Datasets:

OpenCoder-LLM

/

opc-fineweb-math-corpus

like 19

Follow

OpenCoder 70

https://www.worldtransitresearch.info/research/8385/

math

Service operation design in a transit network with congested common lines mode - bus, economics - operating costs, operations - frequency, ridership - behaviour, planning - methods, planning - network design Continuous transit network design, Common line problem, Equilibrium, Surrogate optimization This paper focuses on a transit service operation design problem that primarily determines the optimal frequency settings with explicit consideration of congested common lines in a bus service network. Other than passengers’ transit route choices, the transit service line choices among congested common lines are also specified in the model formulation. A tri-level programming approach is applied to formulate this problem, wherein the upper-level program optimizes the transit frequency to minimize the total operating costs and passengers’ transit costs; the middle-level program describes passengers’ transit routing choices, in which passengers will select a sequence of transfer nodes to minimize their transit costs; and the lower-level program formulates the equilibrium strategy in the common line problem on the route sections (i.e., between two successive transfer nodes), whose equilibrium solution may have multiple strategies depending on the congestion level of the common lines. The tri-level model is then reformulated into a mathematical program with equilibrium constraints. Two solution methods are proposed to solve the problem. One is to transform the model into a mixed-integer linear program so that the global optimal solution of the linearized problem can be guaranteed, and the other employs a surrogate optimization approach to ensure high solution efficiency for large size problems without compromising solution quality. Finally, we conduct extensive numerical examples to demonstrate the validity of our model formulation and the performance of the proposed solution algorithms. Permission to publish the abstract has been given by Elsevier, copyright remains with them. Tian, Q., Wang, D.Z.W., & Lin, Y.H. (2021). Service operation design in a transit network with congested common lines. Transportation Research Part B: Methodological, Vol. 144, pp. 81-102.

s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474808.39/warc/CC-MAIN-20240229103115-20240229133115-00817.warc.gz

CC-MAIN-2024-10

http://quantitativeskills.com/sisa/statistics/rxctablhlp.htm

math

Simple Interactive Statistical Analysis RxC table concerns a basic two dimensional crosstable procedure. The procedure matches the values of two variables and counts the number of occasions that pairs of values occur. It then presents the result in tables and allows for various statistical tests. One case per row individual level data has to be given in two columns, one column for the table rows and one for the table columns. Separators between the two columns can be spaces, returns, semicolons, colons and tabs. Any format within the columns will do. Both numbers, letters and words will be read and classified. Numbers are treated by name, thus 10 and 10.0 are in different categories and 5 is larger than 12. For table input you have to give the number of rows and columns in your table and the table is read unstructured, row after row. The input is presumed to consist of whole counted integer numbers without decimals or scientific notation. Seperators between numbers can be spaces, commas, dots, semicolons, colons, tabs, returns and linefeeds. Show Tables presents the usual cross tables, tables which counts the occurrence of combinations of each row with each column label. Separate tables give the cell, row and column percentages/probabilities of these combinations. List or Flat Table gives in separate rows each unique combination of row and column labels and how often these combinations are counted. In two further columns the cell and column percentages are given. Flat table is the default format for most spreadsheet programs, it forms the basis for the pivot tables in M.S.excel, and it is the mostly preferred format of input for GLM analysis. To do the reverse, change a flat table into a cross table, in SISA or SPSS, enter the flat table (without the sums and totals) into the data input field, two columns of labels followed by a column of counts, and weigh the labels by the counts. For the other orientation, rows first, you have to turn the table. Ordinal pairs form the basis of many analyses of ordinal association, such as Goodman and Kruskal's Gamma and Kendall's Tau. Concordant pairs consist of individuals paired with other individuals who score both lower on the column and lower on the row variable. Discordant pairs consist of individuals paired with other individuals who are lower on the one, and higher on the other variable. Tied pairs are individuals paired with others who have the same score on either the rows or the columns. Chi squares are the usual nominal procedures to determine the likelihood of independence between rows and columns. Goodman and Kruskal's Gamma and Kendall's Tau are based on the ordinal pairs, counted with the option above. You will get the sample standard deviations and p-values for the difference between the observed association and the expected (no) ordinal association of 0 (zero). Gamma is the difference between the number of concordant and discordant pairs divided by the sum of concordant and discordant pairs; Tau-a is the difference between the number of concordant and discordant pairs divided by the total number of pairs. Gamma usually gives a higher value than Tau and is (for other reasons as well) usually considered to be a more satisfactory measure of ordinal association. The p-values are supposed to approach the exact p-value for an ordinal association asymptotically, and the program shows that they generally do that reasonably well. But, beware of small numbers: the p-values for the gamma and Tau become too optimistic! Goodman and Kruskal’s Lambda is an example of a Proportional Reduction in Error (PRE) measure. PRE measures work by taking the ratio of: 1) an error score in predicting someone’s most likely position (in a table) using relatively little information; with: 2) the error score after collecting more information. In the case of Lambda we compare the error made when we only have knowledge of the marginal with the reduced error after we have collected information regarding the inside of the table. Two Lambda’s are produced. First, how much better are we able to predict someone’s position on the column marginal: Lambda A. Second, how much better are we able to predict someone’s position on the row marginal: Lambda B. The program gives the proportional improvement in predicting someone’s score after collecting additional information. To guess the name of a man on the basis of the weighted table below, and the only information we have is the distribution of mans' names in the sample, we would guess John, with a (38+34)/113*100=63.7% chance of an erroneous guess. However, if we know the name of mans' partners, we would guess John if the partner is Liz, with a (10+8)/41*100=43.9% chance of an error, Peter if the partner is Mary (44.7% errors), Steve if the partner is Linda (58.8%). The average reduction in errors in the row marginal, weighted by cell size (Lambda-B), equals 23.6%, the average weighted error rate in guessing a man's name after knowing the women's name, equals 63.7*(1-0.236)=48.7%. This 48.7% can also be calculated as: (10+8+6+11+8+12)/113. With a p-value of 0.00668 we significantly improve our probability of guessing a man's name correctly, after considering the name of the man's partner. Same for guessing a woman's name, only now you have to use the Lambda–A. Lambda is always positive, and the significance test always single sided, because information on the inside of the table will always lead to an improvement compared with knowing only the marginal. Cohen's Kappa is a measure of agreement and takes on the value zero when there are no more cells on the diagonal of an agreement table than can be expected on the basis of chance. Kappa takes on the value 1 if there is perfect agreement, i.e. if all observations are on the diagonal and a row score is perfectly predictive of a column score. It is considered that Kappa values lower than 0.4 represent poor agreement between row and column variable, values between 0.4 and 0.75 fair to good agreement, and values higher than 0.75 excellent agreement. Kappa only works in square tables. Bowker Chi-square tests to see if there is a difference in the scoring pattern between the upper and the lower triangle (excluding the diagonal) of a table. Each cell in the upper triangle is compared with its mirror in the lower triangle, the difference between the two cells is Chi-squared and summed. If cell i,j equals cell j,i the contribution of this comparison to the Bowker Chi square is zero. If the Bowker Chi-square is statistically significant the pattern of scoring in the upper triangle is different from the scoring in the lower triangle beyond chance. Note that the pattern of scoring between the two triangles is dependent on two factors. First, whether there is a 'true' difference in the pattern of scoring. Second, the level of marginal heterogeneity. Marginal heterogeneity means that the marginals are different; this increases the Bowker Chi-square. The Bowker Chi-square is the same as the McNemar Chi-square in a two by two table. Bowker Chi-square only works in square tables. For Read weights a third column is added in the data input field and the third value is the case weight of the previous two values. The case weights in the third column must be numerical, if not the case including its previous two values is ignored. Weighted cross tables are produced and a weighting corrected Chi-square is presented . For a discussion of data weighting and the correction applied please read this paper. Lowercase All. Lowercase all non numerical text characters for both the table rows and columns. Use this option if you want to categorize text data case insensitive. Transpose/Turn Table. Change the rows into columns and the columns into rows. Sort Descending. Sorts the values descending. Separate for rows and columns. Show Rows or Columns limits the number of rows displayed. Particularly relevant if you request a large Table. Can also be used to exclude particularly high or low (after "Sort Descending") (missing) values from the analysis. Solve problems into 99999.9. Change the data sequence -carriage return-line feed-tab- and the sequence -tab-carriage return-line feed- into 99999.9 if labels or delete the case if weigths. Wil mostly solve the problem of system missing values in data copied and pasted from SPSS. Might cause other problems. If you copy and paste the following data into the input field: You get the following table: |Table of Counts| Pearson: 3.111 (p= 0.53941). There is no statistically significant relationship between between boys names and girls names, although this conclusion has to be viewed with care as the table is based on very few observations. You could count (in a flat table) how often each of the pairs of names occurs in a sample, and weigh each of the pairs with these counts. john linda 10 john liz 23 john mary 8 peter linda 6 peter liz 11 peter mary 21 steve linda 14 steve liz 8 steve mary 12 And you get the following table: |Table of Weights| Weighted Pearson: 17.77 (p= 0.00137). After considering how often pairs of names occur in a sample there is a highly significant relation between certain boys and certain girls names. The formatting and tabulating of large data sets might take a while in which case there might be warnings, just select "continue" and in the end the computer will get there. The procedure is meant for relatively small tables. Number of cells is in principle limited to 120, but might be less dependent on your browser and other settings. Is also rather less with weighted data as more info has to be transferred. All software and text copyright by SISA

s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948609.41/warc/CC-MAIN-20230327060940-20230327090940-00478.warc.gz

CC-MAIN-2023-14

http://www.cincinnatipersonalinjury.com/1784/questions-answers/

math

If you have a question relating to an accident where you were not at fault and it resulted in injuries, I have answered many of the common questions I have been asked over the years. Just below, you’ll see a number of categories where the questions and answers are listed. You can click one of the categories that applies to your situation, and all the answers will be presented to you. An even easier way to get an answer to your question is to type the question in the search box to the right…. and every answer that applies to your question will be shown. If by chance I don’t have the answer to your question on this site, call me at 513-621-4775. The call is free and I’ll take the time necessary to answer your question.

s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687324.6/warc/CC-MAIN-20170920142244-20170920162244-00002.warc.gz

CC-MAIN-2017-39

http://indiavidya.com/junior-inter-physics-long-answer-important-questions-for-public-exams/

math

IPE Overall Strategy: The total syllabus of 1st year can be divided into 3 parts. First, the student has to concentrate on Long Answer Questions (LAQ) i.e. Section-C of question paper. We can expect one question from each part of above 3 parts. From part 1, the important questions may be like.. i) State and prove conservation of energy ii) Newton's laws and their applications iii) Conservation of Angular momentum and linear momentum. From part 2, the important questions may be like.. i) Time period simple pendulum derivation ii) Gravitational potential energy, variation of g, determination of universal gravitational constant G iii) Determination of Young's modulus, Bernoulli's theorem, Viscosity. From part 3, i) Newton's laws of cooling ii) Carnot Engine & refrigerator iii) Pressure of an ideal gas. Also Read: NEET Physics Model Questions with Answers Important Long Answer Questions PART - I 1) a) State newton's second law of motion. Hence derive the equation of motion F = ma from it. b) A body is moving along a circular path such that its speed always remains constant. Should there be a force acting on the body? 2) Define angle of friction and angle of repose. Show that angle of friction is equal to angle of repose for a rough inclined plane. A block of repose for a rough inclined plane. A block of mass 4 kg is resting on a rough horizontal plane and is about to move when a horizontal force of 30 N is applied on it. If g = 10 ms−2. Find the total contact force exerted by the plane on the block. 3) Develop the notions of work and kinetic energy and show that it leads to work-energy theorem. 4) What are collisions? Explain the possible types of collisions? Develop the theory of one dimensional elastic collision. 5) State the law of conservation of energy and verify it in case of a freely falling body. What are the conditions under which the law of conservation of energy is applicable? 6) a) State and prove parallel axes theorem. b) For a thin flat circular disk, the radius of gyration about a diameter as axis is k. If the disk is cut along a diameter AB as shown in to two equal pieces, then find the radius of gyration of each piece about AB. 7) a) State and prove perpendicular axes theorem. b) If a thin circular ring and a thin flat circular disk of same mass have same moment of inertia about their respective diameters as axes. Then find the ratio of their radii. 8) State and prove the principle of conservation of angular momentum. Explain the principle of conservation of angular momentum with examples. PART - II 1) Define simple harmonic motion. Show that the motion of (point) projection of a particle performing uniform circular motion, on any diameter, is simple harmonic. 2) Show that the motion of a simple pendulum is simple harmonic and hence derive an equation for its time period. What is seconds pendulum? 3) State Bernoulli's principle. From conservation of energy in a fluid flow through a tube, arrive at Bernoulli's equation. Give an application of Bernoulli's theorem. 4) Define coefficient of viscosity. Explain Stoke's law and explain the conditions under which a rain drop attains terminal velocity 'v1'. Give the expression for 'v1'. PART - III 1) State Boyle's law and Charle's law. Hence, derive ideal gas equation. Which of the two laws is better for the purpose of thermometry and why? 2) Explain thermal conductivity and coefficient of thermal conductivity. A copper bar of thermal conductivity 401 W/mK has one end at 104°C and the other end at 24°C. The length of the bar is 0.10 m and the cross-sectional area is 1.0×10−6 m2. What is the rate of heat conduction, along the bar? 3) State and explain Newton's law of cooling. State the conditions under which Newton's law of cooling is applicable. A body cools down from 60°C to 50°C in 5 minutes and to 40°C in another 8 minutes. Find the temperature of the surroundings. 4) Explain reversible and irreversible processes. Describe the working of Cornot engine. Obtain an expression for the efficiency. 5) State second law of thermodynamics. How is heat engine different from a refrigerator? 6) Derive an expression for the pressure of an ideal gas in a container from Kinetic Theory and hence give Kinetic Interpretation of Temperature.

s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948568283.66/warc/CC-MAIN-20171215095015-20171215115015-00704.warc.gz

CC-MAIN-2017-51

https://www.internet4classrooms.com/common_core/explain_each_step_solving_simple_equation_reasoning_with_equations_and_inequalities_high_school_algebra_math_mathematics.htm

math

CCSS.Math.Content.HSA.REI.A.1 - Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Authors: National Governors Association Center for Best Practices, Council of Chief State School Officers Title: CCSS.Math.Content.HSA.REI.A.1 Explain Each Step In Solving A Simple Equation... Reasoning with Equations & Inequalities - High School Algebra Mathematics Common Core State Standards Publisher: National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C. Copyright Date: 2010 (Page last edited 10/20/2014) - Algebra I Graphing Inequalities - graphing inequalities and testing assertion - Explain Why the x Coordinates of the Points Where the Graphs of the Equations y = f(x) and y = g(x) Intersect Are the Solutions of the Equation f(x) = g(x). - Students should understand that an equation and its graph are just two different representations of the same thing. The graph of the line or curve of a two-variable equation shows in visual form all of the solutions (infinite as they may be) to our equation in written form. When two equations are set to equal one another, their solution is the point at which graphically they intersect one another. Depending on the equations (and the alignment of the planets), there might be one solution, or more, or none at all. A quiz is provided. - Graph the Solutions to a Linear Inequality in Two Variables and Graph the Solution Set to a System of Linear Inequalities in Two Variables - All this is asking us to do is what we already know from the previous standards, plus one simple step. Students should know how to graph a linear inequality. A linear inequality is the same as a linear equation, but instead of an equal sign, we'll have to use the inequality signs (like ?, ?, <, and >). Because we're graphing an inequality and our linear equation is with a different sign now, it'll be shaded above or below the line as part of our solution. If the inequality is greater than or greater than or equal to (using either > or ?), then we shade the upper half of the graph. If the inequality if less than or less than or equal to (using either < or ?), then we shade the lower half of the graph. A quiz is provided. - Graphing Inequalities - Video lesson on Graphing Inequalities - Graphing Inequalities - Video lesson including tips - Graphing Linear Inequalities - Graphing Linear Inequalities in Two Variables - Inequalities - Graph a linear inequality in two variables - Problem solving - Weighted averages: word problems - Properties - Properties of equality - Simple Logical Arguments - simple logical reasoning - Simplifying Expressions - simplifying expressions and word problems - Solving and Graphing Linear Equations - Solving and graphing linear inequalities in two variables - Solving and Graphing Linear Equations - Solving and Graphing Linear Equations: Example 2 - Solving and Graphing Linear Equations: 2 - Solving and Graphing Linear Equations - Systems of linear equations: Graphing - Solve a system of equations by graphing: word problems - Systems of linear equations: Substitution - Solve a system of equations using substitution: word problems

s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474659.73/warc/CC-MAIN-20240226094435-20240226124435-00371.warc.gz

CC-MAIN-2024-10

http://mathhelpforum.com/advanced-applied-math/95775-formulate-linear-programming-problem-print.html

math

Formulate as a Linear programming problem I really need help understanding how to formulate into a LP. here is one of the problems i'm struggling with: A company makes three different products: product A; product B; product C. Each product requires a piece of metal of the size: ∙ 90cm×3m for product A; ∙ 70cm×3m for product B; and ∙ 50cm×3m for product C. The company receives metal sheets with size 2m×3m, which needs to be cut into smaller pieces above. A large order has come in and the company needs to make at least ∙ 300 pieces of product A; ∙ 400 pieces of product B; and ∙ 1000 pieces of product C. The company wants to find out how to cut up the metal sheets so as to minimize waste. (a) There are 6 ways to cut a 2m×3m metal sheet into pieces of sizes 90cm×3m, 70cm×3m and 50cm×3m with a waste having the shorter side smaller than 50cm (so that no other pieces can cut out of the waste). What are they and how much metal does each one waste (list them in the order of most waste to least)? (b) Each of the ways of cutting a metal sheet wastes a certain amount of metal. Obviously we would like to minimize this waste while still producing enough products A, B and C. The cutting machine requires that there is some waste left after the cutting. This leaves only five cutting options. Write this as a linear programming problem. Hint. Use variables x1, . . . , x5, where xi denotes the number metal sheets cut using option i. Your goal is to satisfy the production requirements while minimizing the waste. I'd appreciate any help regarding this problem. @Thine Blood and Captain Black Thanks for sharing the procedure of the LPP's formulation; It really helped:

s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121865.67/warc/CC-MAIN-20170423031201-00504-ip-10-145-167-34.ec2.internal.warc.gz

CC-MAIN-2017-17

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3376765

math

Numerical Approach Determining the Optimal Distance Separating from Two Electric Cables 11 Pages Posted: 24 Apr 2019 Last revised: 31 May 2019 Date Written: April 3, 2019 In this work we present a two-dimensional numerical study of laminar natural convection in a square cavity, containing two circular-shaped heat sources that represent two electric cables in which an electric current produces a Joule effect heat release. These cables are at the edge of an AIRBUS aircraft and their excessive heating can cause material and human damage. The objective of the study, is to determine the optimal distance between these two sources to avoid excessive heating of the installation. Different cases have been examined by varying the distance between the two heat sources and their angle of inclination and this for different values of the Rayleigh number. The governing equations were numerically solved by a FORTRAN calculation code based on the finite volume method. The results obtained show that the heat transfer in the cavity is affected mainly by the spacing between the two heat sources and the variation of the Rayleigh number. Indeed, for the horizontal position (α = 0°), heat sources we note the heat transfer in the cavity is maximum (Nua = 14.69), for a distance between the heat sources equal to 0.7 and a value of Rayleigh 106. In the case where the heat sources are inclined (α = 45°), the heat transfer is better (Numoy = 20.91) for a distance between the two sources equal to 0.4 and a value of Rayleigh 105. In the vertical position of heat sources (α = 90°), there is a maximum heat transfer (Numoy = 47.64) for a distance between the heat sources equal to 0.7 and a Rayleigh value equal to 106. In conclusion we can say that the optimal distance between the two heat sources and which was the object of this study is reached in the case where the sources are in vertical position (α = 90°), and spaced from a distance (d) equal to 0.7. (a) α = 0°, d = 0.7 (b) α = 45°, d = 0.4 (c) α = 90°, d = 0.7. Keywords: natural convection, heat source, optimal distance, finite volume Suggested Citation: Suggested Citation

s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402131412.93/warc/CC-MAIN-20201001112433-20201001142433-00121.warc.gz

CC-MAIN-2020-40

https://www.accountingdetails.com/concept_of_present_value.htm

math

Present Value and Future Value – Explanation of the Concept: - Understand present value concepts and the use of present value tables. - Compute the present value of a single sum and a series of cash flows. A dollar received now is more valuable than a dollar received a year from now for the simple reason that if you have a dollar today, you can put it in the bank an have more than a dollar a year from now. Since dollars today are worth more than dollars in the future, we need some means of weighing cash flows that are received at different times so that they can be compared. Mathematics provides us with the means of making such comparisons. With a few simple calculations, we can adjust the value of a dollar received any number of years from now so that it can be compared with the value of a dollar in hand today. The Mathematics of Interest: If a bank pays 5% interest, than a deposit of $100 today will be worth $105 one year from now. This can be expressed in mathematical terms by means of the following formula or equation: Formula or Equation: F1 = P ( 1 + r ) Where: F1 = the balance at the end of one period, P = the amount invested now, and r = the rate of interest per period. If the investment made now is $100 deposited in a bank saving account that is to earn interest at 5%, than P = $100 and r = 0.05. Under these conditions, F1 = $105, the amount to be received in one year. The $100 present outlay is called the present value of the $105 amount to be received in one year. It is also known as the discounted value of the future $105 receipt. The $100 figure represents the value in present terms of $105 to be received a year from now when the interest rate is 5%. Compound Interest: When if the $105 is left in the bank for a second year? In that case, by the end of the second year the original $100 deposit will have grown to $110.25: |Interest for the first year ($100 × 0.05) |Balance at the end of the first year |Interest for the second year ($105 × 0.05) |Balance at the end of the second year Notice that the interest for the second year is $5.25, as compared to only $5.00 for the first year. The reason for the greater interest earned during the second year is that during second, interest is being paid on interest. That is, the $5.00 interest earned during the first year has been left in the account and has been added to the original $100 deposit when computing interest for the second year. This is known as the compound interest. In this case, the compound is annual. Interest compounded on a semiannual, quarterly, monthly, or even more frequent basis. The more frequently compounding is done, the more rapidly the balance will grow. We can determine the balance in an account after n periods of compounding using the following formula or equation: Fn = P (1 = r)n (1) Where n = number of periods. If n = 2 years and the interest rate is 5% per year, then the balance in two years will be as follows: F2 = $100 ( 1 + 0.05 )2 F2 = $110.25 Computation of Present Value: An investment can be viewed in two ways. It can be viewed either in terms of its future value or in terms of its present value. We have seen from our computations above that if we know the present value of a sum (such as $100 deposit), it is a relatively simple task to compute the sum’s future value in n years by using equation Fn = P (1 = r)n. But what if the the tables are reversed and we know the future value of some amount but we do not know its present value? For example, assume that you are to receive $200 two years from now. You know that the future value of this sum is $200, since this is the amount that you will be receiving after two years. But what is the sum’s present value – what is it worth right now? The present value of any sum to be received in the future can be computed by turning equation Fn = P (1 = r)n. around and solving for P: P = Fn / ( 1 + r )n (2) In our example, F = $200 (the amount to be received in future), r = 0.05 (the annual rate of interest), and n=2 (the number of years in the future that the amount is to be received) P = $200 / (1 + 0.05)n P = $200 / (1 + 0.05)2 P = $200 / 1.1025 P = $181.40 As shown by the computation above, the present value of a $200 amount to be received two years from now is $181.40 if the interest rate is 5%. In effect, $181.40 received right now is equivalent to $200 received two years from now if the rate of return is 5%. The $181.40 and the $200 are just two ways of looking at the same thing. The process of finding the present value of a future cash flow, which we have just completed, is called discounting. We have discounted the $200 to its present value of $181.40 The 5% interest figure that we have used to find this present value is called the discount rate. Discounting future sums to their present value is a common practice in business, particularly in capital budgeting decisions. If you have a power key (yx) on your calculator, the above calculations are fairly easy. However, some of the present value formulas will be using are more complex and difficult to use. Fortunately, tables are available in which many of the calculations have already been done for you. For example, Table 3 at Future Value and Present Value Tables page shows the discounted present value of $1 to be received two periods from now at 5% is 0.907. Since in our example we want to know the present value of $200 rather than just $1, we need multiply the factor in the table by $200: $200 × 0.907 = $181.40 This answer is the same as we obtained earlier using the formula in equation (2). Present Value of a Series of Cash Flow: Although some investments involve a single sum to be received (or paid) at a single point in the future, other investments involve a series of cash flows. A series (or stream) of identical cash flows is known as an annuity. To provide an example, assume that a firm has just purchased some government bonds in order to temporarily invest funds that are being held for future plant expansion. The bonds will yield interest of $15,000 each year and will be held for five years. What is the present value of the stream in interest receipts from the bonds? As shown from the following calculations the present value of this stream is $54,075 if we assume a discount rate of 12% compounded annually. ||Factor at 12% (Future Value and Present Value Tables-Table 3) | Interest Received || Present Value The discount factors used in this calculation have been taken from Future Value and Present Value Table – Table 3. Two points are important in connection with this computation. . First, notice that the present value of the $15,000 received a year from now is $13,395, as compared to only $8,505 for the $15,000 interest payment to be received five years from now. This point simply underscores the fact that money has a time value. The second point is that the computations involved above involve unnecessary work. The same present value of $54,075 could have been obtained more easily by referring to Table 4 at Future Value and Present Value Table. Table 4 contains the present value of $1 to be received each year over a series of years at various interest rates. This table have been derived by simply adding together the factor from Table 3 as follows: The sum of the five factors above is 3.065. Notice from the Table 4 at Future Value and Present Value Tables Page that the factor of $1 to be received each year for five years at 12% is also 3.605. If we use this factor and multiply it by the $15,000 annual cash inflow, then we get the same $54075 present value that we obtained earlier. $15,000 × 3.605 = $54,075 Therefore, when computing the present value of a series (or stream) of equal cash flows that begins at the end of period 1, Table 4 should be used. To summarize, the the present value tables, at Future Value and Present Value Tables Page, should be used as follows: Table 3: This table should be used to find the present value of a single cash flow (such as a single payment or receipt) occurring in future. Table 4: This table should be used to find the present value of a series (or stream) of identical cash flows beginning at the end of the current period and continuing into the future. You may also be interested in other articles from “capital budgeting decisions” chapter: - Capital Budgeting – Definition and Explanation - Typical Capital Budgeting Decisions - Time Value of Money - Screening and Preference Decisions - Present Value and Future Value – Explanation of the Concept - Net Present Value (NPV) Method in Capital Budgeting Decisions - Internal Rate of Return (IRR) Method – Definition and Explanation - Net Present Value (NPV) Method Vs Internal Rate of Return (IRR) Method - Net Present Value (NPV) Method – Comparing the Competing Investment Projects - Least Cost Decisions - Capital Budgeting Decisions With Uncertain Cash Flows - Ranking Investment Projects - Payback Period Method for Capital Budgeting Decisions - Simple rate of Return Method Post Audit of Investment Projects - Inflation and Capital Budgeting Analysis - Income Taxes in Capital Budgeting Decisions - Review Problem 1: Basic Present Value Computations - Review Problem 2: Comparison of Capital Budgeting Methods - Future Value and Present Value Tables

s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710691.77/warc/CC-MAIN-20221129100233-20221129130233-00277.warc.gz

CC-MAIN-2022-49

https://oilandgasguru.com/how-much-power-does-an-electric-car-use

math

Top best answers to the question «How much power does an electric car use» - Your electric car requires 30 kWhs to go 100 miles on a fully charged battery. That would mean it costs $3.60 to charge a depleted battery, which works out to be $0.036 per mile or roughly 1/3 kilowatt-hour per mile (3.3 miles per kWh). But that's not the end of the calculation. - Going that many miles would require 341 kWhs for an EV that gets 3.3 miles per kWh. In this example, the electric car uses 341 kWhs a month for a total cost of $41 in electricity. Of course, this is a very straightforward example. In the real world, there are a lot of variables that can affect the electricity usage and rate that is paid. 6 other answers The consumption of electric cars depends on the model and the manufacturer. Most of the ... An average electric car consumes approximately 0,20 kWh/km. On favourable weather conditions the consumption can be even 0,15 kWh or less, but year-around average in most countries is closer to 0,2 kilowatthours. And what are the average power requirements for the average motor that is used in a converted electric vehicle? Hi, Aaron - In DC electric car conversions, with very simple inexpensive controllers, 6v batteries conserve battery power better than higher voltage batteries, so your range ends up being longer with 6v batteries. In contract, most vehicle manufacturers limit the current drawn from a standard domestic 3 pin socket to 10A or less, which equates to a maximum of 2.3kW. A 7kW home charger therefore delivers approximately three times as much power and is approximately three times as fast as using a domestic socket. Most electric cars can be plugged into a regular (120-volt) outlet, but you'll be much better off using a higher-voltage charger because your charging rate will be woefully slow. For example, a... TIP: click on a vehicle to show full data. Select a cheatsheet: Acceleration Battery Useable Energy Consumption Range Top Speed Towing Weight. Average. 195 Wh/km. Lightyear One. 104. Tesla Model 3 Standard Range Plus. 147.

s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335469.40/warc/CC-MAIN-20220930113830-20220930143830-00494.warc.gz

CC-MAIN-2022-40

http://www.thefullwiki.org/Displacement_(vector)

math

A displacement is a relative motion between two points independent of the path taken. The displacement is thus distinct from the distance traveled by the object along given path. The displacement vector then defines the motion in terms of translation along a straight line. We may compose these acts of motion by adding the displacement vectors. More precisely we define the addition of displacement vectors as the composition of their actions "move this way and then move that way". A position vector expresses the position at a point in space in terms of displacement from a fixed origin. Namely, it indicates both the distance and direction of a point from the reference position (origin). In considering motions of objects over time the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The velocity then is distinct from the instantaneous speed which is the time rate of change of the distance traveled along a specific path. If a fixed origin is defined we may then equivalently define the velocity as the time rate of change of the position vector. However if one considers a time dependent choice of origin as in a moving coordinate system the rate of change of the position vector only defines a relative velocity. For motion over a given interval of time the net displacement divided by the length of the time interval defines the average velocity. Given an origin point defining position vectors then the net displacement vector is the difference between the final and initial position vectors. This difference, divided by the time needed to perform the motion, then also gives the average velocity of the point or particle. In dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement (displacement along a line), while the rotation is called angular displacement. If the position of an object is described by a vector function then the distance traveled as a function of t is described by the integral of one with respect to arc length. The arc length differential is described by the following equation: On a graph representing the position of a particle with respect to time (position vs. time graph), the slope of the straight line joining two points on the graph is the average velocity of the particle during the corresponding time interval, while the slope of the tangent to the graph at a given point is the instantaneous velocity at the corresponding time (first derivative of the particle position). To calculate displacement all vectors and scalars must be taken into consideration. The following formulas can be used to calculate displacement for, s, for an object undergoing constant acceleration.. Height displacement is the distance an object peaks in height vertically. If, for example, a ball was thrown up in the air and fell back into the owner's hand, the displacement would be zero, since displacement over a period of time is defined as the distance between an object's starting and finishing points. However one may use the general equation to calculate overall vertical height. This is modified to for the case of a ball in the presence of gravity. The height h is dependent upon the time t at which it is being measured. g is the acceleration caused by Earth's gravity; it stays constant at approximately . The term is preceded by a minus sign because gravity acts in the opposite direction of h and u, which signify a distance and speed, respectively, away from the Earth's center of mass.

s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607648.39/warc/CC-MAIN-20170523163634-20170523183634-00270.warc.gz

CC-MAIN-2017-22

https://www.teacherspayteachers.com/Product/Multiplication-Division-Unit-for-Grade-2-Ontario-Curriculum-3095007

math

This Grade 2 Multiplication & Division unit contains lesson ideas, worksheets, task cards, and a test based on the Ontario Curriculum Expectations. Adding & Multiplying #1 Adding & Multiplying #2 Adding & Multiplying #3 Write the Multiplication Sentence Draw Equal Groups Multiplication as Repeated Addition Multiplication Task Cards Math Centre (12 cards) Dividing Equally #1 Dividing Equally #2 Dividing Equally #3 Division Math Centre (12 cards) For a better deal, purchase this resource as part of my Grade 2 Math Units FULL YEAR BUNDLE , which covers every unit and expectation for Grade 2 math for the entire year!

s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818686043.16/warc/CC-MAIN-20170919221032-20170920001032-00583.warc.gz

CC-MAIN-2017-39

http://ifeelinfinite.net/15a88f9e846534e706c86e035997b278

math

High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,...System of Equations Calculator - MathPapa Wolfram|Alpha is a great tool for finding polynomial roots and solving systems of equations. It also factors polynomials, plots polynomial solution sets and inequalities and more.Math Equation Solver - Calculator Soup To solve your equation using the Equation Solver, type in your equation like x+4=5. The solver will then show you the steps to help you learn how to solve it on your own. Solving Equations Video Lesson Khan Academy Video: Solving Simple EquationsMath Problem Solver and Calculator | Chegg.com Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE Last post, we talked about linear first order differential equations. In this post, we will talk about separable...Simultaneous Equations Solver - eMathHelp Free quadratic equation calculator - Solve quadratic equations using factoring, complete the square and the quadratic formula step-by-step ... High School Math Solutions – Quadratic Equations Calculator, Part 2. Solving quadratics by factorizing (link to previous post) usually works just fine. But what if the quadratic equation...Mathway | Algebra Problem Solver How to Use the Calculator. Type your algebra problem into the text box. For example, enter 3x+2=14 into the text box to get a step-by-step explanation of how to solve 3x+2=14.. Try this example now! »Equation calculator (linear, quadratic, cubic, linear ... Differential Equation Calculator The calculator will find the solution of the given ODE: first-order, second-order, nth-order, separable, linear, exact, Bernoulli, homogeneous, or inhomogeneous. Initial conditions are also supported.Integer Equation calculator (linear, quadratic, cubic ... Get the free "General Differential Equation Solver" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.Microsoft Math Solver - Math Problem Solver & Calculator Sofsource.com makes available essential advice on ordered pair solution equation calculator, intermediate algebra syllabus and geometry and other algebra topics. Should you require advice on a polynomial as well as systems of linear equations, Sofsource.com is going to be the ideal destination to check out!Solving of differential equations online for free Radical Equation Solver. Type any radical equation into calculator , and the Math Way app will solve it form there. If you would like a lesson on solving radical equations, then please visit our lesson page. To read our review of the Math Way -- which is what fuels this page's calculator, please go here.Inequality Calculator - MathPapa The equation solver allows to solve equations with an unknown with calculation steps : linear equation, quadratic equation, logarithmic equation, differential equation. Syntax : equation_solver(equation;variable), variable parameter may be omitted when there is no ambiguity. Examples : Equation resolution of first degree. equation_solver(`3*x-9 ...Simultaneous Equations Calculator With Steps When you enter an equation into the calculator, the calculator will begin by expanding (simplifying) the problem. Then it will attempt to solve the equation by using one or more of the following: addition, subtraction, division, taking the square root of each side, factoring, and completing the square.Trigonometric Equations Calculator & Solver - Snapxam Equivalent equations are equations that have identical solutions. Thus, 3x + 3 = x + 13, 3x = x + 10, 2x = 10, and x = 5. are equivalent equations, because 5 is the only solution of each of them. Notice in the equation 3x + 3 = x + 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection.Graphing Equations Using Algebra Calculator - MathPapa The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant (b 2 − 4 a c) is less than, greater than or equal to 0. When b 2 − 4 a c = 0 there is one real root. When b 2 − 4 a c > 0 there are two real roots.Calculus Calculator | Microsoft Math Solver QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices.Separable differential equations Calculator & Solver - Snapxam Calculator Use. Use this calculator to solve polynomial equations with an order of 3 such as ax 3 + bx 2 + cx + d = 0 for x including complex solutions.. Enter values for a, b, c and d and solutions for x will be calculated.Differential Equation Calculator - Free Online Calculator Find the value of X, Y and Z calculator to solve the 3 unknown variables X, Y and Z in a set of 3 equations. Each equation has containing the unknown variables X, Y and Z. This 3 equations 3 unknown variables solver computes the output value of the variables X and Y with respect to the input values of X, Y and Z coefficients.Equation Calculator - Free Online Calculator Calculates the solution of a system of two linear equations in two variables and draws the chart. System of 2 linear equations in 2 variables Calculator - High accuracy calculation Welcome, GuestWolfram|Alpha Widgets: "3 Equation System Solver" - Free ... Limit size of fractional solutions to digits in numerator or denominator. CowPi › Math › System Solver › 5×5 › Math › System Solver › 5×5Equation Solver - Free Online Math Equation Calculator Solving systems of linear equations. This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule.Also you can compute a number of solutions in a system of linear equations (analyse the compatibility) using Rouché–Capelli theorem.. Enter coefficients of your system into the input fields.System of 2 linear equations in 2 variables Calculator ... Here you can solve systems of simultaneous linear equations using Cramer's Rule Calculator with complex numbers online for free with a very detailed solution. The key feature of our calculator is that each determinant can be calculated apart and you can also check the exact type of matrix if the determinant of the main matrix is zero.OnSolver.com - Solving mathematical problems online Calculates the solution of simultaneous linear equations with n variables. Variable are allowed input of complex numbers.Solution Dilution Calculator | Sigma-Aldrich Voted as Best Calculator: Percentage Calculator Email . Print . System of Equations Solver. System of equations solver. Solve system of equations, no matter how complicated it is and find all the solutions. Input equations here, in square brackets, separated by commas (","): Equations: ... Solutions Of Equations Calculator The most popular ebook you must read is Solutions Of Equations Calculator. I am sure you will love the Solutions Of Equations Calculator. You can download it to your laptop through easy steps.

s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655879738.16/warc/CC-MAIN-20200702174127-20200702204127-00249.warc.gz

CC-MAIN-2020-29

https://slopeinterceptform.net/whats-the-slope-intercept-form/

math

The Definition, Formula, and Problem Example of the Slope-Intercept Form What’s The Slope Intercept Form – There are many forms used to represent a linear equation the one most frequently seen is the slope intercept form. You may use the formula of the slope-intercept to identify a line equation when that you have the straight line’s slope as well as the y-intercept. This is the point’s y-coordinate at which the y-axis is intersected by the line. Read more about this particular line equation form below. What Is The Slope Intercept Form? There are three basic forms of linear equations: the traditional, slope-intercept, and point-slope. Even though they can provide the same results , when used however, you can get the information line produced faster with the slope intercept form. It is a form that, as the name suggests, this form utilizes a sloped line in which its “steepness” of the line reflects its value. This formula can be used to calculate the slope of straight lines, the y-intercept, also known as x-intercept where you can utilize a variety formulas available. The equation for this line in this formula is y = mx + b. The straight line’s slope is signified in the form of “m”, while its y-intercept is indicated by “b”. Each point of the straight line can be represented using an (x, y). Note that in the y = mx + b equation formula the “x” and the “y” are treated as variables. An Example of Applied Slope Intercept Form in Problems The real-world in the real world, the slope-intercept form is often utilized to depict how an object or issue evolves over the course of time. The value of the vertical axis indicates how the equation tackles the extent of changes over the value provided via the horizontal axis (typically times). A simple example of using this formula is to figure out how many people live in a specific area as the years pass by. Based on the assumption that the population in the area grows each year by a fixed amount, the point worth of horizontal scale increases one point at a time with each passing year and the point value of the vertical axis will rise to represent the growing population by the fixed amount. You can also note the beginning point of a particular problem. The beginning value is at the y-value of the y-intercept. The Y-intercept is the place at which x equals zero. By using the example of the above problem, the starting value would be at the time the population reading starts or when the time tracking begins along with the associated changes. The y-intercept, then, is the place where the population starts to be monitored to the researchers. Let’s say that the researcher is beginning with the calculation or take measurements in 1995. This year will represent”the “base” year, and the x = 0 points would occur in the year 1995. Thus, you could say that the 1995 population will be the “y-intercept. Linear equation problems that use straight-line formulas are nearly always solved this way. The initial value is expressed by the y-intercept and the rate of change is represented in the form of the slope. The main issue with the slope-intercept form generally lies in the horizontal variable interpretation particularly when the variable is linked to one particular year (or any other kind number of units). The first step to solve them is to make sure you comprehend the definitions of variables clearly.

s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571911.5/warc/CC-MAIN-20220813081639-20220813111639-00655.warc.gz

CC-MAIN-2022-33

https://www.mysciencework.com/publication/show/lplq-multipliers-locally-compact-groups-58b1d23a?search=1

math

In this paper we discuss the L-p-L-q boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups G for the range 1 < p <= 2 <= q < infinity. As a consequence of the established Fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the L-p-L-q norms of the heat kernels for general positive unbounded invariant operators on G. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group, as well as for higher order operators. We show that our results imply the known results for L-p-L-q multipliers such as Hormander's Fourier multiplier theorem on R-n or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on advancing the analysis in the group von Neumann algebra and its application to the derivation of the desired multiplier theorems.

s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780053717.37/warc/CC-MAIN-20210916174455-20210916204455-00445.warc.gz

