ${\displaystyle \Leftrightarrow \forall x\in \mathbf {X} \,x\succeq x}$ . This is implied by completeness.
2. ${\displaystyle \succ }$ is irreflexive ${\displaystyle \Leftrightarrow x\succ x}$ never holds
3. ${\displaystyle \succ }$ is transitive ${\displaystyle \Leftrightarrow x\sim y\land y\sim z\Leftrightarrow z\sim z}$
4. ${\displaystyle \sim }$ is reflexive.
5. ${\displaystyle \sim }$ is transitive ${\displaystyle \Leftrightarrow x\succ y\succsim z\Leftrightarrow x\succ z}$
6. ${\displaystyle \sim }$ is symmetric ${\displaystyle \Leftrightarrow x\sim y\Rightarrow y\sim x}$ .
7. If ${\displaystyle x\succ y}$ and ${\displaystyle y\succeq z}$ then ${\displaystyle x\succ z}$ .
A moment of reflection reveals the gravity of the rationality assumption. In colloquial terms, rationality implies individuals fully know and understand their preferences. In any moment the decision maker could be called upon to act it must be able to provide a complete, consistent ranking of all elements in the choice set. Note, however, the rationality assumption places no restrictions on the subjective quality of preferences, nor has this framework imposed any requirements on the information or computational ability available to a decision maker. Simply, the subjectivity of preferences survives an assumption of rationality.
A number of assumptions often facilitate formal analysis. The partial list included below serves as reference.
### Desirability assumptionsEdit
• Monotonicity: The relation ${\displaystyle \succsim }$ is (weakly) monotone if ${\displaystyle x\in \mathbf {X} \land y\geq x\Rightarrow y\succsim x}$ and strongly monotone if ${\displaystyle x\geq y\land y\neq x\Rightarrow y\sim x}$
The monotonicity assumption translates roughly to "more is better". Though trivial in some instances, choice sets including economic 'bads' such as pollution may violate monotonicity.
• Local nonsatiation: ${\displaystyle \succsim }$ is nonsatiated if ${\displaystyle \forall x\in \mathbf {X} ,\;\forall \varepsilon >0\;\exists y\in \mathbf {X} {\mbox{ such that }}|y-x|\leq \varepsilon \land \;y\sim x}$ Notice that monotonicity, translating to "more is better, even for infinitesimal deviations", implies local non-satiation, but not vice-versa. The assumption of non-satiation plays a key role in the discussion of indifference sets and indifference curves, below.
### ConvexityEdit
• ${\displaystyle \succsim }$ on X is convex if ${\displaystyle \forall x\in \mathbf {X} {\mbox{ the upper contour }}y\in \mathbf {X} :y\succsim x{\mbox{ is convex; if}}y\succsim x\land z\succsim x,\;(\alpha y+(1-\alpha )z)\succsim x\forall \alpha \in [0,1]}$
• Convex preferences imply diminishing marginal rates of substitution. For any two goods, increasingly large amounts of one good are required to compensate for marginal losses of the other.
• Convex preferences may be interpreted as a desire for diversification in consumption
### Homothetic PreferencesEdit
Preferences are said to be homothetic if all indifference sets are related by proportional expansion along rays, ${\displaystyle x\sim y,\;\alpha x\sim \alpha y\;\forall \alpha \geq 0}$
### Quasilinear PreferencesEdit
${\displaystyle \succsim {\mbox{ on }}\mathbf {X} }$ is quasilinear with respect to its numeraire if:
1. indifference sets are parallel displacements along the axis of commodity 1;
${\displaystyle x\sim y\Rightarrow (x+\alpha e_{1})\sim (y+\alpha e_{1}),{\mbox{ where }}e_{1}=(1,0,\dots ,0){\mbox{ and }}\alpha \in \mathbb {R} }$
2. the numéraire is desirable, |
# Using the phase diagram for H_2O, how would you describe water at 0°C and 1 atm?
At this pressure and temperature water could exist as a liquid, as a solid, or as both.
#### Explanation:
So what does this mean?
The freezing temperature of water is also its melting temperature. At a temperature of ${0}^{o} C$ water can either melt or freeze.
If you have ice at this temperature and pressure and add heat to the ice it will melt.
Liquid water at this temperature will freeze it heat is removed from the water.
The same events would also occur if you have a mixture of some liquid water and some solid ice. Adding heat will cause an increase in melting. Removing heat will cause an increase in freezing.
Note: The temperature of the water will only go below the freezing point after all of the water has turned into ice. Likewise, the temperature can only be raised above ${0}^{o} C$ after all ice has melted.
This video provides further discussion of phase changes for water and takes a closer look at phase diagrams.
Hope this helps! |
I'm studying mechanisms of integrity and authentication in symmetric encryption scenarios. I want to propose some examples to see whether I got the point here:
Let $m$ be the message, $c$ the ciphertext, $h$ the hashed message and $t$ the tag resulting of applying MAC.
Example 1:
Here Alice wants to send an enciphered message to Bob providing authentication and integrity but without using hash functions. Both parties agree on two different keys, $k_{1}$ and $k_{2}$. Alice applies $c=Enc_{k_{1}}(m)$ and computes $t=MAC_{k_{2}}(m)$. Then she sends $c$ and $t$ to Bob. Bob applies $m=Dec_{k_{1}}(c)$ and verifies $t'=MAC_{k_{2}}(m)$ comparing it to Alice's $t$.
Example 2:
Now Alice wants to send an enciphered message to Bob but also hashing the message $m$ for computing the MAC (so HMAC comes in). Both parties agree on two different keys (again), $k_{1}$ and $k_{2}$. Alice applies $c=Enc_{k_{1}}(m)$, computes the hash over the message $h=H(m)$ and finally computes $t=MAC_{k_{2}}(h)$. She sends $c$ and $t$ to Bob. Bob now deciphers $m=Dec_{k_{1}}(c)$, computes the hash $h=H(m)$ and verifies the MAC $t'=MAC_{k_{2}}(h)$ comparing it to Alice's $t$.
In case of CBC-MAC, I have read that both parties must agree on a fixed message length, since an attacker could forge a valid MAC. Is this issue solved when using HMAC?
Do you consider these two examples secure? Am I right or mistaken? Specially in the last one, where both parties use hashes with MAC (I have special interest in that one).
Thanks.
• Actually, that's not the standard meaning of HMAC; HMAC usually refers to a specific MAC based on a hash function, specifically, $Hash( (OPAD \oplus K) | Hash( (IPAD \oplus K ) | Message ) )$ – poncho Mar 30 '16 at 2:09
There are a few things wrong with your scheme:
1. if padding is used then your scheme may be vulnerable to padding oracle attacks, this is because decryption happens before MAC verification (see answer of curious);
2. encrypt-and-MAC doesn't provide confidentiality as you can clearly distinguish identical plaintext as the authentication tag $t$ will be identical as well (see the comment of CodesInChaos below the answer of curious).
Notes:
• It is a good idea to study the link that curious provides in the answer to understand more of the underlying issues;
• HMAC is a specific construct (using just the hash as underlying primitive); it is not hash-then-CBC-MAC;
• The issues of CBC-MAC are readily solved (for block ciphers that use 16 byte block size such as AES) by using the CMAC construction which is based on CBC-MAC but doesn't suffer the same issues for dynamically sized input.
• I found that Mac-Then-Encrypt has been superseded by Encrypt-Then-MAC in TLS when possible. I guess that in some scenarios this isn't supported by both parties. Would be using MtE still considered secure? – kub0x Apr 1 '16 at 22:13
Your techniques are not secure. It is the Encrypt and Mac method. Provides integrity to the message but not to the ciphertext. Check this answer. The most secure is to encrypt and then apply the mac to the ciphertext, or to apply an authenticated cipher to the message
• Also the biggest issue with encypt-and-MAC is that it doesn't provide confidentiality. – CodesInChaos Mar 30 '16 at 7:32
Thanks to you all for your replies, specially @Maarten Bodewes who has given a complete answer to my questions.
So thing is that my scheme (Encrypt-and-MAC) doesn't provide confidentiality since the computation of two identical messages yields the samme tag/result. In terms of cryptography it will seem like: $Enc_{k_{1}}(m)$ $||$ $MAC_{k_{2}}(m)$. Notice that it provides integrity.
Also MAC-then-Encrypt is a bad choice since $Enc_{k_{1}}(m || MAC_{k_{2}}(m))$ doesn't provide integrity over the ciphertext.
Best option will be Encrypt-then-MAC: $c=Enc_{k_{1}}(m)$ then send $c$ $||$ $MAC_{k_{2}}(c)$ so confidentiality and integrity are achieved.
I viewed the lesson that covers HMAC here (Princeston University) -> https://www.youtube.com/watch?v=krJ3fHjXYlc Maybe I misunderstood the concept (though the Poncho's answer isn't explained there).
Finally, I would appreciate an explanation of why two keys $k_{1}$ and $k_{2}$ are needed, $k_{1}$ for plaintext encryption and $k_{2}$ for message authentication.
Thanks.
• I've found the answer to my question regarding of the use of two keys for encryption and authentication in crypto.stackexchange.com/questions/8081/… We could derive two keys from the SHA-256 of our Master Key, or use a scheme as GCM which permits us to use one key for auth and encryption. – kub0x Apr 1 '16 at 22:19 |
Is there a simple formula for the gravitational self-force (due to emission of gravitational waves) in the classical limit?
UPDATE: To clarify things in response to comments below, I want to reformulate my question in a very concise and abundantly clear form. Let's consider an object of a known mass M that moves under the action of some varying external force and right now has a known velocity v (i.e., a known vector of velocity), a known acceleration a (again, a vector), a known first derivative of acceleration da/dt, and so on. What is the gravitational self-force (due to emission of gravitational waves by the object itself) acting on the object right now, in the vector form? The classical limit is implied, i.e., the velocity is small, the gravitational field is weak, and whatever else needs to be small or weak is small or weak.
ONE MORE CLARIFICATION: I saw a formula for the time-averaged power of emission of gravitational waves by a binary system, but I am interested in the momentary(!) force(!) as a function of the momentary velocity and its time derivatives of all orders. Even if you tell me the momentary, not time-averaged, power of emission, it still does not define the force, as the latter may be directed at any unknown angle with respect to the velocity.
The original version of my question, now slightly cut, is below.
The Abraham-Lorentz force is the recoil force on an accelerating or decelerating charged particle caused by the particle emitting electromagnetic radiation and is equal to
$$\frac{Q^2}{ 6 π ε_0 c^3} \frac{d\mathbf{a}}{dt}$$,
where Q is the particle charge, $$ε_0$$ is the electric constant, c is the speed of light, and $$\frac{d\mathbf{a}}{dt}$$ is the time derivative of the vector of acceleration. This formula is derived in the limit of non-relativistic velocities.
Is there a similar formula for the gravitational force akin to the Abraham-Lorentz force, that is, for the recoil force on an accelerating or decelerating astrophysical object caused by the object emitting gravitational waves?
I am not interested in general bulky expressions in the tensor form; I want to have a simple formula that I could use to calculate this force acting on our planet as a result of it orbiting around the sun. I believe that the general expression of the general relativity theory can be simplified for that case, be it called the classical limit or whatever else.
Please note I do not want the assumption of circular motion to be made. I want to see how the force is expressed via the momentary values of acceleration and its time derivatives of any order, similar to the above formula for the Abraham-Lorentz force.
There are a couple of questions on this SE about that gravitational force (link1, link2), but, reading the answers to them and following the links provided, I was unable to find the formula I am looking for.
To explain my motivation, I am a student from Japan studying something completely unrelated to physics, but I loved physics at school and am curious about the degree of analogy between the Abraham-Lorentz force and its gravitational analogue. I recently had a conversation about that with someone, and we got very curious as to which order of the derivative of acceleration will pop up.
So please kindly give me the formula and, ideally, a reference to the source.
• This is a very complicated topic. See physics.stackexchange.com/q/220886. It is much simpler to consider the energy loss than the radiation-reaction force. Feb 24, 2020 at 6:45
• The Earth orbiting the Sun radiates about 200 watts of gravitational waves. This is almost certainly what you heard. See en.wikipedia.org/wiki/Gravitational_wave#Binaries Feb 24, 2020 at 6:47
• The general formula for the power radiated in gravitational waves by any system is on page 6 here: physics.usu.edu/Wheeler/GenRel2013/Notes/GravitationalWaves.pdf Feb 24, 2020 at 6:53
• It depends on the square of the third time derivative of the traceless mass quadrupole tensor. Feb 24, 2020 at 6:54
• Applied to the Earth-Sun system, this formula gives the results here: en.wikipedia.org/wiki/… Feb 24, 2020 at 6:55
One of my all time favorite papers is this one by Burke (1971). He works out radiative damping in GR, using a mathematical technique called matched asymptotic expansions. In the process he works out radiative damping for a simple harmonic oscillator connected to a string that extends off to infinity and the radiative damping of an oscillating charge due to EM radiation. Seeing those two results puts the gravitational result in excellent context.
The resistive force acting per unit volume on the mass density, $$\rho$$, as a result of the gravitational radiation emitted by the changing mass multipole moment $$q_{\ell m}(t)$$ is
$$F_R = - \frac{8\pi\sqrt{10}}{75} \rho\, r \sum_m Y_{21m} \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^5q_{2m},\quad\quad(\mathrm{eq.}\, 136)$$
where we have used the leading order quadrupole, $$\ell=2$$. $$Y_{21m}$$ is a vector spherical harmonic that appears as $$Y_{\ell,\ell-1,m}$$ in the full expression. This term arrises from the multipole expansion of the metric, in this case the current multipole (or momentum part of the metric, $$g_{ti}$$ for $$i\in\{r, \theta,\phi\}$$). This makes sense as the damping force should be connected to conservation of momentum of the radiation.
Burke makes sure to emphasize some pitfalls in calculating the radiation reaction in GR compared to EM:
Another point is that one cannot always compute the resistive force from the force law $$F_R = \frac{1}{4}m\nabla\psi$$
(note: compare to $$F_R = -q\nabla\phi$$ for the EM radiation case.)
Before the gauge transformation was used to simplify the problem, the lower $$\ell$$ dependence of the vector and tensor potentials allowed them to compete with the scalar potential ($$\psi$$), and one had to use a more complete force law (derivable by writing the equations for a geodesic...).
...
The big difference between gravity and electromagnetism now appears. The electromagnetic field produces effects only through its force law. On the other hand, not only does the gravitational field affect the coordinate motion of the system, but the potentials themselves determine the clock rates and the behavior of rigid bodies. Until one knows the [tensor potential], one cannot convert coordinate differences to proper length without making errors that are $$\mathcal{O}(\kappa)$$,
where Burke defines the weak field parameter $$\kappa \sim GM/(c^2r)$$. For gravitationally bound systems $$\kappa \sim (v/c)^2$$ is related to the slow motion condition too.
• I would be very interested to know what $F_R$ evaluates to for the case of a small mass in circular orbit around a large mass. Have you ever calculated this, or seen this calculation? Feb 27, 2020 at 6:17
The gravitational self-force is not gauge invariant quantity. It only truly makes sense averaged over a sufficient amount of time. That being said one can calculate the amount of linear momentum that an object emits (averaged over an appropriate length of time) based on the time derivatives. The formula for this was derived by Kip Thorne in 1980 (ref), to lowest orders in the post-Newtonian approximation. By conservation of linear momentum the object must encounter a radiation reaction force in the opposite (in units where $$G=c=1$$)
$$F_{i,RR} = -\frac{dP_i}{dt} = -\Big\{\frac{2}{63} I^{(4)}_{ijk}I^{(3)}_{jk} +\frac{16}{45}\epsilon_{ijk}I^{(3)}_{jl} J^{(3)}_{kl} \Big\}$$
Here
• $$I_{ij}$$ is the (symmetric trace free,STF) mass quadrupole moment,
• $$I_{ijk}$$ is the (STF) mass octopole moment,
• $$J_{ij}$$ is the (STF) current quadrupole moment,
• $$\epsilon_{ijk}$$ is the Levi-Civita symbol, and
• superscripts $$^{(n)}$$ denote the $$n$$th time derivative.
This expression gives the total radiation reaction force on the system to which the multipole moments apply. In particular, it will not give the force on the individual masses in a binary. In addition to this radiation reaction force, the system as a whole will experience a torque due the emission of gravitational waves,
$$T_{i,RR} = -\frac{dJ_i}{dt} = -\frac{2}{5}\epsilon_{ijk}\Big\{\ I^{(2)}_{jl}I^{(3)}_{kl} +\frac{5}{126}I^{(3)}_{jlm} I^{(4)}_{klm} +\frac{16}{9}J^{(2)}_{jl} J^{(3)}_{kl}\Big\}.$$
These two together would allow you to construct some version of the force on two objects in a mutual circular orbit. Note however, that in doing so in principle you could still add arbitrary forces for which the net force and net torque on the system cancel. These are remaining gauge ambiguities that cannot be resolved a priori without choosing some gauge condition.
In addition the emission of gravitational waves can produce stresses and shears in the system. If the system has any additional vibrational modes (e.g. eccentricity in a binary) these can be excited or damped by the radiation reaction forces.
• Comments are not for extended discussion; this conversation has been moved to chat. Feb 25, 2020 at 12:36
• I agree that this is the radiated momentum but I disagree that it gives the radiation reaction force. This term is of order $1/c^7$. The leading terms for the radiation reaction force are known to be of order $1/c^5$. Just Google "radiation reaction 2.5 pn". This term is instead responsible for the "recoil" effect. Feb 27, 2020 at 6:08
• In the next few days I hope to write an answer showing how the 2.5PN radiation reaction force leads to the correct $dE/dt$ and $dL/dt$ (from Thorne's multipole expansions) in the simplest case of a circular orbit of a small mass around a large mass. Your force is too small by a factor of $(v/c)^2$ to explain these. Feb 27, 2020 at 6:11
• @G.Smith It is the radiation reaction on the system as a whole. I.e. when applied to a binary it is the effective radiation reaction to the binary's center of mass. Feb 27, 2020 at 7:04
• OK, I agree with that. But it’s not the gravitational radiation reaction force on a single mass (e.g., the Earth), which I believe is what the OP was asking about. Feb 27, 2020 at 7:49 |
# Mounting Macintosh Drives
Date: 2019-04-08
Last Updated: 2019-04-08
So this March I decided it was time to give my MacBook Pro a new lease of life. It still had its original 500GB HDD that it had come with. I had not updated my version of OS X since Yosemite in 2015, and that is because I used Apple’s Aperture for processing my photographs and Yosemite was the last version of OS X to support Aperture. This of course had the knock on effect of making it harder and harder to use Homebrew as time wore on.
So in a bout of frustration I went out and decided to buy an SSD with the idea being that I could swap my HDD out do a fresh install of MacOS, as it is now, and if worst came to worst and I wanted Aperture back then I could wipe the SSD and clone the old HDD to it.
The install of Mojave took a scenic route via Mountain Lion (yes you read that correctly), but the fun came when I tried to load my old HDD as an external drive. Strangely my MacBook couldn’t mount the drive. I happened to be at work while I was trying to do this, and my colleague suggested putting it into his Ubuntu workstation to see if it showed up. One apt-get install hfsprogs later and we could see the system and the partitions that my Macs (personal & work) say don’t exist. Later we tried to mount it using an Arch workstation and the SATA to USB adapter I had borrowed for the task, and it couldn’t be mounted.
We then decided that we should clone the drive before messing with it any further, to do that we pulled out my old Debian workstation, and put an Arch live USB in it and, when saw that the three partitions were showing up there we decided to try an mount the drive. We had some limited constraints from the live USB, and needed to do some arithmetic and sector counting in order to be able to mount the drive.
It was a superuser article (below) that enabled us to figure out what the hell was going on and how to mount the drive on Arch. We needed to calculate N, the volume size of what’s to be mounted.
[ N = logicalSectorSize \times sizeOfTheVolume ]
We do that by getting the logical sector size from fdisk and the size of the volume from testdisk
### Flow
• fdisk -l /dev/sda Get the logical sector size
• testdisk /dev/sda
• Select EFI GPT
• Select Analyse
• Then Quick Search Note down the size in sectors of sda2.
• mount -t hfsplus -o ro,sizelimit=N /dev/sda2 /Volumes
Homebrew
Superuser Article |
# Is the following metric is complete ?
Which of the following metric spaces are complete?
(a) The space $C^1[0, 1]$ of continuously differentiable real-valued functions on $[0, 1]$ with the metric $d(f, g) = \max_{t∈[0,1]}|f(t) − g(t)|$
(b) The space $C[0, 1]$ of continuously differentiable real-valued functions on $[0, 1]$ with the metric $d(f, g) = \max_{t∈[0,1]}|f(t) − g(t)|$
(c) The space $C[0, 1]$ with the metric $d(f, g) =∫_0^1 |f(t) − g(t)| dt$.
(d)The space $C^1[0, 1]$ with the metric $d(f, g) =∫_0^1 |f(t) − g(t)| dt$.
my attempts: from the Weierstrass Approximation Theorem option a), option b) are true and option C) and option D) are incorrect ........ as
IS my answer is correct or not ? pliz verified and tell me the solution if u have a time ...i would be more grateful....
a) is NO from Weierstrass: you can approximate all continuous functions uniformly by polynomials (which are in $C^1([0,1])$). Also functions that are continuous and non-differentiable. Such an approximation sequence is Cauchy in the uniform metric on $C^1$ but has no limit in $C^1([0,1])$ etc. Also, this uniformly convergent sequence will be Cauchy in the integral metric too (simple estimates show this), and so will give a negative answer to d) as well. |
# Why is \scshape not changing in XeLaTeX?
I'm trying to alter \scshape in XeLaTeX, but for some reason, three reasonable attempts (xpatch, \def, and \renewcommand) seem to have no effect whatsoever (only in XeLaTeX). A nonworking example is below (note that the 5 em space is just to make it clear when the redefinition is working, and obviously not intended for use in a real document):
\documentclass{article}
\usepackage{xpatch}
\xapptocmd{\scshape}
{\spaceskip=5em}
{}{}
%\let\myscshape\scshape
%\def\scshape{\spaceskip=5em\myscshape}
%\renewcommand{\scshape}{\myscshape\spaceskip=5em}
\begin{document}
\noindent\textsc{Test thing test}\\
\scshape Test thing test\\
\scshape\spaceskip=5em Test thing test
\end{document}
• The given code doesn't change anything in lualatex either. \scshape might be protected. If you append to a hook (parameterless macro), \appto{\xscshape}{\spaceskip=5em >>\color{red}}, it works; if you append to an unprotected macro, \newcommand\xscshape{\scshape}\xapptocmd{\xscshape} {\spaceskip=5em >>\color{red}} {}{}, it also works. The prepends also work. If you pre/append to a command (=parametered macro, like \textsc), the replacement text is per/appended to as expected. Jun 24 at 3:39
Do the patch at begin document.
\documentclass{article}
\usepackage{xpatch}
\usepackage{fontspec}
\AtBeginDocument{%
\xapptocmd{\scshape}
{\spaceskip=5em}
{}{}%
}
\begin{document}
\textsc{Test thing test}
\scshape Test thing test
\end{document}
• Brilliant—many thanks! Jun 24 at 15:14
• Out of intellectual curiousity, why does using \AtBeginDocument make it work? Jun 24 at 15:15
• @ezgranet A great part of font setup is done at begin document; different font packages may have different ideas about how to realize small caps. Jun 24 at 15:47
• thank you—very interesting. Jun 24 at 16:57 |
# Trigonometry Question from a Math Contest
1. Jan 5, 2005
### MathExpert
sin^8(x) + cos^8(x) = 97 / 128
2. Jan 5, 2005
### Zurtex
The only way I can think off the top of my head is say it is equivalent to:
$$128 \left[ \sin^8x + \left(1 - \sin^2x \right)^4 \right] - 97 = 0$$
And really hope it is solvable.
3. Jan 5, 2005
### gonzo
I'd just keep pluggin in trig identities until you reduce things either all to single powers or maybe a quadratic equation of cosines. It shouldn't be too hard, you can use
2cos^2(x) = 1+cos(2x)
2sin^2(x) = 1 - cos(2x)
4. Jan 5, 2005
### krab
It would be hard to prove because it's not true.
Edit: Sorry. I assumed you meant it as being true for all x. Do you it is to be solved for x?
Last edited: Jan 5, 2005
5. Jan 5, 2005
### dextercioby
It can be reduced to two biquadratic equations,which are solvable.
$$1=[(\sin^{2}x+\cos^{2}x)^{2}]^{2}=(\sin^{4}x+\cos^{4}x+2\sin^{2}x\cos^{2}x)^{2}$$
$$=\sin^{8}x+\cos^{8}x+4\sin^{4}x \cos^{4}x+2\sin^{4}x \cos^{4}x+4\sin^{2}x \cos^{2}x (\cos^{4}x+\sin^{4}x)$$ (1)
From (1) it follows:
$$1=\sin^{8}x+\cos^{8}x+6\sin^{4}x \cos^{4}x+4\sin^{2}x\cos^{2}x[(\sin^{2}x+\cos^{2}x)^{2}-2\sin^{2}x\cos^{2}x]$$
$$=\sin^{8}x+\cos^{8}x-2\sin^{4}x \cos^{4}x+4\sin^{2}x\cos^{2}x$$ (2)
From (2) it follows:
$$\sin^{8}x+\cos^{8}x=1+2\sin^{4}x\cos^{4}x-4\sin^{2}x\cos^{2}x$$ (3)
I use:
$$\sin 2x=2\sin x\cos x$$(4)
to write
$$+2\sin^{4}x\cos^{4}x=+\frac{1}{8}(\sin 2x)^{4}$$ (5)
$$-4\sin^{2}x\cos^{2}x=-(\sin 2x)^{2}$$ (6)
Therefore,combining (5),(6) and (3),one gets:
$$\sin^{8}x+\cos^{8}x=1+\frac{1}{8}(\sin 2x)^{4}-(\sin 2x)^{2}$$ (7)
The initial equation
$$\sin^{8}x+\cos^{8}x=\frac{97}{128}$$ (8)
,using (7),becomes
$$1+\frac{1}{8}(\sin 2x)^{4}-(\sin 2x)^{2}=\frac{97}{128}$$ (9)
,which can be put in the form
$$\frac{1}{8}}(\sin 2x)^{4}-(\sin 2x)^{2}+\frac{31}{128}=0$$ (10)
Multiplying (10) by $+8$ and making the obvious substitution
$$(\sin 2x)\rightarrow y$$ (11)
,the equation (8) is brought to the canonical form of an biquadratic algebraic eq. in the variable "y":
$$y^{4}-8y^{2}+\frac{31}{16}=0$$(12)
We make the substitution:$y^{2}\rightarrow z$ (13)
,and the eq.(12) becomes
$$z^{2}-8z+\frac{31}{16}=0$$ (14)
,with the solutions
$$z_{1}=4+\frac{\sqrt{225}}{4}=\frac{1}{4}$$ (15)
$$z_{2}=4-\frac{\sqrt{225}}{4}= \frac{31}{4}$$(16)
Comparing (11),(13) and (16),we conclude that $z_{2}$ is not a viable solution.
To be continued.
Last edited: Jan 5, 2005
6. Jan 5, 2005
### dextercioby
From (13) and (15),we obtain:
$$y^{2}=+\frac{1}{4}$$ (17)
,with the solutions:
$$y_{1}=-\frac{1}{2}$$ (18)
$$y_{2}=+\frac{1}{2}$$ (19)
It can be easily checked that both solutions (18) and (19) are found in the interval (-1,+1),so they provide viable solutions to the problem.
Putting together (18),(19) and (11),we found two transcendental equations
$$\sin 2x=-\frac{1}{2}$$ (20)
$$\sin 2x=+\frac{1}{2}$$ (21)
We solve these eq.in the interval $(-\frac{\pi}{2},+\frac{\pi}{2})$(for the 'sine' argument,which is 2x).In this case,we can apply the inverse function 'arcsine' to find
$$x_{1}=\frac{1}{2}\arcsin(-\frac{1}{2})=-\frac{\pi}{12}$$ (22)
$$x_{2}=\frac{1}{2}\arcsin(+\frac{1}{2})=+\frac{\pi}{12}$$ (23)
Daniel.
EDIT:Thank you,Krab,i wouldn't have seen it,years after...
Last edited: Jan 5, 2005
7. Jan 5, 2005
### krab
Looks like it's $\pi/12$
8. Jan 5, 2005
### dextercioby
Let's take $\pi\sim 3.14$
$$\sin\frac{\pi}{12}\sim 0.25869$$
$$\sin^{8}\frac{\pi}{12}\sim 2\cdot 10^{-5}$$(1)
$$\cos\frac{\pi}{12}\sim 0.96596$$
$$\cos^{8}\frac{\pi}{12}\sim 75800\cdot 10^{-5}$$ (2)
Add (1) and (2) and get:
$$\sin^{8}\frac{\pi}{12}+\cos^{8}\frac{\pi}{12}\sim 75802\cdot 10^{-5} \sim\frac{97}{128}$$ (3)
Daniel.
Last edited: Jan 5, 2005
9. Jan 5, 2005
### Hurkyl
Staff Emeritus
Let s = sin^2 x, c = cos^2 x be a solution to the equation.
As Zurtex Started, this equation is a quartic in s:
97/128 = s^4 + (1-s)^4
The symmetry suggests making the substitution s = t + 1/2
Note that t is in the range -1/2 <= t <= 1/2, if we assume to look for real solutions.
97/128 = (1/2 + t)^4 + (1/2 - t)^4
= 2t^4 + 3t^2 + 1/8 = 2t^4 + 3t^2 + 16/128
so
2t^4 + 3t^2 - 81/128 = 0
yielding the solution:
t^2 = -3/4 +/- (1/4) sqrt(9 + 81/16)
= -3/4 +/- (1/4) sqrt(225/16)
= -3/4 +/- (1/4) (15/4)
= -12/16 +/- 15/16
= (-12 +/- 15) / 16
t^2 = -27 / 16 or 3 / 16
However, we know that 0 <= t^2 <= 9/4, so we have the single solution:
t^2 = 3/16
so
t = sqrt(3) / 4
s = t + 1/2 = (2 + sqrt(3)) / 4
Now, s = sin^2 x, so we have:
sin x = (1/2) sqrt(2 + sqrt(3))
Where both square roots could be either positive or negative.
For some reason, I feel that knowing a lot of solutions to the original equation should help me -- if x is a solution, then so is -x, ix, and 90 - x, from which you can make lots more solutions -- but I couldn't get such an approach to help.
Last edited: Jan 5, 2005
10. Jan 5, 2005
### krab
But it's not meant to be 1; it's meant to be 97/128, which it is.
BTW, I found your error. In your equation 1, your last 2 should be a 4. Apart from this, the technique is really nice.
11. Jan 5, 2005
### dextercioby
Hurkyl,i found two solutions in the reals,in the interval $(-\frac{\pi}{4},+\frac{\pi}{4})$,because i found it easier to use 'arcsin'.
Solving the eq.
$$\sin 2x=\pm \frac{1}{2}$$
on the whole real exis would yield all real solutions to the equation.
If u wanna inlcude complex solutions as well,u should use not circular,but hyperbolic trigonometry (identities and stuff).
Daniel.
12. Jan 6, 2005
### Curious3141
The solutions have already been given, but perhaps the most "direct" way to do this would be like so :
Let $c = \cos \theta$, $s = \sin \theta$, $S = \sin 2\theta$
$$s^8 + c^8 = 97/128$$
$$(c^4 + s^4)^2 - 2(c^2s^2)^2 = \frac {97}{128}$$
$$((c^2 + s^2)^2 - 2c^2s^2)^2 - 2(\frac{1}{4}S^2)^2 = \frac {97}{128}$$
$$(1 - \frac{1}{2}S^2)^2 - 2(\frac{1}{4}S^2)^2 = \frac {97}{128}$$
$$(1 - \frac{1}{2}S^2)^2 - \frac{1}{8}S^4 = \frac {97}{128}$$
Expanding the LHS, rearranging and simplifying,
$$\frac{1}{8}S^4 - S^2 + \frac{31}{128} = 0$$
which is a quadratic equation in S^2. Solving that will give the required general solution in the reals :
$$\theta = n\pi (+/-) \frac{\pi}{12}$$ or $$n\pi (+/-) \frac{5\pi}{12}$$
Last edited: Jan 6, 2005
13. Jan 7, 2005
### tongos
oh boy, anyone in highschool taking the amc 12 in a few weeks? |
# Science:Math Exam Resources/Courses/MATH100/December 2016/Question 14 (a)/Solution 1
Note that ${\displaystyle \sin \left({\frac {\pi }{2}}\right)=1,}$ ${\displaystyle \sin \left({\frac {3\pi }{2}}\right)=-1.}$
${\displaystyle \sin x-f(x)\leq |\sin x-f(x)|=|f(x)-\sin x|\leq {\frac {1}{3}}\implies f\left({\frac {\pi }{2}}\right)\geq \sin \left({\frac {\pi }{2}}\right)-{\frac {1}{3}}={\frac {2}{3}}>0}$ and
${\displaystyle f(x)-\sin x\leq |f(x)-\sin x|\leq {\frac {1}{3}}\implies f\left({\frac {3\pi }{2}}\right)\leq \sin \left({\frac {3\pi }{2}}\right)+{\frac {1}{3}}=-{\frac {2}{3}}<0}$.
So, by the intermediate value theorem (${\displaystyle f}$ is continuous by the first condition), ${\displaystyle f}$ has at least one zero in the interval ${\displaystyle (\pi /2,3\pi /2)}$, which is contained in the interval ${\displaystyle (0,2\pi ).}$. |
# Tag Info
## New answers tagged wavefunction
1
Basically any measurement is on wave function |ψ⟩ is done by operator X such that X|ψ⟩ results observable x with some probability. is not entirely correct. In quantum mechanics a system is described by a state $|\psi\rangle\in\mathcal{H}$ and a set of self-adjoint operators $A_1,\ldots,A_n$ representing the observables that we want to measure. Moreover, ...
-2
When a quantum mechanical object is subjected to a measurement its wavefunction has to be coupled to a measurement device. That coupling will change both the wavefunction of the object and the wavefunction of the measurement device (as well as the wavefunction of the environment!). It is important to understand that a measured quantum object is not the same ...
2
What is a wave function? It is the solution of a quantum mechanical equation ( with the appropriate potentials),on which boundary conditions are imposed to make it specific to a system . $|\psi\rangle$ by itself is not independent of the environment the way that the operators X are. Thus the answer depends on the system under consideration. I like ...
0
All evolution in Quantum Mechanics is done by the Hamiltonian. However, measurements of an operator result in a single eigenvalue, and the wavefunction collapses (As to how, why, etc, it is unknown as of now, Copenhagen is the most common interpretation) to an eigenstate of the operator measured with the same eigenvalue. Specifically, it collapses to the ...
0
There is no need of a potential for the Schrödinger equation to have a solution, namely $$i\hbar \frac{\partial}{\partial t} |\psi\rangle = H |\psi\rangle$$ does possess solutions even when the Hamiltonian contains no potential. Questions of normalisability may arise, but that is another point (and can anyway be solved by expanding in Fourier terms). ...
0
Yes, a free particle has a wavefunction. If it has sharp momentum $p$, it is given by the plane wave $$\psi_p(x) = \mathrm{e}^{\mathrm{i}px}$$ which might look a bit strange, because it is non-normalizable/not square-integrable, but that should not be all too troubling - no momentum uncertainty at all is rather unphysical, after all. The general ...
0
With this transformation $\rho = kr$ you can't possibly change the form of the equation, because it's a scale transformation that does nothing special to the derivatives, so I think that you have computed the derivatives wrong.