CC-MAIN-2021-39

https://www.hindawi.com/journals/mpe/2014/137616/

math

Recent Theory and Applications on Inverse Problems 2014View this Special Issue Research Article | Open Access Chengzhi Deng, Shengqian Wang, Wei Tian, Zhaoming Wu, Saifeng Hu, "Approximate Sparsity and Nonlocal Total Variation Based Compressive MR Image Reconstruction", Mathematical Problems in Engineering, vol. 2014, Article ID 137616, 13 pages, 2014. https://doi.org/10.1155/2014/137616 Approximate Sparsity and Nonlocal Total Variation Based Compressive MR Image Reconstruction Recent developments in compressive sensing (CS) show that it is possible to accurately reconstruct the magnetic resonance (MR) image from undersampled -space data by solving nonsmooth convex optimization problems, which therefore significantly reduce the scanning time. In this paper, we propose a new MR image reconstruction method based on a compound regularization model associated with the nonlocal total variation (NLTV) and the wavelet approximate sparsity. Nonlocal total variation can restore periodic textures and local geometric information better than total variation. The wavelet approximate sparsity achieves more accurate sparse reconstruction than fixed wavelet and norm. Furthermore, a variable splitting and augmented Lagrangian algorithm is presented to solve the proposed minimization problem. Experimental results on MR image reconstruction demonstrate that the proposed method outperforms many existing MR image reconstruction methods both in quantitative and in visual quality assessment. Magnetic resonance imaging (MRI) is a noninvasive and nonionizing imaging processing. Due to its noninvasive manner and intuitive visualization of both anatomical structure and physiological function, MRI has been widely applied in clinical diagnosis. Imaging speed is important in many MRI applications. However, both scanning and reconstruction speed of MRI will affect the quality of reconstructed image. In spite of advances in hardware and pulse sequences, the speed, at which the data can be collected in MRI, is fundamentally limited by physical and physiological constraints. Therefore many researchers are seeking methods to reduce the amount of acquired data without degrading the image quality [1–3]. In recent years, the compressive sensing (CS) framework has been successfully used to reconstruct MR images from highly undersampled -space data [4–9]. According to CS theory [10, 11], signals/images can be accurately recovered by using significantly fewer measurements than the number of unknowns or than mandated by traditional Nyquist sampling. MR image acquisition can be looked at as a special case of CS where the sampled linear combinations are simply individual Fourier coefficients (-space samples). Therefore, CS is claimed to be able to make accurate reconstructions from a small subset of -space data. In compressive sensing MRI (CSMRI), we can reconstruct a MR image with good quality from only a small number of measurements. Therefore, the application of CS to MRI has potential for significant scan time reductions, with benefits for patients and health care economics. Because of the ill-posed nature of the CSMRI reconstruction problem, regularization terms are required for a reasonable solution. In existing CSMRI models, the most popular regularizers are , sparsity [4, 9, 12] and total variation (TV) [3, 13]. The sparsity regularized CSMRI model can be understood as a penalized least square with norm penalty. It is well known that the complexity of this model is proportional with the number of variables. Particularly when the number is large, solving the model generally is intractable. The regularization problem can be transformed into an equivalent convex quadratic optimization problem and, therefore, can be very efficiently solved. And under some conditions, the resultant solution of regularization coincides with one of the solutions of regularization . Nevertheless, while regularization provides the best convex approximation to regularization and it is computationally efficient, the regularization often introduces extra bias in estimation and cannot reconstruct an image with the least measurements when applied to CSMRI . In recent years, the regularization [16, 17] was introduced into CSMRI, since regularization can assuredly generate much sparser solutions than regularization. Although the regularizations achieve better performance, they always fall into local minima. Moreover, which should yield a best result is also a problem. Trzasko and Manduca proposed a CSMRI paradigm based on homotopic approximation of the quasinorm. Although this method has no guarantee of achieving a global minimum, it achieves accurate MR image reconstructions at higher undersampling rates than regularization. And it was faster than those regularization methods. Recently, Chen and Huang accelerated MRI by introducing the wavelet tree structural sparsity into the CSMRI. Despite high effectiveness in CSMRI recovery, sparsity and TV regularizers often suffer from undesirable visual artifacts and staircase effects. To overcome those drawbacks, some hybrid sparsity and TV regularization methods [5–8] are proposed. In , Huang et al. proposed a new optimization algorithm for MR image reconstruction method, named fast composite splitting algorithm (FCSA), which is based on the combination of variable and operator splitting techniques. In , Yang et al. proposed a variable splitting method (RecPF) to solve hybrid sparsity and TV regularized MR image reconstruction optimization problem. Ma et al. proposed an operator splitting algorithm (TVCMRI) for MR reconstruction. In order to deal with the problem of low and high frequency coefficients measurement, Zhang et al. proposed a new so-called TVWL2-L1 model which measures low frequency coefficients and high frequency coefficients with norm and norm. In , an experimental study on the choice of CSMRI regularizations was given. Although the classical TV regularization performs well in CSMRI reconstruction while preserving edges, especially for cartoon-like MR images, it is well known that TV regularization is not suitable for images with fine details and it often tends to oversmooth image details and textures. Nonlocal TV regularization extends the classical TV regularization by nonlocal means filter and has been shown to outperform the TV in several inverse problems such as image deonising , deconvolution , and compressive sensing [24, 25]. In order to improve the signal-to-noise ratio and preserve the fine details of MR images, Gopi et al. , Huang and Yang , and Liang et al. have proposed nonlocal TV regularization based MR reconstruction and sensitivity encoding reconstruction. In this paper, we proposed a novel compound regularization based compressive MR image reconstruction method, which exploits the nonlocal total variation (NLTV) and the approximate sparsity prior. The approximate sparsity, which is used to replace the traditional regularizer and regularizer of compressive MR image reconstruction model, is sparser and much easier to be solved. The NLTV is much better than TV for preserving the sharp edges and meanwhile recovering the local structure details. In order to compound regularization model, we develop an alternative iterative scheme by using the variable splitting and augmented Lagrangian algorithm. Experimental results show that the proposed method can effectively improve the quality of MR image reconstruction. The rest of the paper is organized as follows. In Section 2 we review the compressive sensing and MRI reconstruction. In Section 3 we propose our model and algorithm. The experimental results and conclusions will be shown in Sections 4 and 5, respectively. 2. Compressive Sensing and MRI Reconstruction Compressive sensing [10, 11], as a new sampling and compression theory, is able to reconstruct an unknown signal from a very limited number of samples. It provides a firm theoretical foundation for the accurate reconstruction of MRI from highly undersampled -space measurements and significantly reduces the MRI scan duration. Suppose is a MR image and is a partial Fourier transform; then the sampling measurement of MR image in -space can be defined as The compressive MR image reconstruction problem is to recover given the measurement and the sampling matrix . Undersampling occurs whenever the number of -space sample is less than the number of unknowns . In that case, the compressive MR image reconstruction is an underdetermined problem. In general, compressive sensing reconstructs the unknowns from the measurements by minimizing the norm of the sparsified image , where represents a sparsity transform for the image. In this paper, we choose the orthonormal wavelet transform as the sparsity transform for the image. Then the typical compressive MR image reconstruction is obtained by solving the following constrained optimization problem [4, 9, 12]: However, in terms of computational complexity, the norm optimization problem (2) is a typical NP-hard problem, and it was difficult to solve. According to the certain condition of the restricted isometric property, the norm can be replaced by the norm. Therefore, the optimization problem (2) is relaxed to alternative convex optimization problem as follows: When the measurements are contaminated with noise, the typical compressive MR image reconstruction problem using relaxation of the norm is formulated as the following unconstrained Lagrangian version: where is a positive parameter. Despite high effectiveness of sparsity regularized compressive MR image reconstruction methods, they often suffer from undesirable visual artifacts such as Gibbs ringing in the result. Due to its desirable ability to preserve edges, total variation (TV) model is successfully used in compressive MR image reconstruction [3, 13]. But the TV regularizer has still some limitations that restrict its performance, which cannot generate good enough results for images with many small structures and often suffers from staircase artifacts. In order to combine the advantages of sparsity-based and TV model and avoid their main drawbacks, a TV regularizer, corresponding to a finite-difference for the sparsifying transform, is typically incorporated into the sparsity regularized compressive MR image reconstruction [5–8]. In this case the optimization problem is written as where is a positive parameter. The TV was defined discretely as , where and are the horizontal and the vertical gradient operators, respectively. The compound optimization model (5) is based on the fact that the piecewise smooth MR images can be sparsely represented by the wavelet and should have small total variations. 3. Proposed Model and Algorithm As mentioned above, joint TV and norm minimization model is a useful way to reconstruct MR images. However, they have still some limitations that restrict their performance. norm needs a combinatorial search for its minimization and its too high sensibility to noise. problems can be very efficiently solved. But the solution is not sparse, which influences the performance of MRI reconstruction. The TV model can preserve edges, but it tends to flatten inhomogeneous areas, such as textures. To overcome those shortcomings, a novel method is proposed for compressive MR imaging based on the wavelet approximate sparsity and nonlocal total variation (NLTV) regularization, named WasNLTV. 3.1. Approximate Sparsity The problems of using norm in compressive MR imaging (i.e., the need for a combinatorial search for its minimization and its too high sensibility to noise) are both due to the fact that the norm of a vector is a discontinuous function of that vector. The same as [29, 30], our idea is to approximate this discontinuous function by a continuous one, named approximate sparsity function, which provides smooth measure of norm and better sparsity than regularizer. The approximate sparsity function is defined as The parameter may be used to control the accuracy with which approximate the Kronecker delta. In mathematical terms, we have Define the continuous multivariate approximate sparsity function as It is clear from (7) that is an indicator of the number of zero-entries in for small values of . Therefore, norm can be approximate by Note that the larger the value of , the smoother the and the worse the approximation to norm; the smaller the value of norm, the closer the behavior of to norm. 3.2. Nonlocal Total Variation Although the classical TV is surprisingly efficient for preserving edges, it is well known that TV is not suitable for images with fine structures, details, and textures which are very important to MR images. The NLTV is a variational extension of the nonlocal means filter proposed by Wang et al. . NLTV uses the whole image information instead of using adjacent pixel information to calculate the gradients in regularization term. The NLTV has been proven to be more efficient than TV for improving the signal-to-noise ratio, on preserving not only sharp edges, but also fine details and repetitive patterns [26–28]. In this paper, we use the NLTV to replace the TV in compound regularization based compressive MR image reconstruction. Let , , be a real function , and let be a weight function. For a given image , the weighted graph gradient is if defined as the vector of all directional derivatives at : The directional derivatives apply to all the nodes since the weight is extended to the whole domain . Let us denote vectors such that ; the nonlocal graph divergence is defined as the adjoint of the nonlocal gradient: Due to being analogous to classical TV, the norm is in general more efficient than the norm for sparse reconstruction. In this paper, we are interested in NLTV. Based on the above definition, the NLTV is defined as follows: The weight function denotes how much the difference between pixels and is penalized in the images, which is calculated by where and denote a small patch in image centering at the coordinates and , respectively. is the normalizing factor. is a filtering parameter. 3.3. The Description of Proposed Model and Algorithm According to the compressive MR image reconstruction models described in Section 2, the proposed WasNLTV model for compressive MR image reconstruction is It should be noted that the optimization problem in (14), although convex, is very hard to solve owing to nonsmooth terms and its huge dimensionality. To solve the problem in (14), we use the variable splitting and augmented Lagrangian algorithm following closely the methodology introduced in . The core idea is to introduce a set of new variables per regularizer and then exploit the alternating direction method of multipliers (ADMM) to solve the resulting constrained optimization problems. By introducing an intermediate variable vector , the problem (14) can be transformed into an equivalent one; that is, The optimization problem (15) can be written in a compact form as follows: where The augmented Lagrangian of problem (16) is where is a positive constant, , and denotes the Lagrangian multipliers associated to the constraint . The basic idea of the augmented Lagrangian method is to seek a saddle point of , which is also the solution of problem (16). By using ADMM algorithm, we solve the problem (16) by iteratively solving the following problems: It is evident that the minimization problem (19) is still hard to solve efficiently in a direct way, since it involves a nonseparable quadratic term and nondifferentiability terms. To solve this problem, a quite useful ADMM algorithm is employed, which alternatively minimizes one variable while fixing the other variables. By using ADMM, the problem (19) can be solved by the following four subproblems with respect to and . (1) subproblem: by fixing and , the optimization problem (19) to be solved is Due to the computational complexity of NLTV, the same as , the NLTV regularization in this paper only runs one time. (2) subproblem: by fixing , , , and , the optimization problem (19) to be solved is Clearly, the problem (24) is a quadratic function; its solution is simply (3) subproblem: by fixing , , , and , the optimization problem (19) to be solved is The same as problem (24), the problem (26) is a quadratic function and its gradient is simplified as . The steepest descent method is desirable to use to solve (26) iteratively by applying (4) subproblem: by fixing , , , and , the optimization problem (19) to be solved is Problem (28) is a norm regularized optimization problem. Its solution is the well-known soft threshold : where denotes the component-wise application of soft-threshold function. 4. Experimental Results In this section, a series of experiments on four 2D MR images (named brain, chest, artery, and cardiac) are implemented to evaluate the proposed and existing methods. Figure 1 shows the test images. All experiments are conducted on a PC with an Intel Core i7-3520M, 2.90 GHz CPU, in MATLAB environment. The proposed method (named WasNLTV) is compared with the existing methods including TVCMRI , RecPF , and FCSA . We evaluate the performance of various methods both visually and qualitatively in signal-to-noise ratio (SNR) and root-mean-square error (RMSE) values. The SNR and RMSE are defined as where and denote the original image and the reconstructed image, respectively, and is the mean function. For fair comparisons, experiment uses the same observation methods with TVCMRI. In the -space, we randomly obtain more samples in low frequencies and fewer samples in higher frequencies. This sampling scheme is widely used for compressed MR image reconstructions. Suppose a MR image has pixels and the partial Fourier transform in problem (1) consists of rows of matrix corresponding to the full 2D discrete Fourier transform. The chosen rows correspond to the sampling measurements . Therefore, the sampling ratio is defined as . In the experiments, the Gaussian white noise generated by in MATLAB is added, where standard deviation . The regularization parameters , , and are set as 0.001, 0.035, and 1, respectively. To be fair to compare the reconstruction MR images of various algorithms, all methods run 50 iterations and the Rice wavelet toolbox is used as the wavelet transform. Table 1 summarizes the average reconstruction accuracy obtained by using different methods at different sampling ratios on the set of test images. From Table 1, it can be seen that the proposed WasNLTV method attains the highest SNR (dB) in all cases. Figure 2 plots the SNR values with sampling ratios for different images. It can also be seen that the WasNLTV method achieves the larger improvement of SNR values. Table 2 gives the RMSE results of reconstructed MRI after applying different algorithms. From Table 2, it can be seen that WasNLTV method attains the lowest RMSE in all cases. As is known, the lower the RMSE is, the better the reconstructed image is. That is to say the MR images reconstructed by WasNLTV have the best visual quality. To illustrate visual quality, reconstructed compressive MR images obtained using different methods with sampling ratios 20% are shown in Figures 3, 4, 5, and 6. For better visual comparison, we zoom in a small patch where the edge and texture are much more abundant. From the figures, it can be observed that the WasNLTV always obtains the best visual effects on all MR images. In particular, the edge of organs and tissues obtained by WasNLTV are much more clear and easy to identify. (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV (b) Cropped TVCMRI (d) Cropped RecPF (f) Cropped FCSA (h) Cropped WasNLTV Figure 7 gives the performance comparisons between different methods with sampling ratios 20% in terms of the CPU time over the SNR. In general, the computational complexity of NLTV is much higher than TV. In order to reduce the computational complexity of WasNLTV, in the experiment, we perform the NLTV regularization once in some iterations. Despite the higher computational complexity of WasNLTV, the WasNLTV obtains the best reconstruction results on all MR images by achieving the highest SNR in less CPU time. In this paper, we propose a new compound regularization based compressive sensing MRI reconstruction model, which exploits the NLTV regularization and wavelet approximate sparsity prior. The approximate sparsity prior is used in compressive MR image reconstruction model instead of or norm, which can produce much sparser results. And the optimization problem is much easier to be solved. Because the NLTV takes advantage of the redundancy and self-similarity in a MR image, it can effectively avoid blocky artifacts caused by traditional TV regularization and keep fine edge of organs and tissues. As for the algorithm, we apply the variable splitting and augmented Lagrangian algorithm to solve the compound regularization minimization problem. Experiments on test images demonstrate that the proposed method leads to high SNR measure and more importantly preserves the details and edges of MR images. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. The authors would like to thank the anonymous referees for their valuable and helpful comments. The work was supported by the National Natural Science Foundation of China under Grants 61162022 and 61362036, the Natural Science Foundation of Jiangxi China under Grant 20132BAB201021, the Jiangxi Science and Technology Research Development Project of China under Grant KJLD12098, and the Jiangxi Science and Technology Research Project of Education Department of China under Grant GJJ12632. - K. P. Pruessmann, “Encoding and reconstruction in parallel MRI,” NMR in Biomedicine, vol. 19, no. 3, pp. 288–299, 2006. - B. Sharif, J. A. Derbyshire, A. Z. Faranesh, and Y. Bresler, “Patient-adaptive reconstruction and acquisition in dynamic imaging with sensitivity encoding (PARADISE),” Magnetic Resonance in Medicine, vol. 64, no. 2, pp. 501–513, 2010. - M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1182–1195, 2007. - D. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Magazine, vol. 2, pp. 72–82, 2008. - J. Huang, S. Zhang, and D. Metaxas, “Efficient MR image reconstruction for compressed MR imaging,” Medical Image Analysis, vol. 15, no. 5, pp. 670–679, 2011. - Z. Zhang, Y. Shi, W. P. Ding, and B. C. Yin, “MR images reconstruction based on TVWL2-L1 model,” Journal of Visual Communication and Image Representation, vol. 2, pp. 187–195, 2013. - A. Majumdar and R. K. Ward, “On the choice of compressed sensing priors and sparsifying transforms for MR image reconstruction: An experimental study,” Signal Processing: Image Communication, vol. 27, no. 9, pp. 1035–1048, 2012. - J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE Journal on Selected Topics in Signal Processing, vol. 4, no. 2, pp. 288–297, 2010. - S. Ravishankar and Y. Bresler, “MR image reconstruction from highly undersampled k-space data by dictionary learning,” IEEE Transactions on Medical Imaging, vol. 30, no. 5, pp. 1028–1041, 2011. - E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006. - D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. - X. Qu, X. Cao, D. Guo, C. Hu, and Z. Chen, “Combined sparsifying transforms for compressed sensing MRI,” Electronics Letters, vol. 46, no. 2, pp. 121–123, 2010. - F. Knoll, K. Bredies, T. Pock, and R. Stollberger, “Second order total generalized variation (TGV) for MRI,” Magnetic Resonance in Medicine, vol. 65, no. 2, pp. 480–491, 2011. - D. Donoho and J. Tanner, “Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing,” Philosophical Transactions of the Royal Society of London A, vol. 367, no. 1906, pp. 4273–4293, 2009. - Z. B. Xu, X. Y. Chang, and F. M. Xu, “L-1/2 regularization: a thresholding representation theory and a fast solver,” IEEE Transactions on Neural Networks and Learning Systems, vol. 7, pp. 1013–1027, 2012. - R. Chartrand, “Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data,” in Proceedings of the 6th IEEE International Conference on Biomedical Imaging: From Nano to Macro (ISBI '09), pp. 262–265, July 2009. - C. Y. Jong, S. Tak, Y. Han, and W. P. Hyun, “Projection reconstruction MR imaging using FOCUSS,” Magnetic Resonance in Medicine, vol. 57, no. 4, pp. 764–775, 2007. - J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Transactions on Medical Imaging, vol. 28, no. 1, pp. 106–121, 2009. - C. Chen and J. Z. Huang, “The benefit of tree sparsity in accelerated MRI,” Medical Image Analysis, vol. 18, pp. 834–842, 2014. - S. Q. Ma, W. T. Yin, Y. Zhang, and A. Chakraborty, “An efficient algorithm for compressed MR imaging using total variation and wavelets,” in Proceeding of the 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR '08, pp. 1–8, Anchorage, Alaska, USA, June 2008. - A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 490–530, 2005. - F. F. Dong, H. L. Zhang, and D. X. Kong, “Nonlocal total variation models for multiplicative noise removal using split Bregman iteration,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 939–954, 2012. - S. Yun and H. Woo, “Linearized proximal alternating minimization algorithm for motion deblurring by nonlocal regularization,” Pattern Recognition, vol. 44, no. 6, pp. 1312–1326, 2011. - X. Zhang, M. Burger, and X. Bresson, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp. 253–276, 2011. - W. Dong, X. Yang, and G. Shi, “Compressive sensing via reweighted TV and nonlocal sparsity regularisation,” Electronics Letters, vol. 49, no. 3, pp. 184–186, 2013. - V. P. Gopi, P. Palanisamy, and K. A. Wahid, “MR image reconstruction based on iterative split Bregman algorithm and nonlocal total variation,” Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 985819, 16 pages, 2013. - J. Huang and F. Yang, “Compressed magnetic resonance imaging based on wavelet sparsity and nonlocal total variation,” in Proceedings of the 9th IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI '12), pp. 968–971, Barcelona, Spain, May 2012. - D. Liang, H. F. Wang, Y. C. Chang, and L. L. Ying, “Sensitivity encoding reconstruction with nonlocal total variation regularization,” Magnetic Resonance in Medicine, vol. 65, no. 5, pp. 1384–1392, 2011. - H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “A fast approach for overcomplete sparse decomposition based on smoothed L0 norm,” IEEE Transactions on Signal Processing, vol. 57, pp. 289–301, 2009. - J.-H. Wang, Z.-T. Huang, Y.-Y. Zhou, and F.-H. Wang, “Robust sparse recovery based on approximate norm,” Acta Electronica Sinica, vol. 40, no. 6, pp. 1185–1189, 2012. - M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Transactions on Image Processing, vol. 20, no. 3, pp. 681–695, 2011. - P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Modeling & Simulation, vol. 4, pp. 1168–1200, 2005. Copyright © 2014 Chengzhi Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662555558.23/warc/CC-MAIN-20220523041156-20220523071156-00084.warc.gz

CC-MAIN-2022-21

https://www.physicsforums.com/threads/statistical-mechanics-why-is-temperature-not-a-mechanical-variable.241922/

math

statistical mechanics -- why is temperature not a mechanical variable Hi, I have heard that temperature is not a mechanical variable. That is, that even if you knew the positions and momenta of all the particles in some system, you still couldn't calculate the temperature, because temperature (and entropy, and free energy, etc) are ensemble variables. Why is that? By the way, one implication of this statement is that temperature is not really the average kinetic energy of a system, at least in some cases. Say you had a dilute (better yet, ideal) system of independent gas (argon) atoms and you knew the mass of any particle (they all have the same mass) and its velocity. You could then calculate kinetic energy (0.5 * m*v*v, right?) and average kinetic energy, therefore kinetic energy (and average kinetic energy) is a mechanical variable. But temperature is not. So temperature is not really average kinetic energy. So, what is temperature? Thanks!

s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676596204.93/warc/CC-MAIN-20180723090751-20180723110751-00308.warc.gz

CC-MAIN-2018-30

https://learn-electrical.com/20733/describe-the-operation-of-a-delta-sigma-modulator

math

A delta-sigma (ΔΣ) modulator, also known as a sigma-delta modulator, is a type of analog-to-digital converter (ADC) that is commonly used to convert analog signals into digital form with high resolution, especially in applications where accuracy and precision are critical, such as audio, instrumentation, and communication systems. The key principle behind a delta-sigma modulator is oversampling, which involves sampling the input signal at a much higher frequency than the desired output digital sampling rate. This oversampling, combined with a feedback loop, helps achieve high-resolution conversion and noise shaping. Here's how a delta-sigma modulator operates: Analog Input: The modulator takes in an analog input signal, which can be any continuous-time signal that needs to be converted into digital form. Oversampling: The modulator samples the analog input signal at a significantly higher frequency than the desired output digital sampling rate. This higher sampling frequency is often referred to as the "oversampling ratio." 1-Bit Quantization: At each sampling instant, the analog input is compared to the previous output of the modulator. The comparison produces a 1-bit quantization result that indicates whether the input has increased or decreased since the last sample. Integration and Feedback Loop: The 1-bit quantization result is passed through an integrator, which accumulates the quantization errors over time. The integrator's output is then subtracted from the input signal to produce a "delta" signal. This delta signal represents the difference between the input signal and the feedback signal, which is an estimate of the original analog signal. Digital Output: The delta signal is then passed through a digital-to-analog converter (DAC) to produce a continuous analog signal. This analog signal is subtracted from the original input signal to generate the quantization error. The quantization error is then quantized to 1 bit again, forming the digital output of the modulator. Noise Shaping: The quantization error is effectively spread out across a higher frequency range due to the continuous feedback loop. This phenomenon is known as noise shaping. The noise energy is moved to higher frequencies, where it can be filtered out more effectively using digital filters. Digital Filtering: The high-frequency noise, which has been shaped by the delta-sigma modulation process, is filtered out using digital filters, usually in a decimation stage. The output of these filters provides the final digital representation of the input signal with increased resolution and significantly reduced quantization noise in the desired frequency band. The oversampling and feedback loop of a delta-sigma modulator effectively convert the original analog signal into a high-resolution digital signal while pushing quantization noise to higher frequencies. This allows for the use of simpler and more effective digital filters to achieve a desired level of signal-to-noise ratio in the output signal. Delta-sigma modulators are commonly used in applications requiring high-resolution conversion with relatively low-speed ADCs.

s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296818999.68/warc/CC-MAIN-20240424014618-20240424044618-00291.warc.gz

CC-MAIN-2024-18

https://www.brightstorm.com/tag/quadratic-equation/page/4

math

How to find the vertex, intercepts, domain and range of a quadratic graph. How to sketch a simple polar curve by plotting points. How to derive the equation for a circle using the distance formula. How to solve equations with the same variable on both sides. How to solve a logarithmic equation. How to solve equations with absolute values. How to write the equation of a graphed line. How to solve a simple rational equation. How to connect linear equations, tables of values, and their graphs. Solving two step equations that involve fractions How to solve for a parameter in a simple linear equation. Review of operations with decimals in order to solve equations with decimals How to solve a quadratic equation by completing the square. How to calculate and interpret the discriminant of a quadratic equation. Solving two step equations with integers How to solve equations using square roots or cube roots. How to find the domain and graph radical equations.

s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703564029.59/warc/CC-MAIN-20210125030118-20210125060118-00345.warc.gz

CC-MAIN-2021-04

https://www.physicsforums.com/threads/multimeter-ohms-exponential-scale-formula.559681/

math

1. The problem statement, all variables and given/known data Hi, I'm asking the usual "what formula gives me these results" question, but with a twist; I'm attempting to make a "virtual" analogue multimeter that replicates a physical piece of equipment, for training purposes. The idea is that a random MegaOhm value will be generated, and displayed by the needle; the user must then determine whether it falls within the pass or fail mark for a piece of equipment being tested. The problem is that the scale is non-linear. It's exponential of some sort. But after a day of intensive online searching, and resorting to throwing numbers at various equations, I can't come up with a formula that gives a reasonable approximation, or even anything remotely close! NB: ANY formula that fits the points is fine by me; I've severely over-engineered this problem, but I'm determined to get it working! 2. Relevant equations Essentially the multimeter needle range is 90degrees of motion, going from zero MOhms to Infinite MOhms in an exponential manner. The following are data points relating MOhms to degrees, very roughly measured from a photo of the multimeter: value, angle (degrees) 0, 0 0.1, 5.5 0.2, 11.25 0.5, 28 1, 39.7 2, 46.5 5, 61.8 10, 68.65 20, 79.2 50, 86 100, 88.35 200, 89.4 infinity, 90 [EDIT] Here's an example multimeter to show the exponential scale (different from mine, but you get the idea) http://www.opamp-electronics.com/tutorials/images/dc/50035.jpg 3. The attempt at a solution I've tried plotting these in Excel and using a trendline, but nothing comes remotely close. Tried creating dynamic formula that I could change various key values of and see the results for all input values, but could never get a curve that came close; always curved too sharply too late, and I just don't know what terminology I should be searching for to fix it! I can SEE a definite trend in the plotted points, but Excel doesn't seem to see what I see. This is intended just as a "cool" factor for some training material, and I've spent way too much time on it, so it's highly frustrating that something that should be so simple has had me (and several of my colleagues) climbing up the walls! Any help would be greatly appreciated!

s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376824601.32/warc/CC-MAIN-20181213080138-20181213101638-00261.warc.gz

CC-MAIN-2018-51

https://www.karisastravel.com/2017-neco-chem-answers-neco-chemistry-expo-obj-theory

math

Presently Neco Chemistry 2017 past questions and answer, Neco expo runz, Neco chemistry final answers for today’s exam paper. No doubt the Senior School Certificate Examination (SSCE) of the National Examination Council (NECO) is ongoing and the Chemistry paper is scheduled to be written on Thursday, 6th July from 10am to 1pm. Finally Neco Chemistry 2017 past questions and answer, Neco expo runz, Neco chemistry final answers for today’s exam paper. Furthermore the Neco chem exam is for Paper III & II: Objective & Essay which will take a total of 3hrs to write. Here, we will be posting out samples of the neco chemistry questions for candidates that will participate in the examination. Any ways this is to inform you all that as a matter of fact the below chemistry questions and answers are from Neco past questions and answers that we feel are likely questions for SSCE preparation. This should in no way discourage you. Believe me when i say all answers will be uploaded three hours before the exam begins this is as a matter of fact. Therefore Keep following this page and make sure you bookmark this site for reference purposes. Above all Please NOTE: Vapor Pressure of Water Chart is on the back of this page. 1. A sample of air collected at STP contains 0.039 moles of N2, 0.010 moles of O2, and 0.001 moles of Ar. (Assume no other gases are present.) a) Find the partial pressure of O2. b) What is the volume of the container? 2. A sample of hydrogen gas (H2) is collected over water at 19°C. a) What are the partial pressures of H2 and water vapor if the total pressure is 756 mm Hg? b) What is the partial pressure of hydrogen gas in atmospheres? 3. If 600. cm3 of H2 at 25°C and 750. mm Hg is compressed to a volume of 480. cm3 at 41°C, what does the pressure become? FINALLY -2017 Neco Chem Answers | Neco Chemistry Expo obj & theory5. a) Write a balanced chemical equation for the reaction of butane gas with oxygen gas to form carbon dioxide and water vapor. b) First and foremost How many liters of oxygen are required to produce 2.0 liters of CO2? c) Similarly How many liters of CO2 are produced from 11.6 g of butane at STP? d) Finally How many molecules of water vapor are produced from 5.6 liters of butane gas at STP? 6. Find the molar volume of a gas at 68°C and 2.00 atmospheres pressure. 7. How many liters of methane are there in 8.00 grams at STP? 8. Calculate the density of carbon dioxide at 546 K and 4.00 atmospheres pressure. 9. What volume of O2 at 710. mm Hg pressure and 36°C is required to react with 6.52 g of CuS? CuS(s) + 2 O2(g) ® CuSO4(s) 10. Meanwhile What is the molar mass of a gas if 7.00 grams occupy 6.20 liters at 29°C and 760. mm Hg pressure? 11. At a particular temperature and pressure, 15.0 g of CO2 occupy 7.16 liters. What is the volume of 12.0 g of CH4 at the same temperature and pressure? 12. Finally To prepare a sample of hydrogen gas, a student reacts 7.78 grams of zinc with acid: Zn(s) + 2 H+ (aq) ® Zn2+(aq) + H2(g) The hydrogen is collected over water at 22°C and the total pressure of gas collected is 750. mm Hg. What is the partial pressure of H2? What volume of wet hydrogen gas is collected?

s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743521.59/warc/CC-MAIN-20181117123417-20181117144648-00025.warc.gz

CC-MAIN-2018-47

http://www.chegg.com/homework-help/questions-and-answers/1-magnitude-buoyant-force-acting-floating-object-compare-weight-object-weight-water-displa-q5704509

math

1. How does the magnitude of the buoyant force acting on the floating object compare to the weight of the object, and to the weight of the water displaced by the floating object? 2. What conditions, in terms of forces, are required for an object to float? What conditions, in terms of the densities of the object and the fluid, are required for the object to float? 3. What is the condition for such a stack having neutral buoyancy, in terms of masses (mH and mL) and volumes (VH and VL) of the blocks and the density ?w of the water? How did you express the condition mathematically?

s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982905736.38/warc/CC-MAIN-20160823200825-00196-ip-10-153-172-175.ec2.internal.warc.gz

CC-MAIN-2016-36

https://www.meritnation.com/ask-answer/popular/class-12-science-physics/gr12-sb4/page:3

math

A dc motor connected to a 460 V supply has an armature resistance of 0.15 ohms.Calculate (1) The value of back emf when the armature current is 120A. (2) The value of armature current when the back emf if 447.4V. if a dielectric slab of dielectric constant K is introduced between the plates of a parallel plate capacitor completely , how does the energy density of the capacitor change? A 3 m long copper wire carrying a current of 3 A.. How long does it take for an electron to drift from one end to other end ? The cross- sectional area of the wire is 2 * 10 ^ -6 m2 and the number of conduction electrons in copper is 8.5 * 10 ^ 28 m-3 . Two point charges 2uC and -2uC are placed at points A and B 6 cm apart . (1) Draw the equipotential surfaces of the system. (2) Why do equipotential surface get closer to each other near the point charges? Copyright © 2022 Aakash EduTech Pvt. Ltd. All rights reserved.

s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103646990.40/warc/CC-MAIN-20220630001553-20220630031553-00583.warc.gz

CC-MAIN-2022-27

https://researchoutput.ncku.edu.tw/zh/publications/nonparametric-bounds-for-european-option-prices

math

There is much research whose efforts have been devoted to discovering the distributional defects in the Black-Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new nonparametric lower bound and provide an alternative interpretation of Ritchken's (1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out-of-the-money (OTM) options where the previous lower bounds perform badly. Moreover, we present that our bounds can be derived from histograms which are completely nonparametric in an empirical study. We first construct histograms from realizations of S & P 500 index returns following Chen, Lin, and Palmon (2006); calculate the dollar beta of the option and expected payoffs of the index and the option; and eventually obtain our bounds. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out-of-the-money calls are substantially overpriced (violate the lower bound). All Science Journal Classification (ASJC) codes - 經濟學、計量經濟學和金融學 (全部)

s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817014.15/warc/CC-MAIN-20240415174104-20240415204104-00582.warc.gz

CC-MAIN-2024-18

http://www.math.ubc.ca/~reichst/423-502S13syll.html

math

Mathematics 423/502, Topics in Algebra January-April 2013, MWF 13:00-13:50, Room MATH 204 J.J. Rotman, Advanced Modern Algebra, Course description : This course is a sequel to It will to cover a range of topics in commutative and homological algebra, including some of the algebraic prerequisites for advanced work in number theory, algebraic geometry and algebraic topology. Topics will include the structure theorem for modules over principal ideal domains, Hilbert Basis Theorem, Noether Normalization Lemma, Hilbert's Nullstellensatz and an introduction to affine algebraic geometry, Groebner bases, tensor products, Office: 1105 Math Annex E-mail: reichst at math.ubc.ca Class projects : Each student will be required to choose a project and submit a paper on it during the term. The paper should include complete statements of the relevant results and complete proofs. Depending on the topic, I will ask some students to present their projects in class. Project 1: Give an example of a principal ideal domain which is not a Euclidean domain. Project 2: Tsen-Lang theory (C_n fields). Project 3: Wedderburn's Little Theorem (about associative division rings of finite order) and Artin-Zorn Theorem (generalization to alternative Project 4: Effective Nullstellensatz. Possible starting point: Project 5: The fibre dimension theorem. (One possible source for this is Basic Algebraic Geometry by Shafarevich.) Project 6: Hilbert's theorem on the finite generation of the ring Project 7: Krull dimension. Project 8: Efficient implementations of Buchsberger's algorithm Possible sources: Becker and Weispfenning MR1213453, Cox, Little and O'Shea MR1189133. Project 9: Application of Grobner bases. Start with section 2.8 of Cox, Little and O'Shea (see the reference above). Project 10: SAGBI = Sabalgebra analogue of Grobner bases for ideals. Possible sources: my paper MR1966757 and the references there.

s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423716.66/warc/CC-MAIN-20170721042214-20170721062214-00679.warc.gz

CC-MAIN-2017-30

https://www.primidi.com/subjective_logic/subjective_opinions/multinomial_opinions

math

Let be a frame, i.e. a set of exhaustive and mutually disjoint propositions . A multinomial opinion over is the composite function, where is a vector of belief masses over the propositions of, is the uncertainty mass, and is a vector of base rate values over the propositions of . These components satisfy and as well as . Visualising multinomial opinions is not trivial. Trinomial opinions could be visualised as points inside a triangular pyramid, but the 2D aspect of computer monitors would make this impractical. Opinions with dimensions larger than trinomial do not lend themselves to traditional visualisation. Dirichlet distributions are normally denoted as where represents its parameters. The Dirichlet distribution of a multinomial opinion is the function where the vector components are given by Famous quotes containing the word opinions: “America owes most of its social prejudices to the exaggerated religious opinions of the different sects which were so instrumental in establishing the colonies.” —James Fenimore Cooper (17891851)

s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534669.47/warc/CC-MAIN-20220520191810-20220520221810-00067.warc.gz

CC-MAIN-2022-21

http://essaysreasy.org/content/essay-on-optimum-age-to-start-the-transformation.html

math

In order to define the optimum age for the start of the transformation it is sufficient to forecast the cash flows for the next 80 years. Thus, it is important not to delay the start of the transformation and not to begin the transformation too early. The discounted values of the later fellings are less likely to make up for the low early revenues obtained from thinning only and this phenomenon is exaggerated with higher discount rates (Shrimpton, 1990). The net revenues from the different starting ages begin to coincide, as these trees approach peak, because the rate of increase in value of the residual trees begins to decrease. In Appendix A, it is explicitly shown that the higher the discount rate, the less the optimum age to start the transformation is. Thus, the optimum age to start the transformation is around age 45, at a discount rate of 3%, while at a discount rate of 7%, it is 35. However, the optimum starting age for higher Yield Classes is earlier for each discount rate. Consequently, delaying the starting age on the higher Yield Class site incurs a higher monetary loss, since a greater amount of timber yielded during any one felling period is obtained from thinnings and therefore put a downward pressure on revenues. By the same token, as the peak for clear felling for Yield Class 8 (one year after planting) site, for instance, approaches later thus thinning volumes do not fall substantially and converting age may be delayed to 40 at 7% discount rate or even beyond 45 at 3% discount rate. Even though the Internal Rate of Return is not the best criterion to use to choose one of silvicultural systems, it shows the similar picture as NPV for each Yield Class (see appendix B). Graphs, charts, curves, figures are at times difficult to deal with. Our paper writers can help with that and with any topic that you need to write an essay or paper on.

s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703518240.40/warc/CC-MAIN-20210119103923-20210119133923-00607.warc.gz

CC-MAIN-2021-04

https://www.mathnasium.com/au/blog/20150207-happy-e-day

math

As the school bells chime and backpacks are packed, it's time to embark on a new academic adventure. At Mathnasium, we're thrilled to welcome students back to school and support them on their journey to success. Whether your child is starting a ne.. February 7th is e Day! While not as famous as Pi Day, e Day celebrates the mathematical constant e, which has a wide range of mathematical applications. Here's a bit of background on e and why it's worthy of celebration! As the history of mathematics has unfolded, mathematicians have discovered a number that, in addition to being useful in number theory, occurs over and over in describing nature in mathematical terms. This number is the mathematical constant e. It is an irrational number (a non–repeating decimal) and also a transcendental number (it is not the solution of any algebraic polynomial). The first 20 decimal places of the numerical value of e are: Historically, e is sometimes called Euler’s Number after the master Swiss mathematician Leonhard Euler. It is also sometimes called Napier’s Number, in honor of John Napier, the creator of logarithms. e is one of the most important numbers in mathematics, along with 0 and 1, the additive and multiplicative identities; i, the imaginary unit; π; and Ï•, the number of the Golden Ratio. Like π, i, and Ï•, e appears unexpectedly in many areas of mathematics. There are many ways to calculate the value of e. Let's take a look at one way of finding e: Check out this fun song tribute to e from Daniel Wedge! While e Day might not have the cool factor π Day has, e is definitely worth celebrating. Make sure to wish someone a Happy e Day today! Mathnasium meets your child where they are and helps them with the customised programme they need, for any level of mathematics.

s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296818711.23/warc/CC-MAIN-20240423130552-20240423160552-00867.warc.gz