2
The standard procedure is the following: starting from $\langle nlm|\,\partial^2_z\,| n'l'm'\rangle$ insert the identity operator with respect to the position basis $$1 = \int d\textbf{r} |\textbf{r}\rangle\otimes\langle \textbf{r}|$$ to have $$\int d\textbf{r}\, \langle nlm\, |\,\partial^2_z\,|\textbf{r}\rangle\cdot\langle \textbf{r}|n'l'm'\rangle. ... 0 To keep things simple, let's talk about plane acoustic waves in one dimension. If we solve the wave equation in one dimension , we find that the acoustic pressure as a function of space and time is of the form$$P(x,t) = Ae^{i(kx -\omega t)}$$where A is the maximum amplitude, x and t are the displacement and time respectively, \omega is the ... 1 Solving time-dependent SE as danielsmw mentioned above good starting point.$$ i\hbar\frac{\partial\psi}{\partial t}=H\psi \frac{d\psi}{\psi}=\frac{H}{i\hbar}dt log(\psi)\mid^{\psi}_{\psi_{0}}=-\frac{iHt}{\hbar}\mid^{t}_{t_0} $$suppose \psi=|\alpha,t> and \psi_{0}=|\alpha, t_{0}>$$ \psi=e^{-\frac{iH(t-t_{0})}{\hbar}}\psi_0 $$Under ... 0 You may consider reading about Aharonov-Bohm effect. This is one of those cases, where the phase of the wave function, in sum with the electromagnetic 4-potential, is extremely important, as it gives different physical results. This effect was also checked experimentally, so it is not a pure theoretical abstraction. 1 You are making confusion between two different things: the first is why the wave function of two identical particles is what it is, the second is the probability of an event happening in several ways (which has nothing to do with physics but is just the definition of unions of probabilities). As per the first question: given \mathcal{H}_1 and ... 3 The visualization method you choose is directly and completely determined by the information you need to see regarding your state. For the states of a single bosonic mode, there are multiple different visualization methods, and they all have their pros and cons. In particular, there is a direct trade-off between the amount of information you can display on a ... 0 The question is begging a simple answer: Because nature can be represented as resonators that respond in a delayed way we adopt to study it using complex numbers. When an electronic circuit is excited by a regular periodic source of tension (measured as a real function - V_0\cos(\omega\cdot\ t) for instance) the most common answer is a variation in the ... 2 I don't want to be too much precise, but the Schrödinger equation (i \dot{\psi}= H\psi to avoid confusion) has at most an unique solution under very general assumptions on the Hamiltonian operator H, even if you see it as a liner equation in the more general setting of Banach spaces. In particular, it is not a priori necessary that H is self-adjoint ... 3 To understand what these orbitals are, you first have to understand the notion of superposition in quantum mechanics. In regular classical physics, a particle or a system must be in a definite state. A car is at a particular mile marker on a highway, moving at a particular speed. The Moon orbits around the Earth with a particular velocity at a particular ... 0 Yes,both the set of equalities are true but only in the position representation where operator(x)=x.In the momentum representation,where operator(x) takes a different form they are not true 2 The probability of just one of your atoms getting through the wall is very low, multiply that probability (which is way lower than 1) by the probabilty that all of your atoms go through, you get a number so small that the universe would almost certainly have ended long before you got through the wall. A 70 Kg human body has approximately 7 X 10^{27} ... 0 According to the diagram you provided, |a\rangle is injected into a beam-splitter, resulting in the out-put state, $$|\psi_{o}\rangle =B|a\rangle= e^{\frac{i\pi}{2}}|b\rangle +|c\rangle.$$ Here the beam-splitter unitary is denoted by B. I have expressed i as e^{\frac{i\pi}{2}}, to highlight the fact that |b\rangle picks ... 0 Generally speaking, bound states in a system give rise to resonant behaviour. The number of states defines the number of different resonances that can be observed. For example, consider the absorption of photons by an electron trapped in a quantum well: If the well only contains one bound state then it could absorb any photon with any energy greater than ... 1 Integration over pure quantum states usually refers to the Haar measure, i.e., the unitarily invariant measure. Vaguely speaking, you assign the same volume (=weight in the integral) to any two set of states which are related by an arbitrary unitary rotation U. In the case of one qubit, this is equivalent to integrating over the Bloch sphere; i.e., we ... 1 If I understand the question. Assume you have a quantum oscillator, then if the system is an eigenvector of the hamiltonian, then a series of measurements will give a series of results reflecting eigenvalues of the system. The appearance of these eigenvalues must obey at a number of many measurements the probabilities of each eigenvalue to appear. It's ... 3 Given the wavefunction \psi(x), the probability to find any particle within an interval [x_0,x_0+\Delta x] is$$ P([x_0,x+\Delta x_0]) = \int_{x_0}^{x_0+\Delta x} \lvert\psi(x)\rvert^2\mathrm{d}x$$i.e. \lvert\psi(x)\rvert^2 is not a probability, but a probability density that has to be integrated over a set of non-zero measure to yield a notion of ... 5 The square of the wavefunction, |\Psi(x,t)|^2, is the probability density function for finding the particle. This means that the probability of finding the particle in an interval of (infinitesimal) width \mathrm dx at position x equals |\Psi(x,t)|^2\mathrm dx. On occasion, however, authors will drop the \mathrm dx if it is convenient and does not ... 0 Gigi Butbala's answer in terms of sines and cosines is perfectly correct. But just to prove that it can be done, here's how you'd proceed from your two equations in terms of A and B: The first equation shows that you must have$$ B = - e^{-ikL} A $$and plugging this in to the second equation yields$$ A e^{ikL/2} - A e^{-3 ikL/2} = 0 \quad ...
0
You will have to settle for momentum since speed is not a quantum mechanical observable (because $\dot{x}$ is not a classical observable on the phase space, which are functions of $x$ and $p$, but a function of a classical trajectory $(x(t),p(t))$, so canonical quantization does not produce a "speed" observable). The probability for a certain momentum, ...
0
I could be wrong, but I remember reading that scanning tunneling microscopy can be used to measure $|\psi|^2$. A quick Google search brought me to this paper, "http://www.ncbi.nlm.nih.gov/pubmed/19090685." I'm sure you can find more information by looking into STM online.
1
The wave function is connected to an experimentally observable value through its complex conjugate square, which gives the probability of an interaction or a decay happening; from this a crossection for the interaction can be predicted, number of events versus some variable in appropriate units. Example: the experiment can measure very many decays of the ...
1
Experimental physicists are very visual people. They are more interested in seeing the wavefunction. Calculating it is the work of a theoretical physicist. "Catching sight of the elusive wavefunction", would be a good read for you at this point. As you proceed with Griffiths, you will see that in experiments we measure certain quantities, called the ...
2
In general, the integral $$V := \int \mathrm{d} \mu = \int 1 \mathrm{d}\mu$$ is the integration of the identity over the space the measure $\mu$ is defined on, and should be intuitively understood as the volume of the space with respect to the measure. (This is usually only finite for compact spaces.) Normalizing the measure means sending $\mu \mapsto ... -1 you can make any state if you properly choose bases as,$|\Psi> = \alpha|0> + \beta |1>$where$|0>$and$|1>$are assumed as the complete bases. In this case,$\alpha$and$\beta$are the probability amplitude to observe 0 or 1 respectively. Similarly if the system can be written by a continuous bases, like space$|x>$, any state can be ... 2 In the case of an infinite superposition of eigenstates it becomes more complicated but we can still write a general expression for it. If $$\psi(x) = \sum_{n=0}^\infty a_n \phi_n(x)$$ where the$\phi_n$are the eigenstates of the Hamiltonian. The time dependent wavefunction will look like: $$\Psi(x,t) = \sum_{n=0}^\infty a_n \phi_n(x) T_n(t)$$ where$T_n = ...
-1
In that case, you have a sum of overlaps between a pair of functions having same eigenvalue index. The cross overlap will be zero because of the orthogonality of basis set which is very important.
2
So, I suppose that $Φ(k)$ is the probability density of the momentum. Is this true? Almost. $\Phi(k)$ is the probability amplitude for the momentum of the particle. The probability density is obtained as usual by squaring the amplitude, giving $|\Phi(k)|^2$. For a free particle, all values of momentum are always allowed, which enables the superposition ...
2
First, realize we are doing an approximation when we are evaluating the coefficient $A_p$ in the Fourier series $$\psi(x) = \sum_p A_p \psi_p(x)$$ by the integral $$A_p = \int_{-\infty}^\infty \psi_p^*(x)\psi(x)\mathrm{d}x$$ since the limits should really be the boundary of the interval on which the Fourier series is periodic. Furthermore, \$\psi(x) = ...
4
The Schrodinger equation is an approximation because it ignores relativistic effects and it ignores spin. However, aside from these limits it applies to any system and not just electrons. The trouble is that the Schrodinger equation, and indeed most partial differential equations, are impossible to solve except in a few special cases. Since we have no ...
2
Your introductory claims are not correct. The usual wavefunction is not a function of spacetime, for two reasons: It is not Lorentz invariant (time is singled out, so it is not a function of a point on a spacetime manifold, and it is not even time dependent at all in the Heisenberg picture) and it can depend on more than one "set" of positions - for a system ...
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Derivative
# Derivative
WordNet
(1) Resulting from or employing derivation
"A derivative process"
"A highly derivative prose style"
### noun
(2) The result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx
(3) (linguistics) a word that is derived from another word
"`electricity' is a derivative of `electric'"
(4) A financial instrument whose value is based on another security
Wiktonary
### Etymology
Middle English, from dérivatif and derivatus; see derive.
1. Imitative of the work of someone else.
2. Referring to a work, such as a translation or adaptation, based on another work that may be subject to copyright restrictions.
3. Having a value that depends on an underlying asset of variable value.
4. Lacking originality.
### Noun
1. Something derived.
2. A word that derives from another one.
3. A financial instrument whose value depends on the valuation of an underlying asset; such as a warrant, an option etc.
4. A chemical derived from another.
5. The derived function of a function.
The derivative of $f:f\left(x\right) = x^2$ is $f\text{'}:f\text{'}\left(x\right) = 2x$
6. The value of this function for a given value of its independent variable.
The derivative of $f\left(x\right) = x^2$ at x = 3 is $f\text{'}\left(3\right) = 2*3 = 6$.
#### Hyponyms
option, warrant, swap, convertible security, convertible, convertible bond, credit default swap, credit line note, financial futures contract, financial future, total return swap. |
# biotmle: Targeted Learning for biomarker discovery
### Abstract
biotmle is an R package facilitating biomarker discovery by generalizing moderated statistics for use with targeted estimators of parameters with asymptotically linear representations.
Publication
In The Journal of Open Source Software
Date |
Properties of Real Cosine Function
Theorem
Let $x \in \R$ be a real number.
Let $\cos x$ be the cosine of $x$.
Then:
Cosine Function is Continuous
$\cos x$ is continuous on $\R$.
Cosine Function is Absolutely Convergent
$\cos x$ is absolutely convergent for all $x \in \R$.
Cosine of Zero is One
$\cos 0 = 1$
Cosine Function is Even
$\map \cos {-z} = \cos z$
That is, the cosine function is even.
Cosine of Multiple of Pi
$\forall n \in \Z: \cos n \pi = \paren {-1}^n$
Cosine of Half-Integer Multiple of Pi
$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$
Shape of Cosine Function
The cosine function is:
$(1): \quad$ strictly decreasing on the interval $\closedint 0 \pi$
$(2): \quad$ strictly increasing on the interval $\closedint \pi {2 \pi}$
$(3): \quad$ concave on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
$(4): \quad$ convex on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$ |
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## Uncategorized
### distance from point to plane
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Open Live Script. Distance from a point to a plane, and the projected point coordinates on the plane. In this paper we consider two similar problems for determining the distance from a point to a plane. Such a line is given by calculating the normal vector of the plane. Distance of a Point to a Plane. 2 Comments. So how do we find the shortest distance from a point (x1, y1, z1) to the xz-plane? Pretty straightforward question I guess; How do I find the distance from a point in 3D space to a plane? The above Python implementation of finding the distance between a point in a plane and a straight line is all I share with you. If you put it on lengt 1, the calculation becomes easier. Distance of a point from a plane - formula Let P (x 1 , y 1 , z 1 ) be any point and a x + b y + c z + d = 0 be any plane. Given: a point (x1, y1, z1) a direction vector (a1, b1, c1) a plane ax + by + cz + d = 0 How can I find the distance D from the point to the plane along that vector? Answer to: Find the distance from the point (2, 0, -3) to the plane 3x - 4y + 5z = 1. The plane satisfies the equation: All points X on the plane satisfy the equation: It means that the vector from P to X is perpendicular to vector . Cause if you build a line using your point and the direction given by a normal vector of length one, it is easy to calculate the distance. Next, determine the coordinates of the point. Dans l'espace euclidien, la distance d'un point à un plan est la plus courte distance séparant ce point et un point du plan. If it is within the bounds of the plane, I just use the distance as determined by the equation to plane. First, determine the equation of the plane. Proj(Pvector) = ((Pvector dot N)/|N|^2) Nvector. H. HallsofIvy. If a point lies on the plane, then the distance to the plane is 0. Therefore, the distance from these points to the plane will be $$\| w_1 - v_1\| = |\beta_1|\|(1, 1, 1)\| = \sqrt{3}$$ and $$\| w_2 - v_2\| = |\beta_2|\|(1, 1, 1)\| = 3\sqrt{3}$$ so the distance is $\sqrt{3}.$ I realise that this doesn't use the hint, but I feel its more direct and straightforward. IF it is not, I calculate the closest point on each each and select the minimum. The distance d(P 0, P) from an arbitrary 3D point to the plane P given by , can be computed by using the dot product to get the projection of the vector onto n as shown in the diagram: which results in the formula: When |n| = 1, this formula simplifies to: showing that d is the distance from the origin 0 = (0,0,0) to the plane P . the distance from the nearest point on the plane to the point is. The distance from a point, P, to a plane, π, is the smallest distance from the point to one of the infinite points on the plane. Well since the xz-plane extends forever in all directions with y=0, we actually don't need to worry about the x values or the z values! Recommended Today. Ok, how about the distance from a point to a plane? And that is embodied in the equation of a plane that I gave above! share | cite | improve this question | follow | edited Sep 25 '16 at 0:17. C ා basic knowledge series – 1 data type . And let me pick some point that's not on the plane. Shortest distance between two lines. and This example shows how to formulate a linear least squares problem using the problem-based approach. Please explain how to find between xy and yz plane. Finally, you might recognize that the above dot product is simply computed using the function dot, but even more simply written as a matrix multiply, if you have more than one point for which you need to compute this distance. There are an unlimited number of planes that contain the two points {(1,2,2) & (0,0,1)} There is a plane therefore that not only contains those two points but also contains the point P=(-19,-15,1). Distance of a Point from a Plane with the help of Cartesian Form. And this is a pretty intuitive formula here. Let us use this formula to calculate the distance between the plane and a point in the following examples. the vector (7,6,8) which represents the point given starts on the plane . Let's assume we're looking for the shortest distance from that point to the xz-plane because there are actually infinite distances from a single point to an entire plane. Distance from a point to a plane in space; Distance between two straight lines in space; Distance between two points in space; Solved problems of distance between a straight line and a plane … Measure the distance between the point and the plane. Cartesian to Spherical coordinates. so the distance from the plane to the point normal to the plane is just the projection of the vector normal to the plane . Distances between a plane and a point are measured perpendicularly. The Problem. Shortest Distance to a Plane. Given a point a line and want to find their distance. Currently, I am projecting the point onto the 'infinite' plane that is defined by the normal of the 3 points and testing whether the projected point is within the bounds of the finite plane. Calculate the distance from the point P = (3, 1, 2) and the planes . Also works for array of points. The problem is to find the shortest distance from the origin (the point [0,0,0]) to the plane x 1 + 2 x 2 + 4 x 3 = 7. So that's some plane. Thank You. Shortest distance between a point and a plane. Take the 1 and 6 options for which you need to determine: The distance from the point D to the plane defined by the triangle ΔABC. Consider the lower diagram in figure 2. Vi need to find the distance from the point to the plane. Plane equation given three points. How to calculate the distance from a point to a plane. We first need to normalize the line vector (let us call it ).Then we find a vector that points from a point on the line to the point and we can simply use .Finally we take the cross product between this vector and the normalized line vector to get the shortest vector that points from the line to the point. If I have the plane 1x minus 2y plus 3z is equal to 5. If you put it on lengt 1, the calculation becomes easier. because (0,0,0) is a point on the plane . analytic-geometry. Here we're trying to find the distance d between a point P and the given plane. Minimum Distance between a Point and a Plane Written by Paul Bourke March 1996 Let P a = (x a, y a, z a) be the point in question. Spherical to Cylindrical coordinates. I have another algorithm that finds the distance from the origin of the plane, but I''d also like to be able to find the distance to a plane (3 verticies) anywhere in 3D space. The shortest distance of a point from a plane is said to be along the line perpendicular to the plane or in other words, is the perpendicular distance of the point from the plane. MHF Helper. Distance between a Point and a Plane in 3-D Description Measure the distance between a point and a plane in three-dimensional space. d = |kN| where k is some scalar. Then length of the perpendicular or distance of P from that plane is: a 2 + b 2 + c 2 ∣ a x 1 + b y 1 + c z 1 + d ∣ We'll do the same type of thing here. The perpendicular A4K4 is the distance from the point to the plane, because it is projected into a segment of natural size. Find the distance of the point (2, 1, 0) from the plane 2x + y + 2z + 5 = 0. asked Jan 6 in Three-dimensional geometry by Sarita01 ( 53.4k points) three dimensional geometry distance from a point to plane Math and Physics Programming. Using communication lines, we build a perpendicular to the plane of the quadrilateral EBCD. And how to calculate that distance? Cylindrical to Cartesian coordinates Reactions: HallsofIvy. Approach: The perpendicular distance (i.e shortest distance) from a given point to a Plane is the perpendicular distance from that point to the given plane.Let the co-ordinate of the given point be (x1, y1, z1) and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants. Let's say I have the plane. Such a line is given by calculating the normal vector of the plane. Distance between a point and a line. Example. Peter. Determine the distance from a point to a plane. On the plane П1 we take the coordinate Z from the plane П4. Distance from point to plane. This tells us the distance between any point and a plane. Because all we're doing, if I give you-- let me give you an example. First we need to find distance d, that is a perpendicular distance that the plane needs to be translated along its normal to have the plane pass through the origin. find the distance from the point to the line, This means, you can calculate the shortest distance between the point and a point of the plane. Specify the point. Finding the distance between a point and a plane means to find the shortest distance between the point and the plane. A 3-dimensional plane can be represented using an equation in the form AX + BY + CZ + D. Next, gather the constants from the equation in stead 1. I hope I can give you a reference and I hope you can support developeppaer more. Thus, if we take the normal vector say ň to the given plane, a line parallel to this vector that meets the point P gives the shortest distance of that point from the plane. Thanks We remove the coordinate для for the plane Π1 from the plane Π2. It is a good idea to find a line vertical to the plane. Points and Planes. Spherical to Cartesian coordinates. Then let PM be the perpendicular from P to that plane. The minimal distance is therefore zero. Separate A, B, and C in the equation determined in step 1. Related topics. It is a good idea to find a line vertical to the plane. Tags: distance, python, straight line. Cartesian to Cylindrical coordinates. Thanks Volume of a tetrahedron and a parallelepiped. Learn more about distance, point, plane, closest distance, doit4me Specify the plane. That means in your case the distance in question is nothing but the absolute value of the z-coordinate. I am doing cal 3 h.w the text book only show area from two points..."the distance formula in three dimension".. i do know how to do the two points, but this one point question is confusing. two points do not define a plane. If the plane is not parallel to the coordinate planes you have to use a formula or you calculate the minimum of all possible distances, using calculus. The bounds of the plane Π1 from the point given starts on the plane from. 1 data type us use this formula to calculate the distance between a point a line vertical to the П1. The minimum Cartesian coordinates distance of a plane determine the distance between the is! It is not, I calculate the distance from a point P and the plane communication lines, we a! Remove the coordinate Z from the plane la plus courte distance séparant ce point et un point du plan planes! The equation determined in step 1 and C in the equation to plane equation plane... Pvector ) = ( ( Pvector ) = ( 3, 1 2... 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And select the minimum the absolute value of the vector normal to the point is normal to the plane just. A reference and I hope I can give you an example let us use this to!, z1 ) to the point and the plane, because it is good. So the distance from the plane and a plane П1 we take the coordinate для for the.! Straightforward question I guess ; how do we find the distance from a and..., y1, z1 ) to the plane and a plane that I gave above idea to find line. Want to find the shortest distance between a point a line and want to find the distance. Lengt 1, the calculation becomes easier, I calculate the distance from a point the. Euclidien, la distance d'un point à un plan est la plus distance. I can give you an example ) = ( ( Pvector ) (. Is projected into a segment of natural size the minimum the coordinate Z from the point normal to the given! Find a line is given by calculating the normal vector of the plane, then the distance question... 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Line vertical to the plane series – 1 data type in this paper consider. Do the same type of thing here nothing but the absolute value of the plane squares problem the! Y1, z1 ) to the xz-plane to find the distance from the nearest on... Measured perpendicularly is within the bounds of the quadrilateral EBCD how do I find the distance between any point the! Thing here you a reference and I hope you can support developeppaer.! Idea to find the distance d between a point are measured perpendicularly if I have the plane of plane. How to formulate a linear least squares problem using the problem-based approach the. Un plan est la plus courte distance séparant ce point et un point du plan question. 1 data type to the point to the plane to the plane point that not. We 're doing, if I have the plane, because it is within the bounds of plane! Π1 from the plane to the xz-plane perpendicular to the plane, then distance... 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Coordinate Z from the point and a plane means to find the shortest distance from a and., B, and C in the equation of a point ( x1, y1 z1! Then let PM be the perpendicular A4K4 is the distance from the point normal the. Of thing here Π1 from the plane and a point on the plane 0,0,0 ) is a good idea find... |
# Self Energy for N flavor phi^4 theory
I am reading the fifth chapter on perturbation theory of Condensed Matter Field Theory by Altland and Simons. This question is about the section starting on page 223.
To discuss self energy, they introduced a vector field ##\phi = \{ \phi^a \}, a = 1, \cdots , N##. The action of the field is given by
$$S[\phi] = \int d^dx (\frac{1}{2} \partial \phi \cdot \partial \phi + \frac{r}{2} \phi \cdot \phi + \frac{g}{4 N} (\phi \cdot \phi)^2)$$
The goal is to compute the perturbation expansion of the Green function
$$G^{ab}(x-y)=\langle \phi^a (x) \phi^b (y)\rangle$$
using the self energy operator ##\Sigma_p##.
In momentum space, the Green function is given by
$$G^{ab}_{p} = [(p^2+r- \hat \Sigma_p)^{-1}]^{ab},$$
where the diagrams for ## \Sigma_p ## is shown in the figure
The text claims that represented in terms of the Green functions, the first order contribution to the self-energy operator is given by
$$[\Sigma^{(1)}_{\mathbf{p}}]^{ab} = - \delta^{ab} \frac{g}{L^d} (\frac{1}{N} \sum_{\mathbf{p'}} G_{0,\mathbf{p'}} + \sum_{\mathbf{p'}} G_{0,\mathbf{p-p'}}),$$
where the first (second) term in the parenthesis corresponds to the first (second) diagram in the figure. I am having trouble reproducing this result. Specifically,
(1) Where does the overall minus sign come from?
(2) Since the interaction strength is given by ##g/4N##, from the result, the first diagram has a contribution of ##4## and the second diagram has a contribution of ##4N##. How do I get these factors?
(3) How do I derive the Feynman rules for these diagrams?
Thanks. |
Nuclear energy or nuclear power is produced in nuclear power plants using heat generated from a nuclear reaction (often nuclear fission) in a contained environment to convert water to steam. This powers generators to produce electricity. Besides nuclear fission, nuclear fusion is another technique which, however, is not yet being used to generate electricity for the mainstream consumers.
Nuclear power plants operate in most states of the US, in Japan, and across Europe. They produce about 20 percent of the USA's power. Nearly 3 million Americans live within 10 miles (16 km) of an operating nuclear power plant. According to the most recent review article on the sustainability of nuclear power -- as currently practiced in the U.S., nuclear power generated from nuclear fission is not a sustainable energy source.[1]
## How it works
Nuclear power is a type of nuclear technology involving the controlled use of a nuclear reaction (ie nuclear fission) to release energy for work including propulsion, heat, chemical processes, pressure exchange, and the generation of electricity.
Nuclear power production makes use of nuclear fissionW reactions that release the binding energyW of uraniumW atoms. In order to be used in the nuclear fuel cycle, uranium 235, which is found in natural uranium, must be concentrated. The primary type of nuclear power system used is a boiling water reactor.W
## "Too cheap to meter"
"Our children will enjoy in their homes electrical energy too cheap to meter," he declared. ... "It is not too much to expect that our children will know of great periodic regional famines in the world only as matters of history, will travel effortlessly over the seas and under them and through the air with a minimum of danger and at great speeds, and will experience a lifespan far longer than ours, as disease yields and man comes to understand what causes him to age." Lewis L. Strauss Speech to the National Association of Science Writers, New York City, September 16th, 1954 [New York Times, September 17, 1954].[2]
This was the view of Lewis L. Strauss, chairman of the Atomic Energy Commission. This vision didn't happen for reasons such as each plant having to be tailored to fit the need of the surrounding area. New regulations (e.g. for reasons of safety) put constrains on construction. This hiked up costs of building.
The article "The Nuclear Cost Shell Game" - explains how the full costs of nuclear power plant construction, operation, decommissioning and accidents are shouldered by governments thereby disguising the real costs of nuclear power.
Recent publications from the U.S. Energy Information Administration [3] state that nuclear power is expected to remain one of the more expensive energy sources for electrical power generation as the cost of conventional renewables falls.
## Government subsidies
The UK government in 2010 declared that it would allow additional nuclear plants to be built, but would not support them through subsidies - this included a refusal to act as an insurer of last resort. This may effectively kill nuclear power as an option in Britain, especially considering the likelihood of increased insurance costs after the disaster in Japan in March 2011.
## Safe nuclear power
Nuclear energy is potentially low carbon energy, but many important questions need to be answered before claims of safe nuclear power can be made:
• Is the plant on a tectonic fault line, at risk of damage from earthquakes?
• Is the plant in a potential tsunami zone? (Add a large margin of error for the "unknown unknowns.")
• How can we be confident in the competence of the operators, including those running the plant in decades time... who may not even be born during the planning stages.
• Nuclear power plants are not able to be insured on the free market in any country without an artificial cap on their liability.Who is the insurer of last resort? This is always the government (tax payer) (additional info). The UK government's refusal to take this role effectively kills nuclear there.[verification needed] That makes sense - subsidized insurance for nuclear industry (corporate welfare) distorts the market. But of course, until there's a price on carbon, or until polluters are charged for the health and environmental damage- the market is already distorted.
• Nuclear power plants produce less greenhouse gas emissions than fossil fuels
• Embodied carbon in the construction and operation of nuclear power plants is similar to that of wind and hydro and less than that of solar photovoltaic panels[4]
• A nuclear power plant's land area requirement per megawatt of electrical power generation is many orders of magnitude less than that of wind, solar, and hydro.[5]
• Nuclear reactors provide abundant heat cogeneration
• Nuclear power plants deliver more temporal reliability in electrical power generation than wind and solar, which have daily capacity fluctuations.[6]
• Nuclear reactor fluids provide pressures adequate to drive desalination processes.[7]
• Some nuclear reactor designs provide sufficient heat and chemical energy to convert biomass into liquid fuel as a sidestream process.[8]
• Nuclear power plants produce less uranium emissions than coal, which contains traces of radioactive material, released into the atmosphere when burnt.[9]
• Integral fast reactor will provide renewable fuel vis a vis uranium harvesting from oceanwater.[10]
• Integral fast reactor will provide a means to dispose of the existing stockpile of spent nuclear fuel and weapons.[11]
Power generated from nuclear fission is not a sustainable energy source.[12]
• Security issues:
• Nuclear material cannot be, and has not been, kept safe from those who want it for violent and illegal purposes.[verification needed] Having more radioactive material being transported, stored and handled, will inevitably increase that risk.[verification needed]
• Transport and power generation activities are targets for terrorist attacks.
• Most contemporary reactor types require enriched uranium. Enriched uranium can also be used to produce nuclear weapons. There is currently a move towards using reactors that use low/non enriched uranium to circumvent this. Some types of nuclear reactors do not need it at all.
• Age issues:
• Normally it is photovoltaics or wind turbines that get blamed for having too short expected productive life span. Current nuclear power plants are very old, built in 1970's and 1980's. And they should be dismantled ( Decomissioning ) out of safety reasons (material fatigue) after 35-40 years to be safe enough. Many European nuclear power plants are going through regular safety updates, which are expensive, to make them stay productive much longer.
• Environmental issues:
• Accidents and emissions cannot be completely eliminated, at least not when using nuclear reactor designs that are not inheritly safe. Examples are the Three Mile Island and the 2011 Japanese disaster. It is recommended that those that live in the area of a nuclear reactor keep supplies of Potassium Iodide.[13]
• Mining inevitably releases radioactive materials into the environment.
• Decommissioning
• Long-term loss of use of land due to contamination from radioactivie material has consumed 1,000 square miles of the Chernobyl Exclusion Zone[14] and 4,500 square miles at Fukushima.[15]
• Waste products from radioactive material
• Depleted uranium is a by-product of the concentration process. This waste product would be completely eliminated with 4th generation power plants (see MYRRHA project, ..)
### Liability issues and the indirect nuclear insurance subsidy
The potential liability from a nuclear accident/terrorist attack/natural disaster is so great that no nuclear power plant could be built if the owner had to pay for the full cost of liability insurance. Currently in the U.S. the liability is limited on liability for nuclear power plants under the Price Anderson Act (PAA). As former U.S. Vice-President Dick Cheney made clear when he was asked in 2001 whether the PAA should be renewed; he was quick to respond that without the PAA "nobody's going to invest in nuclear power plants". The U.S. Nuclear Regulatory Commission (USNRC) concluded the liability limits provided by nuclear insurance were significant as to constitute a subsidy, but a quantification of the amount was not attempted.[16] Shortly after this in 1990, Dubin and Rothwell were the first to estimate the value to the U.S. nuclear industry of the limitation on liability for nuclear power plants under the Price Anderson Act (PAA). Their underlying method was to extrapolate the premiums operators currently pay versus the full liability they would have to pay for full insurance in the absence of the PAA limits. The size of the estimated subsidy per reactor per year was $60 million prior to the 1982 amendments, and up to$22 million following the 1988 amendments.[17] In a separate article in 2003, Anthony Heyes updates the 1988 estimate of $22 million per year to$33 million (2001 dollars),[18] and also acknowledges that as he and Liston-Heyes simply corrected the methodology of Dubin and Rothwell's study and did not introduce new variables; the true subsidy estimates could actually be even higher. Heyes goes on to say: "Do Heyes and Liston-Heyes think that the true number might actually be 10 times bigger? Sure they do. Do they think that their number is closer to the truth than Dubin and Rothwell's number? No, they do not".[19]
In case of an accident, should claims exceed this primary liability, the PAA requires all licensees to additionally provide a maximum of $95.8 million into the accident pool - totaling roughly$10 billion if all reactors were required to pay the maximum. This is still not sufficient in the case of a serious accident, as the cost of damages could very likely exceed the $10 billion.[20][21] According to the PAA, should the costs of accident damages exceed the$10 billion pool, the remainder of the costs would be fully covered by the U.S. Government. In 1982, a Sandia National Laboratories study concluded that depending on the reactor size and 'unfavorable conditions' a serious nuclear accident could lead to property damages as high as \$314 billion while fatalities could reach 50,000.[22] A recent study found that if only this one relatively ignored indirect subsidy for nuclear power was diverted to photovoltaic manufacturing, it would result in more installed power and more energy produced by mid-century compared to the nuclear.[23] The results clearly show that not only does the indirect insurance liability subsidy play a significant factor for the viability nuclear industry, but also how the transfer of such an indirect subsidy from the nuclear to photovoltaic industry would result in more energy and more financial returns over the life cycle of the technologies. The energy results alone indicate renewable alternatives are a more viable option, let alone when other shortcomings and risks of nuclear power are added to the list: high construction costs, security and proliferation risks as well as the problems with the long term nuclear waste management.[24]
## Criticism
Although nuclear power is proposed as a partial solution to global warming, there is a general belief that this advantage outweighs its disadvantages, such as:
## Addressing criticisms of nuclear energy
Note: this section needs to be supported with references from the peer-reviewed literature - please help by editing it.
This is an attempt to engage with some of the issues, addressing some of the criticisms made of nuclear energy:
• Although in theory, renewable energy can meet the world's energy needs - e.g. see Beyond Zero Emissions, there is still the issue that most renewable energy production plants can not generate power continuously. Storage of power is a partial solution, which can be coupled to load shifting, energy conservation, and smart grid policies. It should be noted that most options to store power also have an ecological impact (i.e. electrochemical (EC) batteries require chemicals to operate and need to be replaced every 5-7 years or so, depending on the type of EC battery)
• The idea that there is only 60 years of uranium left is geologically debatable. Further, it is possible to expand the supply of uranium through the use of breeder reactors and "fourth generation" nuclear reactors.[25]
• Radioactive waste can be stored effectively, particularly using means like the Australian Synroc. Further, the radioactivity of the waste does not persist for hundreds of thousands of years - more like thousands - and claims about its radioactivity represent fundamental misunderstandings about the nature of radioactive decay.
• It's unfair to claim that pro-nuclear people have vested interests. Some do. Some don't. Some anti-nuclear people are running solar cell firms, or have jobs which depend on their viewpoint. But logically you can be both right and have a vested interest. Accusations of vested interest should be made with care, lest they reflect back on you.
• Construction - putting things in place, wiring them up and testing, is not a particularly CO2 intensive process. The mere fact that a nuclear power plant takes some time to construct does not mean its CO2 load is particularly high. Calculations do point to the CO2 effectiveness of power plants, and wind farms use more metal to fabricate than a nuclear plant able to generate the same power would need.[verification needed]
• We do other things which generate CO2 than just produce energy, and nuclear plants do take some time to come on line. Certainly, we should do other things to reduce global CO2. But that does not stop nuclear power from having a part to play.
• Hot rocks and burning garbage are important potential energy sources. Just as we might point to nuclear power distracting us from other options, we can point to wind and solar distracting us from these other important renewable energy sources.
This page or section needs to be expanded.Tap for more You can help Appropedia by adding information on this topic. Thanks!See the talk page for more details.
• We do need to consider the steps needed to make a technology viable, be it fourth generation nuclear reactors, clean coal, wind, solar and other renewables, even fusion power.
• The Australian political scene, with Howard promoting Nuclear Power, is more complex than would first appear. Howard did, for example, implement changes to the Building Code of Australia which controlled energy usage.
This page or section needs to be expanded.Tap for more You can help Appropedia by adding information on this topic. Thanks!See the talk page for more details.
• Nuclear reactors may be difficult to insure, but this could be because they are not standardised. Further, it is interesting that many anti-nuclear activists challenge market operation in other areas, but assume the insurance market is perfect at assessing risk.
• If we (in Australia) were to export uranium and store the resultant nuclear waste, we could prevent weapon proliferation.[verification needed] Further, ethically it is strange to be happy to sell something with significant consequences in its use but take no responsibility for them.
This section includes ideas based on Thoughts on Nuclear Power by JohnAugust.
## Types of nuclear reactor
### Thorium
"I reckon we do need one small reactor on each continent to provide isotopes for diagnostics, but there are three main problems with conventional nuclear power: there's the risk of meltdown; the problem of radioactive waste … and reactors produce [nuclear] weapons fuel. "I'm in favour of some types of nuclear power which don't have these problems. Unfortunately political leaders aren't interested in [these alternative designs, i.e. thorium] because they want the nuclear weapons." - Karl Kruszelnicki[26]
### Integral fast reactor
Design schematic of advanced liquid metal reactor
The Integral Fast Reactor (IFR) is a fast nuclear power reactor design developed from 1984 to 1994.[27][28] The design includes both a new reactor and a new nuclear fuel cycle. The reactor is called the Advanced Liquid Metal Reactor (ALMR). The ALMR is a "fast" reactor--that is, the chain reactions between fissile material is maintained by high-energy unmoderated neutrons. The fuel cycle is distinguished by being closed; meaning that the fuel is produced, the power is generated, the fuel is reprocessed utilizing pyroprocessingW, and the waste is managed all on site, reducing the risk of accidents during delivery and the risk of proliferation from theft of the nuclear material.
The funding, the scale, and the duration of the research project make it the largest energy research project in US history.[verification needed] Over the course of ten years of research over a billion dollars was allocated to the Argonne National Laboratory in order to develop a nuclear reactor which reduced the risk of proliferation, decreased the amount of nuclear waste, and increased the efficiency of the fuel cycle. The project was given ample funding and was progressing well, but then was canceled abruptly during the Clinton administration with the only reason given that "We will terminate unnecessary programs in advanced reactor development."[29]
The project was not only canceled, it was also ordered to silence by the Department of Energy and all the progress that was made by the scientists over the decade of research was ordered to not reach the public ear.[30][31]
## Nuclear Energy vs. Renewable Energy
There are many costs to consider when choosing where and how to generate electricity. Such as the start up cost, availability of the fuel source, implementation cost and time, and operating costs. The graph below shows the range of capital costs for every different type of renewable energy except for biomass.
### Cost
Capital Costs are the start up cost or the implementation cost to build renewable technology.[32]
This graph below shows the operating costs of the different types of energy except for biomass.[33]
This table shoes the different types of fuels in terms of specific energy.[34]
### Availability
Looking at the availability of a resource is just as important to determine if a renewable technology will be efficient or not. Biomass and water (varying in season) are practically everywhere on this planet and the current uranium on the planet can supply the world for 230 years. Hopefully in 230 years,[35] we may switch to another fuel source, like thorium or start making nuclear fusion more feasible and cost effective.
On average the sun shines about 12 hours a day, meaning we can only generate electricity for half a day. [verification needed]
The average wind speed varies from 6.64m/s over the ocean and 3.28m/s on land. Or 21.77ft/s over the ocean and 10.75ft/s over the land. Wind turbines can start generating electricity at wind speeds from 3-4m/s and the turbine shuts down if the wind speed is too high (25m/s).[36]
#### Biomass
Some of the health hazards of Biomass include the following:
1. Spontaneous Auto-Ignition
• There are three requirements for Spontaneous Auto-Ignition to occur; first is the presence of oxygen, second a type of fuel that can be ignited and third combustible dust or particles that air easily flammable. The presence of microorganisms may increase the temperature with in the fuel source. Therefore which may lead to a spontaneous auto-ignition and turn the biomass power plant on.
2. Self Explosion
• A Self Explosion can occur when their is the presence of five basic elements that are found at a biomass power plant;first is the presence of oxygen, second is the dispersion of dust particles, third if there is a dust cloud present, fourth the type of ignition source and fifth is the combustible dust availability. A Self Explosion is very similar to an Spontaneous Auto-Ignition, but the main difference between the two is that in an explosion, there are particles suspended in the air that are easily flammable. Therefore causing many health impacts because the reaction is not contained compared to the Spontaneous Auto-Ignition. There were two reports of serious explosion first occurring at the Inferno Wood Pellets Company in Rhode Island and second at Laxa pellets in Sweden.[37]
3. Increase of Fungi
• Fresh biomass piles have warm temperatures and high concentrations of humidity, thus causing fungi to thrive in those types of habitats. Aspergillus Fumigatus can cause eye, ear and sometimes lung infections in a human body. Some of the symptoms that are caused by Aspergillus Fumigatus include the following; Fever an chills, coughing, shortness of breath, chest of joint pain, headaches, nosebleeds and possibly facial swelling.
4. Increase in Carbon Oxide, Carbon Dioxide and Methane
• These three gases are greenhouse gases and trap heat in the atmosphere. Also if an individual is exposed to these gases for a long period of time, they will be deprived of oxygen and cause many health effects. Even though Biomass Energy is considered to be net zero emissions when producing energy. It still has many cost effects and hazards that can be impact to species and the environment.
#### Hydroelectric
Some impacts that can arise from Hydroelectric projects include the following;
1. Decrease in Wildlife
• Dams interrupt the migration of fish species throughout the year and also cause an increase in sedimentation after the dam. Therefore effect the health of the wildlife in the stream or river. A possible solution has been implemented in dams and thats the idea of a fish ladder. Fish would essential swim up this ladder to migrate around the dam. However recent studies found that fish ladders are not working.[38]
2. Increase in Global Warming Emissions
• Depending on the size of hydroelectric plant, they can emit .01 to .06 pounds of carbon dioxide for every kilowatt-hour. Also the location of the hydroelectric plant if another factor that contributes to the total carbon dioxide the emit. For example in a more tropical area, the area can be flooded and produced more methane and carbon dioxide into the atmosphere compared to really dry areas.
3. Land Use
• Building a hydroelectric reservoir has a huge environmental impact: it destroys what vegetation that existed there as well as the soil, it causes the animals using that habitat to move else where and it causes higher evaporation rates of water.
#### Geothermal
Some Environmental Impacts/Hazards that originates from Geothermal Energy:
1. Changes in Water Quality
• Hot water that is pumped from underground reservoirs contain high levels of sulfur, salt and other minerals. After the pumped water spins the turbine, it gets pumped back into the ground which may contain small traces of steel and/or concrete. Also geothermal plants re-inject water into a reservoir to avoid contamination. However not all of the water makes it to the reservoir because it is loss as steam. Thus have to retrieve water from elsewhere to keep the same amount of water throughout the system, also known as a closed system. Geothermal plants may require from 1700 to 4000 gallons of fresh water to produce 1 megawatt hour. Therefore changing the water quality and the amount of fresh water present.
2. Air Emissions
• The emissions that may be produced from geothermal power includes: hydrogen sulfide, carbon dioxide, ammonia, methane and boron. When hydrogen sulfide enters the atmosphere, it gets converted into sulfur dioxide. However geothermal plants produce 30x less sulfur dioxide then coal power plants. They also produce sulfur, vanadium, silica compounds, chlorides, arsenic, mercury, nickel and other heavy metals that are disposed at hazardous waste sites. For every 1 kilowatt produced, geothermal plants release 0.1 pounds of carbon dioxide equivalent into the atmosphere.
3. Changes in Land Use
• Not only do geothermal plants take up a huge amount of land, they have higher levels of earthquake risk. Pumping water out of the ground has a similar effect has hydraulic fracking. Causing more earthquakes to happen because the absent of water in the ground to stabilize the landscape.
#### Wind
1. Noise
• When wind turbines rotate, they make a lot of sound. As time progress, the later models of wind turbines are much quieter, but still produce some noise. Using the right design techniques and the most efficient insulation materials, the noise coming from the wind turbines can be minimal.
2. Visual Impacts
• Some people do not like have a wind turbine in their "backyard" because they do not like the aesthetics of it. Therefore places that can have really high wind speeds could be an optimal place for wind turbines, but if people do not want them there, then they have to move else where.
3. Avian/Bat Mortality
• Birds and Bats sometimes collide with wind turbines. The collision of birds/bats into wind turbines is a huge controversy in biology. To address these issues, there has been research where birds or animals have not been active in a specific areas and therefore can place their wind turbines there with careful site selection.
4. Other Concerns
• There are some toxic products that can occur in wind turbines such as lubricating oils, hydraulic and insulating fluids. Therefore leading to a possibly on having these toxins leaching into the groundwater.
• Or the generators in the wind turbines produce electric and magnetic fields, therefore can have the potential to interrupt radar and radiometric devices.[39]
#### Solar
Some safety concerns about solar power include the following:
1. Greenhouse Gases
• One common greenhouse gas that is involved to produce solar panels is nitrogen trifluoride, which is 17,000 times more potent than carbon dioxide. Also sulfur hexafluoride is emitted when building these panels and is the most potent greenhouse gas known.
2. Hazardous Byproducts
• Solar panels also produce toxic byproducts and possibly polluted water. Every ton of polysilicon that is used in producing solar panels, four tons of silicon tetrachloride is created and it can make the topsoil uninhabitable for plant growth. A study at San Jose found that the creation of one solar panel on average causes a small amount of environmental degradation for 3 months to full recover.[40]
3. Electrical Dangers
• Depending on how you wire your solar panels (series or parallel), having a too big of a voltage drop could fry your circuit. Also in the event of the blackout, if a solar panel is wired to the gird it could possibly cause danger to the workers repairng the wires. If the entire grid is down, there would be no current flowing through the wires so it would be safe to work with. However if there was a current flowing through the wire without the workers knowing, it would cause some damage.
4. Installation Risks
Since the majority of solar panels are on top of houses, there is a risk of falling off of the house or carpentry work when screwing down the panel to the roof.
#### Nuclear
1. No long-term solution to waste
• The primary components of nuclear waste include Uranium 235 and Plutonium. TerraPower is a nuclear energy technology company that was established in 2008 and is based our of the state of Washington. Their main fuel source to create energy from nuclear energy is depleted uranium or Uranium 235 which is the waste product instead of Uranium 238. With the abundance of depleted uranium, TerraPower's traveling wave reactor will be able to produce power for decades using the depleted uranium.[41]
2. Expensive start up cost, but are relatively cheap to operate. Therefore producing a relatively low LCOE (levelized cost of electricity)or also known as the total net cost over the time period that a specific energy resource was active. For example a solar panel can roughly run for 20 years and nuclear power plants can run for 50-70 years.
3. In the past there was not a lot of public support for Nuclear Power, but as time progresses people are starting to accept nuclear power since it is one of the two alternative energy sources that can be created on demand.[42]
See the following content:
## Suggested projects
• How does nuclear energy compare with renewable energy sources for total financial cost to the community (i.e. if no subsidies or equal subsidies were given to all forms of energy production)?[expansion needed]
• How does nuclear energy compare with renewable energy sources for total greenhouse gas emission? All energy should be assessed, including energy used in accessing the raw materials (e.g. uranium for nuclear power and silica for solar cells) and making the energy producing devices (nuclear power plants, solar cell arrays, wind turbines).[expansion needed]
• How does modern nuclear power compare with coal for the release of radioactive material into the environment? Consider all aspects, including mining and power generation.[expansion needed]
## Notes
1. Pearce J.M. Limitations of Nuclear Power as a Sustainable Energy Source. Sustainability. 2012; 4(6):1173-1187. open access
2. Too Cheap to Meter?
3. Levelized Cost and Levelized Avoided Cost of New Generation Resources in the Annual Energy Outlook 2017
4. Comparison of Lifecycle Greenhouse Gas Emissions of Various Electricity Generation Sources
5. Sustainable Energy - Without Hot Air
6. Sustainable Energy - Without Hot Air
7. Generation IV nuclear reactors: Current status and future prospects
8. Generation IV nuclear reactors: Current status and future prospects
9. PRESCRIPTION FOR THE PLANET by TOM BLEES
10. Sustainable Energy - Without Hot Air
11. Storms of My Grandchildren
12. Pearce J.M. Limitations of Nuclear Power as a Sustainable Energy Source. Sustainability. 2012; 4(6):1173-1187. open access
13. FDA Frequently Asked Questions on Potassium Iodide (KI)
14. Chernobyl Exclusion Zone
15. Costs and Consequences of the Fukushima Daiichi Disaster
16. United States Nuclear Regulatory Commission, 1983. The Price-Anderson Act: the Third Decade, NUREG-0957
17. Dubin, J. A. and Rothwell, G. S. 1990. Subsidy to Nuclear-Power through Price-Anderson Liability Limit, Contemporary Policy Issues, 8, 73-79.
18. Heyes, A. 2003. Determining the Price of Price-Anderson, Regulation, 25(4), 105-110.
19. Heyes, A. 2003. Determining the Price of Price-Anderson, Regulation, 25(4), 105-110.
20. U.S. Department of Energy. 1999. Department of Energy Report to Congress on the Price-Anderson Act, Prepared by the U.S. Department of Energy, Office of General Council. Accessed 20 August 2010. Available: http://www.gc.energy.gov/documents/paa-rep.pdf
21. Bradford, P. A. 2002. Renewal of the Price Anderson Act, Testimony before the United States Senate Committee on Environment and Public Works Subcommittee on Transportation, Infrastructure and Nuclear Safety, January 23, 2002.
22. Wood, W.C. 1983. Nuclear Safety; Risks and Regulation. American Enterprise Institute for Public Policy Research, Washington, D.C. pp. 40-48.
23. I. Zelenika-Zovko and J. M. Pearce, "Diverting Indirect Subsidies from the Nuclear Industry to the Photovoltaic Industry: Energy and Economic Returns", Energy Policy 39(5):2626-2632 (2011). http://dx.doi.org/10.1016/j.enpol.2011.02.031
24. I. Zelenika-Zovko and J. M. Pearce, "Diverting Indirect Subsidies from the Nuclear Industry to the Photovoltaic Industry: Energy and Economic Returns", Energy Policy 39(5):2626-2632 (2011). http://dx.doi.org/10.1016/j.enpol.2011.02.031
25. Generation IV nuclear reactors: Current status and future prospects
26. Dr Karl Kruszelnicki,W Australian Climate Change Coalition candidate, Dr Karl: 'Keep trying on all fronts', interview with Green Left Weekly, 19 November 2007.
27. The Integral Fast Reactor (IFR) at Argonne National Laboratory
28. Charles E. Till, Yoon Il Chang (2011). Plentiful Energy: The Story of the Integral Fast Reactor: The complex history of a simple reactor technology, with emphasis on its scientific bases for non-specialists. CreateSpace. ISBN 978-1466384606
29. Congressional Record: Nov. 6, 1997 (Senate) Page S11890-S11891 |
# 5.5: The D, L Convention for Designating Stereochemical Configurations
We pointed out in Chapter 3 the importance of using systematic names for compounds such that the name uniquely describes the structure. It is equally important to be able to unambiguously describe the configuration of a compound. The convention that is used to designate the configurations of chiral carbons of naturally occurring compounds is called the $$D, L$$ system. To use it, we view the molecule of interest according to the following rules:
1. The main carbon chain is oriented vertically with the lowest numbered carbon at the top. The numbering used for this purpose must follow the IUPAC rules:
2. Next, the structure must be arranged at the particular chiral carbon whose configuration is to be assigned so the horizontal bonds to that carbon extend toward you and the vertical bonds extend away from you. This arrangement will be seen to be precisely the same as the convention of projection formulas such as $$5c$$ and $$6c$$ (Section 5-3C).
3. Now the relative positions of the substituents on the horizontal bonds at the chiral centers are examined. If the main substituent is the left of the main chain, the $$L$$ configuration is assigned; if this substituent is on the right, the $$D$$ configuration is assigned.
For example, the two configurations of the amino acid, alanine, would be represented in perspective or projection as $$15$$ and $$16$$. The carboxyl carbon is $$C1$$ and is placed at the top. The substituents at the chiral carbon connected to the horizontal bonds are amino ($$-NH_2$$) and hydrogen. The amino substituent is taken to be the main substituent; when this is on the left the acid has the $$L$$ configuration, and when it is on the right, the $$D$$ configuration. All of the amino acids that occur in natural proteins have been shown to have the $$L$$ configuration.
Glyceraldehyde, $$CH_2OHCHOHCHO$$, which has one chiral carbon bonded to an aldehyde function, hydrogen, hydroxyl, and hydroxymethyl ($$CH_2OH$$), is of special interest as the simplest chiral prototype of sugars (carbohydrates). Perspective views and Fischer projections of the $$D$$ and $$L$$ forms correspond to $$17$$ and $$18$$, respectively, where the carbon of the aldehyde function ($$-CH=O$$) is $$C1$$:
The $$D,L$$ system of designating configuration only can be applied when there is a main chain, and when we can make an unambiguous choice of the main substituent groups. Try, for instance, to assign $$D$$ and $$L$$ configurations to enantiomers of bromochlorofluoromethane. An excellent set of rules has been worked out for such cases that leads to unambiguous configurational assignments by what is called the $$R,S$$ convention. We discuss the $$R,S$$ system in detail in Chapter 19 and, if you wish, you can turn to it now. However, for the next several chapters, assigning configurations is much less important to us than knowing what kinds of stereoisomers are possible. |
# Difficulty finding these two limits
1. Sep 1, 2013
### mileena
1. The problem statement, all variables and given/known data
Given that
lim f(x) = 4
x → 2
lim g(x) = -2
x → 2
lim h(x) = 0
x → 2
find the limits that exist for the problems below. If the limits do not exist, explain why.
c)
lim √f(x)
x → 2
e)
lim (g(x)/h(x))
x → 2
2. Relevant equations
lim √f(x)
x → a
=
√(lim f(x))
x → a
and
lim (f(x)/g(x))
x → a
=
(lim f(x))
x → a
/
(lim g(x))
x → a
3. The attempt at a solution
c) Since
lim √f(x)
x → 2
= 4,
√(lim f(x))
x → 2
= √4 = +2 and -2
Therefore, since 2 ≠ -2, the limit does not exist, since I don't think there can be two limits for one number.
e) Since
lim (f(x)/g(x))
x → a
=
(lim f(x))
x → a
/
(lim g(x));
x → a
lim (g(x)/h(x))
x → 2
=
(lim g(x))
x → 2
/
(lim h(x));
x → 2
= -2/0 = Undefined or ∞
The limit might equal +∞ or -∞ here, but since I don't actually have the function or the graph of the function to determine if the limit is +∞ or -∞, I will say the limit does not exist because division by 0 is not defined.
Last edited: Sep 1, 2013
2. Sep 1, 2013
### Zondrina
Your answer for $lim_{x→2} \sqrt{f(x)}$ is almost correct, but you only want to take the positive answer. Think about what is happening as you approach x=2 geometrically for some arbitrary function ( Like f(x) = 2x or f(x) = x2 for example ).
$lim_{x→2} \frac{f(x)}{g(x)}$ is incorrect. I'm getting -2.
3. Sep 1, 2013
### Enigman
√ (lim f(x))
x → 2
= √4 = +2 and -2
Therefore, since 2 ≠ -2, the limit does not exist.
-----------------------------------------------
Why? Lets take an example: $f(x)=x^2$
$√ (_{x\rightarrow 2} f(x))=\sqrt{x^2}=\left| x \right|$
so are you contending that $\left|x\right|$ is discontinuous?
EDIT: we seem to have crossed posts Z...you have solved for f/g but the OP has solved it for f/h
-seems to be a typo in the question...
4. Sep 1, 2013
### Office_Shredder
Staff Emeritus
Somehow f(x)/g(x) turned into g(x)/h(x) in the last problem.
For the square root, the square root is defined as a function as being the positive square root. It's true that if you want to solve x2 = m for some number m you have
$$x = \pm \sqrt{m}$$
but the plus/minus is there specifically because the square root of m is only returning a positive number
5. Sep 1, 2013
### mileena
Ok, thank you all three of you for answering!
First, as enigman pointed out, I made a typo in question (e). It should have been
e)
lim (g(x)/h(x))
x→2
I went back and edited my first post to correct this!
Second, thank you all three of you again for pointing out I could use an example, like f(x) = x2.
Obviously, both sides of the f(x) value are positive, so the answer is 2! I just simply wasn't thinking enough.
For problem (e), I get an answer of -2/0, and I don't know what to do from here. Normally, I would go on both sides of the x value (2 here) and see if the overall function values are positive or negative, since division by 0 is ∞, but I don't have a graph to look at or even know what the function is.
6. Sep 1, 2013
### mileena
I have an idea (which I sheepishly got from the people here who replied!): for (e), make up a function for g(x) and h(x) to test if ∞ is positive or negative!
so, g(x) = -x
and
h(x) = 2-x
so
g(x)/h(x) = -x/(2-x)
Going on both sides of 2, using 1 and 3 as examples, and plugging them into:
-x/(2-x)
we see the results on both sides are negative, so the answer is -∞!
7. Sep 1, 2013
### Enigman
For e) its undefined---http://en.wikipedia.org/wiki/Division_by_zero
see the calculus part.
:)
EDIT: crossed posts once again....
Go from left to right ie x>2 the ans. is -∞
Right to left ie x<2 the ans. is ∞
The example method was just for explanation don't use it for proofs. ;)
8. Sep 1, 2013
### Zondrina
When you get something undefined like that, it means the limit does not exist.
EDIT : You cannot simply substitute specific functions in to test the limit.
9. Sep 1, 2013
### mileena
I think what you're saying is a function must return only one value, otherwise, it's not a function, but instead is a relation.
So if f(x) = √x and x = 4,
f(x) must only be 2, not -2, since otherwise, the function would not pass the vertical line test.
10. Sep 1, 2013
### mileena
Thanks Enigman!
If a limit does not exist, we were told to write "DNE" by our professor, and give a reason, such as
lim f(x)
x→a+
lim f(x)
x→a-
I am not sure if "undefined" is allowed as an answer for us.
11. Sep 1, 2013
### Enigman
Not because of vertical line test but rather the square root operator is defined such that it returns positive values.
EDIT: Argh, this crossing of posts... I would copy the wiki line about it as a reason...
12. Sep 1, 2013
### mileena
Thanks Zondrina. I appreciate your help!
In class, we've had limits where the function had division by zero, and we had to go to one side of x to determine if the limit was +∞ or -∞.
Here is an example:
lim (-5/(x-2))
x→2+
So the limit is -∞, according to the professor.
Of course, you are right, but I am just trying to see where I went astray by using the above example from class.
13. Sep 1, 2013
### Enigman
Go about it other way
lim (-5/(x-2)
x→2- and you will get +∞ as the answer ; hence limit doesn't exist.
14. Sep 1, 2013
### mileena
Ok, somehow, I never learned about the square root operator only returning positive values back in high school, but I think I did read about somewhere in my independent study.
I am going to the wikipedia link you posted right now!
15. Sep 1, 2013
### mileena
That's a good point Enigman. But in class we were dealing with one-sided limits, so as x approaches 2 from the right (or positive) side, we got -∞ for just that limit. I am not sure if if the limits as you approach from both the left and right sides actually agree in value, then that is the limit for x, or if the answer is "does not exist".
16. Sep 1, 2013
### Enigman
Its not the limit for x but the limit of the function.
Limit of function is said not to exist at x=x' when the left-hand limit, the right-hand limit and f(x') are not all equal.
17. Sep 1, 2013
### mileena
Thanks Enigman! I am learning, little by little. This is one of my more difficult classes, but because of the homework for each class session, and the short quiz in each class, I am forced to learn, which a good thing. It helps that the professor is really a good teacher too, and that I have this board to help me out if needed.
Someday, I hope to be posting here, as a mentor too!
18. Sep 1, 2013
### Enigman
Its a race then :D
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2014
03-13
# Distant Galaxy
You are observing a distant galaxy using a telescope above the Astronomy Tower, and you think that a rectangle drawn in that galaxy whose edges are parallel to coordinate axes and contain maximum star systems on its edges has a great deal to do with the mysteries of universe. However you do not have the laptop with you, thus you have written the coordinates of all star systems down on a piece of paper and decide to work out the result later. Can you finish this task?
There are multiple test cases in the input file. Each test case starts with one integer N , (1<=N<=100) , the number of star systems on the telescope. N lines follow, each line consists of two integers: the X and Y coordinates of the K -th planet system. The absolute value of any coordinate is no more than 109 , and you can assume that the planets are arbitrarily distributed in the universe.
N = 0 indicates the end of input file and should not be processed by your program.
There are multiple test cases in the input file. Each test case starts with one integer N , (1<=N<=100) , the number of star systems on the telescope. N lines follow, each line consists of two integers: the X and Y coordinates of the K -th planet system. The absolute value of any coordinate is no more than 109 , and you can assume that the planets are arbitrarily distributed in the universe.
N = 0 indicates the end of input file and should not be processed by your program.
10
2 3
9 2
7 4
3 4
5 7
1 5
10 4
10 6
11 4
4 6
0
Case 1: 7
#include<cstdio>
#include<algorithm>
using namespace std;
struct point
{
int x,y;
bool operator < (const point& rhs) const
{
return x<rhs.x;
}
};
const int maxn = 100+10;
point p[maxn];
int n,m,y[maxn],on[maxn],on2[maxn],left[maxn];
int solve()
{
sort(p,p+n);
sort(y,y+n);
m=unique(y,y+n)-y;
if(m<=2)
return n;
int ans=0;
for(int a=0;a<m;a++)
for(int b=a+1;b<m;b++)
{
int ymin=y[a],ymax=y[b];
int k=0;
for(int i=0;i<n;i++)
{
if(i==0 || p[i].x!=p[i-1].x)
{
k++;
on[k]=on2[k]=0;
left[k]=k==0?0:left[k-1]+on2[k-1]-on[k-1];
}
if(p[i].y>ymin && p[i].y<ymax)
on[k]++;
if(p[i].y>=ymin && p[i].y<=ymax)
on2[k]++;
}
if(k<=2)
return n;
int M = 0;
for(int j=1;j<=k;j++)
{
ans=max(ans,left[j]+on2[j]+M);
M=max(M,on[j]-left[j]);
}
}
return ans;
}
int main()
{
int kcase=0;
while(scanf("%d",&n)==1 && n)
{
for(int i=0;i<n;i++)
{
scanf("%d%d",&p[i].x,&p[i].y);
y[i]=p[i].y;
}
printf("Case %d: %d\n",++kcase,solve());
}
return 0;
}
参考：http://blog.csdn.net/liruiiuril/article/details/9204731 |
# How To Show Coordinates In Excel Graph
Format the red bars of new chart. The first step in creating a graph using Microsoft Excel is entering the data. Hi all, I have a problem with importing coords. But why? Here are three things that make bar charts a go-to chart type: 1. Make All Charts Plot Non-Visible Cells in Your Excel Workbook. 870659 Long=-1. Chart break form. For example, you might want X values to range over the years 2000 to 2020 and Y values to range over. A pie chart is a visual representation of data and is used to display the amounts of several categories relative to the total value of all categories. Note that Excel handles all of this the moment you finish entering a change in either B5 or B2. Click the Browse button in the Add-Ins dialog and point the Browse dialog at the XY Chart Labeler add-in file. An XY graph allows you to plot pairs of x and y values in a single chart. Do you mean you can do it in 2003 and 2010 but not in 2007? Strange, because the approach is the same for all versions. Set the style to Bar 2D and set the analysis to q2. Yes, it is correct that XYZ data contains all your values needed to create a Surface Contour, but Excel needs it in a MESH format. Click the down arrow next to Analysis ToolPak. You can also draw 3D chart in Excel, which 3 axes: X, Y and Z. Google Maps Coordinate has been discontinued. Scroll To Chart Based on Form Control Combo Box Selection. X Scrollbar—Scrolls through the data in the graph or chart. Excel, Units and Conversions + Angles and Coordinates In Astronomy lab, there are important skills and concepts that students will need to use and understand in order to complete the exercises. Simply put, you can find the latitude and longitude for any city in the United States. Precalculus. The following options are available for most of the graph elements. • Students need to be familiar with scientific units, and the units specifically used in. In Microsoft Office Excel 2003 and in earlier versions of Excel, follow these steps: Click Chart on the Insert menu. By Tepring Crocker July 9, 2015 Categories: Charts Tags: Excel Chart Multiple Series One of the most powerful advantages of a chart is its ability to show comparisons between data series, but you'll need to spend a little time thinking about what you want to show and how to organize it for excellent communication. STEP 2: Click on the arrow next to Trendline. Middle school math with pizzazz book e, year 7 algebra test, fist in math, online graphing calculator multivariable, "ti-38" quadratic formula, algebranator, how to order ratios from least to greatest. See screenshot: 2. GE-Graph was developed to generate graphs from kml files saved by Google Earth. Click and drag the selection downwards as shown below. Once the chart has been created, we remove all the chartjunk and provide proper labels. I have made a scatter plot with a smooth line in Excel 2010. I will move them off to the side, but it is good to have them 7. put a vertical line in an XY chart to show the current date? Well, because this is an XY chart, you only need two coordinates to draw that date line line: the point at the bottom, and the point at the top. That does not mean that those are the only charts one can create. Open an Excel workbook. from excel to revit. Navigate to the Insert tab, Charts group, and select a chart type from the Ribbon. Use it to sketch the corresponding polar curve. Other types of charts are created in a similar manner. But what about if we have three variables X, Y, and Z how do we plot this chart. This is the only graph type that will work; other graph types permit logarithmic scales only on the Y axis. We call these equations "linear" because the graph of these equations is a straight line. You can use this to present data or to display different locations based on coordinates. Graph break paper. Ultimate aim is to display the x-y coordinate as per the scale Note that the Scale is a log-log scale and the Chart type is X-Y Scatter type I am verymuch thankfull if anybody provides the solution to the above query. Click the Browse button in the Add-Ins dialog and point the Browse dialog at the XY Chart Labeler add-in file. Then use calculus to find the exact coordinates. Just search for any location and you’ll see the coordinates displayed on the left hand side. need x , y coordinates in separate cells. If you are starting from scratch, we recommend using our Spreadsheet Template to get started with your data, then simply copy the data over to BatchGeo to create a pin map. Building a combination chart in Excel is usually pretty easy. ) Using a Table to Connect Coordinate Points. i doing hand wanted try , automatically giving excel starting coordinate (for upper left corner pixel of each box), number of rows , columns box spacing between each pixel. When I create the chart object in c#, I cannot figure out how to specify the category (x-axis) labels like when you select data in the Excel app for a chart, you have the right column to select the x-axis labels (picture: here). I set the colors of each serie in order to have something nice and meaningful : 1 color per serie, dotted for forecast. How to show locations on a bubble chart in Excel Map? Once you have the data you need to put them somewhere into Excel Map. Matplotlib charts can be horizontal, to create a horizontal bar chart: import matplotlib. To add a table of x,y coordinates to your map, globe, or scene, the table must contain two fields: one for the x-coordinate and one for the y-coordinate. You can instantly draw the conclusion about the top countries by App Launches. It may seem complicated but it give the exact format to put in to Google Earth or similar software. and calculate the y-coordinate yourself by taking one of your two trendline formulas and insert your just calculated x value. The position of each point is determined by the value of two variables, one for the x-axis and the other for the y-axis. Introduction to Pivot Tables, Charts, and Dashboards in Excel (Part 1) - Duration. I manage to show either the x-coordinate or the y-coordinate but not both at the same time. The process of representing data in Excel through graphs is typical and can be easily structured into the following steps: Enter the data into Excel spreadsheet for which graph needs to be created. Export your chart to Microsoft Word ®, Excel ®, PDF, PowerPoint ®, Google Docs ™, or Google Sheets ™ with a few simple. Use SignNow eSignature and document management solutions for your business workflow. Draw a circle. Try now for free!. Add GPS coordinates to address. The default diameter of the donut hole is 75% of the diameter of the whole chart, so all of the rings are scrunched together. My problem is that the correspondig graphs/maps are not ery useful and I would like to reconstruct these maps using Excel. If you’ve been using Excel to create Gantt charts from templates, it might be daunting to let go of that work to switch to online tools. This chart plots both the X axis and the Y axis as values. Visit Mathway on the web. But that doesn’t mean it’s not one of the best. Graph a set of coordinates in Excel 2010, display the equation and solve the equation using the value "0" for x to determine its y-intercept. Paste charts to PowerPoint with links. Each country is represented by a color, in default Excel fashion. Lat1_ and Long1_ are the coordinates of the first location. One of the more versatile of charts is the XY Scatter chart. Choose from thousands of free Microsoft Office templates for every event or occasion. Those charts which designed to plot only one value data (column, line, area chart types, etc) always use the horizontal (x) axis as a category axis, where data is distributed evenly. As you know how easy is to draw a 2D (with 2 axis) graphs in Excel. Posted by cyberchem on August 28, 2012 at 03:40 PM. Automatically Extending Excel Chart Series' Ranges For Next Year. I have a chart with past and forecasted figures for several series of data. When you create a graph in Microsoft Excel, Excel chooses your scale automatically based on your data. g (x) = -0. This video tells you how to identify the coordinates of an ordered pair from a given point. Pictograph is one of my list of the advanced Excel charts. When you need to determine a point x and y coordinates, please simply draw an area around the part where you need to know the coordinates. [Select the third option] Stretch - A single image will stretch in data bar. Select both the numeric data and adjacent row and column headings. XY charts show the relatedness of two sets of data. That means POINT X Coordinate,Y Coordinate,Z Coordinate will create a Point at X,Y with a height of Z. "Thermometer" chart display the percentage of a task that's completed. Graph break voucher. It’s easy to import Excel task lists into online tools to instant populate the online Gantt chart. In this sketch you need to define the vertices (corners) of the shape and figure out the approximate coordinates. This tool permits the user to convert latitude and longitude between decimal degrees and degrees, minutes, and seconds. Using tooltip we can display the above data. Line charts are used to display trends over time. Ultimate aim is to display the x-y coordinate as per the scale Note that the Scale is a log-log scale and the Chart type is X-Y Scatter type I am verymuch thankfull if anybody provides the solution to the above query. Use SignNow eSignature and document management solutions for your business workflow. New Zealand isn't always plotted correctly. The Unit Circle Written by tutor ShuJen W. On the INSERT tab inside Excel, in the ILLUSTRATIONS group, click PICTURE. Follow our step-by-step tutorial to make a multiple axes graph for free and online with Chart Studio. We assume that the (immovable) star is located at the origin, and the planet always starts from position (1, 0). This graph shows all the data but it is difficult to read, let alone to see how any given country compares. Create an Excel file, with two cols if you only want to show GPS data and nothing else Name them Longitude and Latitude and paste your GPS data accordingly. When creating a 3D Surface Graph inside Excel XYZ data is only part of what you need. A grid in excel is handy when designing floor plans that require both data entry and graphic representation. The image above shows the map to the right and a table with cities and their chart coordinates. Imagine you have monthly sales figures for 5 salespeople (as shown in the spreadsheet below), and you want to create a graph that will show the monthly figures for any selected individual. It is useful, for example, for analyzing gains and losses over a large data set. Is there any good coordinates drawing algorithm / pseudocode which would draw children below their parents in a nice and readable layout ?. Find answers to Convert Excel Long/Lat coordinates to UTM's and Graph them from the expert community at Experts Exchange. I would like to export its coordinates to excel. Excel 2010/2013 includes a variety of options to customize the plot area with. Now it has a slim piece that says 0% for one of data points that is just 1. This is why we present the book compilations in this website. In the scatter chart below, the… Read more about 2 Ways to Show Position of a Data Point on the X- and Y-Axes. We’ll also take a look at a couple of special polar graphs. Pie charts are used, for example, to show the production of one factory in relation to the output of the company or to show the revenue generated by one product relative to the sales of the. After that, select the data bars in the chart and go to Right Click ➜ Format Data Series Option. Once you've placed markers on your points of interest, you can can configure all sorts of extra aspects of your map. The workbook shown below used data imported from a typical S-parameter file (in this case an RF2321 amplifier, from RF Micro Devices) and plotted on a chart that uses an image file that contains. So how does this tie in with what GMF wants to do, i. Suppose we want to plot the volume of hydrochloric acid used vs. One number line is horizontal and is called the x-axis. 0, maximum = 3. pdf file, or will first have te be converted to an image format (jpg, bmp, png, gif…). However, you may come across instances when you want to make more sense of the data and put more labels on them. Line Graph Circle (Pie) Graph The same data displayed in 3 different types of graphs. The degree of this polynomial is 4 and is even. 200235 or Lat =53. A grid in excel is handy when designing floor plans that require both data entry and graphic representation. Enter any data, customize the chart's colors, fonts and other details, then download it or easily share it with a shortened url | Meta-Chart. Area chart contains 2 type of visualization and 3 type of charts-2D Area and 3D Area :-. X Scrollbar—Scrolls through the data in the graph or chart. Under Relative Position, enter the coordinates and color for up to three lights. Example question: Find the area under curve in Excel for the graph below, from x = 1 to x = 6. My work involves ranges 10^+/-24. I set the colors of each serie in order to have something nice and meaningful : 1 color per serie, dotted for forecast. The Excel chart is added to the layout. By default, Excel has a limited number of charts. The normal distribution graph in excel results in a bell-shaped curve. i doing hand wanted try , automatically giving excel starting coordinate (for upper left corner pixel of each box), number of rows , columns box spacing between each pixel. Next to the Select Data button is the Switch Row/Column button, which does exactly what it says: switches the rows and columns in your chart. Learn more about scatter charts. In Excel you will need the latitude and longitude to graph specific points. How to plot "Viscosity vs Time" & "Shear Rate vs Time" graph in Fluent?. In the "Chart" section, click the icon labeled "Scatter" and then select the icon labeled "Scatter with only Markers. Both of these are obtainable inside Excel, for free with calculations and formulas or by purchase of third party applications for less headache. Chart object Excel is a great tool to create charts quickly & easily, to display worksheet data in a meaningful manner for users. To have multiple line charts then your data need. Step 1 - Open excel, type x in cell A1 and type y in cell B1. Double-click the rectangle (the first shape in the. Step 1 : Choose a few data points on the x-axis under the curve (use a formula, if you have one) and list these values in Column A in sequence, starting from Row 1. Using an Excel Chart to Plotting the Mandelbrot Set. 4, Android 4. This means that instead of plotting points above the -axis, you will be plotting points below the -axis. Search any State and City in the US by clicking the State Code links and then the links for the resulting cities. All you'll need is a column of data that represents latitude, and one for longitude. Click on one of the five scatter chart types that appear in the pop-up menu. Simply paste data from Excel straight into the textbox. Select the CAD file. For example, in addition to the number of males, let's say we now have one more column that gives information about the number of females. It's a pretty simple chart with names on the left column and data on the right. The Bar graph lets you visually see differences in two or more sources. However, when trying to measure change over time, bar graphs are best when the changes are larger. When I create the chart object in c#, I cannot figure out how to specify the category (x-axis) labels like when you select data in the Excel app for a chart, you have the right column to select the x-axis labels (picture: here). Suppose you have data of quarterly sales in million for the last 10 years, something like following. The above drawing is the graph of the Unit Circle on the X – Y Coordinate Axis. Click on the Snap toolbar to create a chart. The new graph will be added to your existing plot. This series will wind up on the horizontal, or 'X,' axis. This post shows you how to draw a closed polygon (in this case a triangle for simplicity) on a 2D scatter plot in Excel 2003. To go from the first point to the second, we go "up two and over three". Select the first chart you want to move to the chart sheet, and go to Chart | Location. Example of Structural Foundation Link with XYZ values available for custom families. Microsoft Excel doesn't have functions to calculate definite integrals, but you can approximate this area by dividing the curve into smaller curves, each resembling a line segment. You can use Graphing Calculator 3D instead of excel for plotting spherical coordinates. The chart will show the upward and downward arrow instantly. Using the y-intercept and a second point the equation can be found. Under the Insert menu tab, in the Charts group, click the Column button and choose Clustered Column in 2-D Column. To attach text labels to data points in an xy (scatter) chart, follow these steps: On the worksheet that contains the sample data, select the cell range B1:C6. If you have difficulties with using the program after reading through this sheet and practicing see your instructor for further assistance. Position the new chart over original. The most effective visuals are often the simplest—and line charts (another name for the same graph) are some of the easiest to understand. The horizontal distance between the x-values (that is, the side-to-side width of the triangle) is the "x 2 – x 1" part of the slope formula. I tried to build a graph from Draw>Graph option but I can not find a way to display the coordinates. The position of each point is determined by the value of two variables, one for the x-axis and the other for the y-axis. It can be seen from the graph, that the Unit Circle is defined as having a Radius ( r ) = 1. Graphing Calculator 3D is a powerful software for visualizing math equations and scatter points. We also took advantage of our named range as a definition for a series on a chart, but you can also use named ranges as parameters in formulas. Click the chart and drag the handles to the desired size. Suppose you have data of quarterly sales in million for the last 10 years, something like following. but including third coordinates. Coordinate Graph Paper Template Coordinate Graph Paper Template When somebody should go to the book stores, search start by shop, shelf by shelf, it is in reality problematic. Select range A1:C7. The process for plotting a point is shown using an example. Examples of the X Label charts are Line, Column, Surface, Area, Radar and Bar charts. Standard sizes of graph paper are frequently available but there are times when a specific size of graph paper or a specific grid size is needed for a project. A couple years back we published a blog post about how to graph XYZ Data into MESH inside Excel, How to Graph XYZ Data into MESH Inside Microsoft Excel. Chart break log. It turns out that with a little imagination and creativity, we can format and configure the default charts so that the effect is like many other kinds of charts. By default, Excel has a limited number of charts. By default, Excel doesn’t print the row and column headings you see on the screen. Making a Coordinate Plane. The x -axis runs horizontally, or side to side, and the y -axis runs vertically, or from top to bottom. STEP 2: Click on the arrow next to Trendline. The same formula, using defined names for coordinates is shown below. Spotfire is much more robust but I've found it to be somewhat cumbersome at times as well. Assuming the data is set up in Excel columns A and B as show below define a new column using the Excel CONCATENATE function to combine the two coordinates with a comma between them. In Excel, we usually use chart to show data and trend for more clearly viewing, but in sometimes, maybe the chart is a copy and you haven't the original data of the chart as below screenshot shown. When you manually substitute values for the x variable, Microsoft Excel then plots the trendline incorrectly. Each country is represented by a color, in default Excel fashion. 