CC-MAIN-2024-18

https://www.iz2uuf.net/wp/index.php/2017/07/29/the-myth-of-reflected-power/

math

A common topic among ham radio operators is about power lost due to high VSWR when feeding an untuned antenna. A very frequent explanation about why this should (or should not) be a concern, is more or less like this: The power generated by the transmitter enters the coaxial cable and runs towards the antenna. When it reaches the load (the antenna) it encounter a mismatch; due to this mismatch, some power is transferred to the antenna, while the rest is reflected back and therefore lost. A tuner can be added between the transceiver and the line, but it will just “fool” the transceiver to believe the load is 50Ω: nevertheless the mismatch is still there with all of its consequent losses. The amount of reflected (thus supposedly lost) power is directly related to VSWR and usually quantified in tables like this: The Mismatch Loss in dB is calculated with the formula below: For example, with VSWR=5.85, according to this approach, more than 50% of the power should be lost (-3.021 dB). Where does the energy go? Many sources do not even bother to consider where the “lost power” is supposed to go: simply, it disappears. However we all learned in our high school Physics class that energy can not disappear into nothing. Some more advanced sources, instead, explain that the reflected power runs back into the transmission line until it bangs against the transmitter, whose internal resistance dissipates it. And if it bangs too hard, it can destroy the transmitter, like a train crashing into a wall. According to this theory, the complete process should be: - energy leaves the transmitter and enters the coaxial cable; - while running in the transmission line, some energy is dissipated as heat (all hams are aware of the dBs lost for every 100m/100ft at a given frequency of their favorite coaxial cables); - the surviving energy hits the mismatch point, where the high-VSWR antenna is connected to the coax; - given a VSWR value, a fixed percentage of energy goes to the antenna, while the remaining is “sent back” on the same coax; - the returning energy runs back on the cable and gets dissipated again by the same cable attenuation that it met on its forward run; - finally, the remaining reflected energy hits the transmitter and it is completely dissipated by the generator internal resistance; Let us make an example. We have a cable that has 1dB of attenuation at the frequency in use and we have an antenna presenting VSWR=5.85, thus a Mismatch Loss of 3.021dB: we should expect to have 3.021dB+1dB=4.021dB attenuation, i.e. only 40W out of 100 that go on the air. But… is that true? In order to verify the theory above, I connected my function generator to channel #1 of my oscilloscope; after that, I connected 24.9m of RG-58, then channel #2 of the scope and finally the load resistor representing the antenna. This setup will allow us to see the voltage entering the line and the voltage entering the load after having traversed the entire cable. Knowing the voltage V and the complex impedance Z, we can calculate the resulting power with P=V2/Z. Thus, with this setup and the help of a VNA, we can measure the power entering the coax and the power received by the load without impedance restrictions. The difference will reveal us the real power loss. Before starting the experiments, I carefully measured this test cable with my network analyzer. It resulted having a velocity factor of 0.6636 and, at 5MHz, an attenuation of 0.823dB. Experiment 1: matched load In this experiment, the line is terminated with a 50Ω load, thus it is perfectly matched. In the picture below we can see the function generator sending a single 5MHz sine wave: As expected, we have the generated pulse (yellow) developing on the 50Ω characteristic impedance of the coaxial cable. After 124ns, the same pulse reaches the 50Ω load. Considering that light travels 300mm every 1ns, we have 124 * 300 * 0.6636 = 24686mm = 24.7m, which is fairly close (±1ns) to the measured length of 24.9m. Being R the same on both sides (i.e. 50Ω), we can calculate the power ratio squaring the ratio of peak voltages: (1.12/1.26)2=0.79, which is a loss of 1.02dB, which is the same as the VNA measure ±0.2dB. Now we can set the generator to send a continuous stream of sinewaves at 5MHz: As expected, we obtain the same pattern as before but repeated over and over: voltages and timings are absolutely identical. So far so good. Experiment 2: mismatched load In order to test the behavior of the transmission line when loaded with high VSWR, I prepared a female SMA connector with a 270Ω SMD resistor soldered on it: This load produces VSWR=5.403 and, according to the Mismatch Loss table above, a loss of 2.779dB (53% to the antenna, 47% lost). Let us now send again a single 5MHz pulse and see what happens: What we see now is something a bit different than before. The initial pulse (1) is identical as the one of experiment #1 (1.26V peak). When it arrives to the 270Ω load (2) 124ns later, the voltage is much higher (1.88V peak). Then, after 124ns, a new peak (3) appears on channel 1, the load side. Let’s see what happened. The initial pulse (1) is driven on the transmission line, that at that time appears as a 50Ω load. There should be no surprise to observe that the first pulse is always identical among all the experiments: since information can not travel at infinite speed, the generator can not know instantly that at the end of the line that there is a different load than before. Therefore, the first peak must be identical to the ones we have seen before when we had the 50Ω load – and so it is. The peak power sent by the generator in the coaxial cable is 1.26V on 50Ω (1), which makes 31.75mW. The peak then travels along the line generating heat; when reaches the other end, after 124ns, it should have lost 0.823dB: the power available at (2) should be 26.27mW. At this point the wave encounters the mismatch. The tables say that, due to VSWR=5.403, only 52.7% of this power should be delivered to the load, that is 13.85mW. If we look at the 1.88V peak on 270Ω we have 13.09mW which confirms it. We have now a remainder of 12.42mW that have not been delivered to the 270Ω load. This power is bounced back and travels the coaxial cable in the other direction, loosing again 0.823dB. The power that reaches back the generator should be 10.28mW: the value at point (3) is 0.72V @50Ω, which makes 10.37mW, again perfectly in line with expectations. At this point the returning peak (3) encounters the function generator output port which offers 50Ω, i.e. a perfect match: the returning wave heats up the 50Ω resistor inside the function generator and disappears. So far, the initial theory is perfectly confirmed: the mismatched load has consumed the exact percentage of power and the rest has been bounced back and dissipated in the generator. The power delivered to the load was expected to be attenuated of 0.823dB (cable loss) + 2.779dB (mismatch loss)=3.602dB. Using a script and the binary data downloaded from the oscilloscope, I integrated the energy contained in the driven curve (orange, 3.040429nJ) and the load curve (blue 1.313286nJ): their ratio, 0.4319, accounts to 3.646dB of attenuation, which is almost a perfect match with the expected 3.602dB! Experiment 3: mismatched load and generator This time we shall repeat the experiment 2, but instead of having a 50Ω generator, we shall use a different impedance. In order to attain it, I prepared a matching attenuator with 10.28dB of attenuation and a reverse impedance of 144.5Ω. This is like to have a generator which output impedance is not 50Ω anymore, but 144.5Ω. I increased the function generator voltage to compensate the attenuator so the same 1.26V initial peak was generated again in the transmission line. This is what happened: Here we can see a different story. The initial stimulus (1) is the same as before as predicted; it travels until it reaches the 270Ω load (2) which reacts exactly as in experiment #2, reflecting the 47.3% of the received power. However this time the power coming back finds another mismatch, the 144Ω attenuator pad (3), and it is reflected back again towards the 270Ω load (4). Then it bounces back and forth over and over until all the power is gone. As it appears clearly, this time more energy is delivered to the load, although in multiple steps. Using the energy integration method, I calculated the energy actually delivered to the 270Ω load. This time the loss is only 3.271dB: i.e. the load received 0.37dB more than before. The first cracks in the initial theory begin to appear. The initial claim is founded on a fixed relation VSWR->loss, but a very simple experiment like this shows a case where it does not work. Same identical initial wave, same line, same load, same VSWR, two different results just by changing the impedance of the generator? Experiment 4: let’s the magic begin So far we have seen with that same setup, two different generator impedances feeding exactly the same power can change the amount of power delivered to the load. The experiment above shows that the power not delivered to the load is dissipated as heat by the cable itself and by the internal resistance of the generator. We shall now execute another experiment: this time, we will repeat experiments #2 (50Ω generator, 270Ω load) and #3 (144Ω generator, 270Ω load) but feeding a continuous sine wave. In both tests, the generator is set with the identical voltage level that in the previous tests generated the 1.26V initial peak. Here they are: When feeding the circuit with a continuous sine wave, something weird seems to happen. First we note that by looking at these screenshot, there is no clue of any bouncing anymore: both tests generate a nice yellow sine wave that propagates 124ns ahead to a nice blue sine wave on the load. Even more interesting is that the peak CH1/CH2 voltages, although not identical among the two tests, hold exactly the same ratio: - 1.86/1.24 = 1.5 - 1.68/1.12 = 1.5 Unlike the single-shot tests #2 and #3, the continuously fed lines are delivering exactly the same amount of power, no matter what the generator impedance is. In other words, when the generator sends a single shot, part of the energy is bounced back and dissipated by its internal impedance. As we saw, different generator impedance, different amount of energy dissipated, different amount of energy successfully delivered to the load. But if the generator sends a continuous flow of sine waves, we experience a completely dissimilar behavior: no matter of which is the generator impedance, the very same percentage of the power that enters the coaxial cable is delivered to the load. So, what’s going on? Behavior of a transmission line Without entering into the details, we can have an hint of the reason why a transmission line fed continuously behaves differently from one that receives a single pulse from the picture below: In picture “A” we have a voltage generator Vgen with its internal resistance Rgen feeding a load made of the resistance Rload. What the generator will see is a voltage V1 and a current I1 developing on its terminals: therefore, it will see an impedance Z1=V1/I1 which, in this case, is the same as Rload. The reflected power forms a voltage wave that travels back on the line until reaching the generator. This wave is seen as a voltage generator was added at the feed point (picture “B”). If we calculate the V2 voltage and I2 current we shall see that, due to the contribution of Vload, they will not match I1 and V1 anymore. The generator will see a new impedance value Z2=V2/I2, this time not equal to Rload anymore. In other words, the reflections change the impedance of the transmission line at the feed point. The resulting effect is that the transmission line now acts as a impedance transformer. The power lost in this process is only the one dissipated by the transmission line as heat: no matter what the VSWR is, if we could have a perfect line, all the power would be transferred to the load. Whatever formula that calculates power loss using only VSWR as a parameter, like the one at the beginning, it obviously flawed. Measuring real losses So far, we have established that the Mismatch Loss formula shown at the beginning does not really tell how much power is lost due to mismatch. So, how much power do we really loose? To have an answer, I prepared another experiment of measurement of power entering and exiting a transmission line terminated with a mismatched load (the same 270Ω load). To achieve the best precision, instead of using the oscilloscope, I used a much more accurate Rohde&Schwarz RF millivoltmeter. The test cable was made of 6.22m of RG-58 terminated with SMA connectors. I made two microstrip fixtures that could host the 1GHz probe of the RF millivoltmeter, which adds about 2pF. I then made an S11 and S21 measurement of this setup, including fixtures and probe, to know the impedance values needed to calculate the power levels. At 20MHz my 6.22m test cable has a matched loss of 0.472dB. Then I set my signal generator at 20MHz and measured input and output voltage: The measured impedance at 20MHz is 18.590 -j36.952; on that impedance, a voltage of 241.5mVRMS amounts to 0.634mWRMS (-1.981dBm); the output voltage is 364.1mVRMS on 270Ω, which is 0.491mWRMS (-3.092dBm). The overall power lost in this cable at this frequency is 1.110dB, i.e. only 0.638dB more than the 0.472dB that this cable would have normally dissipated due to line attenuation. This is significantly different than the 2.779dB loss foreseen by the “Mismatch Loss” method. Calculating mismatch losses Is there a formula that allows us to estimate the loss of a mismatched transmission line? Yes, there is. You can find a complete explanation in the very interesting AC6LA’s site. These formulas require some parameters of the transmission line to be measured with a network analyzer. I measured my “Prospecta RG58” with two S11 runs (open/short) and I fed the S11 files to ZPLOT, which gave me back the nominal Zo, nominal VF, K0, K1 and K2 parameters for my line. I fed those parameters to the IZ2UUF Transmission Line calculator, which gave me the following results: The software calculated a matched loss of 0.500dB (I measured 0.472dB) and a total loss of 1.104dB (I measured 1.110dB), which makes it a stunning “perfect match” with only 0.006dB of difference! So far I got very good results comparing real and predicted loss figures up to VHF, with discrepancies of cents of dB. To test higher bands I shall do further work to cancel out the impact of measurement fixtures and probes. Adding a tuner What happens if we add a tuner between the transmitter and the transmission line, as most hams do? In order to verify this, I connected the same 6.22m RG-58 line terminated with the 270Ω load to my MFJ-949E tuner and, with the help of my network analyzer, I tuned it to reach a perfect 50Ω match including the millivoltmeter probe: Then, I connected it to the signal generator and, using the RF millivoltmeter at the feed point of the tuner as a reference, I increased the generator power to compensate the extra cable I added. With 0.4dBm set on the signal generator, I had perfect 0dBm at the perfectly tuned 50Ω tuner input. As far as the signal generator is concerned, it is feeding a perfect load. Let us see the voltage entering the line after the tuner and the voltage reaching the load: We have 301.9mV on the beginning of the line, where the impedance is 18.59-j36.952: solving the complex numbers calculation tells that my tuner is pumping on the line 0.990mW (-0.043dBm). At the end we have 0.454mV, which delivers to the 270+j0 load 0.763mW (-1.173dBm). This means that the line dissipated 1.130dB, which is almost identical to the 1.110dB measured in the previous example (difference is only 0.02dB!) and almost identical the 1.104dB calculated by the online calculator. In these measurements we see that in this case the tuner received 0dBm and produced on its output -0.043dBm, thus dissipating as little as 0.043dB of power (<1%). If we would have fed a perfectly matched 50Ω load with this 6.22m long RG58 line, we would have lost 0.472dB due to normal line attenuation. Feeding the same line with a VSWR>5 load and a tuner, we have lost 1.173dB, which means a net cost of only 0.701dB. Be aware that such a low loss in a tuner is not a general rule, since tuning other impedances could cause greater heat dissipation, but it is very common. Back to the Mismatch Loss After all the experiments above, we have established beyond all reasonable doubt that the Mismatch Loss formula shown at the beginning of the article does not indicate the power lost when feeding a mismatched antenna. So, what is it for? Let us consider these two simple circuits: Picture “A” shows a 100V voltage generator with its internal 50Ω resistance Rgen feeding a 50Ω load Rload. Using Ohm’s law, we can calculate I=V/R=Vgen/(Rgen+Rload)=1A. Given that P=I2R, we can calculate the power dissipated by the load: Pload=I2Rload=50W. The generator itself is generating P=VgenI=100W and 50W are dissipated by the internal resistance Rgen. Now we can do the same calculation on “B”, where Rload is 270Ω. We have that I = Vgen/(Rgen+Rload) = 100/(50+270)=0.3125A. Hence, the power consumed by the load is I2Rload=26.367W. The generator is generating P=VgenI=31.25W and Rgen is dissipating 4.883W. We see that in circuit A the load is receiving more power: 50W vs. 26.367W: due to the maximum power transfer theorem, we get the maximum power (in this case 50W) when Rload=Rgen. For any other value, the power going to the load will be less. The “A” condition is defined as “matched“. If we calculate the ratio of the power delivered on B and the maximum possible delivered power A, we have that 26.367 / 50 = 0.527; if we transform it in dB, we have 2.779dB which is exactly the Mismatch Loss we calculated before for the 270Ω load. The Mismatch Loss value does not tell how much power is actually lost due to other dissipated, but it represents the inability of the generator to generate power due to mismatch. Note also that the Mismatch Loss is not an index of efficiency: with matched load, we got the highest power on the load (50W) but efficiency was at 50% (100W produced, 50W used on the load). In the mismatched circuit, the generator produced 31.25W of which 26.367W were delivered to the load, holding an efficiency of 84.3%! We can see this effect on the power that the R&S SMS2 signal generator has been able to deliver into the mismatched line with or without the tuner: The difference in power between the two is 1.94dB: if we calculate the mismatch for the impedance being fed (note the reference impedance is 18.590 -j36.952 presented at the input of the line, not 270+j0 at load!), we have VSWR=4.3 and Mismatch Loss=2.13dB, again another almost perfect match to the measured values. Without the tuner, due to the mismatch, the signal generator was not able to generate the whole power it would have produced on a matched load: power is not lost, is simply not generated. That is like when a biker is pedaling with the wrong gear: great effort, little performance. The tuner adapts the impedance at the input, exactly like the biker that shifts on the right gear. Mismatch on real transceivers Note that the mismatch effect that prevented the signal generator to generate the full power is mostly due to the fact that laboratory signal generators are designed to behave as close as possible as an ideal 50Ω generator. But being an ideal 50Ω generator, as we have seen, means low efficiency. Real transmitters are indeed designed to work on a 50Ω load, but not necessarily to present back 50Ω impedance when transmitting. Modern transceivers are able to compensate some degree of mismatch by feeding different voltages/currents to make the load happy. My FT-817 sends out the same power no matter of the load: changing the load, changes the voltage but the resulting power is almost the same until the HIGH VSWR protection kicks in by cutting the power. This kind of radio can feed mismatched lines within their VSWR tolerance without suffering loss of power, thus without the need of a tuner (I have planned to write another post reporting on this). - the claim that a given VSWR values gives a fixed loss of power is a myth deriving from a misinterpretation of the concept of “Mismatch Loss”; - if all the people that published such claim would have ever measured input and output power from a mismatched transmission lines, they would have immediately realized that true figures on power loss are most of the times very distant from their forecasts; - the power lost in the transmission line is the result of a function that combines the mismatch and the normal loss of the line in matching conditions; an ideal (lossless) line would have no loss at all no matter of the VSWR; - do not assume that feedline loss due to mismatch is always low: severe mismatches, like feeding a 40m 1/2 wave dipole on the 20m band, may cause very high losses in the transmission line; - a transmission line is an impedance transformer; - unless transmitting single bursts, the impedance of the transmitter has no relevance in the calculation of the power dissipated by the transmission line; - the mismatch between the transmission line and the transmitter might prevent it to generate its maximum power but many transmitters might be able to compensate the mismatch; - a tuner is not fooling the transceiver to believe the antenna is tuned, it is simply adapting two different impedances (after all, not many hams would describe their power supplies as objects fooling the radio to believe that the 220V AC power line is actually 13.8V DC, won’t they?); - tuner is not wasting huge amounts of power as commonly believed: many times its insertion loss is negligible (tenths of dB) even with high VSWR.

s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947473472.21/warc/CC-MAIN-20240221102433-20240221132433-00714.warc.gz

CC-MAIN-2024-10

http://www.justanswer.com/single-problem/6tkuk-firm-net-income-78-000-turnover-1-3-average.html

math

Firm B has net income of $78,000, turnover of 1.3 and average total assets of $950,000. Calcualte the firm's sales, margin and ROI. Round your percentage answer to one decimal.Firm C has net income of $132,000 turnover of 2.1, and ROI of 7.37% Calculate the firms's margin. Round to one decimal.Firm D has net income of $83,700, sales of $2,790,000 and average total assets of $1,395,000. Calculate the firms margin, turnover, and ROI Country/State/Province of question: Ohio Hi,Thanks for the questions.Please click here for the solutions. Hope this helps! If you would like to request me for future posts, please put 'For Bizhelp' at the beginning of them. BA degree and Certified Public Accountant

s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703298047/warc/CC-MAIN-20130516112138-00007-ip-10-60-113-184.ec2.internal.warc.gz

CC-MAIN-2013-20

https://terrilila.web.app/966.html

math

Social network analysis sna is probably the best known application of graph theory for data science. Transcranial direct current stimulation tdcs is an emerging approach for improving capacity in activities of daily living adl and upper limb function after stroke. Therefore, we aim to perform pairwise comparisons of the 6 sglt2 inhibitors. In the simplest possible example of a network metaanalysis, there are two treatments of interest that have been directly compared to a common comparator, but not to each other. The netmeta package in r is based on a novel approach for network metaanalysis that follows the graph theoretical methodology. Methodology from electrical networks and graphic theory also can be used to fit network metaanalysis and is outlined in by rucker rucker 2012. Using generalized linear mixed models to evaluate inconsistency within a network metaanalysis. Sep 29, 2014 the use of network meta analysis has increased dramatically in recent years. One issue with the validity of network meta analysis is inconsistency between direct and indirect evidence within a loop formed by three treatments. Software for network metaanalysis gert van valkenhoef taipei, taiwan, 6 october 20. This network metaanalysis aims to investigate the longterm 12 months efficacy of interventions for ak. In the last decade, a new statistical methodology, namely, network metaanalysis, has been developed to address limitations in traditional pairwise metaanalysis. An assessment of this assumption and of the influence of deviations is fundamental for the validity evaluation. By combining direct and indirect information, it allows assessing all possible pairwise comparisons between treatments, even when, for some pairs of treatments, no headtohead trials are available. This is a readonly mirror of the cran r package repository. A variety of interventions are available for the treatment. In this study, a network metaanalysis was performed to synthesize existing research comparing the effects of different types of cai versus the traditional instruction ti on students learning achievement in taiwan. Dec, 2018 network plot for the network of topical antibiotics without steroids for chronically discharging ears a, comparison graph corresponding to the h xy row of h matrix b, flows f uv with respect to the x versus y network metaanalysis treatment effect are indicated along the edges, streams c and proportion contributions of each direct. Frequentist network metaanalysis using the r package. Network theory provides a set of techniques for analysing graphs complex systems network theory provides techniques for analysing structure in a system of interacting agents, represented as a network applying network theory to a system means using a graph theoretic representation what makes a problem graph like. A typical way of drawing networks, also implemented in statistical software for network meta analysis, is a circular representation, often with many crossing lines. Our aim was to give an overview of the evidence network regarding the efficacy and safety of tdcs and to estimate the effectiveness of the different. This method exploits the analogy between treatment networks and electrical networks to construct the network meta analysis model accounting for the correlated treatment effects in multiarm trials. Network metaanalysis synthesizes direct and indirect evidence in a network of trials that compare multiple interventions and has the potential to rank the competing treatments according to the studied outcome. We show how graph theoretical methods can be applied to network metaanalysis. Terminology in metaanalytic networks and electrical networks. Additionally, a network metaanalysis will be conducted to determine the comparative effectiveness of the treatments with a randomeffects model. A further development in the network metaanalysis is to use a bayesian statistical approach. Network metaanalysis, electrical networks and graph. Graphical tools for network metaanalysis in stata open. Methodology from electrical networks and graphic theory also can be used to fit network metaanalysis and is outlined in by. The netmeta package in r is based on a novel approach for network meta analysis that follows the graph theoretical methodology. Gemtc r package bayesian netmeta r package frequentist. Development of restricted mean survival time difference in network metaanalysis based on data from macnpc update. The absolute and relative effectiveness of the treatments will be provided. Limitations in the design and flaws in the conduct of studies synthesized in network metaanalysis nma reduce the confidence in the results. Advanced statistical methods to model and adjust for bias in metaanalysis version 0. Preoperative hair removal and surgical site infections. Network metaanalysis was performed with r software, version i386 3. The graphtheoretical approach for network metaanalysis uses methods that were originally developed in electrical network theory. A network metaanalysis comparing the efficacy and safety. Which software can create a network metaanalysis for free. Comparative efficacy and acceptability of 21 antidepressant drugs for the acute treatment of adults with major depressive disorder. It aims to combine information from all randomized comparisons among a set of treatments for a given. Unlike r, stata software needs to create relevant ado scripts at. We conducted a network metaanalysis using two approaches. The objective of this study is to describe the general approaches to network metaanalysis that are available for quantitative data synthesis using r software. Development of restricted mean survival time difference in network meta. Graphical tools for network metaanalysis in stata pdf. Development of restricted mean survival time difference in. However, it remains unclear what type of tdcs stimulation is most effective. Network meta analysis also known as multiple treatment comparison or mixed treatment comparison seeks to combine information from all randomised comparisons among a set of treatments for a given medical condition. Winbugs, openbugs, jags bayesian by far most used, most exible meta regression software frequentist multivariate meta analysis software frequentist e. It is used in clustering algorithms specifically kmeans. Methods from graph theory usually used in electrical networks were transferred to nma. Despite these recommendations and the recent development of software statistics. All analyses will be performed using the r software version 3. A randomeffect frequentist network meta analysis model was conducted to assess pfs. The individual patient data from the metaanalysis of chemotherapy in nasopharynx carcinoma database were used to compare all available treatments. An introduction to graph theory and network analysis with. Winbugs, a freely available bayesian software package, has been the most widely used software package to conduct network meta analyses. This package allows to estimate network metaanalysis models within a frequentist framework, with its statistical approach derived from graph theoretical methods developed for electrical networks. Purpose the role of adjuvant chemotherapy ac or induction chemotherapy ic in the treatment of locally advanced nasopharyngeal carcinoma is controversial. We run the simulation study in the freely available software r 2. Briefly, for a network of n interventions and m pairwise comparisons from direct studies a m. A practical guide to network meta analysis with examples and code in the evaluation of healthcare, rigorous methods of quantitative assessment are necessary to establish which interventions are effective and costeffective. Apr 08, 2019 the objective of this study is to describe the general approaches to network meta analysis that are available for quantitative data synthesis using r software. The use of network metaanalysis has increased dramatically in recent years. Free example apply what youve learned discussion 1. Furthermore, critical appraisal of network meta analyses conducted in winbugs can be challenging. Comparing two approaches to multiarm studies in network metaanalysis. Assumes that all interventions included in the network are equally applicable to all populations and contexts of the studies included. Network metaanalysis is a generalisation of pairwise metaanalysis that compares all pairs of treatments within a number of treatments for the same condition. Most complex systems are graphlike friendship network. Winbugs, a freely available bayesian software package, has been the most widely used software package to conduct network metaanalyses. Chaimani a, higgins jpt, mavridis d, spyridonos p, salanti g graphical tools for network metaanalysis in stata anna chaimani 0 julian p. Based thereon, we then show that graph theoretical methods that have been routinely applied to electrical networks also work well in network metaanalysis. Network metaanalysis incorporates all available evidence into a general statistical framework for comparisons of all available treatments. A network metaanalysis on the effects of information and. Network meta analysis compares multiple treatments by incorporating direct and indirect evidence into a general statistical framework. Actinic keratoses ak are common precancerous lesions of the skin due to cumulative sun exposure. Network metaanalysis, a generalization of conventional metaanalysis, allows the simultaneous synthesis of data from networks of trials. This package allows to estimate network meta analysis models within a frequentist framework, with its statistical approach derived from graph theoretical methods developed for electrical networks. Package netmeta the comprehensive r archive network. Network meta analysis for decisionmaking takes an approach to evidence synthesis that is specifically intended for decision making when there are two or more treatment alternatives being evaluated, and assumes that the purpose of every synthesis is to answer the question for this preidentified population of patients, which treatment is best. A graphical tool for locating inconsistency in network meta. A network metaanalysis of nonsmallcell lung cancer. A microsoftexcelbased tool for running and critically. Forest plots are not so easy to draw for networks multiarm trials make everything more complicated but. We illustrate the correspondence between metaanalytic networks and electrical networks, where variance corresponds to resistance, treatment effects to voltage, and weighted. Network metaanalysis, electrical networks and graph theory. Decision making around multiple alternative healthcare interventions is increasingly based on metaanalyses of a network of relevant studies, which contribute direct and indirect evidence to different treatment comparisons 1,2. A network meta analysis looks at indirect comparisons. Despite its usefulness network meta analysis is often criticized for its complexity and for being accessible only to researchers with strong statistical and computational skills. The netmeta package in r is based on a novel approach for network metaanalysis that follows the graphtheoretical methodology. We also demonstrate that the fact that some statistical software commands can remove redundant collinear variables is very useful for specifying the inconsistency parameters in a network metaanalysis involving many designs and. The authors compared egfr tkis in terms of pfs in a network meta analysis. An introduction to network metaanalysis mixed treatment. Because basic metaanalysis software such as revman does not support network metaanalysis, the statistician will have to rely on statistical software packages such as stata, r, winbugs or openbugs for analysis. A graphtheoretical approach to network metaanalysis. All statistical analyses were conducted using sas statistical software version 9. Cipriani, andrea, toshi a furukawa, georgia salanti, anna chaimani, lauren z atkinson, yusuke ogawa, stefan leucht, et al. The pubmed and embase databases and meeting abstracts were screened for relevant studies between january 2009 and november 2017. It has been found to be equivalent to the frequentist approach to network metaanalysis which is based on. A network metaanalysis comparing the efficacy and safety of antipd1 with antipdl1 in nonsmall cell lung cancer. Comparative effectiveness of sodiumglucose cotransporter. A graphical tool for locating inconsistency in network metaanalyses. Methods all randomized trials of radiotherapy rt with or without chemotherapy. A network metaanalysis of nonsmallcell lung cancer patient. Let x k k1,m denote the observed effects and v k the corresponding variances. Network metaanalysis nma is becoming increasingly popular in systematic. A graph theoretical approach to multiarmed studies in frequentist network metaanalysis. Research synthesis methods, 3, 31224 rucker g, schwarzer g 2014. Association of gleason grade with androgen deprivation. Network metaanalysis for decisionmaking statistics in. In network metaanalysis, several alternative treatments can be compared by pooling the evidence of all randomised comparisons made in different studies. It aims to combine information from all randomized comparisons among a set of treatments for a given medical condition. Depends on many factors such as but not limited to. Meta analysis is a statistical technique that allows an analyst to synthesize effect sizes from multiple primary studies. Network metaanalysis is an extension of standard pairwise metaanalysis that can be used to simultaneously compare any number of treatments. Incorporated indirect conclusions require a consistent network of treatment effects. A simulation study to compare different estimation. Automated methods to test connectedness and quantify. In estimating a network metaanalysis model using a bayesian. Based thereon, we then show that graphtheoretical methods that have been routinely applied to electrical networks also work well in network metaanalysis. To estimate meta analysis models, the opensource statistical environment r is quickly becoming a popular choice. Visualizing inconsistency in network metaanalysis by. Rucker g 2012 network metaanalysis, electrical networks and graph theory. We conducted a network meta analysis using two approaches. A network metaanalysis looks at indirect comparisons. Apr 19, 2018 graph theory concepts are used to study and model social networks, fraud patterns, power consumption patterns, virality and influence in social media. A metaanalytic graph consists of vertices treatments and edges randomized comparisons. Software for network meta analysis general purpose software. A simulation study to compare different estimation approaches for. The network metaanalysis will be conducted using the netmeta package in the r software. However the relation between a and b is only known indirectly, and a network metaanalysis looks at such indirect evidence of differences between methods and interventions using statistical method. Terminology in metaanalysis and electrical networks. Network metaanalysis for decision making will be of interest to decision makers, medical statisticians, health economists, and anyone involved in health technology assessment including the pharmaceutical industry. Graph theory concepts are used to study and model social networks, fraud patterns, power consumption patterns, virality and influence in social media. Frequentist network metaanalysis using the r package netmeta. Undertaking network metaanalyses cochrane training. The statistical theory behind network meta analysis is nevertheless complex, so we strongly encourage close collaboration between dental researchers and experienced statisticians when planning and conducting a. A network metaanalysis of nonsmallcell lung cancer patients with an activating egfr mutation. Methods from graph theory usually used in electrical networks were. Chaimani a, higgins jpt, mavridis d, spyridonos p, salanti g graphical tools for network meta analysis in stata anna chaimani 0 julian p. Often a single study will not provide the answers and it is desirable to synthesise evidence from multiple sources, usually randomised controlled trials. Background the conduction and report of network metaanalysis. Network meta analysis synthesizes direct and indirect evidence in a network of trials that compare multiple interventions and has the potential to rank the competing treatments according to the studied outcome. Statistical software has been developed to fit network meta. Requires specialist statistical expertise and software. This method exploits the analogy between treatment networks and electrical networks to construct the network metaanalysis model accounting for the correlated treatment effects in. In more detail, the resulting consistent treatment effects induced in the edges can be estimated via the moorepenrose pseudoinverse of the laplacian matrix. A network metaanalysis toolkit cochrane comparing multiple. Higgins 0 dimitris mavridis 0 panagiota spyridonos 0 georgia salanti 0 0 1 department of hygiene and epidemiology, school of medicine, university of ioannina, ioannina, greece, 2 school of social and community medicine, university of bristol. The graph theoretical approach for network meta analysis uses methods that were originally developed in electrical network theory. A primer on network metaanalysis for dental research. The winbugs software can be called from either r provided r2winbugs as an r package or stata software for network metaanalysis. This approach has been implemented in the r package netmeta rucker and schwarzer 20. Inetworks aregraphs i nodesare treatments i edgesare comparisons between treatments, based on studies ivariances combine like electrical resistancesbailey, 2007 iit is possible to apply methods from electrical network theory to network metaanalysis rucker, 2012. Cappelleri, phd, mph pfizer inc invited oral presentation at the 12th annual scientific meeting of the international society for cns clinical trials and methodology, 1618. Introduction as a new class of glucoselowering drugs, sodiumglucose cotransporter 2 sglt2 inhibitors are effective for controlling hyperglycaemia, however, the relative effectiveness and safety of 6 recently available sglt2 inhibitors have rarely been studied. In the image, a has been analyzed in relation to c and c has been analyzed in relation to b. Network meta analysis is a generalisation of pairwise meta analysis that compares all pairs of treatments within a number of treatments for the same condition. Longterm efficacy of interventions for actinic keratosis. However the relation between a and b is only known indirectly, and a network meta analysis looks at such indirect evidence of differences between methods and interventions using statistical method. Scientific collaboration network business ties in us biotechindustry genetic interaction network proteinprotein interaction networks transportation networks internet. An example was used to demonstrate how to conduct a network meta analysis and the differences between it and traditional meta analysis. The statistical methodology underpinning this technique and software tools for implementing the methods are evolving. Network metaanalysis also known as multiple treatment comparison or mixed treatment comparison seeks to combine information from all randomised comparisons among a set of treatments for a given medical condition. Network metaanalysis was performed with r software version 3. Network metaanalysis nma a statistical technique that allows comparison of multiple treatments in the same metaanalysis simultaneously has become increasingly popular in the medical literature in recent years. However, the learning curve for winbugs can be daunting, especially for new users. This method exploits the analogy between treatment networks and electrical networks to construct the network metaanalysis model accounting for the correlated treatment effects in multiarm trials.1431 1158 323 487 1220 1269 543 551 526 383 1091 210 1449 1016 367 1540 536 1053 672 703 338 185 1304 431 374 641 165 498 559 383 1050 686 1489 237 53 488 847 1003 671 209 202 1397 1349 486

s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446708046.99/warc/CC-MAIN-20221126180719-20221126210719-00476.warc.gz

CC-MAIN-2022-49

https://www.houseofmath.com/encyclopedia/numbers-and-quantities/arithmetic/multiplication/how-to-multiply-by-10

math

There’s a special case you may come across: If the number you’re multiplying by isn’t a decimal number. In that case, there is always an invisible point behind the last digit. By moving the point one place to the right, you get a free place. You can fill this place with the digit . The trick of moving the decimal point when multiplying by 10, is similar to a trick you can use when dividing by 10.

s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710972.37/warc/CC-MAIN-20221204104311-20221204134311-00875.warc.gz

CC-MAIN-2022-49

https://www.unix.com/shell-programming-and-scripting/235959-how-perform-mathematical-operation-within-if-awk.html

math

10 More Discussions You Might Find Interesting 1. UNIX for Beginners Questions & Answers I am trying to generate a data of following order: 4 0 1 642 643 4 642 643 1283 1284 4 1283 1284 1924 1925 4 1924 1925 2565 2566 4 2565 2566 3206 3207 4 3206 3207 3847 3848 4 3847 3848 4488 4489 4 4488 4489 5129 5130 4 1 2 643 644 4 643 644 1284... (6 Replies) Discussion started by: SaPa 2. Shell Programming and Scripting Have three files. Any other approach with regards to file concatenation or splitting, etc is appreciated If column55(billngtype) of file1 contains YMNC or YPBC then pick the value of column13(documentnumber). Now find this documentnumber in column1(Billdoc) of file2 and grep the corresponding... (4 Replies) Discussion started by: as7951 3. Homework & Coursework Questions I am getting two result: string and int in c++ code. That I want to store into database. The request which generates result is very frequent. So each time performing db operation to store the result is costly for me. So how this can be achived using dbms_sql? I dont have any experience and how... (1 Reply) Discussion started by: karimkhan 4. Shell Programming and Scripting I want to display the distinct values in the file and for each distinct value how may occurance or there. I need to display the output like Output (2 Replies) Discussion started by: bbc17484 5. Shell Programming and Scripting Sorry, about this thread - I solved my own problem! Thanks for taking a look. edit by bakunin: no problem, but it would have been a nice touch to actually tell us what the solution was. This would have been slightlich more educating than just knowing that you found it. I changed your title to... (0 Replies) Discussion started by: Blue Solo 6. Shell Programming and Scripting I need to do a mathematical calculation between each data in 3 different files. Output is using formula (A11+B11)/(1+C11). (A11+B11)/(1+C11) (A12+B12)/(1+C12)... (3 Replies) Discussion started by: guns 7. Shell Programming and Scripting I am trying to enter a third column in this file, but the third column should that I call "Math" perform a some math calculations based on the value found in column #2. Here is the input file: Here is the desired output: GERk0203078$ Levir Math Cotete_1... (5 Replies) Discussion started by: Ernst 8. Shell Programming and Scripting I have a txt file having rows and coulmns, i want to perform some operation on a specific coulmn starting from a specific line. 50.000000 1 1 1 3.69797533E-07 871.66394 ... (3 Replies) Discussion started by: shashi792 9. Shell Programming and Scripting I would appreciate if anyone knows how to perform adding to date. As for normal date, i can easily plus with any number. But when it comes to month end say for example 28 Jun, i need to perform a plus with number 3, it will not return 1 Jul. Thanks in advance for your help. (4 Replies) Discussion started by: agathaeleanor 10. UNIX for Dummies Questions & Answers Hello one and all, I have a basic background in UNIX, but I was wondering if there is a way to perform simple mathematical equations (like multiplication, addition)? If so, is there a way to multiply values from a column by a value to produce a column with the answers? :confused: I am... (4 Replies) Discussion started by: VioletFairy

s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679103810.88/warc/CC-MAIN-20231211080606-20231211110606-00848.warc.gz

CC-MAIN-2023-50

https://www.queryhome.com/puzzle/5631/see-the-following-figure-and-find-out-the-area-of-large-square

math

Which Indian cricketer is known as "Brown Bradman"? Which of the countries has had a prime minister whose name featured the name of his country? Which of the gulf city was a dependency of Bombay Presidency under British rule for almost a century? With which field is Enrico Fermi associated? In which state of Australia would you find the area known as the Wheatbelt? What is a Bank called in Hindi? Which British laws introduced by the Importation Act of 1815 and repealed in 1846 are examples of British............... At the end of the 20th century Macau was returned to China by which country? What is the capital of Algeria? Which British sculptor and artist of the 20th century, whose work was said to “exemplify Modernism”? What is the area of the large, black square in the following image? What is the ratio of area of large square to the area of smaller square in the following image? The given figure shows a triangle and a circle enclosed in a square. Find the area of the shaded parts? The given figure shows a circle, centred at O, enclosed in a square. Find the total area of shaded parts? Each side of a square is divided into three equal parts by two points on it, which are connected as shown in the diagram below (that is not drawn to scale). If the smaller, red square has an area of 800, what is the area of the large square? Forgot Your Password? 2018 © Queryhome

s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247494424.70/warc/CC-MAIN-20190220024254-20190220050254-00482.warc.gz

CC-MAIN-2019-09

https://edurev.in/studytube/Stoichiometry-and-Stoichiometric-Calculations-Some/fe5693be-3f41-4847-8045-66fee3369dd9_t

math

Stoichiometry is the calculations of the quantities of reactants and products involved in a chemical reaction. Following methods can be used for solving problems. (a) Mole Method (For Balance reaction) (b) POAC method (Balancing not required but common sense, use it with slight care) (c) Equivalent concept CONCEPT OF LIMITING REAGENT Limiting Reagent: It is very important concept in chemical calculation. It refers to reactant which is present in minimum stoichiometry quantity for a chemical reaction. It is the reactant consumed fully in a chemical reaction. So all calculations related to various products or in sequence of reactions are made on the basis of limiting reagent. It comes into picture when reaction involves two or more reactants. For solving any such reaction, first step is to calculate L.R. Calculation of Limiting Reagent (a) By calculating the required amount by the equation and comparing it with given amount. [Useful when only two reactants are there]. (b) By calculating amount of any one product obtained taking each reactant one by one irrespective of other reactants. The one giving least product is limiting reagent. (c) Divide given moles of each reactant by their stoichiometric coefficient, the one with least ratio is limiting reagent. [Useful when the number of reactants are more than two.] The percentage yield of product The actual amount of any limiting reagent consumed in such incomplete reactions is given by: % yield × given moles of limiting reagent [For reversible reactions]. Example. A compound which contains one atom of X and two atoms of Y for each three atoms of Z is made by mixing 5 gm of X, 1.15 × 1023 atoms of Y and 0.03 mole of Z atoms. Given that only 4.40 gm of compound results. Calculate the atomic weight of Y if atomic weight of X and Z are 60 and 80 respectively. Moles of x = 5/60 = 1/12 = 0.083 Moles of z = 0.03 x + 2y + 3z → xy2z3 For limiting reagent, 0.083/1 = 0.083 0.19/2 = 0.095 , 0.03/3 = 0.01 Hence z is limiting reagent wt of xy2z3 = 4.4 gm = moles × molecular wt. Moles of xy2z3 = 1/3 × 0.03 = 0.01 300 + 2m = 440 ⇒ 2m = 440 - 300 ⇒ m =70 POAC is the simple mass conservation: KClO3 → KClO2 Apply the POAC on K. moles of K in KClO3 = moles of K in KCl 1 × moles of KClO3 = 1 × moles of KCl moles of KClO3 = moles of KCl Apply POAC on O moles O in KClO3 = moles of O in O2 3 × moles of KClO3 = 2 × moles of O2 Example 1. In the gravimetric determination of phosphorous, an aqueous solution of dihydrogen phosphate ion (H2PO4-) is treated with a mix of ammonium & magnesium ions to precipitate magnesium ammonium phosphate MgNH4 PO4.6H2O. This is heated and decomposed to magnesium Pyrophosphate, Mg2P2O7 which is weighted. A solution of H2PO4- yielded 1.054 gm of Mg2P2O7 what weight of NaH2PO4 was present originally. NaH2PO4 → Mg2P2O7 apply POAC on P Let wt of NaH2PO4 = w gm moles of P in NaH2PO4 = moles of P in Mg2P2O7 w/120 × 1 = 1.054/232 × 2 w = 1.054 x 120/232 × 2 = 1.09 gm SOME EXPERIMENTAL METHODS For determination of atomic mass Dulong's and Petit's Law: Atomic weight × specific heat (cal/gm°C) ∝ ≌ 6.4 Gives approximate atomic weight and is applicable for metals only. Take care of units of specific heat. Example 2. 7.5 mL of a hydrocarbon gas was exploded with excess of oxygen. On cooling, it was found to have undergone a contraction of 15 mL. If the vapour density of the hydrocarbon is 14, determine its molecular formula. (C = 12, H = 1) on cooling the volume contraction = 15 ml i.e The volume of H2O (g) = 15 ml V.D of hydrocarbon = 14 Molecular wt. of CxHx = 28 12x + y = 28 ...(1) 12 x + 4 = 28 12 x = 24 x = 2 Hence Hydrocarbon is C2H4. Example 3. Calculate the weight of FeO produced from 2g VO and 5.75 g of Fe2O3. Also report the limiting reagent. VO + Fe2O3 → FeO + V2O5 Solution. Balancing the given equation 2VO + Fe2O3 → 6FeO + V2O5 Mole before reaction = 2/67 = 5.75/160 0.0298 = 0.0359 = 0 = 0 Mole after reaction Mole of FeO formed = 0.0359 × 2 Weight of FeO formed = 0.0359 × 2 × 72 = 5.17 g The limiting reagent is one which is used completely, i.e. Fe2O3 here. Q.1. What is the number of moles of Fe(OH)3 (s) that can be produced by allowing 1 mole of Fe2S3, 2 moles of H2O and 3 moles of O2 to react ? 2Fe2S3 (s) + 6H2O(l) + 3O2(g ) → 4Fe (OH )3(s) + 6S(s) Ans. 1.34 moles Q.2. In a process for producing acetic acid, oxygen gas is bubbled into acetaldehyde containing manganese (II) acetate (catalyst) under pressure at 60ºC. 2CH3CHO + O2 →2CH3COOH In a laboratory test of this reaction, 20 g of CH3CHO and 10 g of O2 were put into a reaction vessel. (a) How many grams of CH3COOH can be produced ? (b) How many grams of the excess reactant remain after the reaction is complete? (a) 27.27 g (b) 2.73 g

s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00302.warc.gz

CC-MAIN-2021-21

https://www.classace.io/learn/math/3rdgrade/multiplying-by-7

math

How to Multiply by 7 7 is a fun number! 🎉 There are 7 days in a week. There are 7 colors in a rainbow. If you like music, then you know that there are 7 musical notes. In Math, multiplying by 7 is really fun! 🎉 🎊 What is multiplication? Multiplication is adding the same number repeatedly. So, what happens when you multiply a number by 7? When you multiply a number by 7, it's just like adding the same number 7 times. Multiply by 7 Take a look at the multiplication table for 7. Did you notice a pattern? 😀 The second digits goes down by 3 each time. There's no secret trick for multiplying by 7, though. You just have to memorize them. Complete the practice to help you master this times table! 😸 You'll have it memorized once you're done.