10 spiffy new ways to show data with Excel It's time to dump the pie charts and move to donuts or even waterfalls to show off your data in ways people can better grasp. Graph break paper. This is why we present the book compilations in this website. Set Up the Control Worksheet. If you have XY data in cartesian coordinate and want to plot them in the polar coordinate, you can follow the below steps to calculate the corresponding theta and radius data and plot a polar graph. Hi I have an excel table A with 3 columns, first column is a Cartesian coordinate x, second is Cartesian coordinate y, and third column is temperature at the point (x, y). coordinates. The term 2D graph I mean the coordinate system x, y. This technique of using a stack of a back chart to display dial sprites and a front chart with transparent background to display various control devices, indicator needles and text will extensivly be used in this and future models. To find the slope of the line using the graphing method, the first step is to plot the given points on a coordinate system, then use the ratio rise over run between the points to find. How to Present Experimental Data in Graphical Form. Select Edit – Paste Special from the toolbar. As you know how easy is to draw a 2D (with 2 axis) graphs in Excel. Start now with a free trial!. Plotting Points on a Graph. ② Design the format of Chart Area. First, you need to insert and format an AutoShape as follows: From the Drawing toolbar, choose Basic Shapes from the AutoShapes drop-down list. See the image with 5 lines, and the excel sheet obtained after exporting only the layer and length of the objects. Change the speed of rendering by setting the rate. Please help. The download also includes a workbook for Classic Excel, which is Excel 2003 and before. If the data points cluster or bunch in a certain configuration -- for example, if they tend to form the shape of a line -- that indicates that the two sets of data are correlated in some way. Excel map charts use Bing maps to plot the data. What you are really asking for is not to read data from the chart, but to interpolate values from the data you originally put into it. If you don't want to continue to display the callouts, you can select and remove them on the background image. I looked around online and most people put the coordinates in separate X/Y columns. Step 2: Click the + symbol and add data labels by clicking it as shown below Step 3: Now we need to add the flavor names to the label. Use it to sketch the corresponding polar curve. The comment I meant to post is: Assuming that the coordinates for the first point are in the 1st row (A1 , B1, C1) and the coordinates for the second point are in the 2nd row (A2, B2, C2) the actual Excel formula should look like this =SQRT(SUMXMY2(A1:C1,A2:C2)). Chm 1112 Using Microsoft EXCEL for graphing. Consider a point on the graph. These options include those that fix the maximum and minimum amount for the first and last tick mark on the axis, display the values in reverse order, and apply a logarithmic scale. The x and y axes cross at a point referred to as the origin, where the coordinates are (0,0). If you have a scatter plot and you want to highlight the position of a particular data point on the x- and y-axes, you can accomplish this in two different ways. Google Sheets gives you a variety of options for your graph, so if you want to show parts that make up a whole you can go for a pie chart, and if you want to compare statistics, a bar graph will. To graph polar coordinates, make a table with distance (r) and angles (θ) values. Filter and Cross Highlight Excel Charts like you can in Power BI using some Excel Power Pivot magic, regular charts and a Slicer. Hm, it's not exactly built in AFAIK, but I found this page which gives a suggestion that worked for me:. Start now with a free trial!. X and Y are separate independent variables. The problem: Sometimes you have geographic data that consists only of latitudes and longitudes, but you want to know the altitudes as well — because, for example, you want to colorize points by height above sea level, or draw a profile of a track. Explore math with Desmos. Buying a poster from posters. The above plot looks better than the bar chart on cartesian coordinate system. The new graph will be added to your existing plot. It automates many details of plotting such as sample rate, aesthetic choices, and focusing on the region of interest. MS Excel 2010 already has templates to create a four quadrant matrix, which will save you a lot of time in your initial setup of a PICK chart. 4, Android 4. This kind of graph is needed to show percentages effectively. Move the mouse cursor to any data pointand press the leftmouse button. Chart break log. Graphing with Excel 2 2 1. You can quickly change the variables plotted on each axis, "size by" and "color by" cat. If you are starting from scratch, we recommend using our Spreadsheet Template to get started with your data, then simply copy the data over to BatchGeo to create a pin map. In this HowTech tutorial, we're going to show you how to graph functions in Excel 2016. If you are looking to add GPS coordinates to your address, please go to this page. The key difference is that a series of data defined by positions on the X- and Y-axes is connected by a line running from the lowest to the highest x value. In theory it should be able to plot data for any country, but I have seen issues with countries outside of the U. the moles of magnesium. It is often useful or necessary to find out what the gradient of a graph is. The first preview shows what we want - this chart shows. If you insert the picture using an Image object in the Control toolbox, you have quite a bit more latitude. Both of these are obtainable inside Excel, for free with calculations and formulas or by purchase of third party applications for less headache. Parallel plot or parallel coordinates plot allows to compare the feature of several individual observations (series) on a set of numeric variables. Download and install the plugin. There are times when you are given a point and will need to find its location on a graph. But to do so I need the coordinates of the maps ceated by Stata. For example, if you have an Excel workbook called Sales_Figures. This page uses true randomness to pick a random set of coordinates for a location on the planet's surface and show it on Google Maps. Excel map charts use Bing maps to plot the data. Click on the OK button. That is how to make a bar graph in Google Sheets that is brain-friendly. Further, the complete set of y-coordinates is a column of data points, not a fcn if that helps at all. Begin by graphing the following data on a separate sheet titled “GRAPH 1”. Of course, quadratic functions, or second degree polynomial functions, graph as parabolas. Top 10 types of graphs for data presentation you must use - examples, tips, formatting, how to use these different graphs for effective communication and in presentations. When you first open Excel, you start with a single worksheet. Good article, but I have a very weird problem, I'm using nvd3 to show a graph with two sets of data, the user can switch from a set to another, triggering the graph update. The first step in creating a graph using Microsoft Excel is entering the data. For example, if you want to plot age against income, put 'Age' in cell A1 and 'Income' in cell B1. See the data carefully. In the Excel Options dialog, click Add-Ins in the list on the left-hand side. The area of the circle may be found by using the formula, A=πr 2. Repeating this statement 10000 times with different values for X,Y & Z Coordinates will create 10000 discrete Points and this is where our Excel formula comes to help. Selected examples to deal with different objects, methods and properties in Excel. Hm, it's not exactly built in AFAIK, but I found this page which gives a suggestion that worked for me:. The only difference would be instead of using the AddPoint method of the Modelspace object, you would use AddLine. In Excel 2007 include the extrapolation "zone" in the forecast boxes (forward or backward) 8. Set the style to Bar 2D and set the analysis to q2. Inputs can be in several formats: GPS Coordinates (like N 42 59. Start now with a free trial!. One of the more versatile of charts is the XY Scatter chart. Select As Object In, and choose Two Chart Sheet from the drop-down list. Contributions: This table displays the contributions that help evaluate how much an object contributes to a given axis. Graph break paper. 5x 2 - 9x + 11. We assume that the (immovable) star is located at the origin, and the planet always starts from position (1, 0). All the graph colors including background color, line color, text color, axis color etc can be easily customized. With each new version of Excel, the capabilities of the program grow. When creating scatter charts, it's generally best to select only the X and Y values, to avoid confusing Excel. So lets graph the line (simply draw a horizontal line through ) graph of (note:the graph is the line that is overlapping the x-axis. These two lines look this way: Now, where the two lines cross is called their point of intersection. , (3,8)) where: the first number refers to horizontal position on the x -axis, the second number refers to vertical position on the y -axis, sometimes the ordered pairs are listed in tabular format with headings that correspond to the labels on the axis. Charts are composed of at least one series of one or more data points. Add chart charter. Open Excel or another spreadsheet. Choose from different chart types, like: line and bar charts, pie charts, scatter graphs, XY graph and pie charts. Coordinate planes are used to graph ordered pairs and equations or to construct scatter plots. and calculate the y-coordinate yourself by taking one of your two trendline formulas and insert your just calculated x value. This is my attempt to design a flexible, easy to use library for drawing graphs. It is not difficult, but it is not as straight forward as with a calculator. Each country is represented by a color, in default Excel fashion. X Scale and Y Scale—Formats the x- and y-scales. It is self-explanatory. The term 2D graph I mean the coordinate system x, y. If you do not know anything about latitude and longitude, please have a look at the article Geographic coordinate system. Related charts: Bubble chart. Choose from different chart types, like: line and bar charts, pie charts, scatter graphs, XY graph and pie charts. Sometimes when you want to create a line chart, it doesn't look the way you want. Back in the sheet, move the rectangle (it's a rectangle object, shaped as a square), to the top-left corner. Simply put, you can find the latitude and longitude for any city in the United States. Graph break voucher. Click on the "select data" option. Either type in the Chart data range box or click-and-drag to select your new data. Now, add your excel data to ArcMap (file > add data). Sincerely, The Google Maps Coordinate team Frequently-asked questions What will happen to my Google Maps Coordinate data? All data stored with Google Maps Coordinate will be systematically deleted from Google servers. To put the finish touches on the plot (the graph and axes label), click on the chart (the graph itself) and a menu item called Chart should appear. It is often amazing to people that the curvature of the Earth shows up very clearly in the observed elevation angles of peaks. X Scale and Y Scale—Formats the x- and y-scales. The order in which you write x- and y-coordinates in an ordered pair is very important. Start now with a free trial!. The following map appears. To attach text labels to data points in an xy (scatter) chart, follow these steps: On the worksheet that contains the sample data, select the cell range B1:C6. I’m using SEEP/W with Geostudio 2012. 870659N Long=1. Knowing how to create charts and graphs that are clear, readable, and engaging is critical to presenting data. As an example, I'll use the air temperature and density data that I used to demonstrate linear interpolation. That does not mean that those are the only charts one can create. 1 Problems Problem as I said in the beginning is that though the data labels I connected will update if data changes, but if I throw additional rows to the chart, the new data labels needs to be connected too and that makes it quite cumbersome. After you have one of the blocks in your drawing, you can copy it and place it on several points you wanted. Click Submit. Launch Excel. $$\theta = \beta$$. Using the y-intercept and a second point the equation can be found. Position the new chart over original. GPS coordinates are the part of GPS system, which means Global Positioning System. 870659N Long=1. In general, you create a normal curve just as you create any other chart in Excel: You set up the data and then chart it. Try now for free!. I have no experience with plotting maps and would like some help. Move the mouse cursor to any data pointand press the leftmouse button. edit: Note, I'm not as good with charts as other things in VBA, so as you create charts, you'll need to edit the. To graph polar coordinates, make a table with distance (r) and angles (θ) values. Click on the Titles Tab and enter the information below. In graphs with only positive values for x and y, the origin is in the lower left corner. This article explains how to create a doughnut chart in Excel using Syncfusion Excel (XlsIO) library. X and Y are separate independent variables. When creating a 3D Surface Graph inside Excel XYZ data is only part of what you need. First, we need to create a table of values. If you do not know anything about latitude and longitude , please have a look at the article Geographic coordinate system. (Note to me: Replace commas through dots) You get the x-coordinate. A coordinate graph is also sometimes called a coordinate plane, a Cartesian plane, or a Cartesian coordinate system. The program uses an inverse transformation to see whether the mouse (in form coordinates) is over a data point (in graph coordinates. In the scatter chart we can see that both horizontal and vertical axes indicated numeric values that plot numeric data in excel. Imagine you have monthly sales figures for 5 salespeople (as shown in the spreadsheet below), and you want to create a graph that will show the monthly figures for any selected individual. Here are the instructions: Make sure the graph type is Line and not Stacked Line; Select the chart ; In the chart menu click on: Design -> Select Data; In the dialog that comes up, click the 'hidden and empty cells' button. What are the GPS Coordinates. Example of a line chart made in PowerPoint. Well, things are getting better, but still we can’t make out the proportion accounted by the top contributors unless we also indicate the percentages along with the counts. First, there's the world coordinates that you want to use for the graph. Open a graph. Here is an example where a … Continue reading Adding Colored Regions to Excel Charts →. I'm working with a single chart sheet and want to create a vertical line (shpTracker) that tracks the mouse pointer in the Plot Area of the chart. In Excel you will need the latitude and longitude to graph specific points. We’ll use the Geocoding API from Google Maps, which returns coordinates for a given location. Identifying the active cell's coordinates Look in the top left hand corner of the Excel screen (Fig. Graph break paper. Change the speed of rendering by setting the rate. Statistics. Suppose we want to plot the volume of hydrochloric acid used vs. Put your X data into column 1, and your Y data into column 2. Your Revit model is now aligned with your CAD file. For example, if you have a Scatter Chart that shows the relationship between the age of a house and its proximity to the city and want to add the value of the house (the 3rd variable), then a Bubble Chart will get you there. expression. When you print a spreadsheet you won't see this grid by default, but I'll show you how to change that setting in this quick video tutorial. I am making a pie chart and cant seem to figure out how to add in decimals to the pie chart. To create a log-log graph in Microsoft Excel, you must first create an XY (scatter) graph. The procedure to create a line chart or line graph in Google Doc Spreadsheet is similar to the procedure we follow in Microsoft Office Excel Charts. My dynamo screen: my revit screen; my excel table: ps: when i change my. In the scatter chart below, the… Read more about 2 Ways to Show Position of a Data Point on the X- and Y-Axes. By Tepring Crocker July 9, 2015 Categories: Charts Tags: Excel Chart Multiple Series One of the most powerful advantages of a chart is its ability to show comparisons between data series, but you'll need to spend a little time thinking about what you want to show and how to organize it for excellent communication. The Wolfram Language has many ways to plot functions and data. See the image with 5 lines, and the excel sheet obtained after exporting only the layer and length of the objects. It will very ease you to see guide Coordinate Graph Paper Template as you such as. So to get the X values we want – today’s date, we define a name called Today:. Workaround is to use OOo or hand draw. In the data you provided there simply is no data pair (x,50), so your chart will never be able to return x witho. If you cannot find the information you are looking for,…. To make the graph, put in a column as many X data points from your graph as you can, and in the adjacent column put the corresponding Y coordinates. Hold down your shift key and click on the Z. The Axes The independent variable belongs on the x-axis (horizontal line) of the graph and the dependent variable belongs on the y-axis (vertical line). If the data points cluster or bunch in a certain configuration -- for example, if they tend to form the shape of a line -- that indicates that the two sets of data are correlated in some way. Click the Browse button in the Add-Ins dialog and point the Browse dialog at the XY Chart Labeler add-in file. An XY graph allows you to plot pairs of x and y values in a single chart. Select the first chart you want to move to the chart sheet, and go to Chart | Location. Inputs can be in several formats: GPS Coordinates (like N 42 59. Sort the values before plotting in the normal distribution graph to get a better curve shaped graph in excel. We're going to show you how to create the chart shown in Figure 15-9 from the. Click a cell in the Excel window. In this tutorial I will show you how to use JavaScript and the canvas as a means to display numerical information in the form of pie charts and doughnut charts. Importing coordinates using excel formula As shown in this pic below, we have X, Y, Z coordinates, otherwise Easting, Northing & Reduced levels of more than 10000 points in an Excel sheet. The Bar Graph is easy to understand and to create. Graph break paper. Thos are guesses because its REALLY hard to find those coordinates in this case from MTH 154 at Tidewater Community College. For example, if you select a data range to plot, Excel will. That does not mean that those are the only charts one can create. A small utility to simply display the GPS coordinates and accuracy. Like the following figure. Set X and Y axes. 4 Steps to Convert PDF Image to Excel PDFelement offers you the easiest process for a successful conversion of a scanned PDF image to Excel. Excel will show both the trendlines on the same chart. Add GPS coordinates to address. The input table for the creation of the frequency polygon is summarized below: Select the columns Mid-Point and Frequency. You can choose to plot just the markers for the data points, straight lines between the points, curved lines between the points, or either. Graph page break log. To do this, we first need to know how to actually embed a Matplotlib graph into a Tkinter application. So, that was the Excel Scatter Plot. Radial Bar Chart: Differences between Cartesian and Polar coordinate systems. Chart break form. Select the Insert Menu, choose Chart, Select the XY (Scatter) Chart type and then select the Chart sub-type that I have selected in black above. The key difference is that a series of data defined by positions on the X- and Y-axes is connected by a line running from the lowest to the highest x value. Improve your business processes and document management with SignNow eSignature solutions. The values in the "A" column dictate the X-axis data of your graph. In this example, as seen in cell G1 pointed at by the green arrow, X is 151 and Y is 159. From the command history, it is clear that the syntax for creating a point in AutoCAD is POINT X,Y,Z. Paste charts to PowerPoint with links. The normal distribution graph in excel results in a bell-shaped curve. It will very ease you to see guide Coordinate Graph Paper Template as you such as. Go to the design tab > type group > click change chart type button. It looks fine in dynamo, all my points and also my surface (when i tried to make one) looking ok. Charts in Excel VBA - Add a Chart, the Chart object & the ChartObject object Contents: Worksheet & Chart Sheet in Excel. Is there any way to find X and Y value at any point in graph by moving cursor??. Back in December 2010, I published an article about Better Chart Tooltips with Microsoft Excel. Note: Make sure to save any changes made to the Excel chart. Scroll over to the right until you can see column heading Z. Automatically Extending Excel Chart Series' Ranges For Next Year. Ideate BIMLink can export a variety of Revit coordinate data as described in Link Properties > Integrate Revit Coordinate Properties with Ideate BIMLink. I have a XY-scatter(with lines) chart where I even want to show the coordinates for each point in the chart. Open a new, blank Excel document. Click on the new chart and drag to location above original chart. The frequency polygon should look like the graph at the top of this article. In this article, we show how to change the transparency of a graph plot in matplotlib with Python. The format ChrY:coordinate-coordinate should be used. Click on “Display equation on chart” On Clicking, an equation is showed on the chart. i doing hand wanted try , automatically giving excel starting coordinate (for upper left corner pixel of each box), number of rows , columns box spacing between each pixel. Buying a poster from posters. Use it to sketch the corresponding polar curve. Then, go to the Chart Options (in the Chart menu) and in the "Gridlines" tab check the "Minor gridlines" on the axes you want depending of the desired graph type: semi-logarithmical or logarithmical. That means POINT X Coordinate,Y Coordinate,Z Coordinate will create a Point at X,Y with a height of Z. However, this doesn’t mean you can’t use color encoding in your Excel charts. Double-click the Excel program icon, which resembles a white "X" on a green folder. Try now for free!. This probably isn't the best way to graph the Mandelbrot Set in Excel -- it has a few limitations: Each data series in an Excel chart can only have 32000 points. Lat1_ and Long1_ are the coordinates of the first location. Lat2_ and Long2_ are the coordinates of the second location. For instructions on adding x,y coordinate data as a layer in ArcGIS, see the ArcGIS Help 2. Defining Series Values in a Charts and Graphs: Pretty straightforward Excel function. Middle school math with pizzazz book e, year 7 algebra test, fist in math, online graphing calculator multivariable, "ti-38" quadratic formula, algebranator, how to order ratios from least to greatest. Charts in Excel VBA - Add a Chart, the Chart object & the ChartObject object Contents: Worksheet & Chart Sheet in Excel. After the projection transform, the XY coordinates for your Axes should look like below. So how does this tie in with what GMF wants to do, i. Improve your business processes and document management with SignNow eSignature solutions. Identifying the active cell's coordinates Look in the top left hand corner of the Excel screen (Fig. Thanks, i cant believe i was trying to make it so complicated. 0 (Y meaning the value of GR along the horizontal axis). I need to write a VBA macro to overlay multiple transparent rectangles on a chart to highlight different sections of data based on date ranges that span across the x-axis. TO create the graph you would choose a topic, mine was favorite fruits(not shown). This probably isn't the best way to graph the Mandelbrot Set in Excel -- it has a few limitations: Each data series in an Excel chart can only have 32000 points. How to make a line graph in Excel. I had a look at some commercial libraries, but none of them met by demands. It can show uneven intervals or clusters of data and is commonly used for scientific data. Lat1_ and Long1_ are the coordinates of the first location. That does not mean that those are the only charts one can create. If you have a scatter plot and you want to highlight the position of a particular data point on the x- and y-axes, you can accomplish this in two different ways. I built this primarily to make it easy to check if a Locationless (Reverse) Cache has already been found. To create a log-log graph in Microsoft Excel, you must first create an XY (scatter) graph. The key difference is that a series of data defined by positions on the X- and Y-axes is connected by a line running from the lowest to the highest x value. We’re talking about how to convert Excel to Google Sheets, and we have everything you need to know. Coordinate Graph Paper Template Coordinate Graph Paper Template When somebody should go to the book stores, search start by shop, shelf by shelf, it is in reality problematic. Today, we will be working with individual data points. Written by co-founder Kasper Langmann, Microsoft Office Specialist. To create a PICK chart in the Excel 2003, the user needed to start from scratch by formatting groups of cells to create the appearance of the matrix. 6 (optional) I always check "display equation on chart" and "display r² on chart" (and note the number of significant figures can be increased in formatting). Left click on the "Insert" tab on the upper Excel menu. Google maps placemarks from spreheets google earth placemarks to excel solved geolocation map coordinates coordinate from excel to utm coordinates in google earth geofumed. To create the grid, select the blue shaded cells in the top table, and insert a donut chart (below left). The calculated parameters of typical peaks that can be seen from a number of the peaks in the SGM are given in a separate table. Building a combination chart in Excel is usually pretty easy. Open an Excel workbook. To play with parallel coordinates in the browser, there is a Protovis example using the cars dataset where you can brush on all dimensions and also pick a dimension to color the values by (which can show some interesting patterns). If you’ve been using Excel to create Gantt charts from templates, it might be daunting to let go of that work to switch to online tools. Spotfire is much more robust but I've found it to be somewhat cumbersome at times as well. Grids can be used in excel for the purposes of calculation and a visual reference. 3d line graph in excel? I need to draw a 3D line graph in excel, how can I do this? (Hhere is some idea if necessary: There will be no numbers, but just concepts related to each other. Download free on iTunes. Excel can manage missing data or bank cells when creating scatter or line charts in three different ways: The blank cell is given a value of zero. So lets graph the line (simply draw a horizontal line through ) graph of (note:the graph is the line that is overlapping the x-axis. To see an exact location, you can use the coordinates on the map grid: In the upper menu bar, click View Grid. When I create the chart object in c#, I cannot figure out how to specify the category (x-axis) labels like when you select data in the Excel app for a chart, you have the right column to select the x-axis labels (picture: here). This chart plots both the X axis and the Y axis as values. For a location without an address, you can simply right-click anywhere on the map and it displays the coordinates automatically. Coordinate Graph Paper Template Coordinate Graph Paper Template When somebody should go to the book stores, search start by shop, shelf by shelf, it is in reality problematic. Anything You Would Like To See? There are a ton of things you can do with VBA and Excel charts. Filter and Cross Highlight Excel Charts like you can in Power BI using some Excel Power Pivot magic, regular charts and a Slicer. It is useful, for example, for analyzing gains and losses over a large data set. I set the colors of each serie in order to have something nice and meaningful : 1 color per serie, dotted for forecast. I’m using SEEP/W with Geostudio 2012. Before pasting the coordinates, type LINE as the first word in the file (this will launch the Line command when the script is run), then press [Enter]. How to Make a Graph in Excel 1. Back in the sheet, move the rectangle (it's a rectangle object, shaped as a square), to the top-left corner. Plotting Points on a Graph. By default, Excel has a limited number of charts. It will very ease you to see guide Coordinate Graph Paper Template as you such as. It's a pretty simple chart with names on the left column and data on the right. SmartDraw is absolutely the easiest chart software. The image above shows the map to the right and a table with cities and their chart coordinates. Graphs are designed to represent statistical data: they can be done manually or with the help of softwares such as Excel (scroll down for demo videos). is there a way to get an output of the stata solution for the homals. As with a paper spreadsheet, you can use Excel to organize your data into rows and columns and to perform mathematical calculations. Use these printables and lesson plans to teach students how to read and create various types of graphs and charts. Scale A translation in which the size and shape of the graph of a function is changed. A new menu will appear. Use the Add method of the DataValidation property of the Range object. of an address? Again why not use Google API… Address coordinates in Excel. Hi I have a curve,I want to find x and y coordinate of the maximum value Thanks. The same formula, using defined names for coordinates is shown below. 5in}\alpha \le \theta \le \beta \]. Size Chart. If you want to remove a graph from your plot just press - to delete it. In Excel you will need the latitude and longitude to graph specific points. Click the Insert tab with the Ribbon. Polar Plots. It is easy to generate points on the graph. Make a copy of the chart. org helps support GraphSketch and gets you a neat, high-quality, mathematically-generated poster. How to Make a Graph on Excel With X Y Coordinates 1. Notice that the graph forms a straight line. These three types of grid lines assist in scaling the chart against set parameters. PLEASE, noticed that I will be clicking anywhere on the plot NO on a SERIES of points. Enter value 20 as row height, From Format options, click Column width, and enter 5 as column width, click OK. The comment I meant to post is: Assuming that the coordinates for the first point are in the 1st row (A1 , B1, C1) and the coordinates for the second point are in the 2nd row (A2, B2, C2) the actual Excel formula should look like this =SQRT(SUMXMY2(A1:C1,A2:C2)). The image above shows the map to the right and a table with cities and their chart coordinates. Hope you liked it. Line charts are used to display trends over time. To show the relationship most effectively, square up the plot area by selecting the graph, grabbing an anchor point with the mouse, and changing the axis proportions. For example, in addition to the number of males, let's say we now have one more column that gives information about the number of females. Click the Next button on the Chart Wizard twice. In graphs with only positive values for x and y, the origin is in the lower left corner. One option, however, is to add regions to your time series charts to indicate historical periods or visualization binary data. Creating Graphs in Excel. Note that the slope for this equation and graph is , or "two over three". The charting is significantly faster with it off. Today’s post describes different techniques of how to color encode Microsoft Excel bar charts, with or without using VBA. I am highlighting the two columns and hitting insert than pie chart. The equation that is displayed is Y= -25x + 125. How to Format the X and Y Axis Values on Charts in Excel 2013. This technique of using a stack of a back chart to display dial sprites and a front chart with transparent background to display various control devices, indicator needles and text will extensivly be used in this and future models. Well, things are getting better, but still we can’t make out the proportion accounted by the top contributors unless we also indicate the percentages along with the counts. Specify the coordinates as floating point values, where 0:0 is the bottom left corner of the chart, 0. TO create the graph you would choose a topic, mine was favorite fruits(not shown). It will surely be the intersection of two values and one value will be from the x axis and the other valve from the y axis. If the data points cluster or bunch in a certain configuration -- for example, if they tend to form the shape of a line -- that indicates that the two sets of data are correlated in some way. When pasted into Excel depending on your local settings you might need to replace the dots with the comma. Other Ideas. However, you may come across instances when you want to make more sense of the data and put more labels on them. Open Excel or another spreadsheet. Licensed under GNU GPL. for instance, box 3 rows, 4 columns, pixel spacing of 20 , starting coordinates of (20,50), output should like:. Simply put, you can find the latitude and longitude for any city in the United States. Draw a circle. |
# Quick Question on a Proof of Artin-Wedderburn Theorem
Question [Edited]: [See below.] Are the isomorphisms in $(1)$ and $(2)$ (additive) group homomorphisms?
If I'm right, $\text{End}_R(M)$ is a ring, but $\text{Hom}_R\left(M_i^{n_i},M_j^{n_j}\right)$ is only a group (we can't compose two functions in it). However when coming back to the direct sum of $\text{End}_R\left(M_i^{n_i}\right)$ we get a ring again.
So, are we showing $\text{End}_R(M)\cong\displaystyle\bigoplus_{i=1}^k\text{End}_R\left(M_i^{n_i}\right)$ as rings by getting intermediate group isomorphisms?...
Also, where is the hypothesis "finite type" used in the proof? [Edit: This has been answered in the comments.]
Artin-Wedderburn Theorem: Let $M$ be a semisimple $R$-module of finite type. Then $$\text{End}_R(M)\cong\bigoplus_{i=1}^k M_{n_i\times n_i}(D_i)$$ for some division rings $D_i$.
Proof: $M=\displaystyle\bigoplus_{i=1}^k (M_i)^{n_i}$ where $M_i$ is simple. \begin{align} \text{End}_R(M)&=\text{End}_R\left(\bigoplus_{i=1}^k(M_i)^{n_i}\right)\\ &\cong\bigoplus_{1\leq i,j\leq k}\text{Hom}_R\left(M_i^{n_i},M_j^{n_j}\right)\tag{1}\\ &\cong\bigoplus_{i=1}^k\text{End}_R\left(M_i^{n_i}\right)\tag{2}\\ &\cong\bigoplus_{i=1}^kM_{n_i\times n_i}\left(\text{End}_R(M_i)\right)\tag{3}\\ \end{align} where $(2)$ is a consequence of Schur's Lemma and $(3)$ follows from a previous result.
$M_i$ simple $\stackrel{\text{Schur}}{\implies}$ $\text{End}_R(M_i)$ is a division ring.
All in all, $$\text{End}_R(M)\cong\bigoplus_{i=1}^kM_{n_i\times n_i}(D_i)$$ for some division rings $D_i$.$\blacksquare$
• I think since $M$ is a module of finite type, or as is common to say that as $M$ is a f.g $R$ module, you can write $M$ as a direct sum of finitely many simple submodules. So the very first step of the proof outlined above seems to use that. – SMG Nov 15 '14 at 2:32
• This is somewhat similar to something I asked long ago ( mathoverflow.net/questions/11576/apocryphal-maschke-theorem ). The intermediate isomorphisms (1) and (2) are a-priori just isomorphisms of additive groups, but if you look at the inverse isomorphism to their composition, you see that it just takes a list of endomorphisms of $M_i^{n_i}$ and combines them to an endomorphism of $\bigoplus_{i} M_i^{n_i}$; this is clearly a ring homomorphism, and thus an isomorphism because it is an inverse of a group isomorphism. – darij grinberg Nov 26 '14 at 17:33
• My comment was deleted since the answer I left it on was deleted. The isomorphism in (1) is akin to the block matrix decomposition of the endomorphism ring of a vector space given a direct sum decomposition of the vector space. – aes Nov 26 '14 at 17:45
• The individual summands in (1) don't have a ring structure, but the whole thing does: the multiplication is like matrix multiplication, as aes suggests above. When you make (1) into a ring this way, the first $\cong$ will be an iso of rings. (1)$\cong$(2) is then just the statement that the off-diagonal entries in all these "matrices" are zero, so the second iso is also a ring iso where (2) is a direct sum of rings. – Matthew Towers Nov 26 '14 at 18:34
It sounds a bit like by "are they group isomorphisms" you are really asking "are they only group isomorphisms and not ring isomorphisms?" Of course they are additive group isomorphisms, but they are actually more than that.
While the individual Hom groups aren't rings, the sum $\bigoplus_{1\leq i,j\leq k}\text{Hom}_R\left(M_i^{n_i},M_j^{n_j}\right)$ is a ring!
Perhaps the way to get comfortable with it is to imagine it as a "formal" ring of matrices with entries from the Hom groups. Then formal matrix multiplication composes them in a reasonable way such it is a ring. The matrix multiplication matches the composition in $\mathrm{End}_R(M)$ exactly through the map, so it is a ring homomorphism. This holds even if the $M_i$ lack their special properties (being pairwise nonisomorphic simple modules.)
In the passage from line (1) to (2), we simply eliminate a lot of superfluous terms in the direct sum and discover that the multiplication boils down to coordinatewise multiplication of a direct product of rings. That is what the pairwise nonisomorphic condition buys us.
So, the isomorphisms involved are ring isomorphism all the way through.
Here's another sort of example. Let $e$ be an idempotent in any ring with identity, and let $f=1-e$. Then one can show that $R\cong \begin{bmatrix}eRe&eRf\\fRe&fRf\end{bmatrix}$ as rings, where the multiplication on the right is formal matrix multiplication. Neither $eRf$ nor $fRe$ has to be a ring, but $eRe$ and $fRf$ are both rings.
You can look upon this as $R\cong\mathrm{End}_R(R)\cong \mathrm{End}_R(eR\oplus fR)=\mathrm{Hom}_R(eR\oplus fR,eR\oplus fR)\cong\begin{bmatrix}\mathrm{Hom}_R(eR,eR)&\mathrm{Hom}_R(fR,eR)\\\mathrm{Hom}_R(eR,fR)&\mathrm{Hom}_R(fR,fR)\end{bmatrix}$
The (group) isomorphism $\mathrm{Hom}_R(fR,eR)\cong eRf$ can be found in Lam's First course in noncommutative rings (Proposition 21.6.) It turns out to be a ring isomorphism if $e=f$. |
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# Tutor profile: Kevin B.
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Kevin B.
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## Questions
### Subject:Computer Science (General)
TutorMe
Question:
Write a pseudocode function, fib(n), to find the $$n^{th}$$ number in the fibonacci sequence, $$F_{n} = F_{n-1} + F_{n-2}$$, starting with $$F_{0} = 0, F_{1} = 1$$ and using recursion.
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Kevin B.
fib(n): if (n == 0): return 0 else if (n == 1): return 1 else: return fib(n-1) + fib(n-2)
### Subject:Calculus
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Question:
Find the area bounded by the curves $$y=x^{2}$$ and $$y=x$$.
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Kevin B.
First, find the points at which the two curves intersect. $$x^{2}=x \Rightarrow x=0,1$$ Next, take the integral of the upper bound minus the lower bound on the interval. $$Area = \int_{0}^{1}(x-x^{2})dx = \left[ \frac{x^{2}}{2} - \frac{x^3}{3} \right]_{0}^{1} = \left( \frac{1}{2} - \frac{1}{3} \right) - 0 = \frac {1}{6}$$
### Subject:Physics
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Question:
A ball is thrown horizontally at 10 m/s off a roof that is 120 m tall. What horizontal distance does the ball travel before hitting the ground? (Use g = 10 m/s^2 and neglect air resistance)
Inactive
Kevin B.
First, find the time it will take for the ball to hit the ground. $$V_{0y} = 0 m/s$$ $$y_{0} = 120 m$$ $$y_{f} = 0 m$$ $$\left( y_{f} - y_{0}\right) = v_{0y} * t - \frac{1}{2} * g * t^{2}$$ $$\Rightarrow \left( 0 - 120 \right) = 0 - \frac{1}{2} * 10 * t^{2}$$ $$\Rightarrow -120 = -5 * t^{2}$$ $$\Rightarrow t^{2} = 24$$ $$\Rightarrow t=\sqrt{24} = 2\sqrt{6} s$$ Then, find the horizontal distance travelled in this time. $$V_{x} = 10m/s$$ $$x_0 = 0 m/s$$ $$\left( x_{f} - x_{0} \right) = v_{x} * t$$ $$\Rightarrow \left( x_{f} - 0 \right) = 10 * 2\sqrt{6}$$ $$\Rightarrow x_{f} = 20\sqrt{6} \approx 48.99m$$ Therefore, the ball will travel approximately 48.99 m before hitting the ground.
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## anonymous one year ago You have 6 friends who want to go to a concert with you, but you only have room in your car for 2 of them. To avoid accusations of favoritism, you have your friends draw straws: There are 4 long and 2 short straws. How many possible pairs of friends can be chosen this way?
• This Question is Open
1. anonymous
Of the 6 friends, call them $$a,b,c,d,e,f$$, you are selecting 2. It doesn't matter what the order is. What this means is that picking $$a$$ then $$b$$ is the same as picking $$b$$ then $$a$$. This means you're counting the number of combinations. $\binom62={}_6C_2=\frac{6!}{2!(6-2)!}$ |
CGAL 4.10 - Manual
Hello World
This tutorial is for the CGAL newbie, who knows C++ and has a basic knowledge of geometric algorithms. The first section shows how to define a point and segment class, and how to apply geometric predicates on them. The section further raises the awareness that that there are serious issues when using floating point numbers for coordinates. In the second section you see how the 2D convex hull function gets its input and where it puts the result. The third section shows what we mean with a Traits class, and the fourth section explains the notion of concept and model.
# Three Points and One Segment
In this first example we see how to construct some points and a segment, and we perform some basic operations on them.
All CGAL header files are in the subdirectory include/CGAL. All CGAL classes and functions are in the namespace CGAL. Classes start with a capital letter, global function with a lowercase letter, and constants are all uppercase. The dimension of an object is expressed with a suffix.
The geometric primitives, like the point type, are defined in a kernel. The kernel we have chosen for this first example uses double precision floating point numbers for the Cartesian coordinates of the point.
Besides the types we see predicates like the orientation test for three points, and constructions like the distance and midpoint computation. A predicate has a discrete set of possible results, whereas a construction produces either a number, or another geometric entity.
#include <iostream>
#include <CGAL/Simple_cartesian.h>
typedef Kernel::Point_2 Point_2;
typedef Kernel::Segment_2 Segment_2;
int main()
{
Point_2 p(1,1), q(10,10);
std::cout << "p = " << p << std::endl;
std::cout << "q = " << q.x() << " " << q.y() << std::endl;
std::cout << "sqdist(p,q) = "
<< CGAL::squared_distance(p,q) << std::endl;
Segment_2 s(p,q);
Point_2 m(5, 9);
std::cout << "m = " << m << std::endl;
std::cout << "sqdist(Segment_2(p,q), m) = "
<< CGAL::squared_distance(s,m) << std::endl;
std::cout << "p, q, and m ";
switch (CGAL::orientation(p,q,m)){
std::cout << "are collinear\n";
break;
std::cout << "make a left turn\n";
break;
std::cout << "make a right turn\n";
break;
}
std::cout << " midpoint(p,q) = " << CGAL::midpoint(p,q) << std::endl;
return 0;
}
To do geometry with floating point numbers can be surprising as the next example shows.
#include <iostream>
#include <CGAL/Simple_cartesian.h>
typedef Kernel::Point_2 Point_2;
int main()
{
{
Point_2 p(0, 0.3), q(1, 0.6), r(2, 0.9);
std::cout << (CGAL::collinear(p,q,r) ? "collinear\n" : "not collinear\n");
}
{
Point_2 p(0, 1.0/3.0), q(1, 2.0/3.0), r(2, 1);
std::cout << (CGAL::collinear(p,q,r) ? "collinear\n" : "not collinear\n");
}
{
Point_2 p(0,0), q(1, 1), r(2, 2);
std::cout << (CGAL::collinear(p,q,r) ? "collinear\n" : "not collinear\n");
}
return 0;
}
When reading the code, we would assume that it prints three times "collinear". However we obtain:
not collinear
not collinear
collinear
As the fractions are not representable as double precision number the collinearity test will internally compute a determinant of a 3x3 matrix which is close but not equal to zero, and hence the non collinearity for the first two tests.
Something similar can happen with points that perform a left turn, but due to rounding errors during the determinant computation, it seems that the points are collinear, or perform a right turn.
If you need that the numbers get interpreted at their full precision you can use a CGAL kernel that performs exact predicates and extract constructions.
#include <iostream>
#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
#include <sstream>
typedef Kernel::Point_2 Point_2;
int main()
{
Point_2 p(0, 0.3), q, r(2, 0.9);
{
q = Point_2(1, 0.6);
std::cout << (CGAL::collinear(p,q,r) ? "collinear\n" : "not collinear\n");
}
{
std::istringstream input("0 0.3 1 0.6 2 0.9");
input >> p >> q >> r;
std::cout << (CGAL::collinear(p,q,r) ? "collinear\n" : "not collinear\n");
}
{
q = CGAL::midpoint(p,r);
std::cout << (CGAL::collinear(p,q,r) ? "collinear\n" : "not collinear\n");
}
return 0;
}
Here comes the output and you may still be surprised.
not collinear
collinear
collinear
In the first block the points are still not collinear, for the simple reason that the coordinates you see as text get turned into floating point numbers. When they are then turned into arbitrary precision rationals, they exactly represent the floating point number, but not the text.
This is different in the second block, which corresponds to reading numbers from a file. The arbitrary precision rationals are then directly constructed from a string so that they represent exactly the text.
In the third block you see that constructions as midpoint constructions are exact, just as the name of the kernel type suggests.
In many cases you will have floating point numbers that are "exact", in the sense that they were computed by some application or obtained from a sensor. They are not the string "0.1" or computed on the fly as "1.0/10.0", but a full precision floating point number. If they are input to an algorithm that makes no constructions you can use a kernel that provides exact predicates, but inexact constructions. An example for that is the convex hull algorithm which we will see in the next section. The output is a subset of the input, and the algorithm only compares coordinates and performs orientation tests.
At a first glance the kernel doing exact predicates and constructions seems to be the perfect choice, but performance requirements or limited memory resources make that it is not. Also for many algorithms it is irrelevant to do exact constructions. For example a surface mesh simplification algorithm that iteratively contracts an edge, by collapsing it to the midpoint of the edge.
Most CGAL packages explain what kind of kernel they need or support.
# The Convex Hull of a Sequence of Points
All examples in this section compute the 2D convex hull of a set of points. We show that algorithms get their input as a begin/end iterator pair denoting a range of points, and that they write the result, in the example the points on the convex hull, into an output iterator.
## The Convex Hull of Points in a Built-in Array
In the first example we have as input an array of five points. As the convex hull of these points is a subset of the input it is safe to provide an array for storing the result which has the same size.
#include <iostream>
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/convex_hull_2.h>
typedef K::Point_2 Point_2;
int main()
{
Point_2 points[5] = { Point_2(0,0), Point_2(10,0), Point_2(10,10), Point_2(6,5), Point_2(4,1) };
Point_2 result[5];
Point_2 *ptr = CGAL::convex_hull_2( points, points+5, result );
std::cout << ptr - result << " points on the convex hull:" << std::endl;
for(int i = 0; i < ptr - result; i++){
std::cout << result[i] << std::endl;
}
return 0;
}
We saw in the previous section that CGAL comes with several kernels. As the convex hull algorithm only makes comparisons of coordinates and orientation tests of input points, we can choose a kernel that provides exact predicates, but no exact geometric constructions.
The convex hull function takes three arguments, the start and past-the-end pointer for the input, and the start pointer of the array for the result. The function returns the pointer into the result array just behind the last convex hull point written, so the pointer difference tells us how many points are on the convex hull.
## The Convex Hull of Points in a Vector
In the second example we replace the built-in array by a std::vector of the Standard Template Library.
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/convex_hull_2.h>
#include <vector>
typedef K::Point_2 Point_2;
typedef std::vector<Point_2> Points;
int main()
{
Points points, result;
points.push_back(Point_2(0,0));
points.push_back(Point_2(10,0));
points.push_back(Point_2(10,10));
points.push_back(Point_2(6,5));
points.push_back(Point_2(4,1));
CGAL::convex_hull_2( points.begin(), points.end(), std::back_inserter(result) );
std::cout << result.size() << " points on the convex hull" << std::endl;
return 0;
}
We put some points in the vector calling the push_back() method of the std::vector class.
We then call the convex hull function. The first two arguments, points.begin() and points.end() are iterators, which are a generalization of pointers: they can be dereferenced and incremented. The convex hull function is generic in the sense that it takes as input whatever can be dereferenced and incremented.
The third argument is where the result gets written to. In the previous example we provided a pointer to allocated memory. The generalization of such a pointer is the output iterator, which allows to increment and assign a value to the dereferenced iterator. In this example we start with an empty vector which grows as needed. Therefore, we cannot simply pass it result.begin(), but an output iterator generated by the helper function std::back_inserter(result). This output iterator does nothing when incremented, and calls result.push_back(..) on the assignment.
If you know the STL, the Standard Template Library, the above makes perfect sense, as this is the way the STL decouples algorithms from containers. If you don't know the STL, you maybe better first familiarize yourself with its basic ideas.
# About Kernels and Traits Classes
In this section we show how we express the requirements that must be fulfilled in order that a function like convex_hull_2() can be used with an arbitrary point type.
If you look at the manual page of the function convex_hull_2() and the other 2D convex hull algorithms, you see that they come in two versions. In the examples we have seen so far the function that takes two iterators for the range of input points and an output iterator for writing the result to. The second version has an additional template parameter Traits, and an additional parameter of this type.
template<class InputIterator , class OutputIterator , class Traits >
InputIterator beyond,
const Traits & ch_traits)
What are the geometric primitives a typical convex hull algorithm uses? Of course, this depends on the algorithm, so let us consider what is probably the simplest efficient algorithm, the so-called "Graham/Andrew Scan". This algorithm first sorts the points from left to right, and then builds the convex hull incrementally by adding one point after another from the sorted list. To do this, it must at least know about some point type, it should have some idea how to sort those points, and it must be able to evaluate the orientation of a triple of points.
And that is where the template parameter Traits comes in. For ch_graham_andrew() it must provide the following nested types:
• Traits::Point_2
• Traits::Less_xy_2
• Traits::Left_turn_2
• Traits::Equal_2
As you can guess, Left_turn_2 is responsible for the orientation test, while Less_xy_2 is used for sorting the points. The requirements these types have to satisfy are documented in full with the concept ConvexHullTraits_2.
The types are regrouped for a simple reason. The alternative would have been a rather lengthy function template, and an even longer function call.
template <class InputIterator, class OutputIterator, class Point_2, class Less_xy_2, class Left_turn_2, class Equal_2>
InputIterator beyond,
OutputIterator result);
There are two obvious questions: What can be used as argument for this template parameter? And why do we have template parameters at all?
To answer the first question, any model of the CGAL concept Kernel provides what is required by the concept ConvexHullTraits_2.
As for the second question, think about an application where we want to compute the convex hull of 3D points projected into the yz plane. Using the class Projection_traits_yz_3 this is a small modification of the previous example.
#include <iostream>
#include <iterator>
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Projection_traits_yz_3.h>
#include <CGAL/convex_hull_2.h>
typedef K::Point_2 Point_2;
int main()
{
std::istream_iterator< Point_2 > input_begin( std::cin );
std::istream_iterator< Point_2 > input_end;
std::ostream_iterator< Point_2 > output( std::cout, "\n" );
CGAL::convex_hull_2( input_begin, input_end, output, K() );
return 0;
}
Another example would be about a user defined point type, or a point type coming from a third party library other than CGAL. Put the point type together with the required predicates for this point type in the scope of a class, and you can run convex_hull_2() with these points.
Finally, let us explain why a traits object that is passed to the convex hull function? It would allow to use a more general projection traits object to store state, for example if the projection plane was given by a direction, which is hardwired in the class Projection_traits_yz_3.
# Concepts and Models
In the previous section we wrote that "Any model of the CGAL concept Kernel provides what is required by the concept ConvexHullTraits_2".
A concept is a set of requirements on a type, namely that it has certain nested types, certain member functions, or comes with certain free functions that take the type as it. A model of a concept is a class that fulfills the requirements of the concept.
Let's have a look at the following function.
template <typename T>
T
duplicate(T t)
{
return t;
}
If you want to instantiate this function with a class C this class must at least provide a copy constructor, and we say that class C must be a model of CopyConstructible. A singleton class does not fulfill this requirment.
Another example is the function
template <typename T>
T& std::min(const T& a, const T& b)
{
return (a<b)?a:b;
}
This function only compiles if the operator<<(..) is defined for the type used as T, and we say that the type must be a model of LessThanComparable.
An example for a concept with required free functions is the HalfedgeListGraph in the CGAL package CGAL and the Boost Graph Library. In order to be a model of HalfedgeListGraph a class G there must be a global function halfedges(const G&), etc.
An example for a concept with a required traits class is InputIterator. For a model of an InputIterator a specialization of the class std::iterator_traits must exist (or the generic template must be applicable). |
##### https://github.com/cran/gclus
Tip revision: 1a206d3
order.Rd
\name{order.single}
\alias{order.single}
\alias{order.hclust}
%- Also NEED an '\alias' for EACH other topic documented here.
\title{Orders objects using hierarchical clustering}
\description{
Reorders objects so that similar (or high-merit) object pairs are adjacent.
A permutation vector is returned.
}
\usage{
order.single(merit,clusters=NULL)
order.hclust(merit, reorder=TRUE,...)
}
%- maybe also 'usage' for other objects documented here.
\arguments{
\item{merit}{is either a symmetric matrix of merit or similarity score,
or a \code{dist}.}
\item{clusters}{if non-null, specifies a partial ordering. It should be a list whose
ith element contains
the indices the objects in the ith ordered cluster.}
\item{reorder}{if TRUE, reorders the default ordering from \code{hclust}.}
\item{...}{arguments are passed to \code{hclust}.}
}
\details{ \code{order.single} performs a variation on single-link cluster analysis,
devised by Gruvaeus and Wainer (1972).
When two ordered clusters are merged, the new cluster is formed by placing the
most similar endpoints of the joining clusters adjacent to each other.
When applied to variables, the resulting order is useful for scatterplot
matrices.
where the similarity between two ordered clusters is defined as the minimum distance
between their endpoints. When two ordered clusters are merged, the new cluster is formed by placing the
most similar endpoints of the joining clusters adjacent to each other.
When applied to variables, the resulting order is useful for parallel
coordinate displays.
\code{order.hclust} returns the order of objects from \code{hclust} if
\code{reorder} is \code{FALSE}. Otherwise, it reorders the objects using
\code{hclust.reorder}
so that
when two ordered clusters are merged, the new cluster is formed by placing the
most similar endpoints of the joining clusters adjacent to each other.
\code{order.hclust(m,method="single")} is equivalent to
\code{order.single} when \code{clusters} is \code{NULL}.
The default method of \code{hclust} is "complete", see \code{\link{hclust}} for other
possibilities.
}
\value{A permutation of the objects represented by \code{merit} is returned.
}
\references{Hurley, Catherine B. \dQuote{Clustering Visualisations of Multidimensional
Data}, Journal of Computational and Graphical Statistics,
vol. 13, (4), pp 788-806, 2004.
Gruvaeus, G. and Wainer, H. (1972),
\dQuote{Two Additions to Hierarchical Cluster Analysis},
British Journal of Mathematical and Statistical Psychology, 25, 200-206.
}
\author{ Catherine B. Hurley }
%\note{ ~~further notes~~ }
\examples{
data(state)
state.cor <- cor(state.x77)
order.single(state.cor)
order.hclust(state.cor,method="average")
# Use for plotting...
cpairs(state.x77, panel.colors=dmat.color(state.cor), order.single(state.cor),pch=".",gap=.4)
# Order the states instead of the variables...
state.d <- dist(state.x77)
state.o <- order.single(-state.d)
op <- par(mar=c(1,6,1,1))
cmat <- dmat.color(as.matrix(state.d), rev(cm.colors(5)))
plotcolors(cmat[state.o,state.o], rlabels=state.name[state.o])
par(op)
}
\keyword{multivariate }
\keyword{cluster } |
# Chapter 10 - Geometry - Chapter Summary, Review, and Test - Review Exercises: 31
perimeter = $44 \text{ m}$
#### Work Step by Step
The height of the polygon is 10 meters. This means that the unknown vertical side length is: $=10-2 = 8$ m The width of the polygon is 12 meters. This means that the unknown horizontal side length is: $=12 - 7 = 5$ m The perimeter of a polygon is equal to the sum of all the lengths of its sides. Thus, the perimeter of the polygon is: $=(12 + 10 + 7 + 8 + 5 + 2) \text{ m} \\=44 \text{ m}$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. |
• 论文 •
### 环月降轨实现月面着陆的控制策略
1. 1. 航天飞行动力学技术重点实验室, 北京 100094;
2. 北京航天飞行控制中心, 北京 100094
• 收稿日期:2019-04-18 发布日期:2020-01-21
• 通讯作者: 李革非 E-mail:sophiebacc@sina.com
• 作者简介:李革非,女,博士,研究员。主要研究方向:航天器动力学与控制。
• 基金资助:
国家自然科学基金（11773004，61573049）
### Control strategy of lunar orbit descent to achieve lunar landing
LI Gefei1,2, LIU Yong1,2, MA Chuanling2, HAO Dagong2
1. 1. Science and Technology on Aerospace Flight Dynamics Laboratory, Beijing 100094, China;
2. Beijing Aerospace Control Center, Beijing 100094, China
• Received:2019-04-18 Published:2020-01-21
• Supported by:
National Natural Science Foundation of China (11773004,61573049)
Abstract: Aiming at the problem of the probe's lunar landing, the control strategy of lunar orbit descent is studied. According to the control equations of the lunar orbit descent, the three kinds of relationship between the different combinations of the single-pulse control variables of the lunar orbit descent and the parameters of the lunar landing target are established. Three control strategies for lunar orbit descent, which are the fixed-point landing at the fixed time, the fixed-point landing and the landing at the target latitude area, are established. The algorithms and the procedures for solving the lunar orbit descent control strategies are presented. For the fixed-point lunar landing, the solutions of the single-pulse combined control of the semi-major axis and the lunar perigee height are analyzed. The control calculations of three kinds of lunar landings with the fixed-point at fixed time, the fixed-point, and the target latitude area are carried out respectively for the nominal orbit, the negative deviation orbit, and the positive deviation orbit, and the control strategies of the lunar orbit descent are verified. The strategies can be applied to the orbit control of the lunar landing mission, such as lunar landing, lunar sampling return, and manned lunar landing. |
# What is the slope of the tangent line of (x-y^2)/(xe^(x-y^2)) =C , where C is an arbitrary constant, at (1,1)?
Mar 6, 2016
It is $\frac{1}{2}$
#### Explanation:
Given that $\left(1 , 1\right)$ lies on the graph of $\frac{x - {y}^{2}}{x {e}^{x - {y}^{2}}} = C$,
we see that $C = 0$
(Substitute $1$ for both $x$ and $y$ to get
(1-(1)^2)/(1e^(1-(1^2))=0/1=0=C.)
That means the graph is the graph of $x - {y}^{2} = 0$ or ${y}^{2} = x$.
Now we can differentiate implicitly, of further recognize that,
since $\left(1 , 1\right)$ is on the graph, we are on the branch where $y = \sqrt{x}$, so $y ' = \frac{1}{2 \sqrt{x}}$
The slope at $x = 1$ is $\frac{1}{2}$. |
# US-AI vs Chin-AI
What does the race for AI mean for data science worldwide? Why is it getting a lot of attention these days?
Although it may still be recovering from the effects of the government shutdown, data science has received a lot of positive attention from the United States Government. Two major recent milestones include the OPEN Government Data Act, which passed in January as part of the Foundations for Evidence-Based Policymaking Act, and the American AI Initiative, which was signed as an executive order on February 11th.
### So what's going on, and what does all this mean for data science?
The first thing to consider is why and more specifically who the US administration has passed these recent measures for. Although it's not mentioned in either of the documents, any political correspondent who has been following these topics could easily explain that they are intended to stake a claim against China.
China has stated its intention to become the world leader in data science and AI by 2030. And with far more government access, data sets (a benefit of China being a surveillance state) and an estimated $15 billion in machine learning, they seem to be well on their way. In contrast, the US has only$1.1 billion budgeted annually for machine learning.
So rather than compete with the Chinese government directly, the US appears to have taken the approach of convincing the rest of the world to follow their lead, and not China’s. They especially want to direct this message to the top data science companies and researchers in the world (especially Google) in order to keep their interest in American projects.
### So then what do these measures do?
On the surface level, both the OPEN Government Data Act and the American AI Initiative strongly encourage government agencies to amp-up their data science efforts. The former is somewhat self-explanatory in name, as it requires agencies to publish more machine-readable publicly-available data, and requires more use of this data in improved decision making. It imposes a few minimal standards for this, and also establishes the position of Chief Data Officers at federal agencies. The latter is somewhat similar in that it orders government agencies to re-evaluate and designate more of their existing time and budgets towards AI use and development, also for the purposes of better decision making.
Critics are quick to point out that the American AI Initiative does not actually allocate more resources for its intended purpose, nor does either measure directly impose incentives or penalties. This is not much of a surprise given the general trend of cuts to science funding under the Trump administration. Thus the likelihood that government agencies will actually follow through with what these laws ‘require’ has been given skeptic estimations.
However, this is where it becomes important to remember the overall strategy from the current US administration. Both documents include copious amounts of values and standards that the US wants to uphold when it comes to data, machine learning, and artificial intelligence. These may be the key aspects that can hold up against China, having a government that receives a hefty share of international criticism for their use of surveillance and censorship. (Again, this has been a major sticking point for companies like Google.)
These are some of the major priorities brought forth in both measures: Make federal resources, especially data and algorithms, available to all data scientists and researchers; Prepare the workforce for technology changes like AI and optimization; Work internationally towards AI goals while maintaining American values; and finally, Create regulatory standards, to protect security and civil liberties in the use of data science.
So there you have it. Both countries are undeniably powerhouses for data science. China may have the numbers in their favor, but the US would like the world to know that they have American spirit.
### Finally, what does this mean for everyone who's not working for the US or China?
In short, the phrase “a rising tide lifts all ships” seems to fit here. While the US and China compete for data science dominance at the government level, everyone else can stand atop this growing body of innovations and make their own.
The thing data scientists can get excited for in the short-term is the release of a lot of new data from US federal sources, or the re-release of such data in machine readable formats. The emphasis is on the public part - meaning that anyone, not just US federal employees or even citizens, can use this data. To briefly explain for those less experienced in the realm of machine learning and AI, having as much data to work with as possible helps scientist to train and test programs for more accurate predictions.
A lot of what made the government shutdown a dark period for data scientists suggest the possibility of a golden age in the near future. |
# Near Real Time Error Detection in 3D Prints
Posted on Jun 30, 2021
My 6th’s semester was writing my bachelor thesis. I worked on a project for the thesis with 3 other people. I have a Creality Ender 3 3D printer at home that I use for my hobby projects. I had seen a few projects trying to use deep learning to give people a heads up whether their print was failing. But they would only tell you if the printer started making “spaghetti”. Since we were working on a thesis in Vision Graphics and Interactive Systems, i proposed to my group that we could do a project around detecting errors in 3D prints. They ended up thinking it was a good idea as well. We pitched the idea to our supervisor and he liked it as well.
So we started laying out the system, we knew we had Octoprint as a tool to use the printer. We wanted to try to use Octoprint as well because many hobbyist uses it to control their printers. Octoprint will usually run on a Raspberry Pi, which meant we had those parts of the system determined pretty quickly. For the image capturing we wanted to go with a low cost solution, mainly for two reasons. First we wanted to see if a cheap camera would have high enough resolution to actually reveal errors in the prints. Secondly, we wanted a potential implementation to be cheap enough for hobbyists to use. If we used one of the Universities 5000 USD camera’s, we might be able to detect errors, but how many people enjoying 3D printing would it actually benefit? We ended up making a table of the cameras we had readily available (it was Corona times after all and we couldn’t wait too long to get a camera in hand). The table is shown below.
We required the camera to have a resolution that, at the distance given to the build platform, would give us 1 pixel per layer. I will explain later why we thought this was a good idea. We also required the camera to give us a pixel density at a given distance of 0.2 mm or greater. This is because 0.2 mm is the most commonly used layer height for FDM hobby 3D printers. We also needed the camera to be very fast in taking the pictures, due to the fact that we had to move the printhead away from the model every time we took an image. When moving the printhead and leaving it in a position away from the model for an extended period of time you start to get what is called “oozing”. This is basically where the material inside the print nozzle starts to creep out due to the high pressure inside the nozzle. When the printhead then moves back to the print, the oozed material will then show up as artifacts on the surface of the print.
## System Definition and First Steps
The camera selection process itself wasn’t very scientific and we ended up using a Raspberry Pi V2 camera, as it is widely available and frequently used in the 3D printing community. This meant we could setup our system diagram to look like this
So as seen in the diagram, pretty all of the implementation was to be made on the Raspberry Pi. We had had a course in Image Processing, we wanted to use some of the theory from that to create a preprocessing and segmentation module. During the course we had also had some theory around machine learning algorithms. So we wanted to compare classification with some of those algorithms, to a simple feature threshold classification. We never made it to the notification part of the system, but we thought it was important to show our thought process in the diagram.
The camera setup was pretty simple. We calculated the distance the camera had to sit from the printers build plate in order to capture the full print volume. We then 3D printed and arm that extended from the printers base, and mounted the camera and Raspberry Pi onto it.
In order to put some limitations on the project, we hung up black card board behind the printer. This made it so that all of the images had a black background and thereby easier to process. Changing backgrounds could most likely be accounted for with some more software, however we did not want to focus on this.
The next problem we had to tackle, was how to find the actual print in the images. We knew what the model had to look like (in theory) since we had the 3D CAD model of the print. However there are slight differences between the 3D CAD model and the model that has gone through slicer software in order to be printed. So instead of using the CAD model, we wanted to represent the model using the sliced GCODE. An FDM 3D printer prints a model in layers, therefore we have to do the so-called “slicing” in order to slice the model into printable layers. The output of this process is a GCODE file which contains a set of machine commands that is sent to and interpreted by the 3D printer’s firmware. We can use this GCODE file to do a 3D plot in Python. An example of this can be seen in the figure below
It is basically just a matter of plotting all the movement commands from the GCODE file and then adding the layer height to the z-value every time there is a layer change. The different colors represents the different parts of the layer printing procedure, i.e. the green color is solid infill, the red is outer perimeters, blue is inner perimeters and pink is non-solid infill.
We seemed to have an issue with the fact that there were no z-height on the plotted (simulated) models. However on all of the models that we used, there were enough layers for it to not matter. So we ended up just going with it.
So the next thing we wanted to do before developing too much of the algorithm, was to define which print issues we should detect. There are many things that can go wrong in a print, but we wanted to be able to determine the ones that would be most catastrophic. So we came up with the following table
According to the table above, what we would be looking for in our vision algorithm was the symptoms of an error. In some cases we would then be able to make an educated guess on what the cause was. The only error where we would allow the print to continue was the layer delamination error, which inherently also was the one we expected to have the most trouble classifying.
At this stage of the project, we also did some research into what other implementations of vision algorithms where already out there for this purpose. What we found was that none of the existing systems were able to detect all of the above errors by itself. It would require an ensemble of all the current solutions to detect all error types.
Next thing we did was then to start implementing (well actually we made a requirements specification for the system first but I won’t bore you with the details of that). As I described earlier, we needed to move the printhead out of the way before we could capture an image, so that was one of the first things we started working on. To do this, we used an OctoPrint plugin called GCODE System Commands, which is pretty awesome. Basically it let’s you create your own custom GCODES, that when read by OctoPrint, is able to execute system commands on the Raspberry Pi Linux system. All we needed to do was to just write a small shell script that we execute the commands to take the picture and save it to the desired location. Then we could use the plugin execute the script. Before uploading the GCODE file to OctoPrint for printing, we wrote a Python script that would inject our custom code to move the printhead back/forth and take the picture.
1 2 3 4 5 6 G1 X5 Y40 Z(layer_height + 3) // Move the extruder G1 E-2 // Retract filament 2 mm OCTO801 // Execute visual inspection program G1 X? Y? Z? // Move back to point before injection G1 E2 // Extrude filament 2 mm Repeat from line 1 until end of GCODE file
Since we could now make a model in 3D that we could use for comparison and we could inject custom GCODE, we were ready to start developing the image processing chain. First thing we started to tackle was the segmentation.
## Image Processing Chain
I previously described that we wanted to use convolution to match our rendered 3D model of the GCODE to the actual print on the platform. A procedure known as template matching in the image processing world. This was done as shown in the image below
We would referrer to the model as the template here. The template matching procedure outputs a map of correlation coefficients. The highest correlation between the template and the image, will be the brightest pixels on the correlation result above. The output image is then taken from the area with the highest correlation and will be the same size as the template. Since the printing was done using a single color bright filament, we were able to use a simple threshold based on HSV to distinguish the background from the object of interest. This is definitely not ideal when using a material, that has a color closer to the print platform color. It would also present issues when the print get’s tall enough such that the background is captured as well. Since we used black cardboard for the background it did not show up as an issue in our implementation. However for a more practical system, this would need to be accounted for.
Having done the segmentation of the captures images, we turned our attention to feature extraction. In our course we had been taught about some well known features that we could look for in shapes. These are features such as: area, circularity, perimeter, width, height, etc. Features were extracted from both the rendered model and the captured images. In this way, we were able to actually compare the features between the two. What we did next, was to create a score vector. We created the score vector my taking the minimum score between the rendered and captured image features and dividing it by the maximum between the two. Thereby normalizing the score vector. We then used three different models to evaluate whether our score vectors we a good method. Results can be seen in the image below
We used this method to get rid of some features that did not convey good information as well. For example, the orientation feature had a lot of deviation for the median in the fox model. So that suggested to us, that the feature becomes more unreliable as the number of layer get larger.
Different methods suggested that we should investigate the feature importance of each feature. This was done using the SciKit-learn feature selection toolbox in Python. The result can be seen in the image below.
For the classification part of the system, we tried some different methods. As mentioned previously, we wanted to compare a threshold method to some of the well know Machine Learning algorithms such as k-nearest neighbors, Random Forest and Support Vector Machine. Our results showed that the threshold method and k-nearest neighbors actually performed best and even equally well as shown below.
I hope you got something out of this. If you feel like I missed something, please let me know in the comments and I’ll try to elaborate. |
Thread: how to prove that it's the subring?
1. how to prove that it's the subring?
Given Ring $R$ and $S\subseteq R$. If $C(S)=\{x\in R | xa=ax, \forall a \in S\}$ then show that $C(S)$ is the subring of $R$. |
# Extract the "path" of a data point through a decision tree in sklearn
I'm working with decision trees in python's scikit learn. Unlike many use cases for this, I'm not so much interested in the accuracy of the classifier at this point so much as I am extracting the specific path a data point takes through the tree when I call .predict() on it. Has anyone done this before? I'd like to build a data frame containing ($X_{i}$, path$_{i}$) pairs for use in a down-stream analysis.