s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00414.warc.gz

CC-MAIN-2023-06

https://www.degruyter.com/document/doi/10.1515/jci-2020-0018/html

math

Understanding the pathways whereby an intervention has an effect on an outcome is a common scientific goal. A rich body of literature provides various decompositions of the total intervention effect into pathway-specific effects. Interventional direct and indirect effects provide one such decomposition. Existing estimators of these effects are based on parametric models with confidence interval estimation facilitated via the nonparametric bootstrap. We provide theory that allows for more flexible, possibly machine learning-based, estimation techniques to be considered. In particular, we establish weak convergence results that facilitate the construction of closed-form confidence intervals and hypothesis tests and prove multiple robustness properties of the proposed estimators. Simulations show that inference based on large-sample theory has adequate small-sample performance. Our work thus provides a means of leveraging modern statistical learning techniques in estimation of interventional mediation effects. Recent advances in causal inference have provided rich frameworks for posing interesting scientific questions pertaining to the mediation of effects through specific biologic pathways (Yuan and MacKinnon , Imai et al. , Valeri and VanderWeele , Pearl , Naimi et al. , Zheng and van der Laan , VanderWeele and Tchetgen Tchetgen , among others). Foremost among these advances is the provision of model-free definitions of mediation parameters, which enables researchers to develop robust estimators of these quantities. A debate in this literature has emerged pertaining to the reliance of methodology on cross-world independence assumptions that are fundamentally untestable even in randomized controlled experiments [8,9,10]. One approach to this problem is to utilize methods that attempt to estimate bounds on effects (Robins and Richardson , Tchetgen Tchetgen and Phiri , among others). A second approach considers seeking alternative definitions of mediation parameters that do not require such cross-world assumptions (VanderWeele et al. , Rudolph et al. , among others). Rather than considering deterministic interventions on mediators (i.e., a hypothetical intervention that fixes every individual mediator to a particular value), these approaches consider stochastic interventions on mediators (i.e., hypothetical interventions where the mediator is drawn from a particular conditional distribution). In this class of approaches, that of Vansteelandt and Daniel is particularly appealing. Building on the prior work of VanderWeele et al. , the authors provide a simple decomposition of the total effect into direct effects and pathway-specific effects via multiple mediators. Interestingly, their decompositions hold even when the structural dependence between mediators is unknown. Vansteelandt and Daniel described two approaches to estimation of the effects using parametric working models for relevant nuisance parameters. In both cases, the nonparametric bootstrap was recommended for inference. A potential limitation of the proposal is that correctly specifying a parametric working model may be difficult in many settings. In these instances, we may rely on flexible estimators of nuisance parameters, for example, based on machine learning. When such techniques are employed, the nonparametric bootstrap does not generally guarantee valid inference . This fact motivates the present work, where we develop nonparametric efficiency theory for the interventional mediation effect parameters. This theory allows us to utilize frameworks for nonparametric efficient inference to develop estimators of the quantities of interest. We propose a one-step and a targeted minimum loss-based estimator and demonstrate that under suitable regularity conditions, both estimators are nonparametric efficient among the class of regular asymptotically linear estimators. The estimators also enjoy a multiple robustness property, which ensures consistency of effect estimates if at least some combinations of nuisance parameters are consistently estimated. Another benefit enjoyed by our estimators is the availability of closed-form confidence intervals and hypothesis tests. 2 Interventional effects Adopting the notation of Vansteelandt and Daniel , suppose the observed data are represented as independent copies of the random variable , where is a vector of confounders, is a binary intervention, and are mediators, and is a relevant outcome. Our developments pertain to both discrete and real-valued mediators, while without loss of generality, we assume . We assume ; that is, any subgroup defined by covariates that is observed with positive probability should have some chance of receiving both interventions. We also assume that for , the probability distribution of given has density with respect to some dominating measures and this density satisfies , where the infimum is taken over . Similarly, we assume that for all . Beyond these conditions, encodes no assumptions about ; however, the efficiency theory that we develop still holds under a model that makes assumptions about , including the possibility that this quantity is known exactly, as in a stratified randomized trial. To define interventional mediation effects, notation for counterfactual random variables is required. For , and , let denote the counterfactual value for the th mediator when is set to . Similarly, let denote the counterfactual outcome under an intervention that sets and . As a point of notation, when introducing quantities whose definition depends on particular components of the random variable , we will use lower case letters to denote the particular value and assume that the definition at hand applies for all values in the support of that random variable. The total effect of intervening to set versus is , where we use to emphasize that we are taking an expectation with respect to a distribution of a counterfactual random variable. The total effect describes the difference in counterfactual outcome considering an intervention where we set and allow the mediators to naturally assume the value that they would under intervention versus an intervention where we set and allow the mediators to vary accordingly. To contrast with forthcoming effects, it is useful to write the total effect in integral form. Specifically, we use to denote the covariate-conditional mean of the counterfactual outcome , to denote the covariate-conditional bivariate cumulative distribution function of , and to denote the marginal distribution of . The total effect can be written as The total effect can be decomposed into interventional direct and indirect effects. The interventional direct effect is the difference in average counterfactual outcome under two population-level interventions. The first intervention sets , and subsequently for individuals with draws mediators from . Thus, on a population level the covariate conditional distribution of mediators in this counterfactual world is the same as it would be in a population where everyone received intervention . This is an example of a stochastic intervention . The second intervention sets and subsequently allows the mediators to naturally assume the value that they would under intervention , so that the population level mediator distribution is again . The interventional direct effect compares the average outcome under these two interventions, For interventional indirect effects, we require definitions for the covariate-conditional distribution of each mediator, which we denote for by . The interventional indirect effect through is As with the direct effect, this effect considers two interventions. Both interventions set . The first intervention draws mediator values independently from the marginal mediator distributions and , while the second intervention draws mediator values independently from the marginal mediator distributions and . The effect thus describes the average impact of shifting the population level distribution of , while holding the population level distribution of fixed. The interventional indirect effect on the outcome through is similarly defined as Note that when defining interventional indirect effects, mediators are drawn independently from marginal mediator distributions. The final effect in the decomposition essentially describes the impact of drawing the mediators from marginal rather than joint distributions. Thus, we term this effect the covariant mediator effect, defined as where . Vansteelandt and Daniel discussed situations where these effects are of primary interest. From the aforementioned definitions, we have the following effect decomposition . These component effects can be identified using the observed data under the following assumptions: the effect of on is unconfounded given , ; the effect of and on is unconfounded given and , ; the effect of on is unconfounded given , . The identifying formula for each effect can now be written as a statistical functional of the observed data distribution by substituting the outcome regression for and the observed-data mediator distributions for the respective counterfactual distributions in the aforementioned integral expressions. We note that the aforementioned assumptions preclude the existence of treatment-induced confounding of the mediator-outcome association. In the Supplementary material, we provide relevant extensions to this setting. 3.1 Efficiency theory In this section, we develop efficiency theory for nonparametric estimation of interventional effects. This theory centers around the efficient influence function of each parameter. The efficient influence function is important for several reasons. First, it allows us to utilize two existing estimation frameworks, one-step estimation [17,18] and targeted minimum loss-based estimation [19,20], to generate estimators that are nonparametric efficient. That is, under suitable regularity conditions, they achieve the smallest asymptotic variance among all regular estimators that, when scaled by , have an asymptotic normal distribution. We discuss how these estimators can be implemented in Section 3.2. The second important feature of the efficient influence function is that its variance equals the variance of the limit distribution of the scaled estimators. Thus, an estimate of the variance of the efficient influence function is a natural standard error estimate, which affords closed-form Wald-style confidence intervals and hypothesis tests (Section 3.3). Finally, the efficient influence function also characterizes robustness properties of our proposed estimators (Section 3.4). To introduce the efficient influence function, several additional definitions are required. For a given distribution , we define , commonly referred to as a propensity score. For and , we introduce the following partially marginalized outcome regressions, . We also introduce notation for the indicator function defined by if and zero otherwise. is similarly defined. Under sampling from , the efficient influence function evaluated on a given observation for the total effect is The efficient influence function for the interventional direct effect is The efficient influence function for the interventional indirect effect through is The efficient influence function for the interventional indirect effect through is The efficient influence function for the covariant interventional effect is . A proof of Theorem 1 is provided in the Supplementary material. We propose estimators of each interventional effect using one-step and targeted minimum loss-based estimation. Both techniques develop along a similar path. We first obtain estimates of the propensity score, outcome regression, and joint mediator distribution; we collectively refer to these quantities as nuisance parameters. With estimated nuisance parameters in hand, we subsequently apply a correction based on the efficient influence function to the nuisance estimates. To estimate the propensity score, we can use any suitable technique for mean regression of the binary outcome onto confounders . Working logistic regression models are commonly used for this purpose, though semi- and nonparametric alternatives would be more in line with our choice of model. We denote by the chosen estimate of . Similarly, the outcome regression can be estimated using mean regression of the outcome onto and . For example, if the study outcome is binary, logistic regression could again be used, though more flexible regression estimators may be preferred. As above, we denote by the estimated outcome regression evaluated under , with providing an estimate of . To estimate the marginal cumulative distribution of , we will use the empirical cumulative distribution function, which we denote by . Estimation of the conditional joint distribution of the mediators is a more challenging proposition, as fewer tools are available for flexible estimation of conditional multivariate distribution functions. We hence focus our developments on the development of approaches for discrete-valued mediators. The approach we adopt could be extended to continuous-valued mediators by considering a fine partitioning of the mediator values. We examine this approach via simulation in Section 4. To develop our density estimators, we use the approach of Dìaz Muñoz and van der Laan , which considers estimation of a conditional density via estimation of discrete conditional hazards. Briefly, consider estimation of the distribution of given and , and, for simplicity, suppose that the support of is . We create a long-form data set, where the number of rows contributed by each individual contribute is equal to their observed value of . An example is illustrated in Table 1. We see that the long-form data set includes an integer-valued column named “bin” that indicates to which value of each row corresponds, as well as a binary column indicating whether the observed value of corresponds to each bin. These long-form data can be used to fit a regression of the binary outcome onto , , and bin. This naturally estimates , the conditional discrete hazard of given and . Let denote the estimated hazard obtained from fitting this regression. An estimate of the density at is Similarly, an estimate of the conditional distribution of given can be obtained. An estimate of the joint conditional density is implied by these estimates, , while an estimate of the marginal distribution of is . An ID is uniquely assigned to each independent data unit and a single confounder is included in the mock data set. In principle, one could reverse the roles of and in the above procedure. That is, we could instead estimate the distribution of given and of given . Cross-validation could be used to pick between the two potential estimators of the joint distribution. Other approaches to conditional density estimation are permitted by our procedure as well. For example, approaches based on working copula models may be particularly appealing in this context, as they allow separate specification of marginal vs joint distributions of the mediators. Given estimates of nuisance parameters, we now illustrate one-step estimation for the interventional direct effect. One-step estimators of other effects can be generated similarly. A plug-in estimate of the conditional interventional direct effect given is the difference between To obtain a plug-in estimate of , we standardize the conditional effect estimate with respect to , the empirical distribution of . Thus, the plug-in estimator of is . The one-step estimator is constructed by adding an efficient influence function-based correction to an initial plug-in estimate. Suppose we are given estimates of all relevant nuisance quantities and let denote any probability distribution in that is compatible with these estimates. The efficient influence function for under sampling from is , and the one-step estimator is . All other effect estimates are generated in this vein: estimated nuisance parameters are plugged in to the efficient influence function, the resultant function is evaluated on each observation, and the empirical average of this quantity is added to the plug-in estimator. While one-step estimators are appealing in their simplicity, the estimators may not obey bounds on the parameter space in finite samples. For example, if the study outcome is binary, then the interventional effects each represent a difference in two probabilities and thus are bounded between and 1. However, one-step estimators may fall outside of this range. This motivates estimation of these quantities using targeted minimum loss-based estimation, a framework for generating plug-in estimators. The implementation of such estimators is generally more involved than that of one-step estimators. In this approach, a second-stage model fitting is used to ensure that nuisance parameter estimates satisfy efficient influence function estimating equations. The approach for this second-stage fitting is dependent on the specific effect parameter considered and the procedure differs subtly for the various effect measures presented here. Supplementary material includes a detailed exposition of how such estimators can be implemented. 3.3 Large sample inference We now present a theorem establishing the joint weak convergence of the proposed estimators to a random variable with a multivariate normal distribution. Because the asymptotic behavior of the one-step and targeted minimum loss estimators (TMLEs) is equivalent, we present a single theorem. A discussion of the differences in regularity conditions required to prove the theorem for one-step versus targeted minimum loss estimation is provided in the Supplementary material. Let denote the vector of (one-step or targeted minimum loss) estimates of and let denote the vector of efficient influence functions defined by In the theorem, we use to denote the -norm, define for any -measurable as . Under sampling from , if for , in probability as , in probability as , , , and in probability as and falls in a -Donsker class with probability tending to 1, The regularity conditions required for Theorem 2 are typical of many problems in semiparametric efficiency theory. We provide conditions in terms of -norm convergence, as this is typical of this literature; however, alternative and potentially weaker conditions are possible to derive. For further discussion, see the supplementary material. As with any nonparametric procedure, there is a concern related to the dimensionality , particularly in situations with real-valued mediators. Minimum loss estimators (MLEs) in certain function classes can attain the requisite convergence rates. For example, an MLE in the class of functions that are right-continuous with left limits (i.e., càdlàg) with variation norm bounded by a constant achieves an convergence rate faster than irrespective of the dimension of the conditioning set . However, this may not allay all concerns pertaining to the curse of dimensionality due to the fact that in moderately high dimensions, these function classes can be restrictive and thus the true function may fall outside this class. Nevertheless, we suggest (and our simulations show) that in spite of concerns pertaining to the curse of dimensionality our procedure will enjoy reasonable finite-sample performance in many settings. The covariance matrix may be estimated by the empirical covariance matrix of the vector applied to the observed data, where is any distribution in the model that is compatible with the estimated nuisance parameters. With the estimated covariance matrix, it is straightforward to construct Wald confidence intervals and hypothesis tests about the individual interventional effects or comparisons between them. For example, a straightforward application of the delta method would allow for a test of the null hypothesis that . 3.4 Robustness properties As with many problems in causal inference, consistent estimation of interventional effects requires consistent estimation only of certain combinations of nuisance parameters. To determine these combinations, we may study the stochastic properties of the efficient influence function. In particular, consider a parameter whose value under is and whose efficient influence function under sample from can be written , where is the value of the parameter of interest under . Then we may study the circumstances under which . This generally entails understanding which parameters of must align with those parameters of to ensure that the influence function has mean zero under sampling from . We present the results of this analysis in a theorem below and refer readers to the Supplementary material for the proof. Locally efficient estimators of the total effect and the intervention direct, indirect, and covariant effects are consistent for their respective target parameters if the following combinations of nuisance parameters are consistently estimated: Total effect: or Interventional direct effect: or or ; Interventional indirect effect through : or or or ; Interventional indirect effect through : or or or ; Interventional covariant effect: or or . The most interesting robustness result is perhaps that pertaining to the indirect effects. The first condition for consistent estimation is expected, as the propensity score plays no role in the definition of the indirect effect. The second condition shows that the joint mediator distribution and propensity score together can compensate for inconsistent estimation of the outcome regression, while the relevant marginal mediator distributions are required to properly marginalize the resultant quantity. The third and fourth conditions show that inconsistent estimation of the marginal distribution one, but not both, of the mediators can be corrected for via the propensity score. We note that Theorem 3 provides sufficient, but not necessary, conditions for consistent estimation of each effect. For example, a consistent estimate of the total effect is implied by a consistent estimate of and , a condition that is generally weaker than requiring consistent estimation of the outcome regression and joint mediator distribution. Because our estimation strategy relies on estimation of the joint mediator distribution, we have described robustness properties in terms of the large sample behavior of estimators of those quantities. In the Supplementary material, we provide relevant extensions to the setting where the mediator–outcome relationship is confounded by measured covariates whose distributions are affected by the treatment. In this case, both the effects of interest and their efficient influence functions involve the conditional distribution of the confounding covariates. We discuss the relevant modifications to the estimation procedures to accommodate this setting in the supplement. Generalization to other effect scales requires only minor modifications. First, we determine the portions of the efficient influence function that pertain to each component of the additive effect. For example, considering , we identify the portions of the efficient influence function that pertain to the mean counterfactual under draws of from and of from versus those portions that pertain to the mean counterfactual under draws of from and of from . We then develop a one-step or TMLE for each of these components separately. Finally, we use the delta method to derive the resulting influence function. In the Supplementary material, we illustrate an extension to a multiplicative scale. Our results can also be extended to estimation of interventional effects for more than two mediators. As discussed in Vansteelandt and Daniel , when there are more than two mediators, say , there are many possible path-specific effects. However, our scientific interest is usually restricted to learning effects that are mediated through each of the mediators, rather than all possible path-specific effects. Moreover, strong untestable assumptions are required to infer all path-specific effects, including assumptions about the direction of the causal effects between mediators. Therefore, it may be of greatest interest to evaluate direct effects such as which describes the effect of setting versus , while drawing all mediators from the joint conditional distribution given , and for , indirect effects such as which describes the effect of setting to the value it would assume under versus while drawing from their respective marginal distributions given and drawing from their marginal distribution given . We provide relevant efficiency theory for these parameters in the Supplementary material. 4.1 Discrete mediators We evaluated the small sample performance of our estimators via Monte Carlo simulation. Data were generated as follows. We simulated by drawing independently from Uniform(0,1) distributions, and independently from Bernoulli distributions with success probability of 0.25 and 0.5, respectively. The treatment variable , given , was drawn from a Bernoulli distribution with and . Here, we consider and . Given , the first mediator was generated by taking draws from a geometric distribution with success probability . Any draw of six or greater was set equal to six. The second mediator was generated from a similarly truncated geometric distribution with success probability . Given , the outcome was drawn from a Bernoulli distribution with success probability . The mediator distribution is visualized for combinations of and in Figure 1. The true total effect is approximately 0.06, which decomposes into a direct effect of 0.05, an indirect effect through of , an indirect effect through of 0.02, and a covariant effect of 0. The nuisance parameters were estimated using regression stacking [23,24], also known as super learning using the SuperLearner package for the R language . We used this package to generate an ensemble of a main-terms logistic regression (as implemented in the SL.glm function in SuperLearner), polynomial multivariate adaptive regression splines (SL.earth), and a random forest (SL.ranger). The ensemble was built by selecting the convex combination of these three estimators that minimized tenfold cross-validated deviance. We evaluated our proposed estimators under this data generating process at sample sizes of 250, 500, 1,000, and 2,000. At each sample size, we simulated 1,000 data sets. Point estimates were compared in terms of their Monte Carlo bias, standard deviation, and mean squared error. We evaluated weak convergence by visualizing the sampling distribution of the estimators after centering at the true parameter value and scaling by an oracle standard error, computed as the Monte Carlo standard deviation of the estimates, as well as scaling by an estimated standard error based on the estimated variance of the efficient influence function. Similarly, we evaluated the coverage probability of a nominal 95% Wald-style confidence interval based on the oracle and estimated standard errors. In terms of estimation, one-step and TMLE behave as expected in large samples (Figure 2). The estimators are approximately unbiased in large samples and have mean squared error appropriately decreasing with sample size. Comparing the two estimation strategies, we see that one-step and TMLEs had comparable performance for the interventional direct effect, while the TMLE had better performance for the indirect effects. However, the one-step was uniformly better for estimating the covariant effect owing to large variability of the TMLE of this quantity. Further examination of the results revealed that the second-stage model fitting required by the targeted minimum loss approach was could be unstable in small samples, leading to extreme results in several data sets. The sampling distributions of the centered and scaled estimators were approximately a standard normal distribution (Figures 3 and 4), excepting the TMLE scaled by an estimated standard error. Confidence intervals based on an oracle standard error came close to nominal coverage in all sample sizes, while those based on an estimated standard error tended to have under-coverage in small samples. 4.2 Continuous mediators We examined the impact of discretization of the mediator distributions when in fact the mediators are continuous valued. To that end, we simulated data as follows. Covariates were simulated as above. The treatment variable given was drawn from a Bernoulli distribution with . Given , and were, respectively, drawn from normal distributions with unit variance and mean values and . As above, Super Learner was used to estimate all nuisance parameters. To accommodate appropriate modeling of the interactions, we replaced the main terms GLM (SL.glm) with a forward stepwise GLM algorithm that included all two-way interactions (SL.step.interaction). The true effect sizes were approximately the same as in the first simulation. We evaluated discretization of each continuous mediator distribution into 5 and 10 evenly spaced bins. For the sake of space, we focus results on the one-step estimator; results for TMLE are included in the supplement. Overall, discretization of the continuous mediator distribution had a greater impact on the performance of indirect effect estimators compared to direct effects (Figures 5 and 6). For the latter effects, oracle confidence intervals for both levels of discretization achieved nominal coverage for all sample sizes considered. For the indirect effects, we found that there was non-negligible bias in the estimates due to the discretization. The impacts in terms of confidence interval coverage were minimal in small sample sizes, but lead to under-coverage in larger sample sizes. Including more bins generally lead to better performance, but these estimates still exhibited bias in the largest sample sizes that impacted coverage. Nevertheless the performance of the indirect effect estimators was reasonable with oracle coverage 90% for all sample sizes. 4.3 Additional simulations In the Supplementary material, we include several additional simulations studying the impact of the number of levels of the discrete mediator, as well as the impact of inconsistent estimation of the various nuisance parameters. For the former, we found that the results of the simulation were robust to number of mediator levels in the setting considered. For the latter, we confirmed the multiple robustness properties of the indirect effect estimators by studying the bias and standard deviation of the estimators in large sample sizes under the various patterns of misspecification given in our theorem. Our simulations demonstrate adequate performance of the proposed nonparametric estimators of interventional mediation effects in settings with relatively low-dimensional covariates (five, in our simulation). In certain settings, it may only be necessary to adjust for a limited number of covariates to adequately control confounding. For example, in the study of the mediating mechanisms of preventive vaccines using data from randomized trials, we need to only adjust for confounders of the mediator/outcome relationship, since other forms of confounding are addressed by the randomized design. Generally, there are few known factors that are likely to impact vaccine-induced immune responses and so nonparametric analyses may be quite feasible in this case. For example, Cowling et al. studied mediating effects of influenza vaccines, adjusting only for age. Thus, we suggest that interventional mediation estimands and nonparametric estimators thereof may be of interest for studying mediating pathways of vaccines. However, in other scenarios, it may be necessary to adjust for a high-dimensional set of confounders. For example, in observational studies of treatments (e.g., through an electronic health records system), we may require control for a high-dimensional set of putative confounders of treatment and outcome. This may raise concerns related to the curse-of-dimensionality when utilizing nonparametric estimators. Studying tradeoffs between the selection of various estimation strategies in this context will be an important area for future research. We have developed an R package intermed with implementations of the proposed methods, which is included in the Supplementary material. The package focuses on implementations for discrete mediators. However, our simulations demonstrate a clear need to extend the software to accommodate adaptive selection of the number of bins in the mediator density estimation procedure for continuous mediators. In small sample sizes, we found that course binning leads to adequate results, but as sample size increased, unsurprisingly there was a need for finer partitioning to reduce bias. In future versions of the software, we will include such adaptive binning strategies, as well as other methods for estimating continuous mediator densities. The behavior of the TMLE of the covariant effect in the simulation is surprising as generally we see comparable or better performance of such estimators relative to one-step estimators. This can likely be attributed to the fact that the targeted minimum loss procedure does not yield a compatible plug-in estimator of the vector , in the sense that there is likely no distribution that is compatible with all of the various nuisance estimators after the second-stage model fitting. A more parsimonious approach could consider either an iterative targeting procedure or a uniformly least favorable submodel that simultaneously targets the joint mediator density and outcome regression. The former is implemented in a concurrent proposal , where one-step and TMLEs of interventional effects are developed for a single mediator when the mediator–outcome relationship is subject to treatment-induced confounding. In their set up, if one can treat the treatment-induced confounder as a second mediator, then their proposal results in an estimate of one component of our indirect effect. In their simulations, they find superior finite-sample performance of the TMLE relative to the one step, suggesting that targeting the mediator densities may be a more robust approach. However, their simulation involved only binary-valued mediators, so further comparison of these approaches is warranted in settings similar to our simulation, where mediators can take many values. We leave these developments to future work. The Donsker class assumptions of our theorem could be removed by considering cross-validated nuisance parameter estimates (also known as cross-fitting) [29,30]. This technique is implemented in our R package, but we leave to future research the examination of its impact of estimation and inference. We hypothesize that this approach will generally improve the anti-conservative confidence intervals in small samples, but will have little impact on performance of point estimates in terms of bias and variance. Funding information: D. Benkeser was funded by National Institutes of Health award R01AHL137808 and National Science Foundation award 2015540. Code to reproduce simulation results is available at https://github.com/benkeser/intermed/tree/master/simulations. Conflict of interest: Prof. David Benkeser is a member of the Editorial Board in the Journal of Causal Inference but was not involved in the review process of this article. Yuan Y, MacKinnon DP. Bayesian mediation analysis. Psychol Methods. 2009;14(4):301. Search in Google Scholar Imai K, Keele L, Tingley D. A general approach to causal mediation analysis. Psychol Methods. 2010;15(4):309. Search in Google Scholar Valeri L, VanderWeele TJ. Mediation analysis allowing for exposure-mediator interactions and causal interpretation: theoretical assumptions and implementation with SAS and SPSS macros. Psychol Methods. 2013;18(2):137. Search in Google Scholar Pearl J. Interpretation and identification of causal mediation. Psychol Methods. 2014;19(4):459. Search in Google Scholar Naimi AI, Schnitzer ME, Moodie EE, Bodnar LM. Mediation analysis for health disparities research. Am J Epidemiol. 2016;184(4):315–24. Search in Google Scholar Zheng W, van der Laan MJ. Longitudinal mediation analysis with time-varying mediators and exposures, with application to survival outcomes. J Causal Infer. 2017;5(2):20160006. Search in Google Scholar VanderWeele TJ, Tchetgen Tchetgen EJ. Mediation analysis with time varying exposures and mediators. J R Stat Soc B (Statistical Methodology). 2017;79(3):917–38. Search in Google Scholar Dawid AP. Causal inference without counterfactuals. J Am Stat Assoc. 2000;95(450):407–24. Search in Google Scholar Pearl J. Direct and indirect effects. In: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence. San Francisco: Morgan Kaufmann; 2001. Search in Google Scholar Robins JM, Richardson TS. Alternative graphical causal models and the identification of direct effects. Causality and psychopathology: Finding the determinants of disorders and their cures. Oxford, New York: Oxford University Press; 2011. p. 103–58. Search in Google Scholar Tchetgen Tchetgen EJ, Phiri K. Bounds for pure direct effect. Epidemiology (Cambridge, Mass.). 2014;25(5):775. Search in Google Scholar VanderWeele TJ, Vansteelandt S, Robins JM. Effect decomposition in the presence of an exposure-induced mediator-outcome confounder. Epidemiology. 2014;25(2):300. Search in Google Scholar Rudolph KE, Sofrygin O, Zheng W, van der Laan MJ. Robust and flexible estimation of stochastic mediation effects: a proposed method and example in a randomized trial setting. Epidemiol Methods. 2018;7(1):2017007. Search in Google Scholar Vansteelandt S, Daniel RM. Interventional effects for mediation analysis with multiple mediators. Epidemiology. 2017;28(2):258. Search in Google Scholar Coyle J, van der Laan MJ. Targeted bootstrap. In Targeted learning for data science. Cham: Springer International Publishing; 2018. p. 523–39. Ch. 28. Search in Google Scholar Muñoz ID, van der Laan MJ. Population intervention causal effects based on stochastic interventions. Biometrics. 2012;68(2):541–9. Search in Google Scholar Ibragimov I, Khasminskii R. Statistical estimation: asymptotic theory. New York: Springer-Verlag; 1981. Search in Google Scholar Bickel P, Klaassen C, Ritov Y, Wellner J. Efficient and adaptive estimation for semiparametric models. Berlin Heidelberg New York: Springer; 1997. Search in Google Scholar van der Laan M, Rubin DB. Targeted maximum likelihood learning. Int J Biostat. 2006;2(1):11. Search in Google Scholar van der Laan M, Rose S. Targeted learning: causal inference for observational and experimental data. Berlin Heidelberg New York: Springer; 2011. Search in Google Scholar Dìaz Muñoz I, van der Laan MJ. Super learner based conditional density estimation with application to marginal structural models. Int J Biostat. 2011;7(1):1–20. Search in Google Scholar Benkeser D, van der Laan MJ. The highly adaptive lasso estimator. In: 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA). IEEE; 2016. p. 689–96. Search in Google Scholar Wolpert DH. Stacked generalization. Neural Netw. 1992;5:241–59. Search in Google Scholar Breiman L. Stacked regressions. Mach Learn. 1996;24:49–64. Search in Google Scholar van der Laan M, Polley E, Hubbard A. Super learner. Stat Appl Genet Mol. 2007;6(1):25. Search in Google Scholar Polley E, LeDell E, Kennedy C, van der Laan MJ. SuperLearner: Super Learner Prediction. R package version 2.0-28; 2013. https://CRAN.R-project.org/package=SuperLearnerSearch in Google Scholar Cowling BJ, Lim WW, Perera RA, Fang VJ, Leung GM, Peiris JM, et al. Influenza hemagglutination-inhibition antibody titer as a mediator of vaccine-induced protection for influenza B. Clin Infect Dis. 2019;68(10):1713–17. Search in Google Scholar Zheng W, van der Laan MJ. Asymptotic theory for cross-validated targeted maximum likelihood estimation. Technical Report 273. Berkeley: Division of Biostatistics, University of California, Berkeley; 2010. Search in Google Scholar Chernozhukov V, Chetverikov D, Demirer M, Duflo E, Hansen C, Newey W, et al. Double/debiased machine learning for treatment and structural parameters. Econom J. 2018;21(1):C1–C68. Search in Google Scholar © 2021 David Benkeser and Jialu Ran, published by De Gruyter This work is licensed under the Creative Commons Attribution 4.0 International License.

s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964361253.38/warc/CC-MAIN-20211202084644-20211202114644-00586.warc.gz

CC-MAIN-2021-49

https://physicalprinciples.wordpress.com/2016/07/29/singularity-at-the-black-hole-horizon-is-physical/

math

1. Newtonian Viewpoint Consider a massive body with mass inside a ball of radius . The Schwarzschild radius is defined by Based on the Newtonian theory, a particle of mass will be trapped inside the ball and cannot escape from the ball, if its kinetic energy, , is smaller than gravitational energy: which implies that In other words, if the radius of the ball is less than or equal to , then all particles inside the ball are permanently trapped inside the ball . It is clear that the main results of the Newton theory of black holes are as follows: - the radius of the black hole may be smaller than the Schwarzschild radius , - all particles inside the ball are permanently trapped inside the ball , and - particles outside of a black hole can be sucked into the black hole . 2. Einstein-Schwarzschild Theory Black-Holes are closed Consequently the black hole enclosed by the event horizon is closed: Nothing gets inside a black hole, and nothing gets out of the black hole either. Black holes are filled We now demonstrate that black holes are filled. Suppose there is a body of matter field with mass trapped inside a ball of radius . Then on the vacuum region , the Schwarzschild solution would be valid, which leads to non-physical imaginary time and nonphysical imaginary distance: Also, when , the TOV metric is given by This observation clearly demonstrates that the black is filled. In fact, we have proved the following black hole theorem: Blackhole Theorem (Ma-Wang, 2014) Assume the validity of the Einstein theory of general relativity, then the following assertions hold true: - black holes are closed: matters can neither enter nor leave their interiors, - black holes are innate: they are neither born to explosion of cosmic objects, nor born to gravitational collapsing, and - black holes are filled and incompressible, and if the matter field is non-homogeneously distributed in a black hole, then there must be sub-blackholes in the interior of the black hole. This theorem leads to drastically different view on the structure and geometry of black holes than the classical theory of black holes. 3. Singularity at is physical A basic mathematical requirement for a partial differential equation system on a Riemannian manifold to generate correct mathematical results is that the local coordinate system that is used to express the system must have no singularity. The Schwarzschild solution is derived from the Einstein equations under the spherical coordinate system, which has no singularity for . Consequently, the singularity of the Schwarzschild solution at must be intrinsic to the Einstein equations, and is not caused by the particular choice of the coordinate system. In other words, the singularity at is real and physical. 4. Mistakes of the classical view Many writings on modern theory of black holes have taken a wrong viewpoint that the singularity at is the coordinate singularity, and is non-physical. This mistake can be viewed in the following two aspects: A. Mathematically forbidden coordinate transformations are used. Classical transformations such as e.g. those by Eddington and Kruskal are singular, and therefore they are not valid for removing the singularity at the Schwarzschild radius. Consider for example, the Kruskal coordinates involving This coordinate transformation is singular at , since becomes infinity when . It is mathematically clear that by using singular coordinate transformations, any singularity can be either removed or created at will. In fact, many people did not realize that what is hidden in the wrong transformations is that all the deduced new coordinate systems, such as the Kruskal coordinates, are themselves singular at : all the coordinate systems, such as the Kruskal and Eddington-Finkelstein coordinates, that are derived by singular coordinate transformations, are singular and are mathematically forbidden. B. Confirmation bias. Another likely reason for the perception that a black hole attracts everything nearby is the fixed thinking (confirmation bias) of Newtonian black hole picture. In their deep minds, people wanted to have the attraction, as produced by the Newtonian theory, and were trying to find the needed “proofs” for what they believe. In summary, the classical theory of black holes is essentially the Newton theory of black holes. The correct theory, following the Einstein theory of relativity, is given in the black hole theorem above.

s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592420.72/warc/CC-MAIN-20180721071046-20180721091046-00351.warc.gz

CC-MAIN-2018-30

https://www.studymode.com/essays/Air-Pressure-Effects-The-Speed-Of-54418.html

math

An object that is falling through the atmosphere is subjected to two external forces. The first force is the gravitational force, expressed as the weight of the object. The weight equation which is weight (W) = mass (M) x gravitational acceleration (A) which is 9.8 meters per square second on the surface of the earth. The gravitational acceleration decreases with the square of the distance from the center of the earth. If the object were falling in a vacuum, this would be the only force acting on the object. But in the atmosphere, the motion of a falling object is opposed by the air resistance or drag. The drag equation tells us that drag is equal to a coefficient times one half the air density (R) times the velocity (V) squared times a reference area on which the drag coefficient is based. The motion of a falling object can be described by Newton's second law of motion, Force = mass x acceleration. Do a little algebra and solve for the acceleration of the object in terms of the net external force and the mass of the object (acceleration = Force / mass). The net external force is equal to the difference between the weight and the drag forces (Force = Weight - Drag). The acceleration of the object then becomes acceleration = (Weight - Drag) / mass. The drag force depends on the square of the velocity. So as the body accelerates, its velocity (and the drag) will increase. It will reach a point where the drag is exactly equal to the weight. When drag is equal to weight, there is no net external force on the object, and the acceleration will become equal to zero. The object will then fall at a constant velocity as described by Newton's first law of motion. The constant velocity is called the terminal velocity. What is aerodynamics? The word comes from two Greek words aerios concerning the air, and dynamis, meaning powerful. Aerodynamics is the study of forces and the resulting motion of objects through the... Bibliography: http://www.dynamicscience.com.au/tester/solutions/flight/parachute%20experiment.htm (10/13/04) Please join StudyMode to read the full document

s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156622.36/warc/CC-MAIN-20180920214659-20180920235059-00003.warc.gz

CC-MAIN-2018-39

https://ldtopology.wordpress.com/2011/01/18/is-knot-theory-obvious-but-hard/

math

Recently, there was a soft question on MathOverflow asking for examples of theorems which are `obvious but hard to prove’. There were three responses concerning pre-1930 knot theory, and I didn’t agree with any of them. This led me to wonder whether there might be a bit of a consensus in the mathematical community that knot theory is really much more difficult than it ought to be; and that good knot theory should be all about combinatorics of knot diagrams. And so knot colouring becomes `good knot theory’ for what I think are all the wrong reasons. Today’s story begins in 1956, when Ralph Fox gave an amazingly good talk to undergraduate students at Haverford College. So good was his talk in fact that it actually changed the history of topology (how I dream of giving `the ultimate talk’!). His talk was about coloured knots- but instead of introducing them via homomorphisms from the knot group onto a dihedral group like we did last time, he introduced knot colourings by physically colouring arcs of knot diagrams red, blue, and green subject to Wirtinger rules (of course he didn’t name them that), and he proved invariance by showing that tricolourability is preserved by Reidemeister moves. Thus was born the Fox n-colouring. It’s quite beautiful and unexpected, and if I were giving an introductory talk to undergrads or to high-school students, surely I would imitate Fox’s approach. Seeing tricolourability introduced as Fox presented it is surely inspiring. On the other hand, successful popular exposition is always a mixed blessing; it’s wonderful when people find a certain facet of our work inspiring, but painful when they take away oversimplified messages which miss the point. In the case of knot colourings, that point is that a colouring isn’t an arbitrary parlour trick, but has sound mathematical basis as a group homomorphism. As such, the concept of a fundamental group is essential for colourings. Otherwise what possible reason would there be for tricolourability to be a knot invariant? None at all. I contend that the mathematical public’s impression of knot theory is heavily influenced by Ralph Fox’s oustanding talk of half a certury ago. The impression is that knot theory should be about arbitrary combinatorial games with knot diagrams, which magically happen to work. And any algebraic topology gets deemed a `hard proof of an obvious fact’. I suppose that the first question really to be asking oneself is what is a knot? I think of a knot as being a smooth (or PL) embedding of a circle into 3-space, modulo ambient isotopy. This is an essentially topological definition, and use of the fundamental group is surely not “over the top” in the study of knots thus defined. Another way of thinking of knots would be as knot diagrams modulo Reidemeister moves. These are combinatorial objects in the plane. Such objects have an operadic nature; anyway, they’re much more complex mathematical objects than (fundamental) groups. I don’t see why one should expect it to be “easy” to distinguish knots by looking at their knot diagrams. Thus, I’m not surprised that I find algebraic topological methods to distinguish knots to be conceptually more natural and easier to understand; although perhaps not quite as much fun as understanding knots using their knot diagrams, especially if we are working in the quantum realm.

s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120349.46/warc/CC-MAIN-20170423031200-00530-ip-10-145-167-34.ec2.internal.warc.gz