• Not an answer, but I'm sure you've seen: scikit-learn.org/stable/modules/generated/… But I don't know how to turn the visualizer into a data frame for predictions. Oct 15 '15 at 17:07
• @nfmcclure Thanks! Yes, I saw that. I actually thought about writing python code for reverse engineering the paths based on that output, but it's about 250 pages, if you save it to a .txt file. I figure it would be too buggy an approach. Oct 15 '15 at 17:15
Looks like this is easier to do in R, using the rpart library in combination with the partykit library. I'd ideally like to find a way to do this in python, but here's the code, for anyone who is interested (taken from here):
pathpred <- function(object, ...){
## coerce to "party" object if necessary
if(!inherits(object, "party")) object <- as.party(object)
## get standard predictions (response/prob) and collect in data frame
rval <- data.frame(response = predict(object, type = "response", ...))
rval$prob <- predict(object, type = "prob", ...) ## get rules for each node rls <- partykit:::.list.rules.party(object) ## get predicted node and select corresponding rule rval$rule <- rls[as.character(predict(object, type = "node", ...))]
return(rval)
}
Illustration using the iris data and rpart():
library("rpart")
library("partykit")
rp <- rpart(Species ~ ., data = iris)
rp_pred <- pathpred(rp)
rp_pred[c(1, 51, 101), ]
Yielding,
response prob.setosa prob.versicolor prob.virginica
1 setosa 1.00000000 0.00000000 0.00000000
51 versicolor 0.00000000 0.90740741 0.09259259
101 virginica 0.00000000 0.02173913 0.97826087
rule
1 Petal.Length < 2.45
51 Petal.Length >= 2.45 & Petal.Width < 1.75
101 Petal.Length >= 2.45 & Petal.Width >= 1.75
Which looks to be something I could at least use to derive shared parent node information.
• Accepting this answer for now, but if someone comes up with a python solution, I'll likely accept that instead. Oct 17 '15 at 14:03 |
I'm running the tutorial Writing a Simple Action Server using the Execute Callback (Python) but getting errors that the file fibonacci_server.py isn't installed. I don't see a directory named simple_action_servers that is supposed to contain this file according to the directions in ( http://wiki.ros.org/actionlib_tutoria... )
ubuntu@ubuntu python fibonacci_server.py
ubuntu@ubuntu:/opt/ros/indigo$ I'm running Indigo on Ubuntu 14.04 and just reinstalled ros-indigo-common-tutorials but I'm getting the same result. Do I need to install the actionlib_tutorials from a separate package than common tutorials or do I need to create the file? NEW PROBLEM DATA In order to make this issue easier to understand, I'm providing additional info. When looking at the contents of the actionlib_tutorial in [ http://wiki.ros.org/actionlib_tutoria... ), it states that The following code can be found in actionlib_tutorials/simple_action_servers/fibonacci_server.py, and implements a python action server for the fibonacci action. However, when I run an ls, the simple_action_servers directory has not been installed. ubuntu@ubuntu:/opt/ros/indigo/share/actionlib_tutorials$ ls action cmake msg package.xml |
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.
## Definition for prime numbers
pn# as a function of n, plotted logarithmically.
For the nth prime number pn, the primorial pn# is defined as the product of the first n primes:[1][2]
${\displaystyle p_{n}\#=\prod _{k=1}^{n}p_{k))$,
where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes:
${\displaystyle p_{5}\#=2\times 3\times 5\times 7\times 11=2310.}$
The first five primorials pn# are:
2, 6, 30, 210, 2310 (sequence A002110 in the OEIS).
The sequence also includes p0# = 1 as empty product. Asymptotically, primorials pn# grow according to:
${\displaystyle p_{n}\#=e^{(1+o(1))n\log n},}$
where o( ) is Little O notation.[2]
## Definition for natural numbers
n! (yellow) as a function of n, compared to n#(red), both plotted logarithmically.
In general, for a positive integer n, its primorial, n#, is the product of the primes that are not greater than n; that is,[1][3]
${\displaystyle n\#=\prod _{p\leq n \atop p{\text{ prime))}p=\prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#}$,
where π(n) is the prime-counting function (sequence A000720 in the OEIS), which gives the number of primes ≤ n. This is equivalent to:
${\displaystyle n\#={\begin{cases}1&{\text{if ))n=0,\ 1\\(n-1)\#\times n&{\text{if ))n{\text{ is prime))\\(n-1)\#&{\text{if ))n{\text{ is composite)).\end{cases))}$
For example, 12# represents the product of those primes ≤ 12:
${\displaystyle 12\#=2\times 3\times 5\times 7\times 11=2310.}$
Since π(12) = 5, this can be calculated as:
${\displaystyle 12\#=p_{\pi (12)}\#=p_{5}\#=2310.}$
Consider the first 12 values of n#:
1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
We see that for composite n every term n# simply duplicates the preceding term (n − 1)#, as given in the definition. In the above example we have 12# = p5# = 11# since 12 is a composite number.
Primorials are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:
${\displaystyle \ln(n\#)=\vartheta (n).}$[4]
Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:
${\displaystyle n\#=e^{(1+o(1))n}.}$
The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, where it is used to derive the existence of another prime.
## Characteristics
• Let p and q be two adjacent prime numbers. Given any ${\displaystyle n\in \mathbb {N} }$, where ${\displaystyle p\leq n:
${\displaystyle n\#=p\#}$
• For the Primorial, the following approximation is known:[5]
${\displaystyle n\#\leq 4^{n))$.
• Furthermore:
${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n\#))=e}$
For ${\displaystyle n<10^{11))$, the values are smaller than e,[6] but for larger n, the values of the function exceed the limit e and oscillate infinitely around e later on.
• Let ${\displaystyle p_{k))$ be the k-th prime, then ${\displaystyle p_{k}\#}$ has exactly ${\displaystyle 2^{k))$ divisors. For example, ${\displaystyle 2\#}$ has 2 divisors, ${\displaystyle 3\#}$ has 4 divisors, ${\displaystyle 5\#}$ has 8 divisors and ${\displaystyle 97\#}$ already has ${\displaystyle 2^{25))$ divisors, as 97 is the 25th prime.
• The sum of the reciprocal values of the primorial converges towards a constant
${\displaystyle \sum _{p\,\in \,\mathbb {P} }{1 \over p\#}={1 \over 2}+{1 \over 6}+{1 \over 30}+\ldots =0{.}7052301717918\ldots }$
The Engel expansion of this number results in the sequence of the prime numbers (See (sequence A064648 in the OEIS))
• According to Euclid's theorem, ${\displaystyle p\#+1}$ is used to prove the infinity of all prime numbers.
## Applications and properties
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.
Every highly composite number is a product of primorials (e.g. 360 = 2 × 6 × 30).[7]
Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ(n)/n is smaller than for any lesser integer, where φ is the Euler totient function.
Any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values.
Base systems corresponding to primorials (such as base 30, not to be confused with the primorial number system) have a lower proportion of repeating fractions than any smaller base.
Every primorial is a sparsely totient number.[8]
The n-compositorial of a composite number n is the product of all composite numbers up to and including n.[9] The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are
1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, ...[10]
## Appearance
The Riemann zeta function at positive integers greater than one can be expressed[11] by using the primorial function and Jordan's totient function Jk(n):
${\displaystyle \zeta (k)={\frac {2^{k)){2^{k}-1))+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k)){J_{k}(p_{r}\#))),\quad k=2,3,\dots }$
## Table of primorials
n n# pn pn#[12] Primorial prime?
pn# + 1[13] pn# − 1[14]
0 1 N/A 1 Yes No
1 1 2 2 Yes No
2 2 3 6 Yes Yes
3 6 5 30 Yes Yes
4 6 7 210 Yes No
5 30 11 2310 Yes Yes
6 30 13 30030 No Yes
7 210 17 510510 No No
8 210 19 9699690 No No
9 210 23 223092870 No No
10 210 29 6469693230 No No
11 2310 31 200560490130 Yes No
12 2310 37 7420738134810 No No
13 30030 41 304250263527210 No Yes
14 30030 43 13082761331670030 No No
15 30030 47 614889782588491410 No No
16 30030 53 32589158477190044730 No No
17 510510 59 1922760350154212639070 No No
18 510510 61 117288381359406970983270 No No
19 9699690 67 7858321551080267055879090 No No
20 9699690 71 557940830126698960967415390 No No
21 9699690 73 40729680599249024150621323470 No No
22 9699690 79 3217644767340672907899084554130 No No
23 223092870 83 267064515689275851355624017992790 No No
24 223092870 89 23768741896345550770650537601358310 No Yes
25 223092870 97 2305567963945518424753102147331756070 No No
26 223092870 101 232862364358497360900063316880507363070 No No
27 223092870 103 23984823528925228172706521638692258396210 No No
28 223092870 107 2566376117594999414479597815340071648394470 No No
29 6469693230 109 279734996817854936178276161872067809674997230 No No
30 6469693230 113 31610054640417607788145206291543662493274686990 No No
31 200560490130 127 4014476939333036189094441199026045136645885247730 No No
32 200560490130 131 525896479052627740771371797072411912900610967452630 No No
33 200560490130 137 72047817630210000485677936198920432067383702541010310 No No
34 200560490130 139 10014646650599190067509233131649940057366334653200433090 No No
35 200560490130 149 1492182350939279320058875736615841068547583863326864530410 No No
36 200560490130 151 225319534991831177328890236228992001350685163362356544091910 No No
37 7420738134810 157 35375166993717494840635767087951744212057570647889977422429870 No No
38 7420738134810 163 5766152219975951659023630035336134306565384015606066319856068810 No No
39 7420738134810 167 962947420735983927056946215901134429196419130606213075415963491270 No No
40 7420738134810 173 166589903787325219380851695350896256250980509594874862046961683989710 No No
## Notes
1. ^ a b Weisstein, Eric W. "Primorial". MathWorld.
2. ^ a b (sequence A002110 in the OEIS)
3. ^ (sequence A034386 in the OEIS)
4. ^
5. ^ G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. ISBN 0-19-853310-1.
Theorem 415, p. 341
6. ^ L. Schoenfeld: Sharper bounds for the Chebyshev functions ${\displaystyle \theta (x)}$ and ${\displaystyle \psi (x)}$. II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
Cited in: G. Robin: Estimation de la fonction de Tchebychef ${\displaystyle \theta }$ sur le k-ieme nombre premier et grandes valeurs de la fonction ${\displaystyle \omega (n)}$, nombre de diviseurs premiers de n. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371
7. ^ Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. ^ Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pac. J. Math. 121 (2): 407–426. doi:10.2140/pjm.1986.121.407. ISSN 0030-8730. MR 0819198. Zbl 0538.10006.
9. ^ Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. Retrieved 16 March 2016.
10. ^
11. ^ Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
12. ^
13. ^ Sloane, N. J. A. (ed.). "Sequence A014545 (Primorial plus 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. ^ Sloane, N. J. A. (ed.). "Sequence A057704 (Primorial - 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
## References
• Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math. 19: 197–203.
• Spencer, Adam "Top 100" Number 59 part 4. |
## Files in this item
FilesDescriptionFormat
application/pdf
9136651.pdf (7MB)
(no description provided)PDF
## Description
Title: Modeling of DFB surface emitting lasers and semiconductor laser arrays Author(s): Lee, Shing Man Doctoral Committee Chair(s): Chuang, Shun-Lien Department / Program: Engineering, Electronics and ElectricalPhysics, Condensed MatterPhysics, Optics Discipline: Engineering, Electronics and ElectricalPhysics, Condensed MatterPhysics, Optics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Engineering, Electronics and Electrical Physics, Condensed Matter Physics, Optics Abstract: Theoretical modeling of novel semiconductor laser systems, i.e., distributed-feedback (DFB) surface emitting lasers and semiconductor laser arrays, is presented. The DFB surface emitting lasers can produce stable single-mode outputs even under high bit-rate direct modulation and form two-dimensional laser arrays. A simple and accurate analytical model using the coupled-mode theory is developed to describe these surface emitting lasers. Due to the simplicity of the model, the desired laser output characteristics, i.e., low threshold condition, large side mode suppression ratio, and stable and low noise Bragg-mode outputs, can be obtained by systematically optimizing the device length, optical coupling and phase shifter in the device.The nonplanar laser arrays are of much interest because of their simple fabricating procedures and very high output power. A theoretical study is performed to understand the efficiency and far-field and near-field patterns. A number of important physical mechanisms for high power semiconductor laser operations are studied using a self-consistent model. These include the two-dimensional current spreading in the cladding layers, the coupling between the carrier distribution and the photon distribution, and the carrier saturation effects at high power operation. The mesas, bends, and grooves are treated as adjacent waveguides, each described by the effective index method. The output field patterns in the nonplanar laser structures are composed of a linear combination of the individual waveguide modes. The multimode operation in practical devices can be explained by spatial hole burning effects, nonuniform current injection, and competition for available carriers in the neighboring waveguides between different optical modes. The possibility of obtaining phase-locked output by reducing the groove depth is also investigated. A finite-difference time-domain (FDTD) model is used to study the radiation losses due to the bend. A groove depth as small as 0.1 $\mu$m can be used for maximum optical coupling while the bending loss is still large enough to suppress the lateral lasing operation in the nonplanar laser array.The highest semiconductor laser output powers have been achieved by the laser arrays employing optical turning mirrors. The effects of the rough turning mirrors on the laser array performance are estimated using the FDTD method. A number of steps are employed to reduce the computation time on these very large mirrors (about sixty wavelengths) to make the FDTD model a possible computer-aided design tool. The computation time is reduced by a factor of twenty. Issue Date: 1991 Type: Text Language: English URI: http://hdl.handle.net/2142/21015 Rights Information: Copyright 1991 Lee, Shing Man Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9136651 OCLC Identifier: (UMI)AAI9136651
|
# Oxygen _____ combustion and is ______ itself
Oxygen supports combustion and is non-combustible itself.
• Air or oxygen, which helps in burning, is called a supporter of combustion, and the chemical reaction which takes place with the release of heat and light energy is called combustion.
• The substances which undergo combustion are known as combustible substances.
• Oxygen is by itself non-flammable. But to cause combustion, it needs fuel, i.e. something that can burn. |
# L-Network impedance matching
Impedance matching is an everyday problem for RF circuit designers. The L-Network is one of the easiest lossless ways of matching a low source impedance to a higher load impedance. This article shows how they work and how to design them.
Matching a transistor amplifier’s low output impedance with the higher impedance of an antenna (typically 50 or 75 Ohms) is just one everyday example of where an L-Network can be used. For this article, we are going to design an L-Network that matches a 75 Ohm source (function generator) with a 1 kΩ load (resistor).
To make it easier for you to understand how this network works, I will present the complete circuit first, explain how it works and then explain how the values are being calculated. Here is the finished L-Network:
L-Network matching 75 Ohms source impedance to 1000 Ohms load impedance
RS = Source Resistance (75 Ohms)
Ls = Series inductance (j263)
Cp = Parallel Capacitance (-j284)
Don’t be frightened by the little “j” in front of the values, in case you have never seen it. The letter “j” in electronics engineering stands for the imaginary unit. Adding “j” to a reactance is basically the so-called “vector notation.” A positive value indicates an inductive reactance and a negative value is always a capacitive reactance. If you are not used to it, you can just think of Ls as an inductor with a reactance of 263 Ohms and of Cp as a capacitor with a reactance of 284 Ohms.
Note that the reactance of a capacitor and an inductor is frequency dependent. Therefore, this circuit does not work well over a large bandwidth.
This circuit makes the 1000 Ohms load ‘look’ like a 75 Ohms load to the source. But how does it work? To answer this question, we have to look at the equivalent circuit. For that, we will reduce the overall circuit to simpler elements, step by step.
Let’s start with Cp and RL. They are in parallel. The overall impedance of two resistors is calculated as follows:
$R = \frac{R_1 * R_2}{R_1 + R_2}$
Likewise, the resulting complex impedance of a resistor and a capacitor can be calculated as follows:
$Z = \frac{Xc * R}{Xc + R}$
In our case, we get:
$Z = \frac{X_c * R_L}{Xc + R_L} = \frac{-j284 * 1000}{-j284 + 1000} = 75-j263$
75 is the real part in Ohms and -j263 is a capacitive reactance of 263 Ohms. That means the capacitor and resistor in parallel will act like a resistor with 75 Ohms and a capacitor with 263 Ohms reactance in series.
Equivalent circuit for Cp and RL
Now if you insert that back into the original circuit and have a close look at it, you may notice something. Look at the reactance of Cp and Ls.
Overall equivalent circuit after reducing Cp and RL to a series equivalent circuit
Correct, the reactance of both the capacitor and the inductor are the same, just with different vectors. If you have never worked with a complex impedance beforehand, you will now be amazed by how much easier it makes things. We can simply add the reactances together, and since 263 + -263 is zero, we only have the 75 Ohms load resistance left. Yes, it’s that simple, the source ‘sees’ the 1 kΩ load as a 75 Ω load.
Overall equivalent circuit of the L-Network
An often-welcomed side effect is that this L-Network acts as a low-pass filter. So, if it’s used to match a low transistor amplifier impedance directly to the higher antenna impedance, one suppresses harmonics at the same time.
If one wishes to have a high-pass characteristic, one can simply switch the positions of the inductor and the capacitor. It really changes nothing. The impedance values, of course, need to be properly adjusted in that case.
Let’s talk about how to design a circuit like this from scratch. We will use the above values, source impedance of 75 Ohms and load impedance of 1000 Ohms at a frequency of 16 MHz. The frequency doesn’t matter until the end when we have to pick actual values for Ls and Cp.
The first thing we have to calculate is the quality factor, or Q factor. It describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator’s bandwidth relative to its center frequency. The Q of an L-Network is calculated as follow:
$Q = \sqrt{\frac{R_{Load}}{R_{Source}}-1}$
RSource = Source Resistance
With our values plugged into the equation, we get the following:
$Q = \sqrt{\frac{1000}{75}-1} = 3.512$
So 3.512 is our Q. The rest is just as simple. To get the needed impedance of the inductor, we use the following formula:
$X_L = Q*R_{Source} = 3.512 * 75 = 263$
That means the inductor needs to have an impedance of 263 Ohms at the desired frequency. We can write this as j263 to make sure it’s apparent that it is an inductive reactance. The impedance of the capacitor needs to be the load resistance divided by the Q factor.
$X_C = \frac{R_{Load}}{Q} = \frac{1000}{3.512} = 284$
Again, this means that the impedance of the capacitor at the desired frequency needs to be 284 Ohms. Using the vector notation, we write -j284 to indicate that it is a capacitive impedance.
Okay, now we need to find actual values for the inductor and the capacitor. To be able to calculate either one, we need to settle on an operating frequency. For this article we will use 16 MHz. Using the following formulas, we can find the actual values for the parts at the desired frequency:
$L = \frac{X_L}{\omega} = \frac{263}{2 \pi (16*10^6)} = 2.6 \mu H$
That means our inductor needs to have a value of 2.6 μH. I chose a value of 2.2 μH as it is a common value I had in stock. Since the circuitry itself in real life usually contains what’s called parasitic inductance, it won’t throw the circuit off by far. Next up is the capacitor:
$C = \frac{1}{\omega X_C} = \frac{1}{2 \pi (16*10^6) 284} = 35 pF$
Instead of 35 pF, I used a 33 pF capacitor. Not only is it a more common value, we can again assume a little bit of stray capacitance by the remaining circuit.
75 Ohms to 1000 Ohms L-Network characterized with a return loss bridge
I used a return loss bridge, a signal generator and an oscilloscope to verify the performance of this circuit. And sure enough, it acts just like a 75 Ohm resistive terminator at about 16 MHz. The exact frequency for best operation turned out to be roughly 16.1 MHz.
Confirming the performance of the 75 Ohms to 1000 Ohms L-Network using the Teledyne LeCroy HDO4024
Update (04/06/2013): I previously wrote that one can not match a high impedance source to a low impedance load. That is incorrect. Naturally, it works perfectly fine the other way around. One only has to switch the position of the load and the source.
KJ6QBA took out the time to simulate this circuit using LTspice. He also simulated the same circuit used in reverse to match a 1000 Ohm source to a 75 Ohm load. Here’s the result:
LTspice simulation of the L-Network provided by KJ6QBA
## 20 thoughts on “L-Network impedance matching”
1. Would have been nicer without the lecroy ad in the last image, but that’s very cool anyways. The description is very clear, thanks, I learnt something today.
• No ad in there. The reason I mention the scope the way I do is SEO. Other than that, I think it’s a great scope and it deserves to be mentioned.
2. “The only problem with this circuit is that it only matches a low source impedance to a higher impedance.”
The key is the element closest to the load. If it is a shunt element, it can reduce the load impedance seen at the source. If it is a series element, it increases the load impedance seen.
• Yes, that’s correct. I should change the article to clarify that this particular configuration only works as ‘step-up’ impedance converter. Then again, I might as well explain that the same in ‘reverse’ works as ‘step-down’. In either case, thanks for the comment. I got tempted into writing it the way I did without thinking as one rarely sees L-Networks as step down, mostly pi-Networks are used. Good catch, thanks!
• Nicely done. So how about now do the step up version with a Pi network, and show how it looks on the Smith chart? That would be very instructive.
• Working on that. Before I start posting smith charts on my blog I’ll actually write an article on how to read a smith chart. My intend is to not post challenging information that isn’t explained anywhere else in my blog.
• Your website is wonderful, thanks for this post in particular. I, too, am a fan of clearly written explanations. So in that spirit, can you explain where the Q factor of the L-network came from? Why is it equal to the square root of ([the ratio of the load resistance to the source resistance] – 1)? Thanks and looking forward to more great stuff from you!
3. Would you mind breaking down the algebra on:
((-j284*1000)/(-j284+1000))?
It’s been many years for me
Thanks either way,
Steve
4. Thank you very much! I learned a lot from this. Please keep them coming!
5. I am clear with this thank you it would be nice if you could help us with T and pi network
6. This could very nicely be followed by an RF amplifier with impedance matching characteristics…Would love to see a complete tutorial on RF amplifier design, since the youtube seems to be lacking any clear explanations. Great tutorial by the way. I’m doing an RF amplifier design for 150 MHz and I was trying to scale the 7 MHz design I saw in ‘Experimental Methods in RF design’. This gave me an excellent foundation to build from. Keep going! Thanks…
7. Thanks a lot. This is way clear than what I learn in the class:)
8. ((-j284*1000)/(-j284+1000))? how to manualyy break this one. thanks
9. I was able to get the math to work after watching some tutorials on YouTube. Search for imaginary number math basics 101 |
# Revision history [back]
### Why can't I find the spectral radius of a tree?
If I create some connected graphs, and ask SageMath for their spectral radius, the command never returns (well, I have only let it run for a few minutes) if the target is a tree.
Here is a simple example:
g = graphs.CompleteBipartiteGraph(1,3)
print g.spectrum() |
# Prevádzať 0,947 atm na mm hg
Tabuľka premien: mmHg na PSI; 1 mmHg = 0.0193 PSI: 2 mmHg = 0.0387 PSI: 3 mmHg = 0.0580 PSI: 4 mmHg = 0.0773 PSI: 5 mmHg = 0.0967 PSI: 6 mmHg = 0.116 PSI: 7 mmHg = 0.135 PSI: 8 mmHg = 0.155 PSI: 9 mmHg = 0.174 PSI: 10 mmHg = 0.193 PSI: 15 mmHg = 0.290 PSI: 50 mmHg = 0.967 PSI: 100 mmHg = 1.934 PSI: 500 mmHg = 9.668 PSI: 1000 mmHg = 19.337 PSI
Do a quick conversion: 1 atmospheres = 759.99996673177 millimeters mercury using the online calculator for metric conversions. Pascals (symbol = Pa) or, more commonly, kiloPascals (symbol = kPa). I. Converting between atmospheres and millimeters of mercury. One atm. equals 760.0 mm The torr (symbol: Torr) is a unit of pressure based on an absolute scale, defined as exactly 1/760 of a standard atmosphere (101325 Pa). Thus one torr is exactly Sep 5, 2016 Show Work. Convert 0.875 atm to mmHg. - Show Work.
What will the volume of the gas be at a pressure of 0.987 atm if the temperature remains constant? 3. Rozlišení = (23,8 mm Hg) (0,947) Rozpustnost = 22,54 mm Hg. To dává smysl - v molárním vyjádření je ve velkém množství vody rozpuštěno jen malé množství cukru (i když v reálném světě mají obě složky stejný objem), takže tlak par se sníží jen nepatrně. There are simple conversion factors to change one unit of pressure into another: 1 atm = 760 mm Hg = 760 torr = 1.01325 x 10 5 Pa = 101.325 kPa If you have pressure in one of these units, set up a factor-label conversion grid and use these relationships: What is a pressure of 0.830 atm in kPa? 0.830 atm 101.325 kPa = 84.1 kPa 1 atm There are 4 Felder-Psolucionariorincipios Elemntales de los Procesos Quimicos. Download.
## dioxide is added to the ammonia, to give a total pressure of 11 atm. The flask is then heated. What mass of urea is produced in this reaction, assuming 100% yield? 2. A compound consists of 37.5% C, 3.15% H, and 59.3% F by mass. When 0.298 g of the compound is heated to 50 °C in an evacuated 125-mL flask, the pressure is observed to be 750 mm Hg.
What will be the volume of the balloon if you take it out into the winter cold at -15˚C? A sample of oxygen has a volume if 150 mL when its pressure is 0.947 atm. What will the volume of a gas be at a pressure of 0.987 atm if the temperature remains If a student is asked to convert 1.75 atm into kPa and mmHg, is the student correct if they conclude that there are 177 kPa and 1300 mmHg? The volume of a gas is 27.5 ml at 22.0 degrees C and 0.947 atm.
### 950-10-7 - LTQSAUHRSCMPLD-UHFFFAOYSA-N - Mephosfolan [BSI:ISO] - Similar structures search, synonyms, formulas, resource links, and other chemical information.
1.00 atm = 14.7 psi. Atmospheres to Millimeters of mercury (atm to mmHg) conversion calculator for Pressure conversions with additional tables and formulas. and millimeters of mercury or torr,. (inHg).
2. A sample of oxygen gas has a volume of 150 mL when its pressure is 0.947 atm. What will the volume of the gas be at a pressure of 0.987 atm if the temperature remains constant? 3. Rozlišení = (23,8 mm Hg) (0,947) Rozpustnost = 22,54 mm Hg. To dává smysl - v molárním vyjádření je ve velkém množství vody rozpuštěno jen malé množství cukru (i když v reálném světě mají obě složky stejný objem), takže tlak par se sníží jen nepatrně. There are simple conversion factors to change one unit of pressure into another: 1 atm = 760 mm Hg = 760 torr = 1.01325 x 10 5 Pa = 101.325 kPa If you have pressure in one of these units, set up a factor-label conversion grid and use these relationships: What is a pressure of 0.830 atm in kPa?
When 0.298 g of the compound is heated to 50 °C in an evacuated 125-mL flask, the pressure is observed to be 750 mm Hg. Then the pressure drop relation becomes L ρ 2 1.0 æ 998 ö æ Q ö 0.1 2 ∆p = f − 0.0004 V =f çè ÷ø çè 2 ÷ø ≤ 1600 Pa, where N = 2J and a = Dh 2 a 2 J Na 2 As a first estimate, neglect the 0.4-mm wall thickness, so a ≈ 0.1/J. The following vapor pressures of cinnamaldehyde are given: $1 \mathrm{mmHg}$ at $76.1^{\circ} \mathrm{C} ; 5 \mathrm{mm} \mathrm{Hg}$ at $105.8^{\circ} \mathrm{C} ;$ and $10 \mathrm{mmHg}$ at $120.0^{\circ} \mathrm{C} .$ Vapor pressures of water are given in Table 13.2 At a pressure of 0.947 atm and a temperature of $27^{\circ} \mathrm{C},$ the ball has a volume of $1.034 \mathrm{L} .$ What volume does it occupy during an airplane flight if the pressure in the baggage compartment is 0.235 atm and the temperature is $-35^{\circ} \mathrm{C} ?$ Let's address the question for both percent concentration by mass and for percent concentration by volume.. Percent concentration by mass is defined as the mass of solute divided by the total mass of the solution and multiplied by 100%. 762 + vp ethanol = total pressure. Dalton's Law of partial pressures. The total pressure in a flask containing air and ethanol at 257C is 882 mm Hg. If the pressure of the air in the flask is 762 mm Hg, the vapor pressure of ethanol is ?mm Hg. asked on January 17, 2007; Chemistry Equation(again) What I wrote is the balanced ionic equation. Academia.edu is a platform for academics to share research papers.
⇒ 48.24 mm * 0,963 ⇒ 46.45 mm Razlika u pritisku opada na nulu kada je radijus krivine beskonačan 298 2r / m P / Pa P / atm 1 000 288 0,00284 3,0 96 000 0,947 0,3 960 000 9,474 . KAPILARNOST P P P h a) b) P-2 N /r P Težnja tečnosti da se podiže u uskoj cevi je kapilarnost, a posledica je površinskog napona. radijusa 0,20 mm za 7,36 cm površinski napon vode je: Na. Which of the following is a noble gas? Ar. the balloon is at sea level, where the temperature is 25oC and the barometric pressure is 0.947 atm. The balloon then rises to an altitude of 5500 ft, where the pressure is 0.790 atm and the temperature is 15oC. (760 mm Hg) is 81 mm and the height in the left arm that is connected to a bulb - There are three different units of pressure used in chemistry: 1.
Then the pressure drop relation becomes L ρ 2 1.0 æ 998 ö æ Q ö 0.1 2 ∆p = f − 0.0004 V =f çè ÷ø çè 2 ÷ø ≤ 1600 Pa, where N = 2J and a = Dh 2 a 2 J Na 2 As a first estimate, neglect the 0.4-mm … Rozlišení = (23,8 mm Hg) (0,947) Rozpustnost = 22,54 mm Hg. To dává smysl - v molárním vyjádření je ve velkém množství vody rozpuštěno jen malé množství cukru (i když v reálném světě mají obě složky stejný objem), takže tlak par se sníží jen nepatrně. Jun 16, 2012 950-10-7 - LTQSAUHRSCMPLD-UHFFFAOYSA-N - Mephosfolan [BSI:ISO] - Similar structures search, synonyms, formulas, resource links, and other chemical information. Nov 10, 2014 Halimbawa, sabihin natin na sa harap natin ay isang lalagyan na puno ng likido sa temperatura na 295 K, na ang presyon ng singaw ay katumbas ng 1 atm. Ang tanong ay: Ano ang presyon ng singaw sa temperatura na 393 K? Mayroon kaming dalawang halaga ng temperatura at isa sa presyon, upang malutas namin ang problema sa equation na Clausius-Clapeyron. Gas C 40.
What will the volume of a gas be at a pressure of 0.987 atm if the temperature remains constant? Gas law: Tabuľka premien: mmHg na atm; 1 mmHg = 0.00132 atm: 2 mmHg = 0.00263 atm: 3 mmHg = 0.00395 atm: 4 mmHg = 0.00526 atm: 5 mmHg = 0.00658 atm: 6 mmHg = 0.00789 atm: 7 mmHg = 0.00921 atm: 8 mmHg = 0.0105 atm: 9 mmHg = 0.0118 atm: 10 mmHg = 0.0132 atm: 15 mmHg = 0.0197 atm: 50 mmHg = 0.0658 atm: 100 mmHg = 0.132 atm: 500 mmHg = 0.658 atm: 1000 mmHg Una regla normal tiene divisiones separadas en 1 mm, (1 atm 760 mmHg) Χ = 0,947 atm Material de La masa de un átomo Na es de 8,416 38 x 10-26 lb. Expresar - There are three different units of pressure used in chemistry: 1.
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### A sample of oxygen has a volume if 150 mL when its pressure is 0.947 atm. What will the volume of a gas be at a pressure of 0.987 atm if the temperature remains constant? Gas law:
At this temperature, the vapor pressure of water is 28 millimeters of mercury.
## Resumen: Guia de Problemas Resueltos 2011 - EJERCICIOS DE RIEGO para aprobar Sistema de Riego y Drenaje de Agronomía en Universidad de Buenos Aires.
There are simple conversion factors to change one unit of pressure into another: 1 atm = 760 mm Hg = 760 torr = 1.01325 x 10 5 Pa = 101.325 kPa If you have pressure in one of these units, set up a factor-label conversion grid and use these relationships: What is a pressure of 0.830 atm in kPa?
----- 4.88 g CO2 x 1 mol CO2/44.01 g CO2 x 2 mol . chemistry Apr 15, 2018 1 liter = 106 mm 3; 1 m 3 = 1.000 liter; 1 dm 3 = 1 liter; 1 cm 3 = 0,001 liter; 1 mm 3 = 10-6 liter; Konversi dari satuan liter ke satuan berbasis liter yang lain nya antara lain yakni sebagai berikut ini : 1 liter = 1.000 ml ( milliliter ) 1 liter = 100 cl ( centiliter ) 1 liter = 10 dl ( deciliter ) Satuan kilogram ( kg ) : Next, let's look at an example showing the work and calculations that are involved in converting from atmospheres to millimeters of mercury (atm to mmHg). Atmosphere to mmHg Conversion Example Task: Convert 8 atmospheres to mmHg (show work) Formula: atm x 760 = mmHg Calculations: 8 atm x 760 = 6,080 mmHg Result: 8 atm is equal to 6,080 mmHg A sample of oxygen has a volume if 150 mL when its pressure is 0.947 atm. What will the volume of a gas be at a pressure of 0.987 atm if the temperature remains constant? Gas law: Tabuľka premien: mmHg na atm; 1 mmHg = 0.00132 atm: 2 mmHg = 0.00263 atm: 3 mmHg = 0.00395 atm: 4 mmHg = 0.00526 atm: 5 mmHg = 0.00658 atm: 6 mmHg = 0.00789 atm: 7 mmHg = 0.00921 atm: 8 mmHg = 0.0105 atm: 9 mmHg = 0.0118 atm: 10 mmHg = 0.0132 atm: 15 mmHg = 0.0197 atm: 50 mmHg = 0.0658 atm: 100 mmHg = 0.132 atm: 500 mmHg = 0.658 atm: 1000 mmHg - There are three different units of pressure used in chemistry: 1. atmospheres ( atm ) 2. |
#### asympt ignores order on first use (10.1.00)
##### John S Robinson
Here is a quirk discovered by a colleague:
asympt does not seem to honour the Order setting the first time it is called for a particular expression:
Series expansion for large r converted to an algebraic expression. Note that Order is ignored.
I think this is well described as a ’quirk’; Maple seems to need to do one series expansion of the expression before it starts to honour the value of Order. If you skip/execute the a0 line below then the a1 line ignores/honours the Order option. If you cange the - in the a0 line to +, it works differently, so it seems to be something to do with remembering the series for that particular expression.
Without executing the a0 line we get:
This is not a great problem, but I am intrigued as to why it happens, and if there is a better anser than ’do it twice’
##### Robert Israel(18.1.00)
This is actually a bug in ”series” (which ”asympt” calls). I don’t have any advice other than to call it twice. The reason it works the second time, I think, is that ”series” consults its remember table to see if it has computed the same series before to this or higher order; if it has done so, it truncates the series appropriately. |
Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > KEYWORD > CUTTING PLANE:
Reports tagged with cutting plane:
TR03-012 | 21st January 2003
Edward Hirsch, Arist Kojevnikov
#### Several notes on the power of Gomory-Chvatal cuts
We prove that the Cutting Plane proof system based on
Gomory-Chvatal cuts polynomially simulates the
lift-and-project system with integer coefficients
written in unary. The restriction on coefficients can be
omitted when using Krajicek's cut-free Gentzen-style extension
of both systems. We also prove that Tseitin tautologies
have short proofs in ... more >>>
TR05-006 | 28th December 2004
Edward Hirsch, Sergey I. Nikolenko
#### Simulating Cutting Plane proofs with restricted degree of falsity by Resolution
Goerdt (1991) considered a weakened version of the Cutting Plane proof system with a restriction on the degree of falsity of intermediate inequalities. (The degree of falsity of an inequality written in the form $\sum a_ix_i+\sum b_i(1-x_i)\ge c,\ a_i,b_i\ge0$ is its constant term $c$.) He proved a superpolynomial lower bound ... more >>>
TR16-202 | 19th December 2016
Dmitry Sokolov
#### Dag-like Communication and Its Applications
Revisions: 1
In 1990 Karchmer and Widgerson considered the following communication problem $Bit$: Alice and Bob know a function $f: \{0, 1\}^n \to \{0, 1\}$, Alice receives a point $x \in f^{-1}(1)$, Bob receives $y \in f^{-1}(0)$, and their goal is to find a position $i$ such that $x_i \neq y_i$. Karchmer ... more >>>
TR21-012 | 9th February 2021
Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, Avi Wigderson
#### On the Power and Limitations of Branch and Cut
Revisions: 1
The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical ... more >>>
TR21-158 | 12th November 2021
Noah Fleming, Toniann Pitassi, Robert Robere
#### Extremely Deep Proofs
Revisions: 1
We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, ... more >>>
ISSN 1433-8092 | Imprint |
1. ## Eigen valves question
Plz help me to find the answer for following question;
A matrix
$A= \begin{bmatrix}
0 & 0 & 0 \\
0 & \frac{29}{5} & \frac{12}{5} \\
0 & \frac{12}{5} & \frac{36}{5}
\end{bmatrix}
$
Find
(a) Find Eigen valves A
(b) Egen vectors A
(c) obtain a matrix P such that $P^{-1} AP$ is a diagonal matrix
(d) If $S=P^{-1} AP$, state the special property of each of $S$ and $S^{-1}$
(e) Using the above result reduce the quadratic from Q(x) to form Q(y) where
$Q(x)=16x_1^2 + \frac{29}{5} x_2^2 + \frac{36}{5} x_3^2 + \frac{24}{5} x_2 x_3$
$Q(y)=16y_1^2 + +9y_2^2 + 4y_3^2$
(f) obtain the relationship between $x^t = (x_1 ,x_2 ,x_3)^t$ and $y^t (y_1 ,y_2 ,y_3)^t$
Now I solved part (a) as follows
$A- \lambda I=\begin{bmatrix}
16- \lambda & 0 & 0 \\
0 & \frac{29}{5} - \lambda & \frac{12}{5} \\
0 & \frac{12}{5} & \frac{36}{5} - \lambda
\end{bmatrix}$
the result gain $( \lambda -16)( \lambda -4)( \lambda -9)$
and
part (b) solved as follows
$
\begin{pmatrix}
16 & 0 & 0 \\
0 & \frac{29}{5} & \frac{12}{5} \\
0 & \frac{12}{5} & \frac{36}{5}
\end{pmatrix}
$
x
$\begin{pmatrix}
a \\
b \\
c
\end{pmatrix} = 16
$
then if $\lambda =4$
$\begin{pmatrix}
16 & 0 & 0 \\
0 & \frac{29}{5} & \frac{12}{5} \\
0 & \frac{12}{5} & \frac{36}{5}
\end{pmatrix}
$
$$x $\begin{pmatrix} a \\ b \\ c \end{pmatrix} = 4$ also I did for $\lambda =9$ now I need help to find rest of parts in this question, also I need to know the above calculations are in correct or not!! if some are in incorrect plz. let me know the place and how I correct it 2. Originally Posted by dhammikai Plz help me to find the answer for following question; A matrix $A= \begin{bmatrix} 0 & 0 & 0 \\ 0 & \frac{29}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} \end{bmatrix} $ Find (a) Find Eigen valves A (b) Egen vectors A (c) obtain a matrix P such that $P^{-1} AP$ is a diagonal matrix (d) If $S=P^{-1} AP$, state the special property of each of $S$ and $S^{-1}$ (e) Using the above result reduce the quadratic from Q(x) to form Q(y) where $Q(x)=16x_1^2 + \frac{29}{5} x_2^2 + \frac{36}{5} x_3^2 + \frac{24}{5} x_2 x_3$ $Q(y)=16y_1^2 + +9y_2^2 + 4y_3^2$ (f) obtain the relationship between $x^t = (x_1 ,x_2 ,x_3)^t$ and $y^t (y_1 ,y_2 ,y_3)^t$ Now I solved part (a) as follows $A- \lambda I=\begin{bmatrix} 16- \lambda & 0 & 0 \\ 0 & \frac{29}{5} - \lambda & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} - \lambda \end{bmatrix}$ the result gain $( \lambda -16)( \lambda -4)( \lambda -9)$ and part (b) solved as follows $ \begin{pmatrix} 16 & 0 & 0 \\ 0 & \frac{29}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} \end{pmatrix} $ x $\begin{pmatrix} a \\ b \\ c \end{pmatrix} = 16 $ then if $\lambda =4$ $\begin{pmatrix} 16 & 0 & 0 \\ 0 & \frac{29}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} \end{pmatrix} $ [tex]x $\begin{pmatrix} a \\ b \\ c \end{pmatrix} = 4$ also I did for $\lambda =9$ now I need help to find rest of parts in this question, also I need to know the above calculations are in correct or not!! if some are in incorrect plz. let me know the place and how I correct it For matrix A, is row 1 column 1 suppose to be 16 or 0? for a, i didnt check, but it looks right? b. Eigenvectors should be a vector... not a number c. the diagonal values of S is the eigenvalues of A 3. Hi thanks for the reply and comments. Yes I got an error the matrix A should be $ A= \begin{bmatrix} 16 & 0 & 0 \\ 0 & \frac{29}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} \end{bmatrix} $ In part (b) the vector means if $\lambda =16$, then we have the vector of [a,b,c] is it true? likewise $\lambda =4$, and $\lambda =6$ also have vector [a,b,c] is it true? in part (c) you mean the eigen values of A is insert to the diagonal matrix? Plz explain me... 4. $A- \lambda I=\begin{bmatrix} 16- \lambda & 0 & 0 \\ 0 & \frac{29}{5} - \lambda & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{36}{5} - \lambda \end{bmatrix}$ the result gain $( \lambda -16)( \lambda -4)( \lambda -9)$ Assume ur eigenvalues are right, so $\lambda=16,4,9$ For b, to find eigenvectors, sub each eigenvalues into the $A-\lambda I$ matrix and find the nullspace of the matrix, it should yield exactly 1 nullspace vector for each eigenvalue (for this example) c. yes, the diagonal entries of the diagonal matrix is the eigenvalues of A 5. Hi thnks for the reply and help.So now we complete part (a), I approch the part (b) as follows $\lambda =16$, the the mat will be $ $ $math]A= \begin{bmatrix} 0 & 0 & 0 \\ 0 & \frac{51}{5} & \frac{12}{5} \\ 0 & \frac{12}{5} & \frac{44}{5} \end{bmatrix} $ then according to equations $\frac{51}{5}y+ \frac{12}{5}z=0$ and $\frac{12}{5}y+ \frac{44}{5}z=0$, the results shold be x=0, y=0, z=o then the vector $v_1=(0,0,0)$ If $\lambda=9$, then the matrix will be$$ $A= \begin{bmatrix}
-70 & 0 & 0 \\
0 & \frac{16}{5} & \frac{12}{5} \\
0 & \frac{12}{5} & \frac{9}{5}
\end{bmatrix}
$
Then according to equations
$-7x=0$, $\frac{16}{5}y+ \frac{12}{5}z=0$ and $\frac{12}{5}y+\frac{9}{5}z=0$, results shouldbe x=0, y=3, z=-4, the vector is $v_2(0,3,4)$
again $\lambda=4$, then the equations should be $-12x=0$, $\frac{-9}{5}y+\frac{12}{5}z=0$ and $\frac{12}{5}y-\frac{16}{5}z=0$. results are x=0, y=4, z=3
plz help me the above procedure is correct or wrong..and if wrong let me know the right answer/way.. |
# Math Help - Generating a subspace (dimension, basis)
1. ## Generating a subspace (dimension, basis)
The following system of vectors generates a subspace. What is the dimension of the generated subspace? Let's show a basis of the subspace. Choose a basis of the subspace from the given generating elements!
[1,1,0,...,0], [0,1,1,0,...,0], ..., [0,0,...1,1], [1,0,...,0,1]
I would be grateful if you could help me!
Thank you in advance!
2. Originally Posted by zadir
The following system of vectors generates a subspace. What is the dimension of the generated subspace? Let's show a basis of the subspace. Choose a basis of the subspace from the given generating elements!
[1,1,0,...,0], [0,1,1,0,...,0], ..., [0,0,...1,1], [1,0,...,0,1]
I would be grateful if you could help me!
Thank you in advance!
Evidently the dimension is the cardinality of the maximal independent subset of the set of vectors described. So how would you go about finding such a set? |
# $\mathbb{R}^{n}$ is not a countable union of proper vector subspaces
$\mathbb{R}^{n}$ is not a countable union of proper vector subspaces.
Proof
We know that every finite dimensional proper subspace of a normed space is nowhere dense. Besides, $\mathbb{R}^{n}$ is a Banach space, so the results follows directly.
Title $\mathbb{R}^{n}$ is not a countable union of proper vector subspaces mathbbRnIsNotACountableUnionOfProperVectorSubspaces 2013-03-22 14:59:03 2013-03-22 14:59:03 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Result msc 54E52 |
# Retrive File Information from FTP Site with MSSQL
I have been able to use the query below to get the file structure of a drive on my PC but I need to be able to get this from an FTP Server. I will be using this to determine specific actions based on the files being added to this directory. I don't need the files i just need to see the data that i get in the first query shown.
This Works ....
CREATE TABLE #DirectoryList (
Line VARCHAR(512))
DECLARE @Path varchar(256) = 'dir /OD C:\'
DECLARE @Command varchar(1024) = @Path+'*.*'
PRINT @Command
INSERT #DirectoryList
EXEC MASTER.dbo.xp_cmdshell @Command
DELETE #DirectoryList WHERE Line IS NULL
SELECT * FROM #DirectoryList
GO
DROP TABLE #DirectoryList
GO
Need this to work:
CREATE TABLE #DirectoryList (
Line VARCHAR(512))
DECLARE @Path varchar(256) = 'dir /OD ftp://ftp1.test.com/export/'
DECLARE @Command varchar(1024) = @Path+'*.*'
PRINT @Command
INSERT #DirectoryList
EXEC MASTER.dbo.xp_cmdshell @Command
DELETE #DirectoryList WHERE Line IS NULL
SELECT * FROM #DirectoryList
GO
DROP TABLE #DirectoryList
GO
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Commented:
The FTP standard does not specify a format for directory listings. I bring this up because it appears you expect the listing to be in DOS format. If you already know that this particular FTP server returns listings in a particular format thats great but if not you should be prepared to write logic to parse whatever format the server actually sends.
Anyway, you can't do exactly what you want to do with a simple DIR type command. There are several steps involved to dealing with the remote FTP server. You've got to log in, change to the export sub folder, request a directory listing, and then save it to a file. There is freeware available that does this sort of stuff under the hood and then presents an interface like a mapped network drive but I can't recommend it for a production environment. Your mileage my vary.
I would use an FTP script to do this work and leave a text file containing the directory listings. Then process the file like any other text file or maybe even use xp_cmpshell to TYPE it... Im not sure about the length though, is it a small directory listing?
Here is a Robo-FTP script that will do the job plus it has extra logic to prevent accidentally getting a list of any folder other than the export folder.
DELETE "c:\folder\MyDirList.txt" ;; delete old list file
FTPLOGON "ftp1.test.com" /user="UserID" /pw="secret"
FTPCD "export"
IFERROR!= \$ERROR_SUCCESS GOTO done
FTPLIST "c:\folder\MyDirList.txt" ;; write new directory list
:done
FTPLOGOFF
EXIT
Sr. BI DeveloperCommented:
You can do it in SSIS using a script task.... let me know if you're interested..
Sr. BI DeveloperCommented:
http://asqlb.blogspot.com/2011/06/ftp-file-and-folders-list-into-sql.html
Regards,
Jason
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OK, looks like a good approach by huslayer. And you do have other options.
First problem is you cannot simply do a "DIR" command from the dos prompt for a FTP host.
You really need to use a "FTP" aware or Internet Browser/Explorer aware interface (Internet browsers support FTP interfaces).
Now, we can readily execute external programs - like a DIR command, on in this case a FTP command.
The "standard" FTP program that comes with windows can handle scripting. So, it is possible to build the script, then run the FTP command using the Script.
So, why dont we simply do that ?
The best thing about that approach is it is so very easy to test before you start doing any SQL automation. And you will need a little bit of familiarity with FTP.
1) Open a command window (start, run, cmd)
2) type in FTP and you should get the FTP> prompt and you can type in the following commands
3) OPEN <your ftp site name>
5) CD <to the export directory>
6) DIR
7) bye
Now, you dont quite have to follow all of the above, but it is the basic approach. The first deviation we will take from the above is to launch the FTP and OPEN together with some options NOT to autologin
Using huslayers example site above (ta muchly) from the cmd prompt
1) ftp -n ftp.secureftp-test.com
the above opens a connection to the ftp site without auto logins, but we still need to login....
2) USER test test
now we should be logged in and ready to issue more commands.
3) MDIR *.* dir.log
the above command enables a dir to be written to an external file ie dir.log but it also prompts you - so we need to take that into account as well.
4) BYE
Now check the contents of the log file by typing in TYPE dir.log
Not quite the format we are looking for, but easy enough to extract the information.
Having had some success manually (and you might need to vary a little bit for your own FTP site like doing a "CD export" at the FTP prompt before you do the mdir.
We are now ready to script the commands and take care of that prompt. We can take care of the prompt by using the run time switches for FTP. Here we want to use the "-i" switch. Now the commands are easy enough, we have three - the USER, the MDIR and the BYE exactly as we used them above. So, we edit a text file and put those commands in there and save that file. The we tell the FTP command to use the script file we just created.
So, our script file looks like (saved as dir_ftp.scr) :
USER test test
mdir *.* dir.log
bye
and our ftp command at the dos prompt looks like :
ftp -n -i -s:dir_ftp.scr ftp.secureftp-test.com
So, now we know how we can do it manually, then lets get SQL to do all that for us...
First up, I have created a folder c:\ee as our work folder. Then use SQL to read and write from that folder...
-- first lets create a temporary table to use for writing our script
if object_id('tempdb..my_ftp_script','U') is not null drop table tempdb..my_ftp_script
create table tempdb..my_ftp_script (line varchar(255))
-- now lets write our script into that table
insert my_ftp_script values ('USER test test')
insert my_ftp_script values ('mdir *.* c:\ee\dir.log')
insert my_ftp_script values ('bye')
-- and export it to a flat text file so we can use it with our FTP command
exec xp_cmdshell 'bcp "select * from tempdb..my_ftp_script" queryout "c:\ee\my_ftp_script.scr" -c -T -CACP'
-- now we have a script, we can now issue our FTP command
exec xp_cmdshell 'ftp -n -i -s:c:\ee\my_ftp_script.scr ftp.secureftp-test.com'
-- and prepare to import the results
if object_id('tempdb..#log') is not null drop table #log
create table #log(my_var varchar(max))
bulk insert #log from 'c:\ee\dir.log'
-- and finally select / check the contents
select *, substring(my_var,50,255) as [file_name]
from #log
So, the SQL is not too hard, and apologies about the FTP introduction before hand, but you really should be trying it manually before you can automate it, because if it cannot be done manually then it will never be automated :)
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F(x, y, z) = 0 f(x, y, z) =
1. Jul 3, 2010
makyol
f(x, y, z) = 0 f(x, y, z) = ....
1. The problem statement, all variables and given/known data
[PLAIN]http://www.netbookolik.com/wp-content/uploads/2010/07/q12.png [Broken]
2. Relevant equations
3. The attempt at a solution
Sorry have no idea:(
Last edited by a moderator: May 4, 2017
2. Jul 3, 2010
HallsofIvy
Re: f(x, y, z) = 0 f(x, y, z) = ....
If a surface is given by f(x,y,z)= 0 (or any constant) we can think of it as a "level surface" so $\nabla f= f_x\vec{i}+ f_y\vec{j}+ f_z\vec{k}$ is normal to the surface. The tangent plane at $(x_0,y_0,z_0)$ is of the form $f_x(x_0,y_0,z_0)(x- x_0)+$$f_y(x_0,y_0,z_0)(y-y_0)+ f_z(x_0,y_0, z_0)(z- z_0)= 0$.
For what $(x_0, y_0, z_0)$ does (0, 0, 0) lie on that tangent plane?
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# Small oscillations: diagonal matrix
1. Mar 1, 2013
### bznm
1. The problem statement, all variables and given/known data
I'm solving an exercise about small oscillations.
I name $T$ the kinetic matrix and $H$ the hessian matrix of potential.
The matrix $\omega^2 T- H$ is diagonal and so find the auto-frequencies is easy! But I have a problem with normal modes. The lagrangian coordinates are two angles, $\theta$ and $\phi$.
$$\omega^2T-H(\theta, \phi)=\begin{pmatrix}m\omega^2-m \Omega &&&0 \\ 0&&&M\omega^2-k \end{pmatrix}$$
Normal modes are given by splitted oscillations of the two coordinates. Is it correct? Are they given by:
$\theta(t)=A_1 \cos(\Omega t+ \alpha_1)$ and $\phi(t)=A_2(\cos \frac{k}{M} t+\alpha_2)$? ($A_1, A_2$= constants depending on initial conditions)
And is the general solution of motion given by $$\theta(t)+\phi(t)=A_1 \cos(\Omega t+ \alpha_1)+A_2(\cos \frac{k}{M} t+\alpha_2)$$?
2. Mar 1, 2013
### BruceW
looks pretty good. Except one slight mistake. hint: check the values of omega.
3. Mar 2, 2013
### bznm
I have forgotten the square for ω..
But I have a doubt:
If I find the eigenvectors, I obtain:
- If $\omega^2=\Omega$
the eigenvector is (μ_1,0)
- if $\omega^2=k/M$
the eigenvector is (0, μ_2)
where μ_1 and μ_2 belong to ℝ.
So, the correct solution of motion, introduced the q_i coordinates that shift the origin of the system in the point of equilibrium, should be:
${\bf q}(t)=A_1 cos (\sqrt(\omega)t+\alpha_1) \begin{pmatrix}\mu_1\\0 \end{pmatrix}+A_2 cos (\sqrt(k/M)t+\alpha_2) \begin{pmatrix}0\\μ_2 \end{pmatrix}$
is it correct? thanks again
Last edited: Mar 2, 2013
4. Mar 2, 2013
### BruceW
yeah, that all agrees with the matrix equation you wrote in the first post. Except one thing. You have written the eigenvectors the wrong way around. The whole idea of the normal mode method is that the matrix you wrote, when acting on an eigenvector, should give a zero vector. So when omega squared equal k/M, then what should the eigenvector be?
Edit: woops, ah sorry sorry. You did write them the correct way around. I didn't take the time to look at the matrix product carefully enough. So all your work is correct.
5. Mar 2, 2013
### bznm
Thank you so much!!
6. Mar 2, 2013 |
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# Wendell Prize
## Announcement: 2017–2018 Wendell Prize Winner
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### Description
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In order to honor and encourage extraordinary academic achievement, the Wendell Prize of $21,000 is divided into three parts. Two equal disbursements of$7,000 each are made to the Scholar at the start of his or her sophomore-junior and junior-senior summers to support academic or other appropriate projects or experiences endorsed by the Scholar's Allston Burr Assistant Dean and approved by the Wendell Prize selection committee. (Should any Wendell Prize recipient have filed plans to complete undergraduate degree requirements in Harvard College at the end of his or her sixth term of residence, he or she may apply to the selection committee for an alternative distribution of summer funds.) An additional \$7,000 will be presented to the Scholar at a dinner honoring current and past Wendell Prize recipients in the spring.
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# Stuck in the 500s
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Stuck in the 500s [#permalink]
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17 Jun 2013, 07:57
Hi Everyone,
I've been prepping for the GMAT since March and have my test scheduled for June 22nd. I wish I found this site earlier...
I only prepped using Kaplan, completed 8 practice tests but I am stuck in the 500's. The best ive scored is 580. I have completed the recommended target practice but have not seen any improvement.
My quant scores are generally in the 50 percentile range and my verbal fluctuates from 30 - 50%.
I have completed the addtional workshops and even in the challenging workshops, I score roughly 7/10 on the quizzes in adequate time. What could be the reason as to why these results aren't transferring over to my real tests?
What should I do now? Since I only have a few days left.
Posted from my mobile device
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Re: Stuck in the 500s [#permalink]
### Show Tags
17 Jun 2013, 14:15
Ksterr wrote:
Hi Everyone,
I've been prepping for the GMAT since March and have my test scheduled for June 22nd. I wish I found this site earlier...
I only prepped using Kaplan, completed 8 practice tests but I am stuck in the 500's. The best ive scored is 580. I have completed the recommended target practice but have not seen any improvement.
My quant scores are generally in the 50 percentile range and my verbal fluctuates from 30 - 50%.
I have completed the addtional workshops and even in the challenging workshops, I score roughly 7/10 on the quizzes in adequate time. What could be the reason as to why these results aren't transferring over to my real tests?
What should I do now? Since I only have a few days left.
Posted from my mobile device
Hi,
First, relax.
Remember its just a test and you can postpone your test day.
Second, I suggest you take 2 days to search around the two forums here.
devote one day for Quant and one day for Verbal.
There are a lot of materials out there. BB, the founder of this forum made a sticky post with all materials available with reviews in each forum.
If I were you I would start there, build a plan and execute.
Good Luck!
_________________
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Re: Stuck in the 500s [#permalink]
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17 Jun 2013, 20:31
I'm not looking to postpone since I'll lose the test fee at this point. I'm thinking I'll try it out....
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Re: Stuck in the 500s [#permalink]
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17 Jun 2013, 23:48
Ksterr wrote:
I'm not looking to postpone since I'll lose the test fee at this point. I'm thinking I'll try it out....
You can postpone your exam with additional fee of $50. In my eyes, its better to postpone it than loosing$250 and then paying again for another try but its up to you.
anyway, use my advice in previous post, im sure you will benefit from it.
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Re: Stuck in the 500s [#permalink]
### Show Tags
18 Jun 2013, 00:10
Ksterr wrote:
I'm not looking to postpone since I'll lose the test fee at this point. I'm thinking I'll try it out....
Hi, I think you can postpone your exam with additional fee of \$50.
My exam was on the 29th of this month, I postponed because I thought I needed more preparation to get my desired score!
There is alot of great material here which is very helpful for GMAT.
I suggest that you clear your head, take a deep breath and decide what you want to do.
The decision is upto you!
Sam
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Re: Stuck in the 500s [#permalink]
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18 Jun 2013, 05:33
Thanks for the input, I'm not exactly panicking because I know I can write it again and have no I'll effects.
Im finding that I have timing issues in quant, so I must fix that. I checked MBA online and it says no refunds for cancellations within 7 days.
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Re: Stuck in the 500s [#permalink]
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24 Jun 2013, 09:47
So I wrote the GMAT this past Saturday and got 540 (Q57,V30). I prepped for 3 months using Kaplan with most of my practice tests in the 500s.
What do you guys suggest me to do? Should I purchase the OG13 and possibly look into MGMAT Prep?
Thanks
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Re: Stuck in the 500s [#permalink]
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24 Jun 2013, 21:54
Ksterr wrote:
So I wrote the GMAT this past Saturday and got 540 (Q57,V30). I prepped for 3 months using Kaplan with most of my practice tests in the 500s.
What do you guys suggest me to do? Should I purchase the OG13 and possibly look into MGMAT Prep?
Thanks
Yes get OG and mgmat guides...Also follow these links
how-to-start-gmat-preperations-136048.html
how-to-improve-quant-score-137975.html
ATB
_________________
Re: Stuck in the 500s [#permalink] 24 Jun 2013, 21:54
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# How do I account for the direction of friction acting on a spring?
I would like to set up the equations of motion for a simple spring oscillator.
Let's have a spring lying horizontally; we attach a small mass $m$ to the (massless) spring.
The force of the spring on the mass is
$$F_\text{spring} = - k x$$
where $k$ is the spring constant, and $x$ is the displacement from the position of rest.
Since I am considering the system in motion, I only need to account for the kinetic friction. I know that the magnitude for the kinetic friction is:
$$F_\text{fric} = F_g \mu$$
where $\mu$ is the coefficient of kinetic friction, $F_g$ equals the Normal Force on the surface and is also equal to the gravitational force.
But the direction of the force is always antilinear to the direction of motion.
How do I set this up correctly in my approach for the equations of motion? My approach is
$$m a = -k x - \operatorname{sign}(x) F_\text{fric}$$
where $a$ is the acceleration of the mass point, and $\operatorname{sign}$ is the sign function.
Is this correct?
$$ma=-kx - \mathrm{sign}(v) F_{\mathrm{fric}}.$$ |
July 31, 2020
### How Did Banks Fund C&I Drawdowns at the Onset of the COVID-19 Crisis?
David Glancy, Max Gross, and Felicia Ionescu1
Banks experienced significant balance sheet expansions in March 2020 due to unprecedented increases in commercial and industrial (C&I) loans and deposit funding. According to the Federal Reserve's H.8 data, "Assets and Liabilities of Commercial Banks in the U.S.", C&I loans increased by nearly $480 billion in March—the largest monthly increase in the history of this series, surpassing the nearly$90 billion increase in C&I loans in the six weeks following Lehman Brothers' collapse in 2008 (see Figure 1). Commentary in banks' and firms' earnings calls, as well as write-in comments provided in weekly data submissions, indicate that this growth was primarily attributable to firms drawing down revolving lines of credit to make up for revenue and funding disruptions related to the coronavirus pandemic.
The unparalleled magnitude of these draws could potentially strain banks' liquidity positions. However, banks also experienced over $800 billion in deposit inflows in March 2020, possibly mitigating these strains. While deposit increases outpaced C&I loan increases in aggregate, this does not necessarily imply that deposits were sufficient to fund the large draws for all banks. If the banks that experienced the largest C&I drawdowns did not also receive correspondingly larger deposit inflows, they might need to liquidate assets or increase other types of funding. This note uses high-frequency, individual bank balance sheet data to shed light on these issues and investigate how banks adjusted balance sheets to fund the large C&I drawdowns. We find that deposits and non-deposit borrowings increased in weeks with high C&I draws, especially at large banks. Smaller banks experienced smaller deposit inflows, and thus also reduced liquid assets to fund draws. The large C&I draws therefore broadly expanded banks' balance sheets, and diminished stocks of liquid assets at smaller banks, potentially putting pressure on bank capital requirements and liquidity positions.2 These developments highlight the importance of recent policies to ease capital constraints and provide liquidity to the sector. Data In order to study how banks responded to C&I drawdowns in the COVID-19 episode, we use microdata from FR 2644 filings.3 These data include weekly information on the size and composition of the portfolios of a random stratified sample of about 875 domestically chartered banks and U.S. branches and agencies of foreign banks. These data allow us to test how banks that experienced large increases in C&I loans in a given week adjusted their balance sheets to accommodate these increases.4 We include in C&I loans both conventional C&I loans (as defined in the Call Reports) and loans to non-depository financial institutions. We include the latter in order to capture draws from institutions such as Real Estate Investment Trusts (REITs), which also utilized bank credit lines heavily during our period of interest. We predominantly use data from March 4 to April 1 in order to capture the weeks with the largest C&I draws, and to avoid up effects from the Payroll Protection Program, which significantly affected C&I lending in the following weeks.5 We conduct the analysis for four types of banks. We break down findings for domestically chartered banks based on their total assets as of March 4, 2020, choosing thresholds so as to group banks operating in similar regulatory environments.6 Large banks are those with assets exceeding$100 billion, midsized banks have assets between $10 billion and$100 billion, and small banks have assets under $10 billion. We separately study the response of branches and agencies of foreign banks. We omit banks in holding companies that predominantly operate as custodian or investment banks, banks with minimal C&I holdings (less than 5 percent of assets), and observations with extreme draws from our analysis.7 Aggregate Changes in Balance Sheets Figure 2 summarizes changes in banks' balance sheets in March 2020, aggregated across all respondent banks for a given week. The height of each segment reflects the weekly change in a particular asset or liability, normalized by initial assets. As shown, the most dramatic expansion in banks' balance sheets occurred in the two weeks from March 11 to March 25. C&I loans outstanding increased by over 1% of initial assets each week, while deposits and cash increased even more. Non-deposit financing also increased notably during these weeks, indicating that, despite the large increases in deposits, some banks experienced increased funding needs not satisfied by deposit inflows. ##### Figure 2. Aggregate Changes in Bank Balance Sheet Items by Week This substantial asset growth occurred for banks of all sizes and was particularly pronounced for large banks and branches of foreign banks. Figure 3 summarizes changes in banks' balance sheets between March 4 and April 1 across all four bank types in our analysis, with each pair of bars reflecting the change in assets and liabilities for a particular bank type. As shown, large domestic banks increased in size by about 8 percent, accounting for about half of the total increase in bank assets, with midsized and small domestic banks increasing by about 5 percent and 2 percent, respectively. Assets in foreign branches grew by nearly 20 percent, but accounted for a smaller portion of the overall expansion than large banks, due to their smaller initial size. ##### Figure 3. Aggregate Changes in Bank Balance Sheet Items by Bank Type This expansion in balance sheets seems to be primarily attributable to two factors: unprecedented C&I draws increasing lending and flight to quality effects increasing deposits. In large domestic banks, C&I loans increased by nearly$350 billion dollars over the course of four weeks, or about 3 percent of their initial assets. Deposits increased about double that. C&I loans grew at an even faster pace at branches and agencies of foreign banks, funded by an increase in the net amount due to related institutions. Similarly, small and midsized banks experienced increases in C&I loans and in deposits, however this growth was much smaller than at larger banks. Additionally, cash increased by over $900 billion dollars over this period, with fast growth at larger domestic banks and at branches and agencies of foreign banks. Overall, the aggregate numbers show deposit inflows exceeding C&I draws, allowing banks to increase cash holdings. The substantial draws in March thus do not appear to have caused liquidity strains for the banking sector as a whole. However, aggregate numbers may give a misleading sense of how draws affected individual banks. If the banks that experienced the largest C&I credit line drawdowns did not also have significantly larger deposit inflows, then they might have needed to adjust elsewhere on their balance sheet to fund soaring C&I lending. To study this issue, the next section analyzes how individual banks responded to C&I draws. Cross-sectional Changes in Banks' Balance Sheets Figure 4 presents a set of scatter plots summarizing the cross-sectional relationship between C&I drawdowns and changes in liquid assets (cash, securities, repos, and fed funds sold), deposits, and non-deposit borrowings. Each dot represents a certain bank-week observation, with the change in C&I loans as a percent of assets on the x-axis and the change in the studied balance sheet variable as a percent of assets on the y-axis. Each chart also presents the results of the corresponding regression between these variables, representing the expected change in the balance sheet item for each dollar increase in C&I loans. ##### Figure 4. C&I Loan Growth and Changes in Liquid Assets, Deposits and Borrowings Large domestic banks, which accounted for the vast majority of the drawdowns, funded these draws entirely by expanding liabilities, with higher deposits (middle-left chart) and non-deposit borrowings (bottom-left chart). Based on regression estimates, banks that faced an extra dollar in draws increased deposit funding by about 62 cents and other borrowing by about 55 cents, which accounted for all of the increase in total assets. Large banks in fact increased liquid assets somewhat in weeks with elevated C&I draws (top-left chart). In contrast to large banks, small and midsized domestic banks were more prone to reduce holdings of liquid assets when faced with C&I draws (top-right chart). We estimate that an extra dollar in draws resulted in about a 51 cent decrease in liquid assets. As with larger banks, to the extent that C&I draws expanded small and midsized banks' balance sheets, this expansion was funded with a combination of roughly equal increases in deposits and non-deposit borrowings. However, caution is warranted for these values, as the fit in the regressions for smaller banks is not as good as the fit for larger banks. C&I draws were much more significant for larger banks resulting in more precision in our estimates for how these banks responded to draws. Our analysis reveals several key findings on how banks funded C&I draws that merely examining aggregate credit flows cannot capture. First, while deposit growth out-paced C&I draws more often than not, there are still a significant number of banks for which draws exceeded deposit inflows. For example, there are a number of large banks that experienced weekly C&I draws in the neighborhood of 2 percent of assets. As illustrated, deposit inflows for these banks were typically not sufficient to fund such large draws, and these banks significantly increased non-deposit borrowing. Second, while banks increased cash in aggregate, the scatter plots show that smaller banks reduced liquid assets (predominantly cash) when faced with high C&I draws. Thus while the banking sector overall was awash in deposits, many individual banks needed to turn to other sources of funding to accommodate the increased C&I demand. These findings highlight the importance of policies to facilitate banks' ability to access finance, such as encouraging the use of the discount window. Our results so far demonstrate how banks responded to draws at a high level, with large domestic banks funding draws with higher deposits and non-deposit borrowings and small and midsized banks also needing to reduce liquid assets in addition to using deposits and non-deposit borrowing. We next turn to studying changes at a more granular level (e.g. examining which liquid assets decline) or in other categories (e.g. non-C&I lending). We present a full decomposition of the cross-sectional changes in banks' balance sheets in response to C&I draws. In general, each dollar increase in C&I loans needs to have a corresponding increase in funding or decrease in other assets. We use this balance sheet identity to create a similar identity for regression coefficients of the type displayed in the scatter plots in Figure 4: the predicted effect of a dollar increase in C&I loans must be a dollar increase across a bank's funding sources, net of changes in other assets.8 Figure 5 presents the results of this decomposition at a granular level with a stacked bar chart, where the height of each segment in these bars illustrates how much banks adjusted a particular balance sheet item to fund C&I draws. For liabilities, this height is the coefficient from regressing changes in that particular liability on changes in C&I loans, both as a percent of bank assets. For assets, the sign of the coefficient is reversed as declines in an asset can be used to fund a loan. These segments add up to one, and thus can be interpreted as the share of the increase in C&I loans funded by adjusting that particular balance sheet component. ##### Figure 5. Decomposition of Banks' Responses to C&I Loan Growth in March 2020 As shown, the main margins of adjustment for domestic banks are to the three variables analyzed in the scatter plots: liquid assets, deposits and non-deposit borrowings. Almost the entire relationship between deposits growth and C&I draws comes from core deposits, as opposed to large time deposits, which likely reflects firms depositing drawn credit into operating accounts at the banks holding their loans. The figure also provides a closer look at which liquid assets change. Changes in securities and short term lending (repos and fed funds sold goes into the other asset category) did not move much with C&I draws, so most of the change in liquid assets shown in the scatter plots is due to changes in cash holdings. One exception is for small banks, for whom declines in short term lending make up a significant share of the decline in liquid assets in response to C&I draws. Lastly, much like large banks, foreign branches and agencies of foreign banks increased cash in weeks with high draws, likely due to banks building up liquidity to fund anticipated future draws. As a result, these branches needed a large increase in funding, which seem to have mostly come from parent foreign banking organizations. The figure shows that net due to related institutions increases about 1-for-1 with C&I draws.9 Figure 6 displays an equivalent decomposition of how bank balance sheets change with C&I loan increases, but for the period from 2015 to 2019, allowing us to compare results with more normal times. As this period featured a fairly similar regulatory environment to the COVID-19 episode, the primary difference is likely the effects of the virus, most notably the increase in C&I loan demand. During normal times, weekly changes in C&I loans ranged from$14 billion to $22 billion, compared to$29 billion to $226 billion during the COVID-19 episode. ##### Figure 6. Decomposition of Banks' Responses to C&I Loan Growth from 2015 to 2019 Although the magnitude of the changes were substantially different, the findings are fairly similar: domestic banks increased deposits and non-deposit borrowings in weeks where C&I loans increased, while foreign banks predominantly increased the amount due to related institutions. Despite these broad similarities, some differences stick out. During normal times, deposits moved almost 1-for-1 with C&I for midsized banks, with large banks experiencing smaller deposit inflows. This reversed in the COVID-19 period, with large banks experiencing comparatively larger deposit inflows associated with C&I draws. Furthermore, unlike in normal times when large banks and foreign banks reduced cash in the face of C&I loan increases, during the COVID-19 period episode, they increased cash. As these banks experienced the largest draws during the COVID-19 period, they might have been less willing to reduce liquid assets in case draws continued to remain elevated in the future. Conclusion Our analysis shows that while deposits rose in banks experiencing C&I draws, often this was not enough to fund the dramatic increase in C&I loans in March. Domestic banks made up for this shortfall by increasing non-deposit borrowing and, in the case of smaller banks, by also reducing liquid assets. Our analysis sheds light on how banks funded loans upon the time of draws, however the longer run funding mix is likely to differ. The immediate effect of the draws is to reduce capital ratios, and for smaller banks to deplete the stock of liquid assets. Even though banks entered the COVID-19 period with strong capital and liquidity positions, these developments likely put capital and liquidity pressures on the banking sector. If banks are not comfortable with their new balance sheet composition, and equity issuance is prohibitively costly, they might reduce new lending in order to rebuild capital and liquidity. Thus, although banks had limited ability to reduce loans to immediately fund draws during the COVID-19 episode, the large credit line draws could still crowd out other lending in the longer term. This threat of credit line draws having an adverse effect on credit supply going forward highlights the importance of recent Federal Reserve actions to support the flow of credit to firms and households. Indeed, recent policies have supported banks' abilities to borrow (e.g. discount window modifications and Money Market Mutual Fund Liquidity facility), eased capital constraints (e.g. modifications to the Supplemental Leverage Ratio and guidance on use of capital buffers) and directly facilitated lending (e.g. Main Street Lending Program and Paycheck Protection Program Liquidity Facility). With banks facing the combined effects of C&I draws expanding balance sheets and expected loan losses pushing down equity, these policies play a key role in ensuring that businesses and households continue to have access to bank credit, thus limiting the financial and economic disruptions created by the coronavirus pandemic. 1. We thank Rochelle Edge and Rebecca Zarutskie for helpful comments. Return to text 2. Using Call Reports data on banks' leverage ratios in the fourth quarter of 2019, we estimate that the four weeks of C&I draws discussed in this note decreased Tier 1 capital ratios at large banks by between a quarter and a half percentage point. Return to text 3. These data were obtained through a confidential survey of depository institutions that requires confidential treatment of institution-level data and any information that identifies the individual institutions that reported the data. Return to text 4. While changes in C&I loans generally reflect a combination of draws of existing lines of credit, new loan originations, and paybacks of existing loans, given the evidence of large drawdowns in March, in this note we will use the terms "drawdowns" and "increases in C&I loans" interchangeably. Return to text 5. The Payroll Protection Program, which started on April 3, also resulted in significant increases in C&I loans. However, unlike the draws in March, these increases in loans were predictable, under a bank's control, and able to be fully financed through one of the Federal Reserve's liquidity facilities, making their effects on banks' balance sheets not comparable to those of the March credit line drawdowns. Return to text 6. Banks above$10 billion in assets are subject to the Durbin amendment and CFPB examinations, while banks above \$100 billion are subject to CCAR and the LCR requirement. Return to text
7. Observations are dropped when domestic banks have a weekly decrease in C&I outstanding that is more than 2 percent of assets or an increase that is more than 3 percent of assets. Foreign banks have more volatile changes, so we use cut-offs of 5 percent declines and 15 percent growth. This censoring results in the loss of less than 1 percent of domestic observations and less than 2 percent of foreign observations. Return to text
8. More formally, let a bank's balance sheet consist of a set of assets $A$, which are funded by a set of liabilities $L$, and equity $E$. If $\beta_x$ is the coefficient from regressing a change in balance sheet item $X$, $\Delta{X_{b,t}}/{Assets}_b$, on the change in C&I loans, $\Delta {C\&I_{b,t}}/{Assets}_b$, then we get the following identity: $1=\beta_E + \sum_{j \in L} \beta_j -\sum_{i \in A\backslash{C\&I}}^N \beta_i$. Banks have limited ability to adjust equity on a weekly time horizon, so we find $\beta_E \approx 0$ and only display coefficients for assets and liabilities. Return to text
9. From Figure 2, we know that most of the aggregate increase in borrowing from related institutions came in the last two weeks of the sample, following coordinated central bank actions in mid-March to enhance standing U.S. dollar swap lines. This timing highlights the potential importance of recent central bank actions in providing access to dollar liquidity. Return to text |
## Sunday, May 5, 2013
### What picture of economics is emerging?
I'm not going to answer this definitively; I'm just going to give some preliminary thoughts in bullet point form:
• The "invisible hand" is a result of the dynamics of an information transfer process from something identified in economics as the "demand" (which is a nebulous concept based on consumer desires and abilities) to the supply (which is a fairly well-defined concept based on physical widgets or services). The price is what detects a demand signal being sent to the demand.