CC-MAIN-2017-17

https://www.smonad.com/books/dot.php

math

The Duality of Time Theory, that results from the Single Monad Model of the Cosmos, explains how multiplicity is emerging from absolute Oneness, at every instance of our normal time! This leads to the Ultimate Symmetry of space and its dynamic formation and breaking into the physical and psychical (supersymmetrical) creations, in orthogonal time directions. General Relativity and Quantum Mechanics are complementary consequences of the Duality of Time Theory, and all the fundamental interactions become properties of the new granular complex-time geometry. - => Conference Talk [Detailed Presentation] Welcome to the Single Monad Model of the Cosmos ( Please log in or register!) This short book presents a brief and concise exploration of the Duality of Time postulate and its consequences on General Relativity and Quantum Mechanics. To make it easier for citing, this book is presented in the form of a scientific paper, which will also make it more accessible and easier to be read by researchers who are interested in the astounding conclusions rather than any exhausting introductions which are provided in the previous books for more general readability. This article explains: Based on the Single Monad Model and Duality-of-Time hypothesis, a dynamic and self-contained space-time is introduced and investigated. It is shown that the resulting “time-time” geometry is genuinely complex, fractal and granular, and that the non-Euclidean space-time continuum is the first global approximation of this complex-time space in which the (complex) momentum and energy become invariant between different inertial and non-inertial frames alike. Therefore, in addition to Lorentz transformation, the equivalence principle is derived directly from the new discrete symmetry. It is argued that according to this postulate, all the principles of relativity and quantum theories can be derived and become complementary. The Single Monad Model provides a profound understanding of time as a complex-scalar quantum field that is necessary and sufficient to obtain exact mathematical derivation of the mass-energy equivalence relation, in addition to solving many persisting problems in physics and cosmology, including the arrow-of-time, super-symmetry, matter-antimatter asymmetry, mass generation, homogeneity, and non-locality problems. It will be also shown that the resulting physical vacuum is a perfect super-fluid that can account for dark matter and dark energy, and diminish the cosmological constant discrepancy by at least 117 orders of magnitude. Relativity, and its classical predecessor, consider space and time to be continuous, and everywhere differentiable, whereas quantum mechanics is based on discrete quanta of energy and fields, albeit they still evolve in continuous background. Although both theories have already passed many rigorous tests, they inevitably produce enormous contradictions when applied together in the same domain. Most scholars believe that this conflict may only be resolved with a successful theory of quantum gravity . In trying to resolve the discrepancy, some space-time theories, such as Causal Dynamical Triangulation , Quantum Einstein Gravity and Scale Relativity , attempted to relax the condition of differentiability, in order to allow for fractal space-time, which was first introduced in 1983 . In addition to the abundance of all kinds of fractal structures in nature, this concept was also supported by many astronomical observations which show that the Universe exhibit a fractal aspect over a fairly wide range of scales , and that large-scale structures are much better described by a scale-dependent fractal dimension , but the theoretical implications of these observations are not very well understood, yet. Nonetheless, the two most celebrated approaches to reconcile Relativity with Quantum Mechanics are Strings Theory and Loop Quantum Gravity (LQG). The first tries to develop an effective quantum field theory of gravity at low energies, by postulating strings instead of point particles, while LQG uses spin networks to obtain granular space that evolves with time. Therefore, while Strings Theory still depends on the background continuum, LQG tries to be background-independent by attempting to quantize space-time itself . In this regard, the author believes that any successful theory of quantum gravity must not rely on either the continuum or discretuum structures of space-time. Rather, these two contrasting and mutually-exclusive views must be the product of such theory, and they must become complementary on the microscopic and macroscopic scales. The only contestant that may fulfill this criterion is “Oneness”, because on the multiplicity level things can only be either discrete or continuous; there is no other way. However, we need first to explain how the apparent physical multiplicity can proceed from this metaphysical oneness, and then exhibit various discrete and continuous impressions. The key to resolve this dilemma is in understanding the “inner levels of time” in which “space” and “matter” are perpetually being “re-created” and layered into the three spatial dimensions, which then kinetically evolve throughout the “outer level of time” that we encounter. This will be fully explained in sections 2 and 4 below. Due to this “dynamic formation of dimensions”, in the inner levels of time, the Duality of Time Theory leads to granular and self-contained space-time with fractal and genuinely-complex structure, which are the key features needed to accommodate both quantum and relativistic phenomena. Many previous studies have already shown how the principles of quantum mechanics can be derived from the fractal structure of space-time [9, 10, 11, 12, 13], but they either do not justify the use of fractals, or they are forced to make new unjustified assertions, such as the relativity of scale, that may lead to fractal space-time. On the other hand, imaginary time had been successfully used in the early formulation of Special Relativity by Poincare , and even Minkowski , but it was later replaced by the Minkowskian four-dimensional space-time, because there were no substantial reasons to treat time as imaginary. Nevertheless, this concept is still essential in current cosmology and quantum field theories, since it is employed by Feynman’s path integral formulation, and it is the only way to avoid singularities which are unavoidable in General Relativity. In the Duality of Time Theory, since the dimensions of space and matter are being re-created in the inner (complete) levels of time, the final dimension becomes multi-fractal and equals to the dynamic ratio of “inner” to “outer” times. Additionally, and for the same reason, space-time becomes “genuinely complex”, since both its “real” and “imaginary” components have the same nature of time, which itself becomes as simple as the “recurrence”, or counting the number of geometrical nodes as they are re-created in one chronological sequence. Without postulating the inner levels of time, both the complex and fractal dimensions would not have any “genuine” meaning, unless both the numerator and denominator of the fraction, and both the real and imaginary parts of the complex number, are all of the same nature (of time). In this manner, normal time is an imaginary and fractional dimension of the complete dimensions of space, which are the real levels of time. Because they are complete integers, the dimensions of space are mutually perpendicular, or spherically orthogonal, on each other, which is what makes (isotropic and homogeneous) Euclidean geometry that can be expressed with normal complex numbers , in which the modulus is given by . In contrast, because it is fractional or non-integer dimension, (normal, or the outer level of) time is hyperbolically orthogonal on the dimensions of space, and thus expressed by the hyperbolic split-complex numbers , in which the modulus is given by . This complex hyperbolic geometry is the fundamental reason behind relativity and Lorentz transformations, and it provides the required tools to express the curvature and topology of space-time, away from Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities. The Duality of Time Theory, and the resulting dynamic re-creation of space and matter, is based on previous research that presented an eccentric conception of time [16, 17, 18, 19, 20, 21, which include other references on the history and philosophical origins of this concept]. For the purpose of this article, this hypothesis can be extracted here into the following postulate: The above postulate means that at every instance of the “real flow of time” there is only one metaphysical point, that is the unit of space-time geometry, and the Universe is a result of its perpetual recurrence in the “inner levels of time”, that is continuously re-creating the dimensions of space and what it may contain of matter particles, which then kinetically evolve throughout the outer (normal) level of time, that we encounter. To understand this complex flow of time, we need to define at least two frames of reference. The first is our normal “space” container which evolves in the outer time, that is the normal time that we encounter. And the second frame is the inner flow of time, that is creating the dimensions of space and matter. This inner frame is also composed of more inner levels to account for the creation of and space, but we shall not discuss them at this point. From our point to view, as observers situated in the outer frame, the creation process is instantaneous, because we only see the Universe after it is created, and we don’t see it in the inner frames when it is being created, or perpetually re-created, at every instance. Nevertheless, the speed of creation, in the innermost level (or real flow) of time, is indefinite, rather than infinite, because there is nothing to compare to it at this level of absolute oneness. We shall show that this speed of creation is the same speed of light, and the reason why individual observers, situated in the outer frame, measure a finite value of it is because they are subject to the time lag during which the spatial dimensions are being re-created. Therefore, in our outer frame, the speed of creation, that is the speed of light, is simply equal to the ratio of the outer to inner times, so it is a unit-less number whose normalized value corresponds to the fractal dimension of the genuinely-complex time-time geometry, rather than space-time, since space itself is created in the inner levels of time. The reason why this cosmological speed is independent of the observer is because creation is occurring in the inner real levels while physical motion is in the outer (normal) time that is flowing in the orthogonal dimension with relation to the real dimensions of space (or inner levels of time). In other words, while the real time is flowing unilaterally in one continuous sequence, creating only one metaphysical point at every instance, individual observers witness only the discrete moments in which they are re-created, and during which they observe the dimensions of space and physical matter that have just been re-created in these particular instances; thus they only observe the collective (physical) evolution as the moments of their own time flows by, and that’s why it becomes imaginary, or latent, with relation to the original real flow of time that is creating space and matter. Therefore, the speed of light in complete vacuum is the speed of its dynamic formation, and it is undefined in its own reference frame (as it can be also inferred from the current understanding of time dilation and space contraction of special relativity), because the physical dimensions are not yet defined at this metaphysical level. Observers in all other frames, when they are re-created, measure a finite value of this speed because they have to wait their turn until the re-creation process is completed, so any minimum action, or unitary motion, they can do is always delayed by an amount of time proportional to the dimensions of vacuum (and its matter contents if they are measuring in any other medium). Hence, this maximum speed that they can measure is also invariant because it is independent of any physical (imaginary) velocity, since their motion is occurring in the outer time dimension that is orthogonal onto the spatial dimensions which are being re-created in the inner (real) flow of time. This also means that all physical properties, including mass, energy, velocity, acceleration and even the dimensions of space, are emergent properties; observable only on the outward level of time, as a result of the temporal coupling between at least two geometrical points or complex-time instances. Moreover, just like the complete dimensions of space, the outer time itself, which is a fractional dimension, is also emerging from the same real flow of time that is the perpetual recurrence of the original geometrical point. This metaphysical entity that is performing this creation is called “the Single Monad”, that has more profound characteristics which we don’t need to analyze in this paper (see [18, Ch. VI] for more details); so we only consider it as a simple abstract or dimensionless point: . It will be shown in section 5 how this single postulate leads at the same time to all the three principles of Special and General Relativity together, since there is no more any difference between inertial and non-inertial frames, because the instantaneous velocity in the imaginary time is always “zero”, whether the object is accelerating or not! This also means that both momentum and energy will be “complex” and “invariant” between all frames, as we shall discuss further in sections 5.3 and 5.4 below. Henceforth, this genuinely-complex time, or time-time geometry will define a profound discrete symmetry that allows expressing the (deceitfully continuous) non-Euclidean space-time in terms of its granular and fractal complex-time space, whose granularity and fractality are expressed through the intrinsic properties of hyperbolic numbers (), i.e. without invoking Riemannian geometry, as discussed further in section 4.1. However, this hidden discrete symmetry is revealed only when we realize the internal chronological re-creation of spatial dimensions; otherwise if we suppose their continuous existence, space-time will still be hyperbolic but not discrete. Discreteness is introduced when the internal re-creation is interrupted to manifest in the outward normal time, because creation is processed sequentially by the perpetual recurrence of one metaphysical point, so the resulting complex-time is flowing either inwardly to create space, or outwardly as the normal time, and not both together. Therefore, in accordance with the correspondence principle, we will see in section 4.3, that semi-Riemannian geometry on is a special approximation of this discrete complex-time geometry on . This approximation is implicitly applied when we consider space and matter to be coexisting together in (and with) time, thus causing the deceptive continuity of physical existence, which is then best expressed by the non-Euclidean Minkowskian space-time continuum of General Relativity, or de Sitter/anti-de Sitter space, depending on the value of cosmological constant. For the same reason, because we ideally consider the dimensions of space to be continuously existing, our observations become time-symmetric, since we can apparently-equally move in opposite directions. Therefore, this erroneous time-symmetry is reflected in most physics laws because they also do not realize the sequential metaphysical re-creation of space, and that is why they fail in various sensitive situations such as the second law of Thermodynamics (the entropic arrow of time), Charge-Parity violations in certain weak interactions, as well as the irreversible collapse of wave-function (or the quantum arrow of time). In the Duality of Time Theory, the autonomous progression of the real flow of time provides a straightforward explanation of this outstanding historical problem. This will be explicitly expressed by equation 1, as discussed further in section 4.2, where we will also see that we can distinguish between three conclusive states for the flow of complex-time: either the imaginary time is larger than the real time, or the opposite, or they are equal. Each of the first two states forms one-directional arrow of time, which then become orthogonal, while the third state forms a two-directional dimension of space, that can be formed by or broken into the orthogonal time directions. This fundamental insight could provide an elegant solution to the problems of super-symmetry and matter-antimatter asymmetry at the same time, as we shall discuss in section 4.2. Additionally, the genuine complex-time flow will be employed in section 5.2 to derive the mass-energy equivalence relation , in its simple and relativistic forms, directly from the principles of Classical Mechanics. This should provide a conclusive evidence to the Duality of Time hypothesis, because it will be shown that an exact derivation of this experimentally verified relation is not possible without considering the inner levels of time, since it incorporates motion at the speed of light which leads to infinities on the physical level. All current derivations of this critical relation suffer from unjustified assumptions or approximations [22, 23, 24, 25, 26], as was also repeatedly acknowledged by Einstein himself [25, 27]. Finally, as an additional support to the Duality of Time Theory, we will show in section 4.4 that the resulting dynamic quintessence will diminish the cosmological constant discrepancy by at least 117 orders of magnitude. This huge difference results simply from realizing that the modes of quantum oscillations of vacuum occur in chronological sequence, and not all at the same time. Therefore, we must divide by the number of modes included in the unit volume, to take the average, rather than the collective summation as it is currently treated in quantum field theories. The remaining small discrepancy could be also settled based on the new structure of physical vacuum, which is shown to be a perfect super-fluid. The Duality of Time Theory, therefore, brings back the same classical concept of aether but in a novel manner that does not require it to affect the speed of light, because it is now the background space itself, being granular and re-created dynamically in time, and not something in a fixed background continuum that used to be called vacuum. On the contrary, this dynamical aether provides a simple ontological reason for the constancy and invariance of the speed of light, which is so far considered an axiom that has not been yet proven in any theoretical sense. The Duality of Time Theory provides a deeper understanding of time as a fundamental complex-scalar quantum field that reveals the discrete symmetry of space-time geometry. This revolutionary concept will have tremendous implications on the foundations of physics, philosophy and mathematics, including geometry and number theory; because complex numbers are now genuinely natural, while the reals are one of their extreme, or unrealistic, approximations. Many major problems in physics and cosmology can be resolved according to the Duality of Time Theory, but it would be too distracting to discuss all that in this introductory article. The homogeneity problem, for example, will instantly cease, since the Universe, no matter how large it could be, is re-created sequentially in the inner levels of time, so all the states are synchronized before they appear as one instance in the normal level. Philosophically also, since space-time is now dynamic and self-contained, causality itself becomes a consequence of the sequential metaphysical creation, and hence the fundamental laws of conservation are simply a consequence of the Universe being a closed system. This will also explain non-local and non-temporal causal effects, without breaking the speed of light limit, in addition to other critical quantum mechanical issues, some of which are outlined in other publications [18, 20, 21]. According to the above Duality of Time postulate, the dynamic Universe is the succession of instantaneous discrete frames of space, that extend in the outward level of time that we normally encounter, but each frame is internally created in one chronological sequence within each inward level of the real flow of time. This is schematically demonstrated in Figure 1, where space is conventionally shown in two dimensions, as the plane, and we will mostly consider the axis only, for simplicity. In reality, however, we can conceive of at least seven levels of time, which curl to make the four dimensions of space-time: , that are the three spatial and one temporal dimensions; since each spatial dimension is formed by two of the six inner levels, as we shall explain further in section 4.2, while the seventh is the outer time that we normally encounter. As it will be explained further in section 4.2 below, each spatial dimension is dynamically formed by the real flow of time, and whenever this flow is interrupted, a new dimension starts, which is achieved by multiplying with the imaginary unit that produces an “abrupt rotation” by , creating a new dimension that is perpendicular on the previous level, or hyperbolically orthogonal on it, to be more precise. This subtle property is what introduces discreteness, as a consequence of the duality nature of time, that is flowing either inwardly or outwardly, not both together. This is what makes space-time geometry genuinely complex and granular, otherwise if we consider all the dimensions to be coexisting together it will appear continuous and real, as we normally “imagine”, which may lead to space-time singularities at extreme conditions. The concept of imaginary time is already being used widely in various mathematical formulations in quantum physics and cosmology, without any actual justification apart from the fact that it is a quite convenient mathematical trick that is useful in solving many problems. As Hawking states: “It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed, but also effects we have not been able to measure yet nevertheless believe in for other reasons.” . Hawking, however, considers the imaginary time as something that is perpendicular to normal time that exists together with space, and that’s how it is usually treated in physics and cosmology. According to the Duality of Time postulate, however, since space is now (dynamically re-created in) the real time, the normal time itself becomes genuinely imaginary, or latent. Employing imaginary time is very useful because it provides a method for connecting quantum mechanics with statistical mechanics by using a Wick rotation, by . In this manner we can find a solution to dynamics problems in dimensions, by transposing their descriptions in dimensions, i.e. by trading one dimension of space for one dimension of time, which means substituting a mathematical problem in Minkowski space-time into a related problem in Euclidean space. Schroedinger equation and the heat equation are also related by Wick rotation. This method is also used in Feynman’s path integral formulation, which was extended in 1966 by DeWitt into gauge invariant functional-integral . For this reason, there has been many attempts to describe quantum gravity in terms of Euclidean geometry [30, 31], because in this way it is possible to avoid singularities which are unavoidable in General Relativity, since it is primarily constructed on curved space-time continuum that uses Riemannian manifolds, in which the geometry becomes ill-defined at the points of singularities. Mathematically, the nested levels of time can be represented by imaginary or complex numbers where space is treated as a plane or spherical wave, and time is the orthogonal imaginary axis. However, in addition to the normal complex number plane: , that can describe Euclidean space, split-complex, or hyperbolic, numbers: , are required to express the relation between space-like and time-like dimensions, which are the inner and outer levels of time, respectively. Normal complex numbers can describe homogeneous or isomorphic space, without (the outer) time, where each number defines a circle, or sphere, because its modulus is given by , while in split-complex numbers the modulus is given by , so , which defines a hyperbola. This negative sign in calculating the modulus of complex time reflects the essential fact that the perpetual re-creation of space and matter particles in the inner levels of time is interrupted and re-newed every instance of the outward time , which produces kinetic motions on the physical level, as dynamic local deformations of the otherwise flat and homogeneous Euclidean space. Therefore the non-Euclidean Minkowski space-time coordinates are an approximation of the complex space-time coordinates , or complex time-time represent the inner In this abstract complex frame, space and time are absolute, or mathematical, just as they had been originally treated in the classical Newtonian Mechanics, but now empty space is The physical vacuum, which is the dynamic aether, is therefore an extreme state which may be achieved when the apparent velocity, or momentum, becomes absolutely zero, both as the object’s total velocity and any vector velocities of its constituents, and this corresponds to absolute zero temperature (). This dynamic vacuum state is therefore a super-fluid, which is a perfect Bose-Einstein condensate (BEC), since it consists of indistinguishable geometrical points that all share the same state. In quantum field theory, complex-scalar fields are employed to describe superconductivity and superfluidity . The Higgs field itself is complex-scalar, and it is the only fundamental scalar quantum field that has been observed in nature, but there are other effective field theories that describe various physical phenomena. Indeed, some cosmological models have already suggested that vacuum could be a kind of yet-unknown super-fluid, which would explain all the four fundamental interactions and provide mass generation mechanism that replaces or alters the Higgs mechanism that only partially solves the problem of mass. In BEC models, masses of elementary particles can arise as a result of interaction with the super-fluid vacuum, similarly to the gap generation mechanism in superconductors , in addition to other anticipated exotic properties that could explain many problems in the current models, including dark matter and dark energy [35, 36]. Therefore, the new complex-time geometry is the natural complex-scalar quantum field that explains the dynamic generation of space, mass and energy. We will discuss the origin of mass in sections 4.5 and 5.2.3 below. Actually, according to this genuinely-complex time-time geometry, there can be four absolute or “super” states: super-mass , super-fluid , super-gas , and super-energy , which can be compared with the classical four elements: earth, water, air, and fire, respectively. These four extreme or elemental states, which the ancient Sumerians employed in their cosmology to explain the complexity of Nature, are formed dynamically, in the inner levels of time, by the Single Monad that is their “quint-essence”. We will see, in section 4.4 below, that this new concept of aether and quintessence is essential for understanding dark matter and energy, and solving the cosmological constant discrepancy. Moreover, the super-fluid and super-gas states, and , are in orthogonal time directions, so if describes matter that is kinetically evolving in the normal level of time with velocity, would similarly describe anti-matter in the orthogonal direction. This could at once solve the problems of super-symmetry and matter-antimatter asymmetry, because fermions in one time direction are bosons in the orthogonal dimension, and vice versa, and of course these two dimensions do not naturally interact because they are mutually orthogonal. This could also provide some handy tests to verify the Duality of Time Theory, but this requires prolonged discussion beyond the scope of this article, as outlined in other literature [20, 21]. Super-symmetry and its breaking will be also discussed further in section 6.1. Discreteness implies interruption or discontinuity, and this is what the outer time is doing to the continuous flow of the inner time that is perpetually re-creating space and matter in one chronological sequence. Mathematically, this is achieved by multiplying with the imaginary unit, which produces an “abrupt rotation” by , creating a new dimension that is orthogonal on the previous level. Multiplying with the imaginary unit again causes time to become real again, i.e. like space. This means that each point of our space-time is the combination of seven dimensions of time, the first six are the real levels which make the three spatial dimensions, , and the seventh is the imaginary level that is the outer time . This outward (normal) level of time, , is interrupting and delaying the real flow of time, , so it can not exceed it, because they both belong to one single existence that is flowing either in the inward levels to form the continuous (real) spatial dimensions, or in the outward level to form the imaginary discrete time, not the two together; otherwise they both would be real as we are normally deceived. As we introduced in section 2 above, the reason for this deception is because we only observe the physical dimensions, in the outer time, after they are created in the inner time, so we “imagine” them to be co-existing continuously, when in fact they are being sequentially re-created. It is not possible otherwise to obtain self-contained and granular space-time, whose geometry could be defined without any previous background topology. Thus, we can write: So because is interrupting and delaying , the actual (net value of) time is always smaller than the real time: , and this is actually the proper time as we shall see in equation 3. However, it should be noted here that, unlike the case for normal complex (Euclidean) plane, the modulus of split-complex numbers is different from the norm, because it is not positive-definite, but it has a metric signature . This means that, although our normal time is flowing only in one direction because it is interrupting the real flow of creation and can not exceed it, it is still possible to have the orthogonal state where the imaginary time is flowing at the speed of creation and the real part is interrupting it, such that: , so: , and then , from our perspective. In this case, the ground state of that vacuum would be , which describes anti-matter as we shall explain further in section 6.1, when we speak about super-symmetry and its breaking. Equivalently, the apparent velocity can not exceed because it is the average of all instantaneous velocities of all individual geometrical points that constitute the object, which are always fluctuating between and ; so by definition is capped by , as expressed by equation 5. Therefore, equation 1 () is also equivalent to: , thus when: , we get: , and if , then ; but both as the total apparent velocity of the object and any vector velocities of its constituents, thus in this case we have flat and infinite Euclidean space without any motion or disturbance, which is the state of vacuum: , or , as we noted in section 4.1. So this imaginary time, , is acting like a resistance against the perpetual re-creation of space, and its interruption, i.e. going in the outward level of time, is what causes physical motion and the inertial mass , which then effectively increases with the imaginary velocity according to: (as we shall derive it in section 5.2, and we shall discuss mass generation in sections 4.5 and 5.2.3), and when the outward imaginary time approaches the inner time, the apparent velocity approaches the speed of creation , and . If this extreme state could ever happen (but not by acceleration, as we shall see further below), the system would be described by , which means that both the real and imaginary parts of complex-time would be continuous, and this describes another homogeneous Euclidean space with one higher dimension than the original vacuum. Actually, the hyperbolic split-complex number is non-invertible null vector that describes the asymptotes, whose modulus equals zero, since both its real and imaginary parts are equal. At the same time, as a normal complex number, describes an isotropic infinite and inert Euclidean space (without time), because its dimensions are continuous, or uninterrupted. The metaphysical entities of the Universe are sequentially oscillating between the two vacuum states (as Euclidean spaces or normal complex numbers): and , while collectively they appear to be evolving according to the physical (hyperbolic) space-time states , as split-complex numbers. Therefore, the vacuum state can be described either as the non-invertible vector in the hyperbolic plane , and that is equal to one absolute point from the time perspective (when we look at the world from outside), or an isotropic Euclidean space as normal complex numbers , but with one lower dimension, and that is the space perspective (when we look from inside). Infinities and singularities occur when we confuse between these two extreme views; because if the observer is situated inside a spatial dimension it will appear to them continuous and infinite, while it forms only one discrete state in the encompassing outer time. As we shall see in section 4.3, General Relativity is the first approximation for inside observers, but since the Universe is evolving we need to describe it by , from the time perspective. So GR is correct every instance of time, because the resulting instantaneous space is continuous, but when the outward time flows these instances will form a series of discrete states that should be described by Quantum Field Theory. If we combine these two descriptions properly, we should be able to eliminate GR singularities and QFT infinities. In other words, the whole homogeneous space forms a single point in the outer time, and our physical Universe is the dynamic combination of these two extreme states, denoted as space-time. This is the same postulated statement that the geometrical points are perpetually and sequentially fluctuating between (for time) and (for space), and no two points can be in the state of (existence in) space at the same real instance of time, so the points of space come into existence in one chronological sequence, and they can not last in this state for more than one single moment of time, thus they are being perpetually re-created. Nonetheless, since it is not possible to accelerate a physical object (to make all its geometrical points) to move at the speed of creation , one alternative way to reach this speed of light, and thus make a new spatial dimension, is to combine the two orthogonal states and , which is the same as matter-antimatter annihilation, and this is a reversible interaction that can be described by the following equation: In conclusion, we can distinguish three conclusive scenarios for the complex flow of time: Therefore, there are two orthogonal arrows of time and , that can combine and split between the states of and , which all together correspond to the four elemental states, or classical elements, whose quintessence is the Single Monad . On the other hand, as we can see from Figure 1, the space-time interval can be obtained from: , or for motion on x-axis only. Alternatively, we can now use the new time-time interval which is the modulus of complex time: , and it is indeed the same proper time, , in Special Relativity: The reason why we are getting the negative signature here is because we exist in the imaginary dimension, and that is why we need some “time extension” to perceive the dimensions of space, that is the real dimension. For example, we need at least three instances to imagine any simple segment; one for each side and one for the relation between them; so we need infinite time to conceive the details of all space. If we exist in the real dimensions of space we would conceive it all at once, as what happens in the event horizon of a black hole. So for us it appears as if time is real and space is imaginary, while the absolute reality is a reflection, and the actual Universe is the dynamic and relative combination of these two extreme states. This essential property, that the outward time is effectively negative with respect to the real flow of time, will be inherited by the velocity, momentum and even energy; all of which will be similarly negative in relation to their real counter part. It is this fundamental property that will enable the derivation of the relativistic momentum-energy relation, the equivalence between inertial and gravitational masses, in addition to allowing energy and mass to become imaginary, negative and even multidimensional. This will be discussed further in sections 5.1, 5.2 and 5.4, respectively. The representation of space-time with imaginary time was used in the early formulation of Special Relativity, by Poincare , and even Minkowski , but because there were no substantial reasons to treat time as imaginary, Minkowski had to introduce the four-dimensional space-time: , with Lorentzian metric , in which time and space are treated equally, except for the minus. This four-dimensional space later became necessary for General Relativity, due to the presence of gravity, which required Riemannian geometry to evaluate space-time curvatures. In the split-complex hyperbolic geometry, Lorentz transformations become rotations in the imaginary plane , and according to the new discrete symmetry of the time-time frame, this transformation will be equally valid between inertial and non-inertial frames alike, because the dynamic relation between the real and imaginary parts of time implies that the instantaneous velocity in the imaginary time is always In the Theory of Relativity, we need to differentiate between inertial and non-inertial frames, because we are considering the “apparent velocity”, since the observer is measuring the change of position (i.e. space coordinates) with respect to time, thus implicitly assuming their real co-existence and continuity; so considering motion to be real transmutation, and that is why space and time are considered continuous and differentiable. The observer is therefore not realizing the fact that the dimensions of space are being sequentially re-created within the inner levels of time, as we described above. This sequential re-creation is what makes space-time complex and granular, in which case the instantaneous velocity is always zero, while the apparent physical velocity is a result of the superposition of all the velocities of the individual geometrical points , which constitute the object of observation, and each of which is either zero, in the outer time, or , in the inner time, as can be calculated by equation 5. So in this hidden discrete symmetry of space, motion is a result of re-creation in the new places rather than gradual and infinitesimal transmutation from one place to the other. Moving objects do not leave their places to occupy new adjacent positions, but they are successively re-created in them, so they are always at rest in any position along the path. When we realize the re-creation of space at the only real speed , and thus consider the apparent velocity of physical objects to be genuinely imaginary, we will automatically obtain Lorentz transformations, equally for velocity, momentum and energy (which will become also complex, as explained further in sections 5.3 and 5.4), without the need for introducing the principle of invariance of physics laws, so we do not need to differentiate between inertial and non-inertial frames, because the instantaneous velocity is zero in either case. As an extra bonus, we will also be able to derive the mass-energy equivalence relation without introducing any approximation or un-mathematical induction, and this relation is indeed the same equivalence between gravitational and inertial masses. All this is treated in section 5 below. Therefore, the non-Euclidean Minkowski space-time continuum is the first global approximation of the metaphysical reality (of Oneness, or sequential re-creation from one single point), just as the Euclidean Minkowski space-time is a local approximation when the effect of gravity is neglected, while the Galilean space is the classical approximation for non-relativistic velocities. These three relative approximations are still serving very well in describing the respective physical phenomena, but they can not describe the actually metaphysical reality of the Universe, which is dynamically re-creating the geometry of space-time itself, and what it contains of matter particles. As Hawking had already noticed: “In fact one could take the attitude that quantum theory, and indeed the whole of physics, is really defined in the Euclidean region and that it is simply a consequence of our perception that we interpret it in the Lorentzian regime.” . The Duality of Time explains exactly that the source of this deceptive perception is the fact that we do not witness the metaphysical perpetual re-creation process, but, being part of it, we always see the Universe after it is re-created, so we “imagine” that this existence is continuous, and thus describe it with the various laws of Calculus and Differential Geometry, that implicitly suppose the continuity of space and the co-existence of matter particles in space and time. In other words, normal observers, since they are part of the Universe, are necessarily approximating the reality, at best in terms of non-Euclidean Minkowskian space, and this approximation is enough to describe the macroscopic physical phenomena from the point of view of observers (necessarily) situated inside the Universe. However, this will inevitably lead to singularities at extreme conditions because, being inside the Universe, observers are trying to fit the surrounding infinite spatial dimensions in one instance of time, which would have been possible only if they are moving at the speed of light, or faster, and in this case a new spatial dimension is formed and the Universe would become confined but now observed from a higher dimension. For example, we normally see the Earth flat and infinite when we are confined to a small region on its surface, but we see it as a finite semi-sphere when we view it from outer space. In this manner, therefore, we always need one higher dimension to describe the (deceptive, and apparently infinite) physical reality, in order to contain the curvatures (whether they are intrinsic or extrinsic), and that is why Riemannian geometry is needed to describe General Relativity. Therefore, since using higher dimensions to describe the reality behind physical existence will always lead to space-time singularities, the Duality of Time Theory is working with this same logic, but backward, by penetrating inside the dimensions of space, as they are dynamically formed in the inner levels of time, down to the origin that is the zero-dimensional metaphysical point, which is the unit of space-time geometry. The Duality of Time Theory is therefore penetrating beyond the apparently-continuous physical existence, into its instantaneous or perpetual dynamic formation through the real flow of time, whose individual discrete instances can accommodate only one metaphysical or geometrical point at a time, that then correlate, or entangle, into physical objects that are kinetically evolving in the normal level of time that we encounter. At the level of this (unreal) physical multiplicity, any attempt to quantize space-time is destined to fail, because we always need a predefined background geometry, or topology, to accommodate multiplicity and define the respective relations between its various entities. In contrast, the background geometry of the Duality of Time Theory is “void”, which is an absolute mathematical vacuum that has no structure or reality, while also explaining how the physical vacuum is dynamically formed by simple chronological recurrence. So, apart from natural counting, the Duality of Time does not rely on any predefined geometrical structure, but it explains how space-time geometry itself is re-created as dynamic and genuinely-complex structure. The fact that each frame of the inner time (which constitutes space) appears as one instance on the outward time is what justifies treating time as imaginary with relation to space, thus orthogonal on it. In this dynamic creation of space in the complex time, the outward time is discrete and imaginary, while space becomes continuous with relation to this outer time, but this is only relative to the dimension in which the observer is situated, so for example: the plane is itself continuous with relation to its inner dimensions but it forms one discrete instance with relation to the flow of time in the encompassing , which then appears internally continuous but discrete with regard to the encompassing outward time. For this reason perhaps, although representing Minkowski space-time in terms of Clifford geometric algebra employing bivectors , or even the spinors of complex vector space , allowed expressing the equations in simple forms, but it could not discover the intrinsic granularity of space-time without any background, since it is still working on the multiplicity level, and not realizing the sequential re-creation process. Aether was described by ancient philosophers as a thin transparent material that fills the upper spheres where planets swim. The concept was also used again in The concept of aether was contradictory because this medium must be invisible, infinite and without any interaction with physical objects. Therefore, after the development of Special Relativity, aether theories became scientifically obsolete, although Einstein himself said that his model could itself be thought of as an aether, since empty space now has its own physical properties . In 1951, Dirac reintroduced the concept of aether in an attempt to address the perceived deficiencies in current models , thus in 1999 one proposed model of dark energy has been named: quintessence , or the fifth fundamental force . Also, as a scalar field, the quintessence is considered as some form of dark energy which could provide an alternative postulate to explain the observed accelerating rate of the expansion of the Universe, rather than Einstein’s original postulate of cosmological constant [44, 45]. The classical concept of aether was rejected because it required ideal properties that could not be attributed to any physical medium that was thought to be filling the otherwise empty space background which was called vacuum. With the new dynamic creation, however, those ideal properties can be explained, because aether is no longer something filling the vacuum, but it is vacuum itself, that is perpetually being re-created at the absolute speed of light. Thus its state is described by: as we explained in section 4.1 above, which indicates infinite and inert space that is the ground state of matter particles that are then described by , whereas the absolutely-empty mathematical space is now called void and its state is . As we already explained above, this state of corresponds to absolute zero temperature, and it is a perfect super-fluid described by Bose-Einstein statistics because its points are non-interacting and absolutely indistinguishable. When this medium is excited or disturbed, matter particles and objects are created as the various kinds of vortices that appear in this super-fluid, and this is what causes the deformation and curvature of what is otherwise described by homogeneous Euclidean geometry. Therefore, the Duality of Time Theory reconciles the classical view of aether with General Relativity and Quantum Field Theory at the same time, because it is now the ground state of particles that are dynamically generated in time. In Quantum Field Theory, the vacuum energy density is due to the zero-point energy of quantized fields, which originates from the quantization of the simple harmonic oscillations of a particle with mass. This zero-point energy of the Klein-Gordon field is infinite, but a cutoff at Planck length is enforced, since it is generally believed that General Relativity does not hold for distances smaller than this length: , which corresponds to Planck energy: . By applying this cutoff we can get: , which gives us: . Comparing this theoretical value with the 1998 observations that produced: orders of magnitude discrepancy, which is known as the vacuum catastrophe [46 The smallness of the cosmological constant became a critical issue after the development of cosmic inflation in the 1980s, because the different inflationary scenarios are very sensitive to the actual value of Many solutions have been suggested in this regard, as it was reviewed by discrepancy is actually many orders of magnitudes larger than the number of all atoms in the Universe, which is called Eddington number According to the Duality of Time postulate, this huge discrepancy in the cosmological constant is diminished, and even eliminated, because the vacuum energy should be calculated from the average of all states, and not their collective summation as it is currently treated in Quantum Field Theory. This means that we should divide the vacuum energy density by the number of modes included in the unit volume. Since we took Planck length as the cutoff, this number is: This will reduce the discrepancy between the observed and predicted values of from into only. The remaining small discrepancy could now be explained according to quintessence models, which is already described by the Duality of Time as the ground state of matter. However, more accurate calculations are needed here because all the current methods are approximate and do not take into account all possible oscillations for all the four fundamental interactions. It is well established in modern physics that mass is an emergent property, and since the Standard Model relies on gauge and chiral symmetry, the observed non-zero masses of elementary particles require spontaneous symmetry breaking, which suggested the existence of the massive Higgs boson, whose own mass is not explained in the model. This Higgs mechanism is part of the Glashow-Weinberg-Salam theory to unify electromagnetic and weak interactions Moreover, the Duality of Time Theory provides an even more fundamental and very simple mechanism for mass generation, in full agreement with the principles of Classical Mechanics, as shown further in section 5.2.3. In general, the fundamental reason for inertial mass is the coupling between the particles that constitute the object, because the binding field enforces specific separations between them, so that when the position of one particle changes, a finite time elapses before other particles move, due to the finite speed of light. This delay is the cause of inertial behavior, and this implies that all massive particles are composed of more sub-particles, and so on until we reach the most fundamental particles which should be massless. This description is fulfilled by the Duality of Time Theory, due to the discrete symmetry of the genuinely-complex time-time geometry as described above. The Duality of Time Theory is based on the Single Monad Model, so the fundamental reason of the granular geometry is the fact that no two geometrical points can exist at the same real instance of time, so they must be re-created in one chronological sequence. This delay is what causes the inertial mass, so physical objects are dynamically formed by the coupling between at least two geometrical points which produces the entangled dimensions. According to the different degrees of freedom in the resulting spatial dimensions, this entanglement is responsible for the various coupling parameters, including charge and mass, which become necessarily quantized because they are proportional to the number of geometrical nodes constituting each state, starting from one individual point for massless bosons. Nevertheless, some bosons might still appear to have heavy masses (in our outer level of time) because they are confined in their lower dimensions in which they are massless, just as the inertial mass of normal objects is exhibited only when they are moved in the outer level of time. Consequently, there is a minimum mass gap above the ground state of vacuum , which is itself also above the void state . This is because each single geometrical node is massless on its own dimensions, while the minimum state above this ground state is composed of two nodes which must have non-zero inertial mass because of the time delay between their sequential creation instances. This important conclusion agrees with Yang-Mills suggestion that the space of intrinsic degrees of freedom of elementary particles depends on the points of space-time. It was already anticipated that proving Yang-Mills conjecture requires the introduction of fundamental new ideas both in physics and in mathematics [ The famous Michelson-Morley experiment in 1887 proved that light travels with the same speed regardless whether it was moving in the direction of the movement of the Earth or perpendicular to it [ Logically, there are two cases under which a quantity does not increase or decrease when we add or subtract something from it. Either this quantity in infinite, or it exists in orthogonal dimension. As we have already introduced in sections 1 and 2 above, according to the Duality of Time postulate, both these cases are equivalent and correct for the absolute speed of light in vacuum, because it is the speed of creation which is the only real speed in nature, and it is intrinsically infinite (or indefinite), but the reason why we measure a finite value of it is because of the sequential re-creation process; so individual observers are subject to the time lag during which the dimensions of space are re-created. Moreover, since the normal time is now genuinely imaginary, the velocities of physical objects are always orthogonal onto this real and infinite speed of creation. As demonstrated in Figure 1 and explained in section 4.1 above, one of the striking conclusions of the sequential re-creation in the inner levels of time is the fact that it conceives of only two primordial states: As the real time flows uniformly in the inner levels, it creates the homogeneous dimensions of vacuum, and whenever it is interrupted or disturbed, it makes a new dimension that appears as a discrete instance on the outer imaginary level which is then described as void, since it does not last for more than one instance, before it is re-created again in a new state that may resemble the previous perished states, which causes the illusion of motion, while in reality it is only a result of successive discrete changes. So the individual geometrical points can either be at rest (in the outer/imaginary time) or at the speed of creation (in the inner/real time), while the apparent limited velocities of physical particles and objects (in the total complex time, which forms the physical space-time dimensions) are the temporal average of this spatial combination that may also dynamically change as they are progressing over the outward ordinary time direction. Therefore, the Universe is always coming to be, perpetually, in “zero time” (on the outward level), and its geometrical points are sequentially fluctuating between existence and nonexistence (or vacuum and void), which means that the actual instantaneous speed of each point in space can only change from to , and vice versa. This instantaneous abrupt change of speed does not contradict the laws of physics, because it is occurring in the inner levels of time before the physical objects are formed. Because they are massless, this fluctuation is the usual process that is encountered by the photons of light on the normal outward level of time, for example when they are absorbed or emitted. Hence, this model of perpetual re-creation is extending this process onto all other massive particles and objects, but on the inner levels of time where each geometrical point is still massless because it is metaphysical, while “space” and “mass” and other physical properties are actually generated from the temporal coupling, or entanglement, of these geometrical points, which is exhibited only on the outward level of time. Accordingly, the normal limited velocity, of physical particles or objects, is a result of the spatial and temporal superposition of these dual-state velocities of its individual points , thus: Individually, each point is massless and it is either at rest or moving at the speed of creation, but collectively they have some non-zero inertial mass , energy , and limited total apparent velocity given by this equation 5 above. Consequently, there is no gradual motion in the common sense that the object leaves its place to occupy new adjacent places, but it is successively re-created in those new places, i.e. motion occurs as a result of discrete change rather than infinitesimal transmutation, so the observed objects are always at rest in the different positions that they appear in (see also Figure 1). This is the same conclusion of the Moving Arrow argument in Zeno’s paradox, which Bertrand Russell described as: “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever.” . This momentous conclusion means that all frames are effectively at rest in the normal (imaginary) level of time, and there is no difference between inertial and non-inertial frames, thus there is even no need to introduce the second principle of Special Relativity (which says that the laws of physics are invariant between inertial frames) neither the equivalence principle that lead to General Relativity. These two principles, which are necessary to derive Lorentz transformations and Einstein’s field equations, are implicit in the Duality of Time postulate and will follow directly from the resulting complex-time geometry as it will be shown in sections 5.1 and 5.3 below. Furthermore, it will be also shown in section 5.2 that this discrete space-time structure that results from the genuinely-complex nature of time is the only way that allows exact mathematical derivation of the mass-energy equivalence relation (). In this manner, the Duality of Time postulate, and the resulting perpetual re-creation in the inner levels of time, can explain at once all the three principles of Special and General Relativity, and transform it into a quantum field theory because it is now based on discrete instances of dynamic space, which is the super-fluid state that is the ground state of matter, while the super-gas state is the ground state of anti-matter, which accounts for super-symmetry and matter-antimatter asymmetry as we shall discuss further in section 6.1. The other fundamental forces could also be interpreted in terms of this new space-time geometry, but in lower dimensions: , , and , while gravity is in . As we noted above, it was originally shown by Poincare that by using the mathematical trick of imaginary time, Lorentz transformation becomes a rotation between inertial frames. For example, if the space coordinates of an event in space-time relative to one frame are , then its (primed) coordinates with respect to another frame, that is moving with uniform velocity with respect to the first frame, are: , where , and since: , then: . In the complex-time frame of the Duality of Time postulate, however, the outer time is the (genuinely) imaginary part, while the real part is the inner time that constitutes space, thus the time coordinates: is used instead of space coordinates: . Therefore, the above rotation equations will still be valid but with time, rather than space, and then the speed of creation will be the ground state, Using the concept of split-complex time, we can easily derive Lorentz factor for example by calculating the proper time as it can be readily seen from Figure 1, which is replicated in Figure 2 that represents complex velocity, for better clarity, and also because we want to stress the fact that the apparent (imaginary) motion in any direction is in fact interrupting the real motion in the inner time that is re-creating space at the absolute speed of light, so that in the end the Lorentz factor is therefore the ratio of the real velocity over the actual velocity , which is equal , as demonstrated in Figure 2: In addition to explaining the constancy and invariance of the speed of light and merging it with the second and third principles of Relativity, the Duality of Time postulate is the only way to explain the equivalence and transmutability between mass and energy Einstein gave various heuristic arguments for this relation without ever being able to prove it in any theoretical way [ It can be readily seen from Figure 3 that the transmutability between mass and energy can only occur in the inner levels of time, because it must involve motion at the speed of light that appears on the normal level of time as instantaneous, hence the same Relativity laws become inapplicable since they prohibit massive particles from moving at the speed of light, in which case , so the mass would be infinite and also the energy. In the inner levels of time, however, this would be the normal behavior because the geometrical points are still massless, and their continuous coupling and decoupling is what generates mass and energy on the inner and outer levels of time, respectively, as explained further in section 5.2.3 below. As we introduced in section 4.1 above, the normal limited velocities of massive physical particles and objects are a result of the spatial and temporal superposition of the various dual-state velocities of their individual points. This superposition occurs in the inner levels of time, where individually each point is massless and it is either at rest or moving at the speed of creation, but collectively they have some non-zero inertial mass , energy , and limited total apparent velocity , which can be calculated from equation 5. We also explained in section 4.3 above, that when we consider this imaginary velocity as being real, the Duality of Time Theory reduces to General Relativity, but when we consider its imaginary character we will uncover the hidden discrete space-time symmetry and we will automatically obtain Lorentz transformation, without introducing the principle of invariance of physics laws. For the same reason, we can see here that the mass-energy equivalence can only be derived based on this profound discreteness that is manifested in dual-state velocity, which then allows the square integration in Figure 3, because the change in speed is occurring abruptly from zero to . Otherwise, when we consider to be real continuous in time, we will get the gradual change which produces the triangular integration with the factor of half that gives the normal kinetic energy . Based on this metaphysical behavior in the inner levels of time, we will provide in the following various exact derivations of the mass-energy equivalence relation, in its simple and relativistic forms, directly from the classical equation of mechanical work . The first two methods, in sections 5.2.2 and 5.2.3, involve integration (or rather: summation) in the inner time when the velocity changes abruptly from zero to , or when the mass is generated (from zero to ) in the inner time. This is obviously not allowed on the normal level of time when dealing with physical objects. The third method, in section 5.2.7, gives the total relativistic energy , by integrating over the inner and outer levels together, while in section 5.2.8 we will derive the relativistic energy-momentum relation directly from the definition of momentum as , also by integrating over the inner and outer levels together and accounting for what happens in each stage. Furthermore, we will see in sections 5.3 and 5.4 that the absolute invariance, and not just covariance, of complex momentum and energy, provide yet other direct derivations because they also lead to , that is equivalent with or as demonstrated in A. Actually, since we have shown previously that the new vacuum is a perfect super-fluid, the mass-energy equivalence relation can be easily derived from the equation of wave propagation in such perfect medium: (), but we will not discuss that further in this article. In normal classical mechanics, the kinetic energy is the work done in accelerating a particle during the infinitesimal time interval , and it is given by the dot product of force and displacement : Now if we assume mass to be constant, so that: (and we will discuss relativistic mass further in section 5.2.4 below), we will get: So in the classical view of apparently continuous existence, when we consider both space and time to be real, i.e. when we consider an infinitesimally continuous and smooth change in speed from zero to , the result of this integration will give the standard equation that describes the kinetic energy of massive particles or objects moving in the normal level of time: The reason why we are getting the factor of “half” in this equation is because the velocity increases gradually with time, which makes the integration equals the area of the triangle as demonstrated by the first arrow in Figure 3. The relativistic energy-momentum relation is derived in section 5.2.8 further below, but the simple mass-energy equivalence relation: (without the “half”) can now be easily obtained from the same integration in equation 9 if, By introducing the duality of time and the resulting perpetual re-creation, this problem is solved because the conversion between mass and energy takes place, sequentially, in the inner levels of time, on all the massless geometrical points that constitute the particle, and this whole process appears as one instance in the outer level, as demonstrated in Figure 1 above. So by integrating equation 9 directly from zero to , which then becomes summation because it is an abrupt change, with only the two states of void and vacuum, corresponding to zero and , respectively, and since the change in the outward time is zero, and here we also consider , since the apparent velocity does not change in this case, but we will also discuss relativistic mass in section 5.2.4 below; thus we obtain: The difference between the above two cases that result in equations 10 and 11 is demonstrated in Figure 3, where in the first case the integration that gives the kinetic is the area of the We explained in section 4.5 above how the Duality of Time Theory provides a fundamental mass generation mechanism in addition to its super-fluid vacuum state where mass can be generated via the interaction with this physical vacuum. Hence we can also arrive to the mass-energy equivalence relation directly from the starting equation 8 in an alternative manner if we consider a sudden decoupling, or disentanglement, of the geometrical points that couple together in order to constitute the physical particle that appears in the outer level of time with inertial mass moving at an apparent (imaginary) velocity , so when these geometrical points are disentangled to remain at their real speed (of light), the mass converts back into energy while the apparent velocity does not change, because this process is happening in the inward levels of time which appears outwardly as instantaneous. Thus, if we put in equation 8 and integrate over mass from to zero (where ), or vice versa, we get: Unlike the classical case in equation 10 where the change in speed occurs in the normal outward level of time, these simple derivations (in equations 11 and 12) would not have been possible without considering the inner levels of time, which appears outwardly as instantaneous. If we want to consider mass to be variable with speed as in early Special Relativity, and distinguish between rest mass: and relativistic mass: , according to the standard equation that uses Lorenz factor: , then we can arrive to the equation by calculating the derivative which in this case will not be equal to zero as we required in equation 9 above. However, the above relativistic equation of mass () is only obtained based on the same mass-energy equivalence relation that we are trying to prove here, so in this case it will be a circular argument. Therefore, the two equations: and are equivalent, and deriving one of them will lead to the other. See also A for how to derive the from . We can conclude, therefore, that on the highest existential level, there is either energy in the form of massless active waves moving at the speed of creation in the inner levels of time, or passive mass in the form of matter particles that are always instantaneously at rest in the outer level of time, not the two together; that is what happens in the real flow of time. The various states of massive objects and particles, as well as thermal radiations and energy, are some spatial and temporal superposition of these two primordial states of their metaphysical constituents, so some particles will be heavier than others, and some will have more kinetic energy. In any closed system, such as an isolated particle, atom, or even objects, the contributions to this superposition state come from all the states in the system that are always fluctuating between mass and energy, or void and vacuum, corresponding to and , respectively, so on average the total state is indeterminate, or determined only as a probability distribution, as far as it is not detected or measured. This wave-particle duality will be discussed further in section 6. Consequently, everything in the Universe is always fluctuating between particle state and wave state, or mass and energy, which can be appropriately written as: . This means that a particle at rest with mass can be excited into a wave with frequency , and the opposite is correct when the wave collapses into particle. Either zero mass at the speed of creation, or (instantaneously) zero energy at rest, or: either energy in the active existence state or mass in the passive nonexistence state. The two cannot happen together on the primary level of time, but a mixture or superposition of various points is what causes all the phenomena of motion and interaction between particles with limited velocities and energy on the outward level of time. Therefore, even when the object is moving at any velocity that could be very close to , its instantaneous velocity is always zero: , at the actual time of measurement, and its mass will still be the same rest mass: , because it is only detected as a particle, while its kinetic energy will be given by its relativistic mass: , and then its total energy equivalence, with relation to an observer moving at a constant (apparent) velocity , will be given by: Thus, with the help of Lorentz factor , we could get rid of the confusion between “rest mass” and “relativistic mass” and just call it mass since the above equation 13 describes energy and not mass: Which means that the mass of any particle is always the rest mass, it is not relativistic, but its energy is relativistic, primarily because energy is related to time and motion or velocity. However, since we have been using all over this article, we will keep using it as the rest mass, and as the effective mass, unless stated otherwise. The total relativistic energy in equation 13 can also be obtained by integrating the starting equation 8 over the inner and outer time together, since in the inner time the rest energy , or mass , is generated at the speed of creation as a result of the instantaneous coupling between the geometrical points which constitute the particle of mass (thus and in the inner time), and in the outer time the kinetic energy is generated as this mass moves gradually so its apparent velocity changes by , which corresponds to increasing the effective mass from to , thus we can integrate: Thus we get the same equation 13: This equation can also be given in the general form that relates the relativistic energy and momentum (see A for how to convert between these two equations): This last equation, that is equivalent to equation 16, will be also derived in section 5.2.8 starting from the definition of momentum as , but because it is genuinely complex, and hyperbolic, the imaginary part of momentum will have negative contribution just as we have seen for the outer time when we discussed the arrow of time in section 4.2 above. Again, however, a fundamental derivation of this relativistic energy-momentum equation 17 is not possible without the Duality of Time postulate. All the current derivations in the literature rely on the effective mass relation: , which is equivalent to the same relation we are trying to derive (see above and also A), while finding this equation from the four-momentum expression, or space-time symmetry, relies on induction rather than rigorous mathematical formulation. The equation: , of the relativistic energy-momentum, can be also derived directly from the fundamental definition of momentum: , when we include the metaphysical creation of mass in the inner levels of time, in addition to its physical motion in the outer level, and by taking into account the complex character of time. Thus we need to integrate over the inner and outer levels, according to what happens in each stage; first by integrating between and on the inner real levels of time where the particle is created, or being perpetually re-created, at the speed of creation , and this term makes the real part of the complex momentum . Then we integrate between and on the outer imaginary level of time where the particle whose mass is gains an apparent velocity , and thus its effective mass increases from to , and this term makes the imaginary part of the complex momentum : The first term gives us the real momentum , while the second term gives the imaginary momentum . So the total complex momentum is: . Hence the modulus of this total complex momentum is given by: Again, we notice here that the contribution of the imaginary momentum is negative with relation to the real momentum , just like the case of and as we have seen in sections 4.2 and 5.1, and as it will be also the case for complex energy as we shall see in section 5.4 further below. All this is because the normal time, or physical motion, is interrupting the real creation, which is causing the disturbance and curvature of the otherwise infinite homogeneous Euclidean space that describes the vacuum state . Therefore, to obtain the relativistic energy-momentum relation from equation 19, we simply multiply by : These equations 19 and 20 above, with the negative sign, do not contradict the equation in current Relativity: which treats energy as scalar and do not realize its complex dimensions (see also section 5.4 below). Practically, in any mass-energy interaction or conversion, the negative term will be converted back to positive because when the potential energy is released, in nuclear interactions for example, this means that it has been released from the inner levels of time where it is captured as mass, into the outer level to become kinetic energy or radiation. In other words: the absorption and emission of energy or radiation, nuclear interactions, or even the acceleration and deceleration of mass, are simply conversions between the inner and outer levels of time, or space and time, respectively. Eight centuries ago, Ibn al-Arabi described this amazing observation by saying: “Space is rigid time, and time is liquid space.” . This derivation of the relativistic energy-momentum relation from the fundamental definition of momentum is based on the Duality of Time concept, by taking into account the complex nature of time, as hyperbolic numbers, which is why the contribution of the imaginary term appears here as negative in equation 20. As we discussed in section 4.3, when we do not realize the discrete structure of space-time geometry that results from this genuinely-complex nature of time, the Duality of Time Theory is reduced to General Relativity, which considers both space and time to be real, and then we take the apparent rather than the complex velocity whose instantaneous value is always zero; so this negative sign in equations 19 and 20 above will appear positive, as if we are treating space-time to be spherical () rather than hyperbolic (). Therefore, when we take into account the complex nature of time as we described in sections 4.1 and 5.1 above (or Figures: 1 and 2), energy and momentum will be also complex and hyperbolic. This significant conclusion, that is a result of the new discrete symmetry, will introduce an essential modification on the relativistic energy-momentum equation which will lead to the derivation of the equivalence principle and allows energy to be imaginary, negative and even multidimensional, as it will be discussed further in sections 5.3 and 5.4. In moving from Special to General Relativity, Einstein observed the equivalence between the gravitational force and the inertial force experienced by an observer in a non-inertial frame of reference. This is roughly the same as the equivalence between active gravitational and passive inertial masses, which has been later accurately tested in many experiments [ When Einstein combined this equivalence principle with the two principles of Special Relativity, he was able to predict the curved geometry of space-time, which is directly related to its contents of energy and momentum of matter and radiation, through a system of partial differential equations known as Einstein field equations. We explained in section 5.2 above that an exact derivation of the mass-energy equivalence relation is not possible without postulating the inner levels of time, and that is why there is yet no single exact derivation of this celebrated equation. For the same reason indeed, there is also no mathematical derivation of the equivalence principle that relates gravitation with geometry, because it is actually equivalent to the same relation that reflects the fact that space and matter are always being perpetually re-created in the inner time, i.e. fluctuating between the particle state and wave state , thus causing space-time deformation and curvature. Due to the discrete structure of the genuinely-complex time-time geometry, as illustrated in Figure 1, the complex momentum should be invariant between inertial and non-inertial frames alike, because effectively all objects are always at rest in the outer level of time, as we explained in section 4 above. This means that complex momentum is always conserved This invariance of momentum between non-inertial frames is conceivable, because it means that as the velocity increases (for example), the gain in kinetic momentum (that is the imaginary part) is compensated by the increase in the effective mass: due to acceleration, which causes the real part also to increase, but since is hyperbolic, thus its modulus remains invariant, and this what makes the geometry of space (manifested here as ) dynamic, because it must react to balance the change in effective mass. Therefore, a closed system is closed only when we include all its contents of mass and energy (including kinetic and radiation) as well as the background space itself, which is the vacuum state , and the momentum of all these constituents is either , when they are re-created in the inner levels, or for physical objects moving in the normal level of time. For such a conclusive system, the complex momentum is absolutely invariant. Actually, without this exotic property of momentum it is not possible at all to obtain an exact derivation of as we mentioned in section 5.2 above, and also in A below. These experimentally verified equations are correct if, Since this previous equation 21 is equivalent to: , therefore, in addition to the previous methods in equations 11 and 12, and the relativistic energy-momentum relation in equation 20, the mass-energy equivalence relation: can now be deduced from equation 21 as it is shown in A below, because, as we mentioned in section 5.2.4 above, the equations: and are equivalent, and the derivation of one of them leads to the other, while there is no exact derivation of either form in the current formulation of Special or General Relativity. This absolute conservation of complex momentum under acceleration leads directly to the equivalence between active and passive masses, because it means that the total (complex) force: must have two components; one that is related to acceleration as changes in the outer time , which is the imaginary part, and this causes the acceleration: , so here is the passive mass, while the other force is related to the change in effective mass , or its equivalent energy , which is manifested as the deformation of space which is being re-created in the inner levels of time , and this change or deformation is causing the gravitational force that is associated with the active mass; and these two components must be equivalent so that the total resulting complex momentum remains conserved. Therefore, gravitation is a reaction against the disturbance of space from the ground state of bosonic vacuum to the state of fermionic particles , the first is associated with the active mass in the real momentum , and the second is associated with the passive mass in the imaginary momentum . However, as discussed further in section 6, because of the fractal dimensions of the new complex-time discrete geometry, performing the differentiation of this complex requires non-standard analysis because space-time is no longer everywhere From this conservation of complex momentum we should be able to find the law of gravitation and the stress-energy-momentum tensor which leads to the field equations of General Relativity. Moreover, since empty space is now described as the dynamic aether, gravitational waves become the longitudinal vibrations in this ideal medium, and the graviton will be simply the moment of time , just as photons are the quanta of electromagnetic radiations and they are transverse waves in this vacuum, or the moments of space . This means that equivalence principle is essentially between photons and gravitons, or between space and time, while electrons and some other particles could be described as standing waves in the space-time; with complex momentum , and the reason why we have three generations of fermions is due to the three dimensions of space. This important conclusion requires further investigation, but we should also notice here that the equivalence principle should apply equally to all fundamental forces and not only to gravity, because it is a property of space-time geometry in all dimensions, and not only the where gravity is exhibited, as it is also outlined in another publication . Since it is intimately related to time, energy has to have complex, and even multiple intersecting dimensions in accordance with the dimensions of space and matter which are generated in the inner levels of time before they evolve throughout the outer level. We must notice straightforward, however, that not all these levels of energy are equivalent to mass which is only a property of space. In lower dimensions, energy should rather be associated with the corresponding coupling property, such as the electric and color charges. Therefore, it is expected that negative mass is only possible in spatial dimensions, as it has been already anticipated before [ It is clear initially that, just like time, velocity and momentum that were discussed above, when we take the complex nature of time into account, the kinetic energy in equation 13, or in the relativistic energy-momentum equation 17, becomes negative with relation to the potential energy stored in mass . Therefore the energy in equation 15 becomes complex with real and imaginary parts. The real part represents re-creation through the change in mass , and the imaginary part represents the kinetic evolution of this mass in the outer time through the change in the apparent velocity : The real part is and the imaginary part is , thus we get: This negative contribution of the kinetic energy, however, does not falsify the current equations 13 and 17, but it means that the potential energy and the kinetic energy are in different orthogonal levels of time and the conversion of potential energy into kinetic energy is like the conversion from the inner time into the outer time, so when they are in the outer time they are added together as in the previous equations because they become both in the same level of time. Again, just as it is the case with the absolute conservation of momentum that we have seen in section 5.3 above, energy is also always conserved, even when the apparent velocity changes, since the instantaneous velocity in the outer level of time is always zero, as we have seen in section 4 and Figure 1 above. As it is the case for momentum, this absolute conservation of energy is conceivable because it means that as the velocity changes, the change in kinetic energy (that is the imaginary part) is compensated by the change in the effective mass: due to motion, which causes the real part of energy also to change accordingly, but since is hyperbolic, thus its modulus remains invariant between all inertial or non-inertial frames. This means that: This equation provides even an additional method to derive the mass-energy equivalence, because the left side in this equation can be reduced to : Soon after the discovery of fractals, fractal structures of space-time were suggested in 1983 , as an attempt to find a geometric analogue for relativistic quantum mechanics, in accordance with Feynman’s path integral formulation, where the typical infinite number of paths of quantum-mechanical particles are characterized as being non-differentiable and fractal [ Accordingly, some theories were constructed based on fractal space-time, including Causal Dynamical Triangulation and Scale Relativity , which also share some fundamental characteristics with Loop quantum gravity, that is trying to quantize space-time itself . Actually, there are many studies that have successfully demonstrated how the principles of quantum mechanics can be derived from the fractal structure of space-time [9, 10, 11, 12, 13], but there is yet no complete understanding of how the dimensionality of space-time evolved to the current Universe. Some multiverse and eternal-inflation theories exhibit fractality at scales larger than the observable Universe [ In this regard, based on the concept of re-creation according to the Duality of Time Theory, the Universe is constantly being re-created from one geometrical point, that is , from which all the current dimensions of space and matter are re-created in the inner levels of time before they evolve in the outer time. Therefore, the total dimension of the Universe becomes naturally multi-fractal and equals to the dynamic ratio of “inner” to “outer” times, because spatial dimensions alone, as an empty homogeneous space, are complete integers, while fractality arises when this super-fluid vacuum, as described in section 4.1 above, starts oscillating in the outer time, which causes all types of vortices that we denote as elementary particles. So we can see how this notion, of space-time having fractal dimensions, would not have any “genuine” meaning unless both the numerator and denominator of the fraction are both of the same nature of time, and this can only be fulfilled by interpreting the complete dimensions of space as inner levels of time. In the absolute sense, the ratio of inner to outer times is the same as the speed of light, which only needs to be “normalized” in order to express the fractality of space-time; to become time-time. For example, if the re-creation process, that is occurring in the inner levels of time, is not interrupted in the outer time, i.e. when the outer time is zero, this corresponds to absolute vacuum, that is an isotropic and homogeneous Euclidean space, with complete integer dimensions, that is in our normal perception, and it is expected to be on large cosmological scales. So the speed of light, in the time-time frame, is a unit-less constant that is equal to the number of dimensions that are ideally for a perfect three-dimensional vacuum, which corresponds to the state of super-energy , as described in section 4.1, but it may condense down to for void, which is absolute darkness that is the super-mass state . The standard value of the speed of light in vacuum is now considered a universal physical constant, and its exact value is meters per second. Since 1983, the length of the Meter has been defined from this constant, as well as the international standard for time. However, this experimentally measured value corresponds to the speed of light in actual vacuum that is in fact not exactly empty. The true speed that should be considered as the invariant Speed of Creation is the speed of light in absolute “void” rather than “vacuum”, which still has some energy that may interact with the photons, and delay them, but void is real “nothing”. Of course, even high vacuum is very hard to achieve in labs, so void is absolutely impossible. Because we naturally distinguish between space and time, this speed must be measured in terms of meters per second, and it should be therefore exactly equal to . The difference between this theoretical value and the standard measured value is what accounts for the quantum foam, in contrast to the absolute void that cannot be excited. Of course all this depends also on the actual definition of the meter, and also the second, which may appear to be conventional, but in fact they are based on the same ancient Sumerian tradition, included in their sexagesimal system which seems to be fundamentally related to the structure of space-time [20, Ch. VII]. Therefore, the actual physical dimensions of the (local) Universe are less than three, and they change according to the medium, and they are expected to be more than three in extra-galactic space, to accommodate negative mass and super-symmetry. For example the fractional dimensions of the actual vacuum is simply , and the fractional dimensions of water would be , and so on for all transparent mediums according to their relative refraction index. Opaque materials could be also treated in the same manner according to their refraction index, but for other light wave-lengths that they may transfer. Dimensionality is a relative and dynamic property, so the Universe is ultimately described by multi-fractal dimensions that change according to the medium, or the inner dimensions (of space), and also wavelength, that is the outer dimension (or time). As we noted above, many previous studies have successfully derived the principles of quantum mechanics from the fractality of space-time, but we want in the remaining of this section to outline an alternative description based on the new complex-time geometry. This has been explained with more details in other publications [18, 20, 21, 19], but a detailed study is required based on the new findings. As a result of perpetual re-creation, matter in the Universe is alternating between the two primordial states of void and vacuum, which correspond to the two states of super-mass and super-energy , respectively. Since is real void or absolute “nothing”, it remains only the state of vacuum , which is a perfectly homogeneous three-dimensional space, according to our normal perception. Therefore, the Universe, as a whole, is in this perfect state of Bose-Einstein Condensation, which is a state of “Oneness”, because its geometrical points are indistinguishable and non-interacting, so it is a perfectly symmetrical and homogeneous or isotropic space. Multiplicity appeared out of this Oneness as a result of breaking the symmetry of the real existence, the super-energy , and its imaginable non-existence, the super-mass , into the two states of super-fluid and super-gas , which correspond to particles and anti-particles, which are perpetually, and sequentially, annihilating back into energy and splitting again, as described by equation 2 above. This process is occurring every moment of time, and this is actually what defines the moments of time, and causes our physical perception and consciousness. If existence remained in the bosonic state, no physical particles will appear, and no “time”, since no change or motion can be conceived. Normal (or the outer level of) time starts when the super-fluid state , which is the aether, is excited into , which describes physical particles, or fermionic states, while at the same time the orthogonal super-gas state is excited into , which describes anti-particles that are also fermions in their own time, but bosons in our time reference, and vice-versa, because these opposite time arrows are orthogonal, as we described in section 4.2. Therefore, physical existence happened as a result of splitting this ideal space, which introduced the outer level of time in which fermions started to move and take various different (discrete) states. The fundamental reason behind the quantum behavior, or why these states are discrete, is because no two particles, or fermionic states, can exist simultaneously in the outer time, which is the very fact that caused them to become multiple and make the physical matter, so their re-creation must be processed sequentially, and this is the ontological reason behind the exclusion principle. Therefore, since all fermions are kinetically moving in the outer time, which is imaginary, they must exist in different states, because we are observing them from orthogonal time direction, otherwise we would not see them multiple and in various dimensions. In contrast to that, because bosons are in the real level of time with respect to the observer, they all appear in the same state even though they may be many. On the other hand, if we suppose the particle is composed of individual geometrical points, each of which is either in the inner or outer levels of time, so their individual speeds are either zero or , but collectively appear to be moving at the limited apparent velocity , that can be calculated from equation 5; thus the particle is described by . Therefore, because only one point actually exists in the real flow of time, the position of this point is completely undetermined, because its velocity is equal to , while the rest have been already defined, because they are now in the past, and their velocities had been sequentially and abruptly collapsed from to zero, after they made their corresponding specific contribution to the total quantum state which defines the position of the particle with relation to the observer. When the number is very large, as it is the case with large objects and heavy particles, the uncertainty will be very small, because only one point is completely uncertain at the real instance of time. But for small particles, such as the electron, the uncertainty could become considerably large, because it is inversely proportional to : . This uncertainty in position will also increase with (the imaginary) velocity , or momentum , because higher physical velocity means that on average more and more points are becoming in the real speed , rather than rest, as can also inferred from equation 5. Moreover, we can now give an exact account of the collapse of the wave function, since the superposition state of a system of individual points comes from averaging their dual-state of zero or , all of which had already made its contribution except the real current one at the very real instance of time of measurement, which is going to be determined right in the following instance. Therefore, because the state of any individual point automatically collapses into zero after it makes its contribution to the total quantum state, once the moment passes, all states are determined automatically, although their eigenstate may remain unknown, as far as it is not measured. So, as in the original Copenhagen interpretation, the act of measurement only provides knowledge of the state. However, if the number of points in a system is very small, and since the observer is necessarily part of the system, the observation may have a large impact on determining the final eigenstate. Accordingly, the state of Schroedinger’s cat, after the box is closed, is either dead or alive; so it is already determined, but we only know that after we open the box, provided that the consciousness of the observer did not interfere during the measurement. Any kind of measurement or detection, necessarily means that the observer, or the measuring device, at this particular instance of measurement, is the subject that is acting on the system; and since there is only one state of vacuum and one state of void, at this real instance of time, the system must necessarily collapse into the passive state, i.e. it becomes the object or particle, because at this particular instance of time the observer is taking on the active state. Of course, this collapsing is not fatal, otherwise particles and objects will disappear forever, but they are re-created or excited again into a new state right after this instantaneous collapse, at which time the observer now would have moved back into an indeterminate state, and becomes an object amongst other objects. The uncertainty and non-locality of quantum mechanical phenomena result from the process of sequential re-creation, or the recurrence of only one geometrical point, which is flowing either in the inward or outward levels of time, which respectively produce the normal spatial entanglement as well as the temporal entanglement. Therefore, entanglement is the general underlying principle that connects all parts of the Universe in space and as well as in time, but it is mostly reduced into simple coherence, which may also dissipate quickly as soon as the system becomes complex. In other words: spatial and temporal entanglement is what defines space-time structure, rather than direct proximity. In this deeper sense, the speed of light is never surpassed even in extreme cases, such as the EPR and quantum tunneling, since there is no transmutation, but the object is re-created in new places which could be at the other end of the Universe, and even in a delayed future time. Consequently, whether the two particles are separated in space or in time, they can still interfere with each other in the same way because they are described by the same wave function either as one single entangled state or two coherent states. In this way we can explain normal as well as single particle interference, since the wave behavior of particles in each case is a result of the instantaneous uncertainty in determining their final physical properties, such as position or momentum, as they are sequentially re-created. Spatial entanglement occurs between the points in the internal level of time, while temporal entanglement is between the points of the outer level, so in reality it is all temporal since all the points of space and time are generated in one chronological order that first spreads spatially in the inner metaphysical level and then temporally in the outer physical level. On the other hand, since the whole Universe is self-contained in space, all changes in it are necessarily internal changes only, because it is a closed system. Therefore, any change in any part of the Universe will inevitably cause other synchronizing change(s) in other parts. In normal cases the effect of the ongoing process of cosmic re-creation is not noticeable because of the many possible changes that could happen in any part of the complex system, and the corresponding distraction of our limited means of attention and perception. This means that causality is no more directly related to space or even time; because the re-creation allows non-local and even non-temporal causal interactions. In regular macroscopic situations, the perturbation causes gradual or smooth, but still discrete, motion or change; because of the vast number of neighboring individual points, so the effect of any perturbation will be limited to adjacent points, and will dissipate very quickly after short distance, when energy is consumed. This kind of apparent motion is limited by the speed of light, because the change can appear infinitesimally continuous in space. In the special case when a small closed system is isolated as a small part of the Universe, and this isolation is not necessarily spatial isolation, as it is the case of the two entangled particles in the EPR, then the effect of any perturbation will appear instantaneous because it will be transferred only through a small number of points, irrespective of their positions in space, or even in time. The Duality of Time Theory exposes a deeper understanding of time, that reveals the discrete symmetry of space-time geometry, according to which the dimensions of space are dynamically being re-created in one chronological sequence at every instance of the outer level of time that we encounter. In this hidden discrete symmetry, motion is a result of re-creation in the new places rather than gradual and infinitesimal transmutation from one place to the other. When we approximate this discrete motion in terms of the apparent (average) velocity, this theory will reduce to General Relativity. We have shown that the resulting space-time is dynamic, granular, self-contained without any background, genuinely-complex and fractal, which are the key features needed to accommodate quantum and relativistic phenomena. Accordingly, many major problems in physics and cosmology can be easily solved, including the arrow of time, non-locality, homogeneity, dark energy, matter-antimatter asymmetry and super-symmetry, in addition to providing the ontological reason behind the constancy and invariance of the speed of light, that is currently considered an axiom. We have demonstrated, by simple mathematical formulation, how all the principles of Special and General Relativity can be derived from the Duality of Time postulate, in addition to exact mathematical derivation of the mass-energy equivalence relation, directly from the principles of Classical Mechanics, as well deriving the equivalence principle that lead to General Relativity. Previous studies have already demonstrated how the principles of Quantum Mechanics can be derived from the fractal structure of space-time, but we have also provided realistic explanation of quantum behavior, such as the wave-particle duality, the exclusion principle, uncertainty, the effect of observers and the collapse of wave function. We also showed that, in addition to being a perfect super-fluid, the resulting dynamic quintessence could reduce the cosmological constant discrepancy by at least 117 orders of magnitude. Starting from equation 8 above: and we can find by differentiating , with respect to : From this equation we find: , and by replacing in equation 26 we get: This method, however, can not be considered a mathematical validation of the mass-energy equivalence relation , because the starting equation is not derived by any other fundamental method from current Relativity, other than being analogous to the equations of time dilation and length contraction: , . Using the same equation with ; thus: , we can also derive the relativistic energy-momentum relation, by squaring and applying some modifications: From this equation we get: , thus , or: Again, since this derivation relies originally on the equation: , it can not be considered a mathematical validation of the mass-energy equivalence relation. O. Lauscher and M. Reuter. Asymptotic Safety in Quantum Einstein Gravity: nonperturbative renormalizability and fractal M. Joyce, F. Sylos Labini, A. Gabrielli, M. Montuori, and L. Pietronero. Basic properties of galaxy clustering in the light of recent results from the sloan digital sky survey. David W. Hogg, Daniel J. Eisenstein, Michael R. Blanton, Neta A. Bahcall, J. Brinkmann, James E. Gunn, and Donald P. Schneider. Cosmic homogeneity demonstrated with luminous red Laurent Nottale and Marie-Noëlle Célérier. Derivation of the postulates of quantum mechanics from the first principles of Mohamed Ali Haj Yousef. Mohamed Ali Haj Yousef. Zeno’s paradoxes and the reality of motion according to ibn al-arabi’s single monad model of the cosmos. In Sotiris Mitralexis, editor, Mohamed Ali Haj Yousef. Albert Einstein. Über die gültigkeitsgrenze des satzes vom thermodynamischen gleichgewicht und über die möglichkeit einer neuen bestimmung der elementarquanta.

s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100308.37/warc/CC-MAIN-20231201215122-20231202005122-00009.warc.gz

CC-MAIN-2023-50

https://infoscience.epfl.ch/record/140339

math

This thesis studies the phenomenon of jamming in granular media, such as when a salt shaker gets clogged. We use modern instrumentation, like X-ray synchrotron tomography, to look inside real jamming experiments. High performance computers allow simulating mathematical models of jamming, but we are also able to treat some of them just using paper and pencil. One main part of this thesis consists of an experimental validation of the distinct-element-method (DEM). In this model, grains are modeled separately, their trajectories obey Newton's laws of motion and a model of the contacts between grains is given. Real experiments of jamming of glass beads flowing out of a container were carried out. 3D snapshots of the interior of the media were taken using X-ray synchrotron tomography. These snapshots were computer processed using state of the art image analysis. It was found that 3D DEM is capable of predicting quite well the final positions of the grains of the real experiments. Indeed, in cases of instant jamming (jamming without a substantial previous flow of beads) the simulations agree well with the real experiments. However, in cases of non instant jamming, because of chaotic behavior of the model and the system, the results do not agree. Furthermore, a sensitivity analysis to grain location and size perturbations was carried out. In a second part, we describe results on 2D DEM simulations of jamming in a hopper. We focus on the jamming probability J, the average time T before jamming and the average number ψ of beads falling through the hole when jamming occurs. These quantities were related to global parameters such as the number of grains, the hole size, the friction coefficient, grain length or the angle of the hopper (in opposition to fine-scale parameters that are the positions and radii of the grains). In agreement with intuition, a monotonic behavior of J and ψ as a function of the number of grains, the hole size, the friction coefficient was found. However, surprising results were also found such as the non-monotonicity of the average number of beads falling through the hole when jamming occurs as a function of the grain length and the hopper angle. In the third part, we study simple probabilistic 2D models called SPM, in which non-interacting particles move with constant speed towards the center of a circular sector. Formulas giving the jamming probability or the average time before jamming when jamming occurs as a function of global parameters were found. SPM and 2D DEM were compared and a locally good correspondence between the global parameters of the two was established. SPM led us to study some combinatorial problems, in particular two bi-indexed recurrence sequences. One gives the number of ways of placing identical balls in fixed-size numbered urns and the other the number of subsets of a given ordered set without a certain number of consecutive elements. Several different ways of computing the sequences, each advantageous in certain cases, were found.

s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178350846.9/warc/CC-MAIN-20210225065836-20210225095836-00245.warc.gz

CC-MAIN-2021-10

http://www.fact-index.com/t/ti/ti_92.html

math

The Texas Instruments TI-92 calculator is an unusual calculator, quite large and with a QWERTY keyboard. A programmable calculator, it comes with a computer algebra system (CAS), and was one of the first calculators to offer 3D graphing. Unfortunately, the TI-92 was not allowed on most standardized tests due mostly to its QWERTY keyboard. Its larger size was also rather cumbersome compared to other graphing calculators. In response to these concerns, Texas Instruments introduced the TI-89 which is functionally almost identical to the TI-92, but smaller and without the QWERTY keyboard.

s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233510214.81/warc/CC-MAIN-20230926143354-20230926173354-00578.warc.gz

CC-MAIN-2023-40

https://www.knjv-venlo.nl/difference-between-payback-period-and-discounted/

math

Acting as a simple risk analysis, the payback period formula is easy to understand. It gives a quick overview of how quickly you can expect to recover your initial investment. This tab allows you to compare the economic merits of the current system and a base case system. The window displays cash flow graphs and a table of economic metrics. Some investments take time to bring in potentially higher cash inflows, but they will be overlooked when using the payback method alone. The payback period is the amount of time required for cash inflows generated by a project to offset its initial cash outflow. This calculation is useful for risk reduction analysis, since a project that generates a quick return is less risky than one that generates the same return over a longer period of time. There are two ways to calculate the payback period, which are described below. The shorter a discounted payback period is, means the sooner a project or investment will generate cash flows to cover the initial cost. The internal rate of return is understood as the discount rate, which ensures equal present values of expected cash outflows and expected cash inflows. An implicit assumption in the use of payback period is that returns to the investment continue after the payback period. nominal payback Payback period does not specify any required comparison to other investments or even to not making an investment. Present Value Vs Internal Rate Of Return Also, high liquidity is translated as a low level of risk. Finally, when there is an uncertain estimation and forecast of future cashflows, the payback period method is useful. All investors, investment managers, and business organizations have limited resources. Therefore, the need to make sound business decisions while selecting between a pool of investments is required. The payback period also facilitates side-by-side analysis of two competing projects. If one has a longer payback period than the other, it might not be the better option. The point after that is when cash flows will be above the initial cost. For the sake of simplicity, let’s assume the cost of capital is 10% (as your one and only investor can turn 10% on this money elsewhere and it is their required rate of return). If this is the case, each cash flow would have to be $2,638 to break even within 5 years. At your expected $2,000 each year, it will take over 7 years for full pay back. The payback period is the amount of time it would take for an investor to recover a project’s initial cost. It’s closely related to the break-even point of an investment. The payback period formula is also known as the payback method. Note that in both cases, the calculation is based on cash flows, not accounting net income (which is subject to non-cash adjustments). The payback period disregards the time value of money and is determined by counting the number of years it takes to recover the funds invested. For example, if it takes five years to recover the cost of an investment, the payback period is five years. That is, the profitability of each year is fixed, but the valuation of that particular amount will be placed overtime the period. Thus the payback period fails to capture the diminishing value of currency over increasing time. The concept does not consider the presence of any additional cash flows that may arise from an investment in the periods after full payback has been achieved. The payback period refers to the amount of time it takes to recover the cost of an investment or how long it takes for an investor to hit breakeven. To begin, the periodic cash flows of a project must be estimated and shown by each period in a table or spreadsheet. These cash flows are then reduced by their present value factor to reflect the discounting process. This can be done using the present value function and a table in a spreadsheet program. Example Of The Discounted Payback Period This may involve accepting both or neither of the projects depending on the size of the Threshold Rate of Return. For the Discounted Payback Period and the Net Present Value analysis, the discount rate is used for both the compounding and discounting analysis. So only the discounting from the time of the cash flow to the present time is relevant. - The discounted payback period is a capital budgeting procedure used to determine the profitability of a project. - Last but not the least, there is a payback rule which is also called the payback period, and it basically calculates the length of time which is required to recover the cost of investment. - Average cash flows represent the money going into and out of the investment. - So in the business environment, a lower payback period indicates higher profitability from the particular project. - For heat capacity flows below 4kW/ K, the optimisation resulted in no investment into solar thermal installations. - The resulting profitability indices are always positive. Both proposals are for similar products and both are expected to operate for four years. A project https://business-accounting.net/ has an initial outlay of $1 million and generates net receipts of $250,000 for 10 years. Modified Internal Rate Of Return It is an equal sum of money to be paid in each period forever. Thus we can compute the future value of what Vo will accumulate to in n years when it is compounded annually at the same rate of r by using the above formula. Future value is the value in dollars at some point in the future of one or more investments. Small projects may be approved by departmental managers. More careful analysis and Board of Directors’ approval is needed for large projects of, say, half a million dollars or more. Is the net cash flow at time t, USD; and t is time of cash flow. - The Net Present Value is the amount by which the present value of the cash inflows exceeds the present value of the cash outflows. - Thus, its use is more at the tactical level than at the strategic level. - The profitability index adjusts for the time value of money. - This is the cheapest way for the rich countries to delay climate change. - It can be used by homeowners and businesses to calculate the return on energy-efficient technologies such as solar panels and insulation, including maintenance and upgrades. - This firm’s forward looking PE ratio is equal to the expected payback period, which is the time it will take for the sum of the cash flows to equal the share price. So, based on this criterion, it’s going to take longer before the original investment is recovered. This is because they factor in the time value of money, working opportunity cost into the formula for a more detailed and accurate assessment. Another option is to use the discounted payback period formula instead, which adds time value of money into the equation. These two calculations, although similar, may not return the same result due to the discounting of cash flows. For example, projects with higher cash flows toward the end of a project’s life will experience greater discounting due to compound interest. For this reason, the payback period may return a positive figure, while the discounted payback period returns a negative figure. Capital Budgeting Basics Choosing the proper discount rate is important for an accurate Net Present Value analysis. Over the long run, capital budgeting and conventional profit-and-loss analysis will lend to similar net values. However, capital budgeting methods include adjustments for the time value of money (discussed in AgDM File C5-96, Understanding the Time Value of Money). Capital investments create cash flows that are often spread over several years into the future. To accurately assess the value of a capital investment, the timing of the future cash flows are taken into account and converted to the current time period . Suppose a situation where investment X has a net present value of 10% more than its initial investment and investment Y has a net present value of triple its initial investment. At first glance, investment Y may seem the reasonable choice, but suppose that the payback period for investment X is 1 year and investment Y is 10 years. - Both proposals are for similar products and both are expected to operate for four years. - The payback period can be found by dividing the initial investment costs of $100,000 by the annual profits of $25,000, for a payback period of 4 years. - However, to accurately discount a future cash flow, it must be analyzed over the entire five year time period. - If the project is accepted then the market value of the firm’s assets will fall by $1m. - This method does not require lots of assumptions and inputs. The crossover point is the rate or return that sets two mutually exclusive projects’ NPVs equal to zero. Both the IRR and the profitability index account for scale. The dividend growth rate cannot be greater than the cost of equity. The CIMA defines payback as ‘the time it takes the cash inflows from a capital investment project to equal the cash outflows, usually expressed in years’. When deciding between two or more competing projects, the usual decision is to accept the one with the shortest payback. Another issue with the formula for period payback is that it does not factor in the time value of money. This method totally ignores the solvency II the liquidity of the business. Payback period doesn’t take time value of money into account. The insurance companies are paying out a lot of money to settle their claims. The company paid off as many workers as it could before bankruptcy. Payback Method With Uneven Cash Flow: Discounted payback implies that the company will not accept any _______ NPV conventional projects. Managers use a variety of decision-making rules when evaluating long-term assets in the capital budgeting process. There is no one perfect rule; all have strengths and weaknesses. One thing all the rules have in common is that the firm must begin the capital budgeting process by estimating a project’s_____ ______. The quality of those estimates is critical to the sound application of the rules we will consider. Assume that the initial $11m cost is funded using the your firm’s existing cash so no new equity or debt will be raised. So whatever happened in the project is not going to be reflected in the payback period. This method only concentrates on the earnings of the company and ignores capital wastage and several other factors like inflation depreciation etc. Calculate the pay back period of buying the stock and holding onto it forever, assuming that the dividends are received as at each time, not smoothly over each year. The project has a positive net present value of $30,540, so Keymer Farm should go ahead with the project. ___________ _________ ________ Subtract a project’s discounted future cash flows from its initial cost until the firm recovers the initial investment. Accept the project if the discounted payback period is less than a predetermined time limit. An advantage of the discounted payback rule is that it adjusts for the time value of money, which is a failing of ordinary payback, though the advantage comes at the cost of increased complexity. Discounted payback implies that the company will not accept any negative NPV conventional projects. Beyond the TVM considerations, discounted payback has the same benefits and shortcomings as nominal payback. Cons Of Payback Period Analysis The shorter time scale project also would appear to have a higher profit rate in this situation, making it better for that reason as well. As a tool of analysis, the payback method is often used because it is easy to apply and understand for most individuals, regardless of academic training or field of endeavor. When used carefully to compare similar investments, it can be quite useful. As a stand-alone tool to compare an investment, the payback method has no explicit criteria for decision-making except, perhaps, that the payback period should be less than infinity. The payback method is a method of evaluating a project by measuring the time it will take to recover the initial investment. If the present value of a project’s cash inflows is greater than the present value of its cash outflows, then accept the project. If a project’s IRR is greater than the rate of return on the next best investment of similar risk, accept the project.

s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662541747.38/warc/CC-MAIN-20220521205757-20220521235757-00242.warc.gz

CC-MAIN-2022-21

https://musescore.org/en/node/268034

math

copy and paste over multiple staves I mostly arrange for a band with 5 horns. Sometimes, I would like to copy, for example, the lead trumpet part and give the same line to the other horns, so that all are playing in unison. I can do that very easily in Sibelius by copying the measures then clicking on the first measure of the next horn, then shift click on the first measure of the last horn in the score, then control V. That doesn't seem to work in Musescore 2. Also, lets say that I want to change the whole rest of a particular measure to quarter rests in every instrument, is there a way to do that all at once? Thanks for your help. I'm new to Musescore, although I'm finding it to be a great program, and similar to Sibelius in many ways.

s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103640328.37/warc/CC-MAIN-20220629150145-20220629180145-00277.warc.gz

CC-MAIN-2022-27

https://www.jiskha.com/display.cgi?id=1178678035

math

posted by Jim . Can you please help with this problem. I don't know how to solve it at all. A uniform solid disk with a mass of 32.3 kg and a radius of 0.464 m is free to rotate about a frictionless axle. Forces of 90.0 N and 125 N are applied to the disk. (a) What is the net torque produced by the two forces? (Assume counterclockwise is the positive direction.) (b) What is the angular acceleration of the disk? The torque about the axle requires further information on where and in what direction the forces are applied. Was there supposed to be a figure with this question? I know the answer... but its probably too late for you.

s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105955.66/warc/CC-MAIN-20170819235943-20170820015943-00371.warc.gz

CC-MAIN-2017-34

https://www.teacherspayteachers.com/Product/Graphing-Accurate-Geometry-Shapes_Linked-to-Common-Core-Math-Standards-406229

math

Common Core: Coordinate Graphing and Classifying 2D Figures Subject: Math Grades: 4 – 8 Objective: Creating two-dimensional shapes on a coordinate plane. Linked to Common Core Standards Review with students how to graph on a coordinate plane using coordinates. Then have students create mathematically accurate drawings of the listed two dimensional shapes. The last step is to give the addresses of the shapes vertices. There are several ways to label the graph axis depending on the ability of the student. There are two versions of the worksheet included. One worksheet has four quadrants and the other has just the first quadrant. Geometric Art Worksheet with 4 Quadrant Grid Geometric Art Worksheet with X,Y Axis Grid Geometric Art Sample Student Work

s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863830.1/warc/CC-MAIN-20180620163310-20180620183310-00388.warc.gz

CC-MAIN-2018-26

http://www.protocol-online.org/biology-forums/posts/430.html

math

3D Analysis of peptide - (Jan/17/2001 ) I want to do 3D analysis on a 500 aa long peptide(Sequence already known). Can you tell me where I can down load software to do this. Thank you very much. Would You like to try this URL: http://www.expasy.ch/swissmod/? This model system need an existing template of which the 3D structure is already known. And your peptide need to be had a homology with it at least 25%

s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864387.54/warc/CC-MAIN-20180622084714-20180622104714-00568.warc.gz

CC-MAIN-2018-26

https://www.convertunits.com/from/decibar/to/yoctopascal

math

How many decibar in 1 yoctopascal? The answer is 1.0E-28. We assume you are converting between decibar and yoctopascal. You can view more details on each measurement unit: decibar or yoctopascal The SI derived unit for pressure is the pascal. 1 pascal is equal to 0.0001 decibar, or 1.0E+24 yoctopascal. Note that rounding errors may occur, so always check the results. Use this page to learn how to convert between decibars and yoctopascals. Type in your own numbers in the form to convert the units! 1 decibar to yoctopascal = 1.0E+28 yoctopascal 2 decibar to yoctopascal = 2.0E+28 yoctopascal 3 decibar to yoctopascal = 3.0E+28 yoctopascal 4 decibar to yoctopascal = 4.0E+28 yoctopascal 5 decibar to yoctopascal = 5.0E+28 yoctopascal 6 decibar to yoctopascal = 6.0E+28 yoctopascal 7 decibar to yoctopascal = 7.0E+28 yoctopascal 8 decibar to yoctopascal = 8.0E+28 yoctopascal 9 decibar to yoctopascal = 9.0E+28 yoctopascal 10 decibar to yoctopascal = 1.0E+29 yoctopascal You can do the reverse unit conversion from yoctopascal to decibar, or enter any two units below: decibar to pound/square inch decibar to kilogram-force/square millimeter decibar to terapascal decibar to zeptopascal decibar to millimeter of water decibar to foot of air decibar to kilogram/square centimeter decibar to nanobar decibar to megabar decibar to poundal/square foot The SI prefix "deci" represents a factor of 10-1, or in exponential notation, 1E-1. So 1 decibar = 10-1 bars. The definition of a bar is as follows: The bar is a measurement unit of pressure, equal to 1,000,000 dynes per square centimetre (baryes), or 100,000 newtons per square metre (pascals). The word bar is of Greek origin, báros meaning weight. Its official symbol is "bar"; the earlier "b" is now deprecated, but still often seen especially as "mb" rather than the proper "mbar" for millibars. The SI prefix "yocto" represents a factor of 10-24, or in exponential notation, 1E-24. So 1 yoctopascal = 10-24 pascals. The definition of a pascal is as follows: The pascal (symbol Pa) is the SI unit of pressure.It is equivalent to one newton per square metre. The unit is named after Blaise Pascal, the eminent French mathematician, physicist and philosopher. ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!

s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662562106.58/warc/CC-MAIN-20220523224456-20220524014456-00719.warc.gz

CC-MAIN-2022-21

http://mathhelpforum.com/statistics/204514-maths.html

math

For each Australia dollar, Amy receives 80 United States cents, I.e $A1 = US80c. She wants $US600 for a trip, how much should she receive in Australian dollars? A)$480 B)$800 C)$750 (D)$720 For each Australian dollar, Amy receives 4/5 U.S. dollars. Letting x be the number of Australian dollars, we can multiply this by 4/5 and equate to 600 to find the number of Australian dollars that is equivalent to 600 U.S. dollars. Now solve for x.

s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122955.76/warc/CC-MAIN-20170423031202-00244-ip-10-145-167-34.ec2.internal.warc.gz