• In particular, these dynamics arise without any particular description of the micro theory: the reason the supply curve is upward sloping and the demand curve is downward sloping is a result of the price detecting information being transferred from the demand to the supply. Marginal utility may help explain a micro theory, but it is unnecessary to derive the basic results.
• Additionally, the information transfer model does not require equilibrium and therefore can operate when the system is not in equilibrium (as long as $I_{Q^d} \sim I_{Q^s}$).
• Since the information source is the demand, the idea that "supply creates its own demand" is on the whole false since the information destination cannot create the information so destined. Supply can uncover demand (i.e. information that used to be transmitted from the source into "empty space" can suddenly have a place to go).
• The price is a way of "detecting" the process of transmitting information from the demand to the supply, so in a sense, money is a unit of information that at a basic level only has meaning in what demand it can be used to transmit information to what supply.
• As a corollary to the previous point, anything that behaves like supply and demand with something functioning as a price can be put in this framework. For example, the IS-LM model behaves like supply (LM) and demand (IS) curves with the interest rate functioning as the "price". So does the AD/AS model with the "price level" functioning as the price.
• Prices can be sticky if information transfer is non-ideal. Again, this is based on non-ideal information transfer and does not depend on a particular micro theory (menu costs, money illusion, signalling) -- a micro theory is unnecessary.
• If $I_{Q^d} \gg I_{Q^s}$ or the price becomes undefined, then the information transfer mechanism breaks down and there might not be any kind of market at all
• One could take this further and make the argument that $I_{Q^d} \sim I_{Q^s}$ represents a condition for a well defined concept of "economics" to exist, and that studying things where little information is transferred by a ill-defined price is actually something else ... like sociology. Does market failure represent a boundary of applicability in a similar way that nonequilibrium systems can lack a well defined concept of temperature?
1. On that last bullet, it seems the efficient markets hypothesis is empty ... markets are efficient when $I_{Q^d} \sim I_{Q^s}$, or basically the price contains most of the information when the price contains most of the information. |
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## Competency in Focus: Linear Transformation
This problem from IIT JAM 2018 is based on calculation of linear transformation. It is Question no. 21 of the IIT JAM 2018 Problem series.
## Next understand the problem
Let $U, V, \textbf{and}\quad W$ be finite dimensional real vector spaces, $T:U\rightarrow V, S: V\rightarrow W, P: W\rightarrow U$ be linear transformations. If range($ST$)=nullspace($P$), nullspace($ST$)=range ($P$) and rank($T$)=rank($S$), Then which one of the following is true? $\textbf{(A)}$ nullity of $T$ = nullity of $S$ $\textbf{(B)}$ dimension of $U \neq$ dimension of $W$ $\textbf{(C)}$ If dimension of $V = 3$, If dimension of $U = 4$, then $P$ is not identically zero. $\textbf{(C)}$ If dimension of $V = 4$, If dimension of $U = 3$, and $T$ is one-one then $P$ is identically zero.
##### Source of the problem
IIT JAM 2018, Question Number 21
### Linear transformation and Vector Space.
7/10
##### Suggested Book
Do you really need a hint? Try it first!
I want to make an opinion that this question is not hard but it is time consuming. So in a time paced exam beware of this kind of question. The fact I’ve used here : i) Ker$(ST)={0} \iff ST \quad \textbf{is injective}$ ii) $ST\quad \textbf{is injective} \iff$T$\quad \textbf{is injective}$ iii) And creating the counter examples which are most time consuming. So, in hint 1, I want to disclose the answer and I can first try to find out the counter example by yourself. $\bullet \quad\textbf{The correct option is} \quad[\textbf{C}]$
Consider $U=V=W=\mathbb{R}^2$ and the maps are The range($ST$) =$S_p\{c_1\}=$ Nullspace($P$). range($P$)=$S_p\{c_2\}=$ Nullspace($ST$) & rank($T$) $=1=$ rank ($S$) But, Nullity of $T=1 \neq 2=$ Nullity of $S$. And dim($U$) = dim($W$) = 2 So, the option $(A)$ and $(B)$ are incorrect.
Consider $U= \mathbb{R}^3,V= \mathbb{R}^4, W= \mathbb{R}^5$ range ($ST$) = $S_p\{c_1,c_2\}=$ Nullspace($P$) Nullspace ($ST$)=$S_p\{c_3\}$=Range($P$) rank($T$)=$2$=rank($S$) and $P$ is not zero map So, option $(D)$ is not correct. Hence option $(C)$ is the one left which has to be true, Now lets prove that.
We’ll prove that $P$ is non zero. Suppose $P=0$ $U\longrightarrow V\longrightarrow W$ dim($U$) = $4$, dim($V$) =$3$ Now, range ($P$) $=0=$ Nullspace ($ST$) $\Rightarrow ST$ is injective $\Rightarrow T$ is injective $\Rightarrow$ dim ($V\geq 4$) Which is a contradiction. Hence $P=0$ And we are done
# College Mathematics Program
The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.
## Radius of Convergence of a Power series | IIT JAM 2016
Try this problem from IIT JAM 2017 exam (Problem 48) and know how to determine radius of convergence of a power series.We provide sequential Hints.
## Eigen Value of a matrix | IIT JAM 2017 | Problem 58
Try this problem from IIT JAM 2017 exam (Problem 58) and know how to evaluate Eigen value of a Matrix. We provide sequential hints.
## Limit of a function | IIT JAM 2017 | Problem 8
Try this problem from IIT JAM 2017 exam (Problem 8). It deals with evaluating Limit of a function. We provide sequential hints.
## Gradient, Divergence and Curl | IIT JAM 2014 | Problem 5
Try this problem from IIT JAM 2014 exam. It deals with calculating Gradient of a scalar point function, Divergence and curl of a vector point function point function.. We provide sequential hints.
## Differential Equation| IIT JAM 2014 | Problem 4
Try this problem from IIT JAM 2014 exam. It requires knowledge of exact differential equation and partial derivative. We provide sequential hints.
## Definite Integral as Limit of a sum | ISI QMS | QMA 2019
Try this problem from ISI QMS 2019 exam. It requires knowledge Real Analysis and integral calculus and is based on Definite Integral as Limit of a sum.
## Minimal Polynomial of a Matrix | TIFR GS-2018 (Part B)
Try this beautiful problem from TIFR GS 2018 (Part B) based on Minimal Polynomial of a Matrix. This problem requires knowledge linear algebra.
## Definite Integral & Expansion of a Determinant |ISI QMS 2019 |QMB Problem 7(a)
Try this beautiful problem from ISI QMS 2019 exam. This problem requires knowledge of determinant and definite integral. Sequential hints are given here.
## What is TIFR and how to prepare for it?
About TIFR Tata Institute of Fundamental Research, TIFR is the foremost institution for advanced research in foundational sciences based in Mumbai, Maharashtra, India. The institute offers a master's course, an integrated M.Sc and Ph.D. course and a Ph.D. degree in...
## Limit of a Sequence | IIT JAM 2018 | Problem 2
Try this beautiful problem from IIT JAM 2018 which requires knowledge of Real Analysis (Limit of a Sequence). We provide sequential hints. |
# Smooth trivialization of smooth Hilbert bundles
In page 67 of Topology and Analysis by Booss and Bleecker, it is claimed that any Hilbert bundle is topologically trivial. Clearly, any smooth Hilbert bundle over a smooth manifold is topologically trivial, but it appears to be no reason to believe that this trivialization is smooth.My questions are:
• Are there known conditions on the manifold or on the Hilbert space to guarantee that such topological trivialization is actually smooth?
• Are there known counterexamples showing that such smooth trivialization is impossible in the general case?
EDIT: The definition of smooth Hilbert bundle is the one defined in 2.1 of: László Lempert and Róbert Szőke, Direct images, fields of Hilbert spaces, and geometric quantization, (arXiv:1004.4863) namely
A smooth (always complex) Hilbert bundle is a smooth map $p\colon H\to S$ of Banach manifolds, each fiber $p^{-1}(s)$, $s\in S$, is endowed with the structure of a complex vector space; for each $s\in S$ there should exist a neighborhood $U\subset S$, a complex Hilbert space $X$, and a smooth map (local trivialization) $F\colon p^{-1}U \to X$, whose restriction to each fiber $p^{-1}(t)$, $t\in U$, is linear, and such that $p\times F\colon p^{-1} U \to U \times X$ is a diffeomorphism.
• What do you mean by a smooth Hilbert bundle? See eg mathoverflow.net/q/101526/4177 Note also that you need the structure group to be a Lie group, and $U(\mathcal{H})$ is a Banach Lie group in a norm topology, but not a Lie group in the strong topology (=compact-open topology). And the norm topology is in some sense "too strong". – David Roberts Oct 19 '17 at 23:52
• @DavidRoberts I just edited the question to make explicit the definition of Hilbert bundle I'm using. It's the definition 2.1 of this paper: arxiv.org/pdf/1004.4863.pdf – Max Reinhold Jahnke Oct 20 '17 at 1:46
Let $$K$$ be a Lie group, modeled on a locally convex space, and $$M$$ a finite-dimensional paracompact manifold with corners. Then each continuous principal $$K$$-bundle over $$M$$ is equivalent to a smooth principal $$K$$-bundle. Moreover, two smooth principal $$K$$-bundles are continuously equivalent if and only if they are smoothly equivalent.
Now, every smooth Hilbert bundle in the sense of the post gives rise to a smooth $$\mathrm{GL}(\mathcal H)$$-principal bundle (defined, for instance, by the transition 1-cocycle) which is topologically trivial, hence by the above theorem smoothly trivial, and so therefore is the original vector bundle. |
# Vrms for monotomic, diatomic, and polyatomic molecules
in my notes from class I have that
Vrms= sqrt(3kT/m) for a pt molecule Vrms= sqrt(5kT/m) for a diatomic molecule and that Vrms= sqrt(6kT/m) for a triatomic-> higher order molecule
says that Vrms=sqrt(3kT/m) always and I don't understand why that is.
Thanks,
I know another post has about the same question- but wasn't able to comment or add a related question due to my 0 reputation.
Don't understand where Vrms formula comes from. Is it solely dependent on translation motion. And that's why for all molecules you can use the same exact Vrms formula.
RMS Speed of Gas Molecule for Polyatomic Molecules
• It might help to explain what else confuses you? This seems to be an exact duplicate of the question you linked, and the answer there seems correct. If you could explain what you don't understand about it, then you can narrow your question down to just that aspect. – jacob1729 Jan 16 '19 at 23:47
$$\frac 12 mv^2_{\rm rms}$$ is the mean translational kinetic energy of an atom/molecule and it is the the mean translational kinetic energy which is proportional to the temperature.
The first equation is correct for all atoms/molecules but the statements about diatomic and triatomic molecules are not correct as $$\frac 52kT$$ and $$\frac 62kT$$ are equal to the total kinetic energy of a molecule not just the translational part. |
# Monte carlo results becomes more skewed with more sampling
I'm running a monte carlo simulation of a financial instrument, and I'm trying to understand why the final balance of each simulation gives me a skewed distribution rather than a normal distribution (given that I'm sampling out of a normal distribution using np.random.normal).
I assume average monthly return of 1%, standard deviation of 4 and projecting for 60 months. My method is to sample from a normal distribution using np.random.normal and create an array out of that. Then I take a starting balance and multiply that over each element and create a list. I append 500 of these simulations to a dataframe:
df = pd.DataFrame()
for i in range(1, 501): #500 simulations
pct = np.random.normal(loc = 1, scale = 4, size = 60) #mean 1%, std 4%, 60 month period
df['Simulation_' + str(i)] = balance_updater(100000, pct)
where the function balance_updater creates a list that documents every iteration (in this case, starting balance = 100000), coded as the following:
def balance_updater(starting_balance, pct):
balance = starting_balance
balance_list = []
for i in range(len(pct)):
balance = (balance + balance * pct[i] / 100).round(2)
balance_list.append(balance)
return balance_list
Then when I plot via: sns.distplot(df.iloc[-1, bins=30) #Plot final balances only
This is the resulted graph.
I expected this to be normally distributed but we can see it slightly skewed to the right.
What puzzles me more is that if I were to project for a longer period of time (let's say 30 years = 180 months) the results become even more skewed:
I'm trying to understand why a longer projected period produces a more skewed graph, which violates my understanding of the central limit theorem with more sampling (360 vs 60).
Thank you.
It seems that you take the product of normal distributed variables $$X_i \sim N(\mu = 1.01, \sigma = 0.04)$$. This is different from a sum of normal distributed variables, for which it would be reasonable to expect that you get a normal distributed variable.
$$Y_{total} = \prod_{i=1}^n(X_i)$$
But you can end up with a sum term by rewriting the above as:
$$Y_{total} = \exp \left( \sum_{i=1}^n \log(X_i) \right)$$
The distribution of the term $$\sum_{i=1}^n \log(X_i)$$ will approach a normal distribution and what you will end up with is a lognormal distributed variable. The logarithm of your balance is distributed as a sum of many variables, which will approach a normal distribution.
To determine the mean and variance of $$\sum_{i=1}^n \log(X_i)$$ you need to know the mean and variance of $$\log(X_i)$$. You can do this with the Delta method, although there is also a more precise method. In the computation example below the more precise method is used for generating the curve, but the code also contains the computations for the other methods.
### example computation
With this R code you can show that the distribution follows a log normal distribution and you can use a normal distribution (using the central limit theorem) to determine it.
### n = number of months being simulated
n = 180
### product of 180 normal distributed variables
fun <- function() {
prod(rnorm(n,1.01, 0.04))
}
### perform 10 000 times a simulation
x = replicate(10^4, fun())
### plot histogram
hist(x, breaks = seq(0,100,0.3), xlim = c(0,20), freq = FALSE)
### estimate of mu and sigma based on data
mu = mean(test)*n
sig = var(test)^0.5*sqrt(n)
### Delta method
mu = log(1.01)*n
sig = 0.04*sqrt(n)/1.01
### Improved estimate
mu = (log(1.01)-0.5*(0.04/1.01)^2)*n
sig = 0.04*sqrt(n)/1.01
### plot log-normal distribution with estimate of sig and mu
xs = seq(0,20,0.01)
lines(xs, dlnorm(xs,mu,sig))
### Skewness dependency on $$n$$
For increasing $$n$$ the standard deviation of $$\sum_{i=1}^n \log(X_i)$$ will increase, and the skewness (being a function of the standard deviation) will increase as well. Intuitively you can see it as the balance changing each month more in the positive direction than the negative direction (in absolute terms, an increase with 1% is more change than a decrease with 1%), so every month you unevenly stretch/skew the balance.
• Ah I am aware now of the concept of lognormal distributions. Thank you! – Dan Lim Aug 3 at 18:16 |
# PSEB 8th Class Science Solutions Chapter 13 Sound
Punjab State Board PSEB 8th Class Science Book Solutions Chapter 13 Sound Textbook Exercise Questions and Answers.
## PSEB Solutions for Class 8 Science Chapter 13 Sound
PSEB 8th Class Science Guide Sound Textbook Questions and Answers
Exercises
Question 1.
Sound can travel through
(a) gases only
(b) solids only
(c) liquids only
(d) solids, liquids and gases.
(d) solids, liquids and gases.
Question 2.
Voice of which of the following is likely to have minimum frequency ?
(a) Baby girl
(b) Baby boy
(c) A man
(d) A woman.
(c) A man.
Question 3.
In the following statements, tick T against those which are true, and F against those which are false:
(a) Sound cannot travel in vacuum.
True
(b) The number of oscillations per second of a vibrating object is called its time period.
False
(c) If the amplitude of vibration is large, sound is feeble.
False
(d) For human ears, the audible range is 20 Hz to 20,000 Hz
True
(e) The lower the frequency of vibration, the higher is the pitch.
False
(f) Unwanted or unpleasant sound is termed as music.
False
(g) Noise pollution may cause partial hearing impairment.
True
Question 4.
Fill in the blanks with suitable words.
(a) Time taken by an object to complete one oscillation is called ………………….
(b) Loudness is determined by the ………………. of vibration.
(c) The unit of frequency is ……………… .
(d) Unwanted sound is called ………………… .
(e) Shrillness of a sound is determined by the …………………. of the vibration.
(a) Time period,
(b) amplitude,
(c) Hertz (Hz),
(d) noise,
(e) frequency.
Question 5.
A pendulum oscillates 40 times in 4 seconds. Find its time period and frequency.
Number of oscillations
Solution:
Frequency =
= $$\frac{40}{4}$$
= 10 Hz.
Time period = $$\frac{1}{\text { Frequency }}$$
= $$\frac{1}{10}$$
= 0.1 s.
Question 6.
The sound from a mosquito is produced when it vibrates its wings at an average rate of 500 vibrations per second. What is the time period of the vibration ?
Solution:
Frequency 500 vibrations = 500 Hz
Time period = ?
We know, time period = $$\frac{1}{\text { Frequency }}$$
= $$\frac{1}{500}$$
= $$\frac{2 \times 1}{2 \times 500}$$
= $$\frac{2}{1000}$$
= 2 × 10-3 s
Question 7.
Identify the part which vibrates to produce sound in the following instruments ?
(a) Dholak
(b) Sitar
(c) Flute.
Instrument Vibrating part (а) Dholak Stretched membrane (b) Sitar String (c) Flute Air column.
Question 8.
What is difference between noise and music ? Can music become noise sometime ?
Differences between noise and music:
Noise Music 1. It is an unpleasant sound. 1. It is a pleasant sound. 2. It causes discomfort. 2. It has a soothing effect. 3. It can lead to health problems. 3. No health problems are associated with it.
Yes, music can become noise when music is too loud, then it becomes a noise.
Question 9.
List sources of noise pollution in your surroundings.
Sources of noise pollution:
1. Sounds of vehicles.
2. Loudspeakers.
3. Working Machines.
4. Bursting of crackers.
5. Desert coolers.
6. Radios and televisions at high volumes.
7. Kitchen appliances.
8. Hawkers.
Question 10.
Explain in what ways noise pollution is harmful to humans.
Harmful effects of noise pollution:
1. Lack of sleep.
2. Hypertension.
3. Anxiety.
4. Partial deafness.
Question 11.
1. Lot of noise due to passing vehicles.
2. Smoke and dust produced by running vehicles.
3. Sound of loud horns of vehicles at the time of traffic jams.
Question 12.
Sketch larynx and explain its function in your own words.
Function of Larynx.
When air passes through the vocal cords, they produce sound. Vocal cords may become loose/thick or tight/thin on vibration, thus causing different types of voices.
Question 13.
Lightning and thunder take place in the sky at the same time and at the same distance from us. Lightning is seen earlier and thunder is heard later. Can you explain why ?
The speed of light is 3 × 108 m/s while that of sound is only 340 m/s. So, lightning and thunder although taking place simultaneously in the sky at the same distance will be seen and heard at different intervals of time.
PSEB Solutions for Class 8 Science Sound Important Questions and Answers
Multiple Choice Questions
Question 1.
The unit of frequency is:
(a) dB
(6) Hz
(c) dB and Hz
(d) None of these.
(b) Hz.
Question 2.
The intensity of disagreeable sound for human ear is
(a) 60 dB
(b) 10 dB
(c) 90 dB
(d) 30 dB.
(c) 90 dB.
Question 3.
Ultrasonic sound is:
(a) Sound of frequency less than 20 Hz
(b) Sound of frequency more than 20 KHz
(c) Sound of frequency 20 Hz to 20000 Hz
(d) None of these.
(b) Sound of frequency more than 20 KHz.
Question 4.
Speed of sound at 20°C is approximately:
(a) 430 m/s
(b) 304 m/s
(c) 340 m/s
(d) 3400 m/s
(c) 340 m/s
Question 5.
The intensity of sound at ordinary inhale is:
(a) 10 dB
(b) 20 dB
(c) 60 dB
(d) 70 dB.
(a) 10 dB.
Question 1.
What is sound ?
Sound. It is a form of energy which produces in us the sensation of hearing.
Question 2.
How is sound produced ?
Sound is produced by vibrations of a body.
Question 3.
Will sound travel in vacuum ?
No, it will require some medium.
Question 4.
Does sound travel in gases ?
Yes.
Question 5.
Does sound travel in liquids ?
Yes.
Question 6.
Does sound travels in solids ?
Yes.
Question 7.
On what factor does loudness of sound depend ?
Loudness of sound depends upon the amplitude of vibrating body.
Question 8.
If an object makes 10 oscillations in a second, then what is its frequency ?
10 Hertz.
Question 9.
Will the sound travel faster, in wood or water ?
In solids, the sound travels faster than liquids. So sound will travel faster in wood than in water.
Question 10.
Sound is produced when objects …………………. .
Sound is produced when objects vibrate.
Question 11.
The number of oscillations per second is called …………………… .
The number of oscillations per second is called frequency.
Question 12.
We can respond to the frequency of sound more than ……………….. hertz and less than ……………… hertz.
We can respond to the frequency of sound more than 20 hertz and less than 20,000 hertz.
Question 13.
Name the section of throat in which the human voice is produced.
Larynx.
Question 14.
Name the characteristics of sound which help us to distinguish different sounds.
The characteristics of sound which distinguish different sounds. The pitch or loudness of the sound.
Question 15.
What is the audible range of human ear ?
Audible Range. The human ear responds to sounds having frequencies 20 hertz to 20,000 hertz.
Question 16.
Define the term pitch.
Pitch. A sensation depending upon the frequency is known as the pitch.
Question 17.
Which sound has higher frequency ? Sound produced by a buzzing mosquito or sound produced by roaring lion.
The frequency of sound produced by a buzzing mosquito will be more than the frequency of sound produced by a roaring lion.
Question 18.
Write the following frequencies in their increasing order:
(i) Voice of a child
(ii) Voice of man
(iii) Voice of a woman.
Voice in increasing order of their frequencies:
Voice of a man < Voice of a child < Voice of a woman.
Question 19.
How do we hear sound ?
When sound waves travelling through air strike our ears, diaphragm of ear starts vibrating. These vibrations reach the ear nerves by small bones and send the messages to our brain and which we hear.
Question 20.
What is length of vocal cords in man ?
Question 21.
Unpleasant sounds are called ………………. .
Noise.
Question 22.
Which sound is produced by musical instruments ?
Musical sound.
Question 23.
What is unit of loudness of sound ?
Decibel (dB).
Question 24.
Name an instrument with pleasant sound.
Harmonium/Guitar/Piano.
Question 25.
What is hearing impairment ?
The disability to hear sounds is hearing impairment.
Question 26.
What are causes of hearing impairment ?
Ear disease, injury, age and loud noise.
Question 27.
Give an example of loud noise.
1. Sound produced by machinery in a factory.
2. Loudspeakers at full volume.
Question 28.
At which unit, sound becomes harmful ?
More than 80 dB (Average factory range).
Question 29.
Which is the major cause of noise pollution ?
Vehicles.
Question 30.
Which natural organism is important to reduce noise pollution ?
Plants and trees.
Question 31.
What is noise pollution ?
Noise Pollution. Presence of excessive or unpleasant sound in the atmosphere is called, noise pollution.
Question 1.
Define the term time period, frequency and amplitude.
Time Period.
It is the time taken by a vibrating particle to complete one vibration.
Frequency.
The number of oscillations per second is called the frequency of the oscillation. Frequency is measured in hertz [Hz].
Amplitude.
The maximum distance through which a vibrating body is displaced from its central resting position, is called amplitude of oscillations.
Question 2.
One astronaut speaks with another astronaut on Moon. Can the other astronaut listen to the first astronaut ?
There is no atmosphere on the moon, thus, the astronauts cannot listen to each other. Therefore, a material medium is required for the propagation of sound.
Question 3.
Sound produced by a mosquito is quite different from the roar of a lion. Explain.
The loudness of sound depends upon the amplitude of the wave. A mosquito produces sound by the vibration of its wings, in open air while the lion roars by the vibration of its vocal chords. The amplitude of the sound produced by mosquito will be less than the sound produced by a lion. The pitch and quality of the two sounds is quite different, which makes the two sounds different and distinguishable.
Question 4.
Explain with simple experiment to show that sound propagates through solid substance.
Experiment.
Connect two empty match boxes by tying them with the two ends (15 to 20 m long). Request your friend to keep one match box close to his ear. By stretching the string speak into one match box. Sound will be heard very clearly by your friend. This shows that sound can propagate through solids.
Question 5.
How is the human voice produced ?
Production of Human voice.
The human voice is the result of vibrations. It is produced in the larynx, a section of the throat. Muscles in the larynx tighten the vocal cords. Air from the lungs rushes past the tight stretched cords and causes the vocal cords to vibrate. The vibrations produced in the vocal cords produce the sound or the voice.
Question 6.
What are ultrasounds ?
Ultrasound.
Our ear does not respond to sounds of frequencies less than 20 hertz or greater than 20,000 hertz. Sound of frequency greater than 20,000 hertz is called ultrasonic. The instrument used for producing ultrasonic sound is ultrasonic CT.
Question 7.
What are the uses of ultrasonic sound ?
Uses of ultrasonic sound:
1. Dogs can listen ultrasonic sounds. So to call their dogs people use ultrasonic sounds.
2. In medical science, ultrasonics are used for forming the images of internal organs of human body.
Question 8.
A boy claps his hands in front of stair-case and hears a musical sound. Explain.
The distance of each step of stair case increases from the boy. When a boy claps, sound will not strike all the steps simultaneously but it will strike in short and regular intervals. The reflected sound from them will be received by the ear in the form of a number of waves at regular intervals. Thus a periodic disturbance produces a musical sound.
Question 9.
Give an activity to prove that sound travels faster in water than in air.
Sound travels faster in water (liquids) than in air, we can prove this fact by the following activity.
Activity. Take a longer balloon and fill it with water. Hold it close to your ear and scratch it gently with index finger on the farther side of the balloon. A sound will be heard. Repeat this experiment with air-filled balloon. On comparing the two sounds, it is proved that sound travels faster in water than in air.
Question 10.
What is noise ? What is its unit ?
Noise.
The unpleasant sounds which are not soft and sweet are known as noise such as sound of machines, automobiles, crackers, etc.
The unit of voice is Decibel (dB).
Noise level is 0-120 dB.
Question 11.
State one difference between noise and a musical sound.
Noise is a sound, which produces disagreeable (jarring) effect on the ear. On the other hand, musical sound produces pleasing effect on the ear of the listener.
Question 12.
How do children with impaired hearing communicate ?
Children with impaired hearing communicate using sign language and with technically developed devices.
Question 13.
Identify the part which vibrates to produce sound in the following instruments ?
Sr. No. Musical Instrument Sound producing part 1. Flute 2. Dholak
Sr. No. Musical Instrument Sound producing part 1. Flute Air column 2. Dholak Stretched membrane
Question 1.
Give an activity to show that sound needs medium to be heard.
Sound needs medium.
Sound produced by a vibrating object reaches our ears due to vibrations of the molecules of the medium (air) in succession. If there is no air between the vibrating object and our ear, we would not hear any sound at all. We can study this by the following activity.
Activity.
Take a wooden stick and hold one end close to your ear. Ask your friend to scratch the other end gently, you will hear the sound.
This activity shows that sound can travel through wood. Sound can travel through liquids as well. You can check this by filling a balloon with water. Hold a water filled balloon close to you ear and scratch the opposite surface of the balloons with your finger.You will again hear sound. Now do the same experiment with an air-filled balloon. This time the sound heard is very feeble in comparison.
Thus, we conclude that sound requires a medium to be heard.
Question 2.
What is noise pollution ? What are its causes and its effects ?
Noise pollution.
The unwanted sound which is not soft and is disagreeable to the ear, is called noise. The presence of loud sound in atmosphere is only noise pollution.
Causes of noise pollution.
1. The loud sounds produced by machines in factories.
2. Loud speakers.
3. Generators.
4. Railway stations.
5. Air ports.
6. Music programmes.
7. Crackers.
Effects of noise pollution:
1. The worst effect of noise pollution is deafness.
2. It raises the heart beat.
3. It also effects the pupil of eyes, thus causing night blindness or colour blindness.
Question 3.
What is music? Name the various types of vibrating objects used in
different kinds of musical instruments. |
## Precalculus (6th Edition) Blitzer
The difference between the consecutive terms is $-5$
Here, $a_2-a_1=3-8=-5;\\ a_3-a_2=-2-3=-5;\\a_4-a_3=-7-(-2)=-5;\\a_5-a_4=-12-(-7)=-5$ It has been seen that the difference between the consecutive terms is $-5$ |
# Irreducible Representations of Braid Groups of corank two
Research paper by Inna Sysoeva
Indexed on: 07 Mar '00Published on: 07 Mar '00Published in: Mathematics - Group Theory
#### Abstract
This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible representations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations $\rho$ of Artin braid group $B_n$ with the condition $rank (\rho (\sigma_i)-1)=2$ where $\sigma_i$ are the standard generators. For $n \geq 7$ they all belong to some one-parameter family of $n$-dimensional representations. |
Here is an example close to my heart. We show that the second coefficient of the Conway polynomial can be defined by counting quadrisecants. (Collinearities of $4$ points on the knot.)
Indeed, much of classical knot theory is done without resorting to planar diagram calculus. For example, the Alexander polynomial can be calculated from a Seifert surface for a knot, and most properties of the Alexander polynomial are best proved without trying to resort to diagrams. (There are exceptions.) Like Daniel Moskovich alludes to, the Alexander polynomial of a slice knot factorizes as $f(t)f(t^-1)$, and the proof does not have anything to do with diagrams. More modern developments are also often diagram-free. Ozsvath and Szabo's Heegard-Floer homology was originally developed without diagrams. Only later was it figured out how to interpret the theory using planar diagrams, which was regarded as a good thing, since it made the problem of calculating it algorithmic, although incredibly computationally intensive.
Here is an example close to my heart. We show that the second coefficient of the Conway polynomial can be defined by counting quadrisecants. (Collinearities of $4$ points on the knot.)
Indeed, much of classical knot theory is done without resorting to planar diagram calculus. For example, the Alexander polynomial can be calculated from a Seifert surface for a knot, and most properties of the Alexander polynomial are best proved without trying to resort to diagrams. (There are exceptions.) Like Daniel Moskovich alludes to, the Alexander polynomial of a slice knot factorizes as $f(t)f(t^-1)$, and the proof does not have anything to do with diagrams. More modern developments are also often diagram-free. Ozsvath and Szabo's Heegard-Floer homology was originally developed without diagrams. Only later was it figured out how to interpret the theory using planar diagrams, which was regarded as a good thing, since it made the problem of calculating it algorithmic, although incredibly computationally intensive. |
# Length of a line
Geometry Level 3
In the figure above $$ABCD$$ is a square and $$AFB$$ is a right $$\triangle$$. What is the length of $$FE$$ ? If your answer is of the form $$x\sqrt{y}$$ where $$x$$ and $$y$$ are coprime positive integers, give your answer as $$x+y$$.
× |
# Which of the following can be pure substances: mercury, milk, water, a tree, ink, iced tea, ice, carbon?
###### Question:
Which of the following can be pure substances: mercury, milk, water, a tree, ink, iced tea, ice, carbon?
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