CC-MAIN-2017-17

https://www.impan.pl/en/activities/banach-center/conferences/21-juliasets3/video_recordings

math

Monday, 27th September Nuria Fagella (Universitat de Barcelona), "Virtual Centers in the Parameter Space of Meromorphic Maps" Xavier Jarque (University of Barcelona), "On the basins of attraction of a one-dimensional family of root finding algorithms: From Newton to Traub" Weiwei Cui (Lund University), "Hausdorff dimension of escaping sets for Speiser functions with few singular values" Anna Miriam Benini (Università di Parma), "Mañé-Sad-Sullivan and meromorphic dynamics" Dan Nicks (University of Nottingham), "Orbits and bungee sets" Dzmitry Dudko (Stony Brook University), "Expanding and relatively expanding Thurston maps" Athanasios Tsantaris (University of Nottingham), "Explosion points of Zorich maps" Dan Alexandru Paraschiv (University of Barcelona), "Newton-like behaviour in the Cebyshev-Halley family of degree n polynomials" Anna Jové Campabadal (University of Barcelona), "Dynamics on the boundary of Fatou components" Tuesday, 28th September Walter Bergweiler (Kiel University), "The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class" Gwyneth Stallard (The Open University), "Boundary dynamics of wandering domains: overview" Phil Rippon (The Open University), "Boundary dynamics of wandering domains: sufficient conditions for uniform behaviour 1" Vasiliki Evdoridou (The Open University), "Boundary dynamics of wandering domains: sufficient conditions for uniform behaviour 2" Luka Boc Thaler (University of Ljubljana), "On the geometry of simply connected wandering domains" James Waterman (Stony Brook University), "Wandering Lakes of Wada" Arnaud Chéritat (Institut de Mathématiques de Toulouse), "Bounded type Siegel disks of finite type maps with few singular values" Kostiantyn Drach (Aix-Marseille Université), "How to use box mappings as “black boxes”" Wednesday, 29th September Mitsuhiro Shishikura (Kyoto University), "Multiply connected Fatou components" Han Peters (University of Amsterdam), "Zeros of the Independence polynomial for graphs of large degrees" Antonio Garijo (Universitat Rovira i Virgili), "Dynamics of the secant map" Christopher Bishop (Stony Brook University), "Dimensions of transcendental Julia sets" Thursday, 30th September Oleg Ivrii (Tel Aviv University), "Shapes of trees" Łukasz Pawelec (SGH - Warsaw School of Economics), "The important set for non-autonomous exponential map" Dierk Schleicher (Aix-Marseille Université), "Postsingularly finite entire functions: combinatorics, complexity, Thurston theory" Davoud Cheraghi (Imperial College London), "Analytic symmetries of parabolic and elliptic elements" Daniel Meyer (University of Liverpool), "Quasisymmetric Uniformization, quasi-visual approximations, and Thurston maps" Michael Yampolsky (University of Toronto), "Harmonic measures, Julia sets, and computability" Gustavo Rodrigues Ferreira (The Open University), "Uniformity in internal dynamics of wandering domains" Robert Florido Llinàs (Universitat de Barcelona), "Projecting Newton maps of Entire functions via the Exponential map" Andrew Brown (University of Liverpool), "Eremenko’s Conjecture, Devaney’s Hairs, and the Growth of Counterexamples" Friday, 1st October Fabrizio Bianchi (CNRS - Université de Lille), "Higher bifurcations for polynomial skew products" Michel Zinsmeister (Université d'Orléans), "Integral means spectrum" Genadi Levin (Hebrew University of Jerusalem), "On hyperbolic sets of polynomials, II"

s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233505362.29/warc/CC-MAIN-20230921073711-20230921103711-00321.warc.gz

CC-MAIN-2023-40

https://ecomsutra.com/glossary/average-order-value/

math

The AOV measures your customers’ total average amount when they perform a successful purchase in your store. An increase in AOV indicates the success of conversion rate optimization and relevant product recommendation strategy. The formula to calculate AOV is as follows: AOV = Revenue / No.of orders For instance, if your revenue is $5000 and the total number of orders was 100, then the AOV will be ($5000/100) which is $50. This means that, on average, a customer will spend $50 for each transaction in your store.

s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517485.8/warc/CC-MAIN-20220517130706-20220517160706-00567.warc.gz

CC-MAIN-2022-21

https://www.clutchprep.com/chemistry/practice-problems/57534/give-the-number-of-possible-orbitals-in-an-h-atom-with-the-values-n-3l-l-1-a-1-b

math

Give the number of possible orbitals in an H atom with the values: n = 3 l, l = 1 e. an infinite number Frequently Asked Questions What scientific concept do you need to know in order to solve this problem? Our tutors have indicated that to solve this problem you will need to apply the Introduction to Quantum Mechanics concept. You can view video lessons to learn Introduction to Quantum Mechanics. Or if you need more Introduction to Quantum Mechanics practice, you can also practice Introduction to Quantum Mechanics practice problems. What is the difficulty of this problem? Our tutors rated the difficulty ofGive the number of possible orbitals in an H atom with the v...as medium difficulty. How long does this problem take to solve? Our expert Chemistry tutor, Dasha took 1 minute and 11 seconds to solve this problem. You can follow their steps in the video explanation above.

s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107900200.97/warc/CC-MAIN-20201028162226-20201028192226-00620.warc.gz

CC-MAIN-2020-45

https://dspace.sunyconnect.suny.edu/handle/1951/69636/discover?filtertype_0=subject&filter_relational_operator_0=equals&filter_0=random+walk&filtertype=subject&filter_relational_operator=equals&filter=Amenable+group

math

Now showing items 1-1 of 1 Amenability and superharmonic functions (Proceedings of the American Mathematical Society, 1993) Let G be a countable group and u a symmetric and aperiodic probability measure on G . We show that G is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets ...

s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987829507.97/warc/CC-MAIN-20191023071040-20191023094540-00181.warc.gz

CC-MAIN-2019-43

https://nrich.maths.org/public/leg.php?code=71&cl=4&cldcmpid=2377

math

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. By proving these particular identities, prove the existence of general cases. Find all the solutions to the this equation. With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even? Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods. What is the largest number of intersection points that a triangle and a quadrilateral can have? An article which gives an account of some properties of magic squares. An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes. Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics. Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n. In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)). We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions. Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number. In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot. Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry Can you discover whether this is a fair game? Some diagrammatic 'proofs' of algebraic identities and inequalities. Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . . This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself. When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators? Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started. This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition. Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. This is the second article on right-angled triangles whose edge lengths are whole numbers. Follow the hints and prove Pick's Theorem. The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it! A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices? It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more. Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions. To find the integral of a polynomial, evaluate it at some special points and add multiples of these values. A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers? The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers. Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions. What can you say about the common difference of an AP where every term is prime? This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats. Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. Kyle and his teacher disagree about his test score - who is right? What fractions can you divide the diagonal of a square into by simple folding? When is it impossible to make number sandwiches? If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle? A introduction to how patterns can be deceiving, and what is and is not a proof. Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials. These proofs are wrong. Can you see why? Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die? Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way? Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524502.23/warc/CC-MAIN-20190716035206-20190716061206-00073.warc.gz

CC-MAIN-2019-30

https://www.jiskha.com/display.cgi?id=1236223419

math

posted by Jake . Calculate the molar concentration of acetic acid (CH3COOH) in a 5.00-mL sample of vinegar (density is 1.00 g/mL) if it is titrated with 25.00 mL of NaOH. Determine moles acetic acid. moles NaOH = M NaOH x L NaOH. Moles acetic acid = moles NaOH. M = # moles/L of solution. Thank you very much. The molarity of the NaOH solution is 0.162. So I did the following: 25.00 mL NaOH*(1 L / 1000 mL)*(0.160 mol NaOH / 1 L NaOH)*(1 mol CH3COOH / 1 mol NaOH) = 0.004 mol CH3COOH (0.004 mol CH3COOH / 5.00 mL) * (1000 mL / 1 L) = 0.8 M CH3COOH Now I'm supposed to determine the mass percent of the acetic acid in the vinegar sample. I assume this is why I was given the density... how do I use the concentration of acetic acid and the density of vinegar to determine the mass percent? "The molarity of the NaOH solution is 0.160." The number in the calculations is correct. moles CH3COOH x molar mass acetic acid = grams CH3COOH. (g CH3COOH/100 g soln)*100 = mass percent. which you can do one of two ways. I have 0.24 g/5 g so how much is in 100? That will be 0.24 x 100/5 = 0.24 x 20 = 4.8 which is 4.8%. The other way, I think, is simpler:

s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889542.47/warc/CC-MAIN-20180120083038-20180120103038-00388.warc.gz

CC-MAIN-2018-05

https://oeis.org/A190634

math

Table of n, a(n) for n=1..6. Cf. A103431 (Gaussian primes in first quadrant), A190635 (index of same prime as Gaussian prime), A190637 (primes == 3 mod 4). Sequence in context: A032419 A225163 A319540 * A130421 A227403 A156736 Adjacent sequences: A190631 A190632 A190633 * A190635 A190636 A190637 Sven Simon, May 15 2011 a(5)-a(6) from Sven Simon, Jun 19 2011

s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039490226.78/warc/CC-MAIN-20210420183658-20210420213658-00455.warc.gz

CC-MAIN-2021-17

https://www.hackmath.net/en/math-problems/multiplication

math

Multiplication - problems Same type of medicament produces a number of manufacturers in a variety of packages with different content of active substance. Pack 1: includes 60 pills of 1200 mg of active substance per pack cost 18 Eur. Pack 2: includes 30 pills of 1000 mg of active s - No. of divisors How many different divisors has number ?? - Unknown x If we add to unknown number 72, then divide by 2 and then subtract 29, we get back an unknown number. What is this unknown number? - Milk package Milk is sold in a box with dimensions of 9.5 cm; 16.5 cm and 6.5 cm. Determine the maximum amount of milk that can fit into a box. Coating thickness is negligible. - Iron fence One field of iron fence consists of 20 iron rods with square cross-section of 1.5 cm side and a long 1 meter. How much weight a field if the density of iron is 7800 kg/m3. Result of the product of the numbers 5, 1, 7, 2, 6, 1, 0, 6, 6, 5, 1, 9, 6 is: Francisco got birthday 4 colour pens in different colors. How many ways he can give them side by side in pencil? Kate paid for the package on delivery of four meters textile 688 CZK. What is the price of one meter textile when shipping and handling amounted to 48 CZK? By how many is the difference of numbers 8 and 34 less than its product? - Heart as pump Heart pumps out 5.17 liters of blood in 1 minute. How many liters of blood pumped per hour and how much per day? Michal, Peter, John and Lenka got together 2,400 euros. They share an amount in ratio 2:6:4:3. How many got each of them? - Hexadecimal number What will be hexadecimal number 240 as decimal number? Determine the value of the following exspressions: a) (23-25)·(4-5) b) (97-123):(18+8) - Value of teeth To know value to the healthy teeth we discover only when we take care about tooth caries followed by loss of teeth and think how to replace missing teeth... Calculate the value (cost of money) healthy teeth, if we assume that man has with 32 teeth and rep - Eight palm There grows 8 palms by the sea. At the first sitting one parrot, on second two, on third sits four parrots on each other twice the previous parrots sitting on a previous palm. How many parrots sitting on eighth palm? Mum bought 16 cartons of milk. One carton of milk weight 0.925 kg. How many weight all purchase? Write time in minutes rounded to one decimal place: 1 h 42 m 22 s. Determine the number of all positive integers less than 2937474 if each is divisible by 11, 23, 17. What is its sum? How many times should be broken chocolate consisting of 7 × 10 pieces to get the 70 parts? - Phone numbers How many 9-digit telephone numbers can be compiled from the digits 0,1,2,..,8,9 that no digit is repeated?

s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320040.36/warc/CC-MAIN-20170623082050-20170623102050-00690.warc.gz

CC-MAIN-2017-26

https://seo1.in.th/page.php?tag=6ee66d-what-is-a-non-example-in-science

math

is a non-example. (Entry 1 of 2) : something (such as a discipline) that is not a science. One of the most beautiful examples of this is the periodic table of elements. Is a non-example of linear a curve. Top Answer. This is an odd question. 1 2. So, a seashell is not the axis. Everything that has mass and takes up space is matter, yet some things do not consist of matter. Answer. Anything that is NOT the Earth's axis. That would be my first thought but not sure non examples are anything made by a human on land that is not natural Asked in Math and Arithmetic , Scientific Notation Numbers to write in scientific notation ? Definition of nonscience. Wiki User. A rigorous science is able to make testable predictions. Dmitri Mendeleev, a Russian chemist, successfully predicted the properties of missing elements on the table – that is, elements that had not been discovered yet. Here is a list of 10 examples of non-matter. It is possible to give an example of non-linear, but I have no idea what a non-example is. 2013-10-17 11:53:15.

s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402130531.89/warc/CC-MAIN-20200930235415-20201001025415-00759.warc.gz

CC-MAIN-2020-40

https://au.mathworks.com/matlabcentral/profile/authors/17778101

math

Error in Simulink: Undefined function or variable 'uniqueOutput' Hello I get the same error mentioned in this question https://de.mathworks.com/matlabcentral/answers/381476-error-in-simulink-u... 10 months ago | 1 answer | 0 nlarx model initial conditions Hello i want to use a nlarx model with focus on simulation to model a system. The results i get with the nlarx command are good... 12 months ago | 1 answer | 0

s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00640.warc.gz

CC-MAIN-2021-31

http://mwtermpaperysbt.designheroes.us/determining-the-density-of-aluminum-pellets-through-its-mass-and-volume.html

math

Alternatively, if the density of a substance is known, and is uniform, the volume can be calculated using its weight this calculator computes volumes for some of the most common simple shapes sphere. Density is the measurement of the amount of mass per unit of volumein order to calculate density, you need to know the mass and volume of the item the mass is usually the easy part while volume can be tricky. Abstract have you ever wondered how a ship made of steel can float or better yet, how can a steel ship carry a heavy load without sinking in this science project you will make little boats out of aluminum foil to investigate how their size and shape affects much weight they can carry and how this relates to the density of water. Explain how objects of similar mass can have differing volume, and how objects of similar volume can have differing mass explain why changing an object's mass or volume does not affect its density (ie, understand density as an intensive property. The car's weight and volume change, but not its mass a ball falling through the air calculate the density with the correct number of significant figures of a. Repeat steps 3 through 8 for the cylindrical copper mass and the spherical lead mass calculate the density of each of the objects and enter it in table 1 assume the density of water is r f = 10 3 kg/m 3. The density of aluminum is 270 g/cm3, and the average mass for one aluminum atom is 448×10-23 g five identical aluminum coins are found to displace a total of 250 ml of water when immersed in a graduated cylinder containing water. Density is the mass of an object divided by its volume density often has units of grams per cubic centimeter (g/cm 3 ) remember, grams is a mass and cubic centimeters is a volume (the same volume as 1 milliliter. It is common to use the density of water at 4 o c (39 o f) as a reference since water at this point has its highest density of 1000 kg/m 3 or 1940 slugs/ft 3 since specific gravity - sg - is dimensionless, it has the same value in the si system and the imperial english system (bg. The mass of water in the pycnometer at this tempearature will be determined by using mass of pycnometer, m w minus mass of empty pycnometer, m let d be the density of water at t°c, so that the volume of pycnometer, v at this temperature can be expressed in the term as. The mass of atoms, their size, and how they are arranged determine the density of a substance density equals the mass of the substance divided by its volume d = m/v objects with the same volume but different mass have different densities. The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume the symbol most often used for density is ρ (the lower case greek letter rho ), although the latin letter d can also be used. Density is defined as the ratio of an object's mass to its volume, as shown in the equation above because it is a ratio, the density of a material remains the same without regard to how much of that material is present. Density = mass / volume you know the density of aluminum and you know the peice of aluminum's mass, therefore you should be able to re-write the density equation in such a way to solve for volume imagine the square of aluminum foil as a very thin block of aluminum. If you have a pure liquid or a solid, you use its density to calculate its mass and then divide the mass by the molar mass if you have a solution, you multiply the molarity by the volume in litres. In the above equation, m is mass, ρ is density, and v is volume the si unit for density is kilogram per cubic meter, or kg/m 3 , while volume is expressed in m 3 , and mass in kg this is a rearrangement of the density equation. Density is the mass per unit of volume of a substance the density equation is: #density# = #mass/volume# to solve the equation for mass, rearrange the equation by multiplying both sides times volume in order to isolate mass, then plug in your known values (density and volume. Since sg = s / w, and since w = 1 gram/cm 3), one can determine the density of the object by measuring its mass and volume directly for a liquid the volumetric flask (or pycnometer ) has a hollow stem stopper that allows one to prepare equal volumes of fluids very reproducibly. Density 1 density the mass density or density of a material is defined as its mass per unit volumethe symbol most often used for density is ρ (the greek letter rho) in some cases (for instance, in the united states oil and gas industry), density is. Since density is mass per unit volume,the density of a metal can be calculated by submerging it in a known amount of water and measuring how much the water risesthis rise is the volume of the metal its mass can be measured using a scale. Method 1: determination of density by direct measurement of volume the object you have is a cube of metal the volume of a cube can be found from the formula v=a 3 , where a is the length of one edge in centimeters. The independent variable, volume, always goes on the x-axis through your data points using your graph, determine the mass of 100 ml of material. The density used in the calculations will appear in the density box in g/cc if you prefer to see the density in other units, just click the units drop-down box next to the density, and the value will be converted for you automatically. A little aluminum boat (mass of 1450 g) has a volume of 45000 cm 3 the boat is place in a small pool of water and carefully filled with pennies the boat is place in a small pool of water and carefully filled with pennies. The iron brick has twice the mass, but its volume compared to the block of wood depends on the density of the wood calculate the density with the correct number.

s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742567.46/warc/CC-MAIN-20181115054518-20181115075738-00006.warc.gz

CC-MAIN-2018-47

https://brilliant.org/discussions/thread/hole-and-square-puzzle/

math

Was wondering if there is a solution to this puzzle ! 3 years, 6 months ago Log in to reply Can you describe the puzzle? Or can you provide a link to explain it?

s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823997.21/warc/CC-MAIN-20171020082720-20171020102720-00221.warc.gz

CC-MAIN-2017-43

http://pigstyave.blogspot.com/2009/08/hard-rain-gonna-fall-note-to-self.html

math

Bob Dylan - A Hard Rain's A-Gonna Fall This would be good for teaching tenses. Or maybe just reviewing them in an end of term lesson. The way different tenses are used throughout. You could teach the concept of aspect with it: where's he standing as the narrrative progresses. And how his tense use depends on the question prompt at the start of each verse. Here's the text of the lyrics. EDIT: you could use a power point or wtf to look at a time line. Image of Bob, different points on the time line. Images to illustrate the lyrics.

s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864776.82/warc/CC-MAIN-20180622182027-20180622202027-00525.warc.gz

CC-MAIN-2018-26

http://www.cas.mcmaster.ca/~qiao/courses/cas708/index.html

math

CAS708/CSE700 Scientific Computation Dr. Sanzheng Qiao ITB246, ext. 27234, email@example.com Monday and Wednesday 9:00--10:30, ITB/222 Floating-point arithmetic, solutions of systems linear equations by direct and iterative methods, sparse matrix algorithms, solving systems nonlinear equations, integration, differentiation, eigenvalue problems, methods for initial value problems in ordinary differential equations. By the end of this course students will be able to midterm (closed book, 1 hr) (20%) final examination (open book and notes, 2 hrs) (50%). MATLAB Programming Style References at the end of each chapter Midterm, March 5, Wednesday, 9:15-10:15, ITB/222 No books and notes. Bring your standard calculators. Final, April 15, Tuesday, 10:00-12:00, ITB/222 Open book and notes, no electronics Office hours: April 14, Monday, 13:00-15:00 Back to S. Qiao's home page

s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823731.36/warc/CC-MAIN-20171020044747-20171020064747-00778.warc.gz

CC-MAIN-2017-43

https://pwntestprep.com/tag/parabolas/

math

Can you explain this Q #29 from Calculator section of Oct 2022 PSAT: A quadratic function can be used to model the height, in feet, of an object above the ground in terms of time, in seconds, after the object was launched. According to the model, an object was launched into the air from a height of 0 feet and reached its maximum height of 3136 feet 14 seconds after it was launched. Based on the model, what was the height, in feet, of the object 1 second after it was launched? HI Mike…Thanks for all your help ! Here’s another question: When a buffet restaurant charges $12.00 per meal, the number of meals it sells per day is 400. For each $0.50 increase to the price per meal, the number of meals sold per day decreases by 10. What is the price per meal that results in the greatest sales, in dollars, from meals each day? COLLEGE BOARD Test 9 Math Section 3 #13 Could you suggest a shortcut or fast way to solve this? All the answers are written in vertex form, so we can quickly eliminate two of them, as the coordinates of the vertex, as indicated by the graph provided, must be (3,1). That leaves choices A and C. Is there a quick way to solve from there without plugging in values from the graph? I do not understand question #4 pg 156 advanced systems of equations? y=c(x^2) + d In the systems of equations above, c and d are constants . For which of the following values of c and does the system of equations have no real solutions? A) c=-6, d=6 B) c=-5, d=4 C) c=6, d=4 D) c=6, d=5 Please help, thank you. Test 10 section 4 number 28 Is there a way to solve this system of equations without using the quadratic formula (or graphing)? f(x) = –2/3 x + 4 g(x) = 3(x + 2)^2 – 4 How many solutions does the system above have? Huge shortcut here if you just know that for a parabola in standard ax^2 + bx + c form, the x-coordinate of the vertex will be at –b/(2a). In this case, that means it’s at –3/(2(–6)) = 3/12 = ¼. from Tumblr https://ift.tt/2CRW93a PWN the SAT Parabolas drill explanation p. 325 #10: The final way to solve: If we are seeking x=y, since the point is (a,a), why can you set f(x) = 0? You start out with the original equation in vertex form, making y=a and x=a, but halfway through you change to y=0 (while x is still = a). How can we be solving the equation when we no longer have a for both x and y? Basically, the question is: how many seconds is h greater than 21? (This tennis ball is being thrown on a planet other than Earth, by the way. I challenge anyone to throw a tennis ball that stays in the air anywhere near as long as this one does.) To figure it out, solve for the (more…) A question from the May 2018 SAT (Section 4 #18) kx + y = 1 y = -x² + k In the system of equations above, k is a constant. When the equations are graphed in the xy-plane, the graphs intersect at exactly two points. Which of the following CANNOT be the value of k? Hi Mike, Can you please explain Question 11, Test 6, section 3 ? I know the parabola opens downward, but I’m confused after that. Thanks. In PWN p. 159 (p. 157 later printing) #8 In the xy-plane, where a and b are constants, the graphs … The question does not specify that a and b are positive values. If one or both were negative, wouldn’t that change the answer?

s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948867.32/warc/CC-MAIN-20230328135732-20230328165732-00441.warc.gz

CC-MAIN-2023-14

https://cmde.tabrizu.ac.ir/article_6816.html

math

Document Type : Research Paper Faculty of Mathematical Sciences and Statistics, Malayer University, P. O. Box 65719-95863, Malayer, Iran Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran In this paper, we apply Legendre wavelet collocation method to obtain the approximate solution of nonlinear Stratonovich Volterra integral equations. The main advantage of this method is that Legendre wavelet has orthogonality property and therefore coefficients of expansion are easily calculated. By using this method, the solution of nonlinear Stratonovich Volterra integral equation reduces to the nonlinear system of algebraic equations which can be solved by using a suitable numerical method such as Newton’s method. Convergence analysis with error estimate are given with full discussion. Also, we provide an upper error bound under weak assumptions. Finally, accuracy of this scheme is checked with two numerical examples. The obtained results reveal efficiency and capability of the proposed method.

s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488559139.95/warc/CC-MAIN-20210624202437-20210624232437-00597.warc.gz

CC-MAIN-2021-25

https://rumahhijabaqila.com/edition-pdf/18909-thermodynamics-kinetic-theory-and-statistical-thermodynamics-3rd-edition-pdf-492-411.php

math

Thermodynamics, Kinetic Theory, and Statistical Thermodynamics 3rd Edition Thermodynamics, Kinetic Theory, and Statistical Thermodynamics Thermodynamics, Kinetic Theory and Statistical Thermodynamics is designed to help undergraduate students of engineering and physics who have basic experience in the subject of calculus. The book provides the reader with useful information on classical thermodynamics and also educates him on the microscopic properties of a system. Each chapter in the book concludes with a number of problems that give the user ample practice to make him proficient in the subject. Narosa Book Publishing House was established in the year It has, over the years, published books for students of various subjects like Mathematics, Computers, Physics and Management. Certified Buyer , Barasat. Additional order info. The general approach has been unaltered and the level remains much the same, perhaps being increased somewhat by greater coverage. The text is particularly useful for advanced undergraduates in physics and engineering who have some familiarity with calculus. Fundamental Concepts. Equations of State. The First Law of Thermodynamics. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium , or is undergoing a fictive reversible process. Teaching PHYS And therefore, two approaches are used to teach thermal physics at the undergraduate level: A traditional approach two separate courses : a semester of "Thermodynamics" followed by a semester of "Statistical Mechanics. Older textbooks present the two approaches in two separated parts. Newer books mix up the two methods. At the graduate level, thermal physics is normally taught through Statistical Mechanics approach, with some brief review of the laws of Thermodynamics. Textbooks F. From the publisher M.

s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780053493.41/warc/CC-MAIN-20210916094919-20210916124919-00545.warc.gz

CC-MAIN-2021-39

https://www.tes.com/news/number-crunching

math

As with many disciplines, the language of mathematics contributes both to its power and to the difficulty some students find in studying it. Access to a good technical dictionary can help, although all too often definitions are couched in the same arcane language which drove one to seek help in the first place. There are important distinctions to be made between the needs, say, of a typical 11-year-old, a primary school teacher and an undergraduate mathematician. All too often, existing mathematical dictionaries, of which there are many, have sacrificed usefulness by trying to serve too wide an audience. The new Collins Educational School Mathematics Dictionary has been designed specifically for secondary school pupils aged 11 to 16. Its attractive design with spacious layout and large typefaces will improve its appeal to young people. With about 900 entries, it is less comprehensive than many of its competitors, but it represents a pretty thorough coverage of the main terms which will be encountered by pupils in key stages 3 and 4. It is very business-like with relatively concise definitions supported by examples and simple diagrams. There is occasional help on pronunciation and the derivations of some words are given. The entries include a number of brief biographies of famous mathematicians, although the criteria for selection are by no means clear (R A Fisher is there, Alan Turing is not). References to units of measurement are staunchly metric - there are no signs of miles, pints or acres. The influence of recent developments in school mathematics is only partial. Pentomino and hexomino are defined but the more general polyomino is not, perfect numbers are explained but happy and sad numbers are not recognised. Some computing terms get in, including byte, LOGO and program. A dictionary of this sort inevitably sacrifices completeness to provide accessibility. However, the assertion under tessellation that "octagons cannot tessellate the plane" is inexcusable. Many can. Overall, this is a dictionary which meets its brief well and deserves a place in all secondary mathematics classrooms where, I predict, it will prove as useful to teachers as to students.

s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000894.72/warc/CC-MAIN-20190627055431-20190627081431-00380.warc.gz

CC-MAIN-2019-26

https://elifemeds1.com/mathhelp-834

math

College algebra answers to questions This College algebra answers to questions helps to quickly and easily solve any math problems. Our website can solve math problems for you. The Best College algebra answers to questions College algebra answers to questions can be found online or in math books. Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. Trigonometry is used in many areas of science, engineering, and construction. Trigonometry can be used to find the height of a building, the length of a bridge, or the slope of a hill. Trigonometry can also be used to calculate the amount of material needed for a project, or to determine the angle of a sunbeam. Trigonometry is an essential tool for many businesses and industries. Trigonometry can be used to calculate interest rates, measure snow depth, or determine the size of a room. Trigonometry can also be used to aid in navigation, calculate distances, and predict tides. Trigonometry is a powerful tool that can be used to solve many problems. Trigonometry can be difficult, but there are many resources available to help students learn trigonometry. There are online tutorials, textbooks, and video lessons. Trigonometry can be learned in a classroom setting, or at home with online resources. Trigonometry is a challenging but rewarding subject. With practice and patience, anyone can learn trigonometry. How to solve perfect square trinomial. First, identify a, b, and c. Second, determine if a is positive or negative. Third, find two factors of ac that add to b. Fourth, write as the square of a binomial. Fifth, expand the binomial. Sixth, simplify the perfect square trinomial 7 eighth, graph the function to check for extraneous solutions. How to solve perfect square trinomial is an algebraic way to set up and solve equations that end in a squared term. The steps are simple and easy to follow so that you will be able to confidently solve equations on your own! Hard math equations with answers are difficult to find. However, there are a few websites that have a compilation of hard math equations with answers. These websites have a variety of equations, ranging from algebra to calculus. In addition, the answers are provided for each equation. This is extremely helpful for students who are struggling with a particular equation. Hard math equations with answers can be very challenging, but by using these websites, students can get the help they need to succeed. Next, take the square root of each coefficient. Finally, add or subtract the results to find the answer. This method may seem daunting at first, but with a little practice it can be mastered. Perfect square trinomials may not be the most exciting type of math problem, but being able to solve them is a valuable skill. With a little patience and persistence, anyone can learn how to solve perfect square trinomials.

s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500288.69/warc/CC-MAIN-20230205193202-20230205223202-00648.warc.gz

CC-MAIN-2023-06

https://www.wyzant.com/Long_Island_City_calculus_tutors.aspx

math

29 Calculus lessons She has lots of practice problems for every calculus concept and tips on how to efficiently solve difficult problems. She was able to explain any confusion I had in my BC Calculus class. She knows her calculus very well. Although I did not take Calculus AB before taking Calculus BC, I still received a 5 on the 2021 BC exam, and a 5 on my Calc AB subscore, thanks to the help I got from Diana. Highly recommend her for any math tutoring you might need. 1 Calculus lesson The only Calculus tutor at her school, is her current teacher. My daughter requested assistance, but her teacher's approach to Calculus was not being understood well by my daughter. She requested assistance from other peers, but the ones who did understand it couldn't explain it well. So I found a tutor on the internet. The first one was not a good fit...he had taken the summer off from college, and just lectured her in a lofty manner about Calculus in general - wasted $50 in my opinion! 2 Calculus lessons I had to learn Calculus in a WEEK to pass this math exam to get into my top university. I didn't realize the exam had calculus on it until last minute, because I was so focused on my other classes. My dream school ranks along BROWN AND DARTMOUTH and this math exam was VERY difficult, and included calculus. I had learned pre-calculus in high school, but that was two years ago for me, and I forgot a lot of the material.

s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662515466.5/warc/CC-MAIN-20220516235937-20220517025937-00349.warc.gz

CC-MAIN-2022-21

http://sehomeworkyplk.agorisme.info/chapter-6-logic.html

math

Chapter 6 logic Unsurpassed for its clarity and comprehensiveness, hurley's, a concise introduction to logic is the #1 introductory logic. Chapter 6 threshold logic logic design of sw functions constructed of electronic gates different type of switching element : threshold element. Tue, 26 mar 2013 23:59:00 gmt chapter 6 logic and pdf - view and download spectra logic t-series spectra t120 user manual online t-series spectra. Cis 103 homework assignment # all problems from the text book programming logic and design do the following exercises at the end of chapter 6. Section 63, part i: 1 documents similar to exercise answers, hurley, 11th edition logic chapter 5 uploaded by raymond ruther. View homework help - practical logic chapter 7 answers from phil 201 at loyola new orleans exercise 71 part i 1 g 1, 2, mt 6 s 2, 3, mt 2 m 1, 2, mp 7 f d 1, 3. 1 basic concepts of logic 1 what is logic exercises for chapter 1 6 hardegree, symbolic logic tively correct. The logic book 6 th edition chapter 6: sentential logic: metatheory 61 mathematical induction 62 truth-functional completeness 6. Starting out with programming logic and design, chapter 4 decision structures and boolean logic 115 chapter 6 functions 217. Logic chapter 6 test logic chapter 6 test logic chapter 6 test logic name:_____ chapter 6 test except for the truth table. Teacher manual for introduction to logic (except that chapter 10 depends only on chapters 6 and 7, and chapter 11 isn’t required for chapters 12 to 14. Chapter 4 informal fallacies the starred items are also contained in the answer key in the back of the power of logic 6 appeal to ignorance.Gary hardegree : philosophy 110 - introduction to logic: umass amherst: gary hardegree: chapter 5: derivations in sentential logic: chapter 6:. [21:16 2003/5/31 demicco-ch06tex] demicco: fuzzy logic in geology page: 153 153–190 chapter 6 fuzzy logic in hydrology and. 61 introduction propositional logic allows us to talk about relationships among individual propositions, and it gives us the machinery to derive logical conclusions. Introduction to logic offers one of the most clear, interesting and accessible introductions to what has long been considered one of the most challenging subjects in. Nomes ink questioned, tilting his skull in confusion yeah, i see them often around where i came from, (y/n) responded and where are yo u from a different. Chapter 6 propositional logic: brazil has a huge foreign debt therefore, either brazil or argentina has a sjsu philosophy chapter 6 05/06/03. Register free to download files | file name : a concise introduction to logic answers chapter 6 pdf satisfied is finishing reading this book and getting the message of. Carl von clausewitz book 8 • chapter 6 it has certainly a grammar of its own, but its logic is not peculiar to itself. Predicate logic : chapter 6-7 15 07/29/09 – w - translating english into predicate logic : week 4 introduction to logic author: rodrigo martins borges. 2 chapter 6 problem set the circuit is given in the next figure 3 consider the circuit of figure 61 a what is the logic function implemented by the cmos. Chapter 6: naturalism and virtue ethics 88, c1 declaration of independence: major thinker in: ethics, logic, metaphysics, also biology and physics. Access introduction to logic design 3rd edition chapter 6 solutions now our solutions are written by chegg experts so you can be assured of the highest quality. Far too many authors of contemporary texts in informal logic logical reasoning has been enjoyable for me, c h a p t e r 6 writing to convince others.

s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156252.62/warc/CC-MAIN-20180919161420-20180919181420-00213.warc.gz

CC-MAIN-2018-39

http://www.questionotd.com/2008/10/fire-above.html

math

I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same answer as someone else, feel free to put up your answer, too! Thursday, October 16, 2008 Fire is often maintained above me, & if you remove my first letter, you will find the home shared by everyone you have ever known.

s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499919.70/warc/CC-MAIN-20230201081311-20230201111311-00833.warc.gz

CC-MAIN-2023-06

https://parentingandmentalhealth.com/qa/question-what-is-rpm-formula.html

math

How do I calculate motor rpm? RPM = (120 * Frequency) / # of poles in the motor….By formula:((Synchronous Speed – Rated Full Load Speed) / (Synchronous Speed)) * 100% = Slip Rating.((1800RPM-1750RPM) / 1800RPM) * 100%= (50RPM/ 1800RPM) * 100%(50RPM/ 1800RPM) * 100%= . 027 * 100%. 027 * 100% = 2.7%Slip Rating = 2.7%. How is RPM shaft calculated? (shaft diameter(mm)/19108) X r.p.m. (shaft diameter(in.)/3.82) X r.p.m. What is the rpm of 1 hp motor? Let’s look a two 1 HP motors. The 1800 RPM, 1HP motor produces 3 ft. lbs of torque at 1800 RPM. The 3600 RPM, 1HP motor produces 1.5 ft. How is RPM related to speed? While RPM stands for revolutions per minute and it very much describes the Rotations of the Engine’s crankshaft per unit time i.e. A minute here. … But to be accurate, there isn’t any direct relationship with the RPM and Speed. The higher the RPM there, the faster the crankshaft in the engine is spinning. What is the formula for calculating rpm? RPM = a/360 * fz * 60 RPM = Revolutions per minute. Example 1: Drive step resolution is set for 1000 steps per revolution. What is RPM value? CARS.COM — RPM stands for revolutions per minute, and it’s used as a measure of how fast any machine is operating at a given time. In cars, rpm measures how many times the engine’s crankshaft makes one full rotation every minute, and along with it, how many times each piston goes up and down in its cylinder.

s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400202007.15/warc/CC-MAIN-20200921175057-20200921205057-00731.warc.gz

CC-MAIN-2020-40

https://bondmatt.wordpress.com/2010/02/

math

February 23, 2010 In other news, I’ve been thinking about writing another essay, similar to Convolutions and the Weierstrass Approximation Theorem. This one will be about the Fundamental Theorem of Algebra. It won’t be a totally rigorous paper – the intent will be to present a very simple idea that a bright high schooler should be able to understand: Euler’s formula for e^(x+iy) tells us that polynomials in a complex variable z = x+iy grow in all directions as |z| increases. It’s such a simple idea that that quickly leads to at least a convincing heuristic argument that it’s a shame that texts often opt to emphasize how weird it is that e^(i*pi) + 1 = 0, instead, while saying nothing more on the FToA than that Gauss proved it seven different ways because he was smart, and don’t you wish you could be smart like Gauss too? It will be a heuristic proof, though maybe I will formally reduce it to several assertions, leaving a very specific hole in a very plausible-looking theorem unfilled by a fundamental group argument. This is just one of those things I could’ve understood much sooner than I actually did. This stuff is supposed to make sense, after all.

s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125654.80/warc/CC-MAIN-20170423031205-00284-ip-10-145-167-34.ec2.internal.warc.gz

CC-MAIN-2017-17

https://www.statisticshowto.com/one-to-one/

math

The Horizontal Line Test for a One to One Function - One easy way of determining whether or not a mapping is injective is the horizontal line test. Graph the function. - Draw a horizontal line over that graph. - If any horizontal line intersects the graph of the function more than once, the function is not one to one. But if there isn’t any straight horizontal line that can be drawn to cross the function more than once, the function is one to one. Below are images of two different mappings. On the first graph, you could draw a horizontal line that cuts through the hump and crosses the mapping twice. On the second graph, though, there is no way we can draw a horizontal straight line that crosses twice. The function in this second graph is therefore one to one. Properties of One to One Functions Here are some useful properties of injective functions: - If two functions f and g are both one to one, so is f · g. - If g · f is one to one, f is as well. - If f is injective, it has an inverse function (a function that undoes it). He, Jiwen. Inverses and More. Lecture 1, Section 7.1 Notes. Retrieved from https://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture01_handout.pdf Oldham, K. (2008). An Atlas of Functions. Springer. University of Toronto Math. Preparing for Calculus: Functions and Their Inverses. Retrieved from https://www.math.toronto.edu/preparing-for-calculus/4_functions/we_3_one_to_one.html on June 15, 2019.

s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296819273.90/warc/CC-MAIN-20240424112049-20240424142049-00565.warc.gz

CC-MAIN-2024-18

http://www.chegg.com/homework-help/questions-and-answers/mass-8kg-stiffness-90n-m-initial-displacement-01m-initial-velocity-09m-s-assume-underdampe-q3687442

math

System Dynamics-Time Domain Analysis of Dynamic Systems 300 pts endedThis question is closed. No points were awarded. mass =8kg stiffness =90N/m initial displacement =0.1m initial velocity =0.9m/s Assume the system is underdamped, and the damping ratio is 0.5. Find the corresponding damping coefficient c. Identify the system parameters such as the natural frequency and damped frequency. Derive the mathematical expression of the displacement as a function of time with the given initial condition. Plot your result I need with this problem i would greatly appreciate the help.

s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999664205/warc/CC-MAIN-20140305060744-00099-ip-10-183-142-35.ec2.internal.warc.gz

CC-MAIN-2014-10

https://www.splashlearn.com/math-vocabulary/divisibility-rules

math

What Are Divisibility Rules? Divisibility rules are simple tips and tricks that are used to check or to test whether a number is divisible by another number. Consider an example. Imagine that you have 13 candy bars. Can you divide them equally among 3 friends? How would you check? You can check if 13 is “divisible by” 3. In other words, you can check if 13 appears in the table of 3 or not! Now, what if you wish to check if you can divide 221 candies equally among 6 friends? When we are dealing with large numbers, it can be very time-consuming to find 221 in the multiplication table of 6. What do you think? To solve problems like these in no time, we use divisibility rules. With divisibility rules at your fingertips, you can answer easily without doing too much calculation! Divisibility Rules: Definition Divisibility rules are a set of general rules that are often used to determine whether or not a number is absolutely divisible by another number. Note that “divisible by” means a number divides the given number, without any remainder and the answer is a whole number. Divisibility Test (Division Rules in Math) Mathematical tests for divisibility or division rules help you employ a quick check to determine whether a number will be totally divisible by another number. What are the divisibility rules? Let’s learn divisibility rules 1-13. Divisibility Rule of 1 Every number ever is divisible by 1. Divisibility Rule of 2 Every even number is divisible by 2. That is, any number that ends with 2, 4, 6, 8, or 0 will give 0 as the remainder when divided by 2. For example, 12, 46, and 780 are all divisible by 2. Divisibility Rules of 3 A number is completely divisible by 3 if the sum of its digits is divisible by 3. You can also repeat this rule, until you get a single digit sum. Example 1: Check whether 93 is divisible by 3 or not. Sum of the digits $= 9 + 3 = 12$ If the sum is a multiple of 3, then the original number is also divisible by 3. Here, as 12 is divisible by 3, 93 is also divisible by 3. Example 2: 45,609 To make the process even easier, you can also find the sum of the digits until you get a single digit. Sum of digits $= 4 + 5 + 6 + 9 + 0 = 24$ Adding further, we get $2 + 4 = 6$ 6 is divisible by 3. Thus, 45609 is divisible by 3. Divisibility Rule of 4 If the number formed by the last two digits of a number is divisible by 4, then that number is divisible by 4. Numbers having 00 as their last digits are also divisible by 4. Example 1: Consider the number 284. Check the last two digits. The last two digits of the number form the number 84. As 84 is divisible by 4, the original number 284 is also divisible by 4. Example 2: 1328 Thus, 1328 is also 4. Divisibility Rule of 5 If a number ends with 0 or 5, it is divisible by 5. For example, 35, 790, and 55 are all divisible by 5. Divisibility Rule of 6 If a number is divisible by 2 and 3 both, it will be divisible by 6 as well. For example, the numbers 6, 12, 18 are divisible by both 2 and 3. So, they are divisible by 6 as well. Divisibility Rules of 7 If subtracting twice of the last digit from the number formed by remaining digits is 0 or divisible by 7, the number is divisible by 7. This one is a little tricky. Let’s understand with an example. Example: Check whether 905 is divisible by 7 or not. Step 1: Check the last digit and double it. Last digit $= 5$ Multiply it by 2. $5 \times 2 = 10$ Step 2: Subtract this product from the rest of the number. Here, the remaining number $= 90$ $90 \;-\; 10 = 80$ Step 3: If this number is 0 or multiple of 7, then the original number is also divisible by 7. 80 is not divisible by 7. So, 905 is also not divisible by 7. Divisibility Rule of 8 If the number formed by the last three digits of a number is divisible by 8, we say that the number is divisible by 8. Example 1: In the number 4176, the last 3 digits are 176. If we divide 176 by 8, we get: Since 176 is divisible by 8, 4176 is also divisible by 8. Thus, 12,920 is divisible by 8. Divisibility Rule of 9 If the sum of digits of the number is divisible by 9, then the number itself is divisible by 9. You can keep adding further by repeating the rule. If the single-digit sum is 9, the number is divisible by 9. Example 1: Consider 189. The sum of its digits$ = (1+8+9) = 18$, which is divisible by 9, hence 189 is divisible by 9. Example 2: 12,897 Sum of digits $= 1 + 2 + 8 + 9 + 7 = 27$ Adding further, $2 + 7 = 9$ Thus, 12897 is divisible by 9. Divisibility Rule of 10 Any number whose last digit is 0 is divisible by 10. Example: 10, 20, 30, 100, 2000, 40,000, etc. Divisibility Rule for 11 If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11. Example 1: Consider the number. 2846767. First, understand the digit positions. We find two sums: the sum of digits at the even places and the sum of digits at the odd places. Sum of digits at even places (From right) $= 8 + 6 + 6 = 20$ Sum of digits at odd places (From right) $= 7 + 7 + 4 + 2 = 20$ Difference $= 20 – 20 = 0$ Difference is divisible by 11. Thus, 2846767 is divisible by 11. Example 2: Is 61809 divisible by 11? Group digits that are in odd places together and digits in even places together. Alt tag: identify digits alternate places: divisibility rule of 11 Here, $6 + 8 + 9 = 23$ and $0 + 1 = 1$ Difference $= 23 \;-\; 1 = 22$ 22 is divisible by 11. Thus, the given number is divisible by 11. Another Divisibility Rule For 11 There’s another simple divisibility rule for 11. Subtract the last digits from the remaining number. Keep doing this until we get a two-digit number. If the number obtained is divisible by 11, the original number is divisible by 11. $174\;-\;9 = 165$ $16\;-\;5 = 11$ … divisible by 11 Thus, 1749 is divisible by 11. Divisibility Rule of 12 If the number is divisible by both 3 and 4, then the number is divisible by 12 Sum of the digits $= 4 + 8 + 8 + 0 = 20$ (not a multiple of 3) Last two digits $= 80$ (divisible by 4) The given number 4880 is divisible by 4 but not by 3. Thus, 4880 is not divisible by 12. Divisibility Rules of 13 To check if it is divisible by 13, we add 4 times of the last digit of the remaining number and repeat the process until we get a two-digit number. If that two-digit number is divisible by 13, then the given number is divisible by 13. Example: Is 4186 divisible by 13? - $418 + (6 \times 4) = 418 + 24 = 442$ - $44 + (2 \times 4) = 44 + 8 = 52$ 52 is divisible by 13 since $13 \times 4 = 52$. Thus, 4186 is divisible by 13. Divisibility Rules: Chart |Divisibility Rules Chart| |Divisibility by 1||Every number is divisible by 1.| |Divisibility by 2||When the last digit is 0, 2, 4, 6, or 8| |Divisibility by 3||When the sum of digits is divisible by 3| |Divisibility by 4||When the last two digits of any dividend are divisible by 4 (NOTE: Numbers having 00 as their last digits are also divisible by 4.)| |Divisibility by 5||When the last digit is either 0 or 5| |Divisibility by 6||When the number is divisible by both 2 and 3| |Divisibility by 7||When the last digit is subtracted twice from the remaining digits and gives the multiple of 7| |Divisibility by 8||When the last three digits are divisible by 8(NOTE: Numbers having 000 as their last digits are also divisible by 8.)| |Divisibility by 9||When the sum of all digits is divisible by 9| |Divisibility by 10||When the last digit is 0| |Divisibility by 11||When the difference between the sums of the alternative digits is divisible by 11| |Divisibility by 12||When a number is both divisible by 3 and 4| |Divisibility by 13||Multiply 4 with the last digit and add this product to the remaining number. Continue till a two-digit number is found. If the 2-digit number is divisible by 13, the number is divisible by 13.| Facts about Divisibility Rules - “Divisible” means a number is able to be divided evenly with another number with NO remainders. - Divisibility rule is a shortcut to analyze whether an integer is completely divisible by a number without actually doing the calculation. - Zero is divisible by any number (except by itself), so it gets a “yes” to all these tests. - When a number is divisible by another number, it is also divisible by each of the factors of that number. For instance, a number divisible by 6 will also be divisible by 2 and 3. A number divisible by 10 is also divisible by 5 and 2. - Numbers that have two zeros at the end are divisible by 4. Numbers with three zeros at the end are divisible by 8. - The number 2,520 is the smallest number that is divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10. In this article, we have learned divisibility rules and charts with examples. Let’s solve some divisibility rules examples to understand it better. Solved Examples for Divisibility Rules 1. If a number is divisible by 6, can we say it is divisible by 2 as well? Yes, because 6 is divisible by 2. If a number is divisible by some numbers, say x, that number is also divisible by factors of x. For example: 480 is divisible by 6. $480 \div 6 = 80$. $480 \div 2 = 240$. Also $480 \div 3 = 160$ Thus, If a number is divisible by 6, can we say it is divisible by 2 and 3 as well, as 2 and 3 are factors of 6. 2. Use divisibility rules to check whether 642 is divisible by 4 and 3. Divisibility rule for 4: If the last two digits of a number are divisible by 4, then that number is divisible by 4. The last two digits of $642 = 42$, which is not divisible by 4. Thus, 542 is not divisible by 4. Divisibility rule of 3: If the sum of digits is divisible by 3, we say that the original number is divisible by 3. Sum of digits $= 6 + 4 + 2 = 12$ 12 is divisible by 3. So, 542 is not divisible by 8. 3. Check on 3640 for divisibility by 13. The last number of the given number is 0, Multiply 4 by 0 and add to the rest of the number. $364 + (0 \times 4) = 364$. Again multiply 4 by the last digit of the obtained three-digit number and add to the rest of the digits as: $36 + (4 \times 4) = 52$ Now, a two-digit number 52 is obtained, which is divisible by 13. $52 = 4 \times 13$. Hence, 3640 is divisible by 13. Practice Problems on Divisibility Rules Rules of Divisibility: Definition, Chart, Examples Which number is not divisible by 5? According to divisibility rule of 5, If the last digit of a number is 5 or 0, the number is always divisible by 5. So, 680 is divisible by 5. Which of the following numbers is divisible by 2? All even numbers are divisible by 2. Which one of the following numbers is divisible by 6? According to the rule of divisibility by 6, the number that is divisible by both 2 and 3 is also divisible by 6. Only 18 of all the given numbers are divisible by 2 as well as 3. Identify a number divisible by 9. Sum of digits in the number $117 = 1 + 1 + 7 = 9$. The sum is divisible by 9. Thus, 117 is divisible by 9. For all the other options, the sum of digits in a number is not divisible by 9. Frequently Asked Questions on Divisibility Rules What are co-primes and their divisibility rules? Co-primes are the pair of numbers that have 1 as the common factor. If the number is divisible by such co-primes, the number is also a divisible by-product of the co-primes. For example, 14 is divisible by both 2 and 7. They are co-primes that have only 1 as the common factor, so the number is divided by 14, the product of 2 and 7. When is a number said to be a factor of another number? A number x is said to be a factor of number y if y is divisible by x. For example, 10 is divisible by 2, so 2 is a factor of 10. Where do we use divisibility rules in real life? Divisibility rules are the quickest way to determine if a number is divisible by another number. It saves the time required to perform the actual division. Divisibility rules also give you a number sense when it comes to division and multiplication of two or more numbers. What are composite numbers? In math, composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers. Factors of $4 = 1, 2, 4$, i.e., Since 4 has more than two factors, 4 is a composite number. How many divisibility rules are there? We often have standards for divisibility from 1 to 20. However, if we were able to recognize the pattern of multiples of integers, we could develop further tests for divisibility. For example, the divisibility rule of 21 states that a number must be divisible by both 3 and 7. It is because 21 is a multiple of two prime numbers 3 and 7, so all the multiples of 21 will definitely have 3 and 7 as their common factors.

s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224653631.71/warc/CC-MAIN-20230607074914-20230607104914-00043.warc.gz

CC-MAIN-2023-23

https://tvprofil.com/ks/seriale/11555100/historia-e-matematikes

math

The history of mathematics from ancient times to the present day. Narrated by Oxford mathematics professor Marcus du Sautoy, the series covers the seminal moments and people in the development of maths. Mathematical problems became spectator sports in the 16th century, with generous prizes given to the winners. In such a competitive atmosphere, it’s not surprising that mathematicians would jealously guard their knowledge – and in some cases, behave very badly. Gerolamo Cardano appeared to solve a problem known as the cubic equation, but he had stolen the solution, from a rival mathematician – Niccolò Tartaglia. France began to challenge Italian mathematical domination with Rene Descartes, who linked algebra and geometry – a decisive step that would change the course of the discipline forever. He was followed by a maths prodigy, Blaise Pascal, who proved that the sum of the angles of a triangle were equal to two right angles at just 12 years old. Pascal went on to invent a mechanical calculator and proved that a vacuum could exist. In England Isaac Newton developed calculus, which could account for the orbits of the planets, but spent the rest of his life embroiled in a dispute with the German mathematical genius Gottfried Leibniz over who developed it first.

s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100800.25/warc/CC-MAIN-20231209040008-20231209070008-00755.warc.gz

CC-MAIN-2023-50

http://www.expertsmind.com/questions/intrinsic-semiconductor-30137308.aspx

math

Draw a graph illustrating how resistivity varies with temperature for an intrinsic semiconductor. b) Gallium nitride, GaN, has an energy gap of 3.36 eV at 300 K. Calculate the wavelength of the light emitted by a GaN LED at this temperature. c) State the standard SI units for: (ii) diffusion coefficient (or diffusivity) (iii) current density. d) The p-type side of a pn junction diode is at a potential of -0.4 V and the n-type side at a potential of +0.3 V. A hole moves from the n-type side to the p-type side. What is the change in the potential energy of the hole in electron volts? e) State three advantages of ion implantation over diffusion, as a means of introducing additional dopant into a wafer. f) Explain why the gate current in an MOS transistor is extremely small and give a typical value for this current. g) The terminal voltages for an n-channel enhancement MOS transistor with VT = 0.5 V are: VG = 5 V VS = 3 V VD = 4 V Determine whether the channel in the device is pinched off in this situation.

s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190754.6/warc/CC-MAIN-20170322212950-00122-ip-10-233-31-227.ec2.internal.warc.gz

CC-MAIN-2017-13

https://pure.york.ac.uk/portal/en/publications/fractional-stochastic-active-scalar-equations-generalizing-the-mu

math

We prove the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations. This class includes the stochastic, dD-quasi-geostrophic equation, $ d\geq 1$, fractional Burgers equation on the circle, fractional nonlocal transport equation and the 2D-fractional vorticity Navier-Stokes equation. We consider the multiplicative noise with locally Lipschitz diffusion term in both, the free and no free divergence modes. The random noise is given by an $Q-$Wiener process with the covariance $Q$ being either of finite or infinite trace. In particular, we prove the existence and uniqueness of a global mild solution for the free divergence mode in the subcritical regime ($\alpha>\alpha_0(d)\geq 1$), martingale solutions in the general regime ($\alpha\in (0, 2)$) and free divergence mode, and a local mild solution for the general mode and subcritical regime. Different kinds of regularity are also established for these solutions. The method used here is also valid for other equations like fractional stochastic velocity Navier-Stokes equations (work is in progress). The full paper will be published in Arxiv after a sufficient progress for these equations. |Publication status||Unpublished - 14 Aug 2012|

s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100705.19/warc/CC-MAIN-20231207221604-20231208011604-00813.warc.gz

CC-MAIN-2023-50

https://www.stevens.edu/events/2021-stevens-math-olympiad

math

The Stevens Math Olympiad will be virtual in 2021. The Stevens Math Olympiad is a free mathematics competition for students in grades 3-12 that entails solving mathematical and logical problems, as well as demonstrating the joy and excitement of mathematics. The Olympiad has the following goals: To stimulate enthusiasm and a love for mathematics To introduce important mathematical concepts to students To strengthen mathematical intuition and creativity Students are offered 15 problems to solve in five divisions: grades 3-4, grades 5-6, grades 7-8, grades 9-10 and grades 11-12. Math Olympiad testing will take place from 10:15 a.m.-12:30 p.m. Following the conclusion of Math Olympiad testing, Dr. Andrey Nikolaev of the Department of Mathematical Sciences will give a lecture, Picture Hanging Puzzles, open to all participants starting at 12:45 p.m.

s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039617701.99/warc/CC-MAIN-20210423101141-20210423131141-00333.warc.gz

CC-MAIN-2021-17

https://fastretrieve.com/kitesurfing/does-a-kite-equal-360-degrees.html

math

A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal: degrees degrees degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles. How many degrees is a kite? Do all quadrilaterals have 360 degrees? You would find that for every quadrilateral, the sum of the interior angles will always be 360°. … Since the sum of the interior angles of any triangle is 180° and there are two triangles in a quadrilateral, the sum of the angles for each quadrilateral is 360°. Are opposite angles in a kite equal? A kite is a quadrilateral in which two disjoint pairs of consecutive sides are congruent (“disjoint pairs” means that one side can’t be used in both pairs). … The opposite angles at the endpoints of the cross diagonal are congruent (angle J and angle L). Which angles in a kite are congruent? The angles between the congruent sides are called vertex angles. The other angles are called non-vertex angles. If we draw the diagonal through the vertex angles, we would have two congruent triangles. Theorem: The non-vertex angles of a kite are congruent. Is a kite a rhombus yes or no? A kite is a quadrilateral (four sided shape) where the four sides can be grouped into two pairs of adjacent (next to/connected) sides that are equal length. So, if all sides are equal, we have a rhombus. … A kite is not always a rhombus. A rhombus is not always a square. What are the 5 properties of a kite? Kite properties include (1) two pairs of consecutive, congruent sides, (2) congruent non-vertex angles and (3) perpendicular diagonals. Other important polygon properties to be familiar with include trapezoid properties, parallelogram properties, rhombus properties, and rectangle and square properties. How do you prove a quadrilateral has 360 degrees? A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°. How many degrees is a hexagon? Do all interior angles add up to 360? All sides are the same length (congruent) and all interior angles are the same size (congruent). To find the measure of the interior angles, we know that the sum of all the angles is 360 degrees (from above)… And there are four angles… Can a kite have a right angle? Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals (quadrilaterals with both a circumcircle and an incircle), since all kites have an incircle. Do diagonals bisect each other in a kite? If two distinct pairs of consecutive sides of the quadrilateral are congruent, then it’s a kite. If one of the diagonals bisects the other diagonal at a perpendicular angle, it’s a kite. Do the diagonals of a kite bisect each other at 90 degrees? The intersection of the diagonals of a kite form 90 degree (right) angles. This means that they are perpendicular. The longer diagonal of a kite bisects the shorter one. This means that the longer diagonal cuts the shorter one in half. How do you prove a kite? How to Prove that a Quadrilateral Is a Kite - If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it’s a kite (reverse of the kite definition). - If one of the diagonals of a quadrilateral is the perpendicular bisector of the other, then it’s a kite (converse of a property). Does a kite have congruent sides? A kite is a quadrilateral shape with two pairs of adjacent (touching), congruent (equal-length) sides. What does congruent mean? Congruent means same shape and same size. So congruent has to do with comparing two figures, and equivalent means two expressions are equal. So to say two line segments are congruent relates to the measures of the two lines are equal.

s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703514121.8/warc/CC-MAIN-20210118030549-20210118060549-00719.warc.gz

CC-MAIN-2021-04

https://carlson-wagonlit.mynewsdesk.com/blog_posts/tag/data

math

Blog posts • Dec 07, 2018 08:43 UTC “If you can look into the seeds of time and say which grain will grow and which will not, speak then unto me.” Since the beginning of times, humankind has been obsessed with predictions. Fortunately, nowadays we don't need to rely on oracles or witches of any sort. We have powerful algorithms that allow us to predict the future and even change it. Blog posts • Aug 02, 2018 07:02 UTC Blog posts • May 31, 2018 09:46 UTC “You have to capture the data.” At the BTN Tech Talk in Chicago in May, Norm Rose of Travel Tech Consulting reiterated a point that many travel managers know all too well. When travelers book outside of your hotel program, you have to find a way to capture the data. Blog posts • Apr 10, 2018 09:32 UTC It’s tempting to dismiss Bitcoin, the world’s first digital currency, as a passing fad, and nothing to do with the world of business travel, but what if the technology upon which it is built could make the fear of forgetting your passport, a problem of the past? Well, now there’s a brave new world. Blog posts • Mar 28, 2018 10:30 UTC Large, complex travel programs often require significant data processing, integration and analytics. Many integrate travel, card and expense data to understand expenditure, tackle off-channel spend and manage suppliers. Sometimes travel managers will look to systems integration services and/or analytics tools for help. But TMCs – like CWT – can actually be a better option. Blog posts • Mar 23, 2018 10:29 UTC In Part One, I looked at the concept of predictive analytics and some common pitfalls. Now I’m going to give you some real-world examples of predictive analytics for travel managers. Blog posts • Mar 22, 2018 10:30 UTC Predictive Analytics is like the afterlife – everybody likes the idea of it but nobody knows what it is. I’d like to help clear this up. To do so, I will show you how travel managers can use predictive analytics. In this first instalment, I’ll walk you around the challenges of predictive analytics. Next, we’ll get into how corporate travel managers can use it to improve travel programs.

s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00580.warc.gz

CC-MAIN-2021-39

http://www.freedictionary.org/?Query=Frictional

math

1. pertaining to or worked or produced by friction; - Example: "frictional electricity" - Example: "frictional heat" - Example: "frictional gearing" The Collaborative International Dictionary of English v.0.48: Frictional \Fric"tion*al\, a. Relating to friction; moved by friction; produced by friction; as, frictional electricity. Frictional gearing, wheels which transmit motion by surface friction instead of teeth. The faces are sometimes made more or less V-shaped to increase or decrease friction, as WordNet (r) 3.0 (2006): adj 1: pertaining to or worked or produced by friction; "frictional electricity"; "frictional heat"; "frictional

s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103324665.17/warc/CC-MAIN-20220627012807-20220627042807-00746.warc.gz

CC-MAIN-2022-27

http://rockridgebrothers.com/lib/an-introduction-to-computational-fluid-dynamics-the-finite-volume-method-2-nd

math

By H. Versteeg, W. Malalasekera This entire textual content provides the basics of laptop Fluid Dynamics easily and obviously. Read or Download An Introduction to Computational Fluid Dynamics: The Finite Volume Method (2nd Edition) PDF Similar fluid dynamics books "The textual content can be utilized because the foundation for a graduate path in any of numerous disciplines which are curious about clever fabric modeling, together with physics, fabrics technological know-how, electromechanical layout, regulate platforms, and utilized arithmetic. .. [T]his well-written and rigorous textual content could be beneficial for someone attracted to particular clever fabrics in addition to basic modeling and keep an eye on of smart-material habit. "This e-book is meant to give for the 1st time experimental tips on how to degree equilibria states of natural and combined gases being adsorbed at the floor of strong fabrics. it's been written for engineers and scientists from and academia who're attracted to adsorption-based gasoline separation tactics and/or in utilizing gasoline adsorption for characterization of the porosity of good fabrics. Der Band stellt als Erg? nzung zum eingef? hrten Grundlagenbuch Str? mungslehre eine tiefergehende Behandlung des Vorlesungsstoffes dar. Die Einteilung der Kapitel entspricht im wesentlichen der im Band Grundlagen: Hydrostatik, Kinematik, Impulssatz, NAVIER-STOKES-Bewegungsgleichung, Potential-, Wirbel- und Grenzschichtstr? For the fluctuations round the potential yet fairly fluctuations, and showing within the following incompressible process of equations: on any wall; at preliminary time, and are assumed identified. This contribution arose from dialogue with J. P. Guiraud on makes an attempt to push ahead our final co-signed paper (1986) and the most inspiration is to place a stochastic constitution on fluctuations and to spot the big eddies with part of the likelihood house. - Statistical Physics of Fluids: Basic Concepts and Applications - Hydraulik : Grundlagen, Komponenten, Schaltungen - Liquid Sloshing Dynamics Theory and Applications - Worked examples in nonlinear continuum mechanics for finite element analysis Additional resources for An Introduction to Computational Fluid Dynamics: The Finite Volume Method (2nd Edition) Initial conditions are needed in the entire rod and conditions on all its boundaries are required for all times t > 0. This type of problem is termed an initial– boundary-value problem. e. ). The solutions move forward in time and diffuse in space. The occurrence of diffusive effects ensures that the solutions are always smooth in the interior at times t > 0 even if the initial conditions contain discontinuities. The steady state is reached as time t → ∞ and is elliptic. 46). The governing equation is now equal to the one governing the steady temperature distribution in the rod. This complicates matters greatly when flows around and above M = 1 are to be computed. Such flows may contain shockwave discontinuities and regions of subsonic (elliptic) flow and supersonic (hyperbolic) flow, whose exact locations are not known a priori. 11 is a sketch of the flow around an aerofoil at a Mach number somewhat greater than 1. 5 Auxiliary conditions for viscous fluid flow equations The complicated mixture of elliptic, parabolic and hyperbolic behaviours has implications for the way in which boundary conditions enter into a flow problem, in particular at locations where flows are bounded by fluid boundaries. 9, which is again bounded by the characteristics. 10a shows the situation for the vibrations of a string fixed at x = 0 and x = L. For points very close to the x-axis the domain of dependence is enclosed by two characteristics, which originate at points on the x-axis. The characteristics through points such as P intersect the problem boundaries. The domain of dependence of P is bounded by these two characteristics and the lines t = 0, x = 0 and x = L. 10b and c) in parabolic and elliptic problems is different because the speed of information travel is assumed to be infinite.

s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826840.85/warc/CC-MAIN-20171023221059-20171024001059-00862.warc.gz

CC-MAIN-2017-43

http://homebrew.stackexchange.com/tags/chili/new

math

New answers tagged chili This MSDS sheet lists Capsaicin as mutagenic in bacteria and yeast, although it doesn't state specifically which strains and by how much. I think the relevant question is, how much is too much? Clearly chili beers can be made, but perhaps the levels of Capsaicin were not significant enough to hinder yeast activity. Given the amount and specific variety of ... Top 50 recent answers are included

s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010638293/warc/CC-MAIN-20140305091038-00049-ip-10-183-142-35.ec2.internal.warc.gz

CC-MAIN-2014-10

http://nqcourseworkjhfz.presidentialpolls.us/a-lab-experiment-on-conducting-a-direct-shear-test-to-determine-the-angle-of-internal-friction-of-dr.html

math

The direct shear test is generally conducted on sandy soils as a consolidated engineering properties of based laboratory testing158direct video showing the basics of the direct shear test along with explanations of soil dilation and principal plane rotation during a direct shear test. The purpose of direct shear test is to get the ultimate shear resistance, peak shear resistance, cohesion, angle of shearing resistance and stress-strain characteristics of the soils. Direct shear tests were performed to determine the internal friction angles of the density and model sands direct shear test results, in terms of shear stress versus horizontal displacement curves, are presented summary of interface shear test results on density sand pile/material dr (%. Instructor: dr george mylonakis lab experiment #7: direct shear test introduction the shear φ where, σ' = effective normal stress φ = angle of friction of soil φ = f( d r , d , e , an ) where, d procedure 1 measure the internal diameter of the cylindrical cell 2 balance the counter weight. Laboratory testfor determination of shear strength parameters a direct shear test b triaxial test c direct simple shear test d plane strain triaxial test e torsional rins shear test in many foundation designproblems,one must determine the angle of fric- tion between the soil and the. In this laboratory, a direct shear device will be used to determine the shear strength of a cohesionless soil (ie angle of internal friction 13 170 direct shear test data sheet date tested: tested by: project name: sample number: visual classification: shear box inside diameter. In an ordinary laboratory direct shear test, with the applied shearing force monitored on the y axis and strain on the x axis, the area under the data curve represents work done on the soil sample in this way, higher values of sample shear strength correlate approximately with higher amounts of work involved shearing the sample to its ultimate. Angle of internal friction (friction angle) a measure of the ability of a unit of rock or soil to withstand a shear stress it is the angle (φ), measured between the normal force (n) and resultant force (r), that is attained when failure just occurs in response to a shearing stress (s) its tangent (s/n. From this test, coulomb parameters, including cohesion and internal friction angle, as well as, bekker parameters can be infferred it has been observed that the inclination angle of particles during an avalanche is consistently higher than the angle of repose for granular materials. Direct shear test, triaxial test and unconfined compression test the shear strength value can be determined as shown, where φ = angle of internal friction c = cohesive stress or adhesion stress. Angle of internal friction, , can be determined in the laboratory by the direct shear test or the triaxial stress test typical relationships for estimating the angle of internal friction, , are as follows: empirical values for , of granular soils based on the standard penetration number. The direct shear test used for soil (powers 1968) can be performed with fresh concrete to assess the cohesive strength of a concrete mixture the test provides additional information, namely the angle of internal friction, not available from most conventional tests. A lab experiment on conducting a direct shear test to determine the angle of internal friction of dry sand. Direct shear test analysis of test results note: cross-sectional area of the sample changes with the horizontal displacement 50 interface tests on direct shear apparatus in many foundation design problems and retaining wall problems, it is required to determine the angle of internal friction. These test results further indicate that owing to the influence of the frictional force of the upper shear box, a higher internal angle of friction was measured in the conventional direct shear test on dilative sample without any improvements. Direct shear test to determine angle of internal friction of fine sand having different dry densities and mix compositions with different percentage by weight of ceramic tile waste material 3. For the current lab, a granular soil is used (dry swelih sand), and to determine the shearing strength of the soil using the direct shear apparatus :general disscussion. This recommendation is based on the relationship between friction angle and dilation angle for all aggregates tested in this program and assumes a 95-percent confidence interval the proposed new default of 39° corresponds to the peak friction angle of 49° (see figure 7), less two standard deviations. In australia, q181c test method of direct shear testing to estimate the effective angle of internal friction at constant volume conditions for granular the results show that accurate effective friction parameter measurements for coarse grained, granular backfill soils require the use of fresh soil. A direct shear test is a laboratory or field test used by geotechnical engineers to measure the shear strength properties of soil or rock several specimens are tested at varying confining stresses to determine the shear strength parameters, the soil cohesion (c) and the angle of internal friction. A direct shear test is a laboratory test used by geotechnical engineers to find the shear strength parameters of soil several specimens are tested at varying confining stresses to determine the shear strength parameters, the soil cohesion (c) and the angle of internal friction (commonly. 18 civil engineering - texas tech university direct shear test direct shear test is quick and inexpensive shortcoming is that it fails the soil on a designated plane which may not be the weakest one used to determine the shear strength of both cohesive as well as non-cohesive soils. Direct shear box test contents introduction objective apparatus description of test results calculations relevance to geotechnics soil objective to determine the angle of shearing resistance of a sample of sand the test may be carried out either dry or fully saturated but not. Dr h c e meyer-peter, and the chief of the soil mechanics laboratory, professor dr ing r haefeli, for permission to carry out the tests and for valuable support during the work.

s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741764.35/warc/CC-MAIN-20181114082713-20181114104713-00065.warc.gz

CC-MAIN-2018-47

https://math.answers.com/Q/What_is_negative_one_half_to_the_sixth_power

math

256 to the negative one-half power = 0.0625 Negative one half to the fourth power equals 0.0625 1.5 to the power of six is 11.39 Answer #1:Negative one-half to the negative 4th power, ie -((1/2)-4) equals -16Answer #2:Negative one-half to the negative 4th power, ie (-1/2)-4 equals +16 -1/6. Negative one sixth. No, any number raised to an even power (6, in this case) is positive or zero. So, the sixth root of minus one can only be a complex number. Negative two thirds is smaller than negative one sixth. To have more of a negative is to have less. One half = three sixth, so the answer is two sixth. One sixth, one third and one half.

s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00744.warc.gz

CC-MAIN-2023-14

https://www.freelancer.com.au/jobs/microbiology/

math

i handmade nail polish using mica and nail polish suspension. i've made just about every kind of nail polish. Thermal color changing, neon, magnetic etc.... I need someone to create a new nail polish formula for me. all ideas are welcome. when u bid please submit a possible idea so i know u understand this project. you don't have to tell me the entire concept i just need to know u unders... nail polish formula 2 days left $299 (Avg Bid) $299 Avg Bid

s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158691.56/warc/CC-MAIN-20180922201637-20180922222037-00215.warc.gz

CC-MAIN-2018-39

http://freecomputerbooks.pickatutorial.com/category/mathematics_5.htm

math

|Categories||Free Downloadable Mathematics eBooks!| |LAPACK95 Users' Guide LAPACK Users' Guide, Third Edition LAPACK Users' Guide provides an introduction to the design of the LAPACK package, a detailed description of its contents, reference manuals for the leading comments of the routines, and example programs Linear Algebra - Theorems and Applications Linear algebra occupies a central place in modern mathematics. Also, it is a beautiful and mature field of mathematics, and mathematicians have developed highly effective methods for solving its problems. It is a subject well worth studying for its own sake. This book proceeds beyond the representation theory of compact Lie groups and offers a carefully chosen range of material designed to give readers the bigger picture. It explores compact Lie groups through a number of proofs and culminates in a 'topics' section that takes the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as unifying them. Linear Partial Differential Equations and Fourier Theory This highly visual introductory textbook presents an in-depth treatment suitable for undergraduates in mathematics and physics, gradually introducing abstraction while always keeping the link to physical motivation. Designed for lecturers as well as students, downloadable files for all figures, exercises, and practice problems are available online, as are solutions. Semantics - Advances in Theories and Mathematical Models The current book is a nice blend of number of great ideas, theories, mathematical models, and practical systems in the domain of Semantics. The book has been divided into two volumes. The current one is the first volume which highlights the advances in theories and mathematical models in the domain of Semantics. Notes on Diffy Qs: Differential Equations for Engineers This book is an one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence. Mathematical Tools for Physics This book provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference. The definitive treatment of analytic combinatorics. This self-contained text covers the mathematics underlying the analysis of discrete structures, with thorough treatment of a large number of applications. Exercises, examples, appendices and notes aid understanding: ideal for individual self-study or for advanced undergraduate or graduate courses. Precalculus: An Investigation of Functions This book is a college level text in precalculus and trigonometry.

s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823114.39/warc/CC-MAIN-20171018195607-20171018215607-00360.warc.gz

CC-MAIN-2017-43

http://axessayqrtv.vatsa.info/problem-formulation.html

math

Lesson 2: problem formulation “the mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or. Introduction to problem formulation introduction to problem formulation. Or-notes j e beasley or-notes linear programming - formulation we consider below some specific examples of the types of problem. Von hippel, eric, and georg von krogh (2016) “identifying viable ‘need-solution pairs’:problem solving without problem formulation” organization science. Strategic problem formulation 1981 volkema, 1986), most efforts have lacked theoretical grounding our effort represents one of the first attempts to theoretically. Problem formulation research methodology problem formulation objective at the end of the session, participants will be able to • identify elements of good problem. Typically, the newspaper vehicle routing problem is of the node-routing variety the miller-tucker-zemlin formulation of the tsp , and apply a subtour. Formulating research problems r esearch problems are questions that indicate gaps in the scope or the role of theory in problem formulation, and how the multimethod. Cognitive-behavioral case formulation jacqueline b persons michael a tompkins problem, and case for example, the symptom of auditory hallucinations. Early stages of any unstructured problem-formulation task outlines can be useful in organizing the components of a model and ultimately in laying out the rows of a. Ot assessment and case formulation in an attempt to develop a more ot- focussed approach to assessment and case formulation in hoarding- i've problem solving. The process: policy development process steps—issues framing, agenda setting, and policy formulation once a problem requiring a policy. Define formulation formulation synonyms, approach, plan of attack, attack - ideas or actions intended to deal with a problem or situation. Casuist research problem-- this type of problem relates to the determination of right and william mk problem formulation research methods knowledge base. Formulation definition, to express in precise form state definitely or systematically: he finds it extremely difficult to formulate his new theory see more. The purpose of this worksheet is to help the risk assessor identify the components of the risk assessment use this worksheet to think through all parts of the. Mathematical formulation of linear programming problems there are mainly four steps in the mathematical formulation of linear programming problem. Research is an investigation or experimentation that is aimed at a discovery and interpretation of facts, revision of theories or laws or practical application of the. 1 a theory of strategic problem formulation markus baer olin business school washington university in st louis one brookings drive. 2 linear programming problem formulation we are not going to be concerned in this class with the question of how lp problems are solved instead, we will focus on. Dual problem usually the term dual problem refers to the lagrangian dual problem but other dual problems are used, for example, the wolfe dual problem and the. Learn how to use more than 25 different problem solving techniques to solve simple and complex problems. Stuck in this problem for quite a while anyone can offer some help the problem is as follows: fred has $5000 to invest over the next five years at the beginning of. Method of innovation strategies formulation and example inventive problem solving, strategy formulation, systematic problem solving with triz methodology and. Problem-solving agents: find sequence of actions that achieve goals problem formulation: we already know what the set of all possible states is. P1 until now the theory of problem-solving (eg newell and simon, 1972) has mainly emphasized the search of solutions within a problem space. Summary: the objective of the quadratic assignment problem (qap) is to assign \(n\) facilities to \(n\) locations in such a way as to minimize the assignment cost. An introduction to the basic transportation problem and its linear programming formulation: transportation network model objective function constraints. Topology optimization- problem formulation and pragmatic outcomes by integration of tosca and cae tools waqas saleem, hu lu, fan yuqing abstract—structural. The travelling salesman problem an equivalent formulation in terms of graph theory is: finding a shortest travelling salesman tour is npo-complete.

s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510853.25/warc/CC-MAIN-20181016155643-20181016181143-00353.warc.gz

CC-MAIN-2018-43

https://coursestar.com/qbank-question/problem-11-17-project-a-and-project-b/

math

Problem 11.17 – Project A & Project B Your numbers will vary. Difficulty – Hard Given two cashflow timelines, you are asked to determine each project's NPV, IRR, and MIRR. You are also asked to construct an NPV profile graph and calculate the crossover rate where the two projects' NPVs are equal. Experts Have Solved This Problem Please login or register to access this content.

s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100724.48/warc/CC-MAIN-20231208045320-20231208075320-00553.warc.gz

CC-MAIN-2023-50

https://masepoxies.answerbase.com/?page=4

math

What is blush? Asked 1 year ago Products > MAS LV Resin, Products > MAS FLAG Resin Can this be used to stabilize spalted maple prior to surface finishing? Asked 3 months ago Shop > Specialty Woodworking Products > Penetrating Epoxy Sealer What is the IOR value of this product? (Index of Refraction) Shop > Specialty Woodworking Products > Deep Pour X Epoxy Resin What is the temperature range for layup? What's the lower limit? Shop > Non Blushing System > FLAG Resin what is the mix ratio of deep pour by WEIGHT? Shop > Specialty Woodworking Products > Deep Pour Epoxy Is this a 0 VOC resin? Shop > Art Epoxy > Art Pro Epoxy Will your resin soften in a vehicle Asked 20 days ago why did my mas penetrating epoxy sealer start to boil in the bucket after I was half way through installing Asked 17 days ago Can I use your epoxy over an oil painting? Products > Art Pro, Products > Table Top Pro

s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103271864.14/warc/CC-MAIN-20220626192142-20220626222142-00082.warc.gz

CC-MAIN-2022-27

https://allnurses.com/nursing-student-assistance/fevers-in-the-122733.html

math

Quote from EMTandNurse2B I have a quick question about fevers in the elderly. When I was in EMT training, we were told that the elderly don't always run fevers even with an acute infection because of the age-related changes in temperature control. Now, when I was studying for my last test in Physical Assessment, my textbook said the same thing, that the elderly don't always run a fever because of the age-related cahnges in temp control. Then, I went to take my test and I answered my test question according to the book. I got it wrong. I politely brought my book to the professor and showed her the sentance in the book that completely contradicted the "correct" test answer. She was real nice about it, but told me that she was right, the elderly run fevers MORE often than younger people do. However because their baseline temp is lower, the fevers are harder to detect. Does somebody with experience have any input on this? Thanks! I think, IMHO, this instructor is incorrect in her rationale. True, the body temp of the elderly are often subnormal. Now what is normal? This is know as 98.6. So, if the elderly individual has a "normal for them" body temp of 96 and their temp is 98, then they are febrile, but, because "normal" body temp is 98.6, then the individual is not febrile. Kinda confusing. That is why we cannot factor these things individually. There is a standard, in this case ..... a standard "normal" body temp. So, yes, I can see her point, but, the elderly do not often times exhibit elevated temps due to the rationale your book reflects. If I were the instructor, I would throw that question out .....again, IMHO.

s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221214538.44/warc/CC-MAIN-20180819012213-20180819032213-00176.warc.gz

CC-MAIN-2018-34

https://studysoup.com/tsg/41899/contemporary-abstract-algebra-8th-edition-chapter-8-problem-35se

math

In S10, let ? = (13)(17)(265)(289). Find an element in S10 that commutes with ? but is not a power of ?. In S10, let = (13)(17)(265)(289). Find an element in S10 Problem 35SE Chapter 8 Contemporary Abstract Algebra | 8th Edition - 2901 Step-by-step solutions solved by professors and subject experts - Get 24/7 help from StudySoup virtual teaching assistants

s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670151.97/warc/CC-MAIN-20191119121339-20191119145339-00432.warc.gz

CC-MAIN-2019-47

https://edu-answer.com/mathematics/question12401870

math

During the spring and summer months, the happy home motel sees a rise in travelers. the manager of the motel turned in this report of how many travelers stayed at the happy home motel. month number of travelers march 217 april 232 may 252 june 277 july 307 august ? if the happy home motel continues to follow this pattern, how many travelers will stay during august? Let's see the difference is each successive values: 232 - 217 = 15 252 - 232 = 20 277 - 252 = 25 307 - 277 = 30 As we can see, each month the increase is 5 more than previous months' increase. Clearly from the pattern we can surmise that the next month (August), it will increase by 35 from the value of July. Number of visitors in August = Number of Visitors in July + 35 = 307 + 35 = 342 i just had this question the answer is pretty simple just minus 17 from every number and well you get 152 for the missing number oops wrong site sorry my bad lol! if you get a question that involves fish that's the answer 342 on plato

s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00675.warc.gz

CC-MAIN-2021-43

https://digitalcommons.aaru.edu.jo/pfda/vol1/iss4/4/

math

Progress in Fractional Differentiation & Applications Numerical Scheme for Solving the Space-Time Variable Order Nonlinear Fractional Wave Equation In this paper, the space-time variable order fractional wave equation with a nonlinear source term is considered. The derivative is defined in the Caputo sense. The non-standard finite difference method is proposed for solving the variable order fractional wave equation. Special attention is given to study the stability analysis and the truncation error of the method. Some numerical test examples are presented, and the results demonstrate the effectiveness of the method. The obtained results are compared with exact solutions and the standard finite difference solutions. Hassan Sweilam, Nasser and Abdulrahman Assiri, Taghreed "Numerical Scheme for Solving the Space-Time Variable Order Nonlinear Fractional Wave Equation," Progress in Fractional Differentiation & Applications: Vol. 1: 4, Article 4. Available at: https://digitalcommons.aaru.edu.jo/pfda/vol1/iss4/4

s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944452.97/warc/CC-MAIN-20230322211955-20230323001955-00661.warc.gz

CC-MAIN-2023-14

https://fliphtml5.com/tags/represents

math

About 13 results. Which of the following graph correctly represents velocity-time relationship for a particle released from rest to fall freely under gravity ? (A) (B) (C) (D) v v ` v ` v t t t t ... 4) The diagram below represents a ray of light moving from air through substance B, through substance C, and back into the air. ... Verb Correlations—MathematicsProblem Solving REMEMBER UNDERSTAND APPLY ANALYZENumber Sense Represents Applies Analyzes Counts Determines Selects Counts on Compares ... Write an equation that represents the distance d in meters that you7cmp06se_MS1. A simplified meaning for the colours represents growth throughskills and passion into golden years. ... there are many Catholicsymbols present, it is important for each class to create a Prayer Table as thefocus of their class and represents them. ... dft N −1 =N −1 dft N −1 dftwhere χ2t represents the observed χ2 test statistic for the target model, dft represents thedegrees of freedom for the target model, N represents ... If h represents any nonzero number, then the quotient f (x + h) − f (x) , h π 0, represents the slope of the line through (x, f(x)) ... The Master represents the pillar of wisdom, which is symbolized bythe Ionic, the Senior Warden represents the pillar of Strength, which is symbolized ... Whenused, it represents blood, war or valor.

s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034930.3/warc/CC-MAIN-20220625095705-20220625125705-00178.warc.gz

CC-MAIN-2022-27

http://www.chegg.com/etextbooks/the-complete-lab-manual-for-electricity-3rd-edition-9781111803544-1111803544?ii=22&trackid=6679f9a1&omre_ir=1&omre_sp=

math

Details about The Complete Lab Manual for Electricity: The Complete Laboratory Manual for Electricity, 3rd Edition is a valuable tool designed to fit into any basic electrical program that incorporates lab experience. This updated edition will enhance your lab practices and the understanding of electrical concepts. From basic electricity through AC theory, transformers, and motor controls, all aspects of a typical electrical curriculum are explored in a single volume. Each lab features an explanation of the circuit to be connected, with examples of the calculations necessary to complete the exercise and step-by-step procedures for conducting the experiment. Hands-on experiments that acquaint readers with the theory and application of electrical concepts offer valuable experience in constructing a multitude of circuits such as series, parallel, combination, RL series and parallel, RC series and parallel, and RLC series and parallel circuits. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. Sample questions asked in the 3rd edition of The Complete Lab Manual for Electricity: In the circuit shown in Figure 9–34 , assume that R1 has a resistance of 2.5 ? and R 2 has a resistance of 16 ?. Power source E S has a voltage of 20 V. As measured across Terminals A and B, what would be the equivalent Norton current for this circuit? INORTON = __________ An inductor and resistor are connected in parallel to a 120-V, 60-Hz line. The resistor has a resistance of 50 ohms, and the inductor has an inductance of 0.2 H. What is the total current flow through the circuit? If the inductor and capacitor described in Question 4 were to be reconnected in parallel to form a tank circuit, what would be the resonant frequency of the tank circuit? Question 4 - A 20- ?F capacitor is connected in series with a 1.2-mH coil. What is the resonant frequency of this connection? Back to top Rent The Complete Lab Manual for Electricity 3rd edition today, or search our site for other textbooks by Herman. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning.

s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549426234.82/warc/CC-MAIN-20170726162158-20170726182158-00555.warc.gz

CC-MAIN-2017-30

https://highgradeessay.com/2020/10/22/2000-solved-problems-in-discrete-mathematics_za/

math

2,500 solved problems in college algebra 2000 solved problems in discrete mathematics and trigonometry by philip a. 0:29. reviews: read online 2000 solved problems in discrete mathematics 2000 solved problems in discrete mathematics and download 2000 solved problems in discrete steps to write a persuasive essay mathematics book full in pdf formats 2000 solved insurance agent business plan problems in discrete mathematics (schaum’s solved problems essays on crime and punishment series) paperback – feel free: essays free letter writing app import, 30 june 1990 by seymour lipschutz (author) reviews: dec 14, 2013 · 2000 solved problems in discrete mathematics. 25 format: find an apple store or other retailer near you. each year, thousands of students improve their test scores how to quote a definition in an essay and final grades unc charlotte sat essay required with these indispensable cornell university essay prompt …. master discrete mathematics with schaum’s–the high-performance solved-problem guide. 991 these problem may be used to supplement those in the course textbook.

s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141729522.82/warc/CC-MAIN-20201203155433-20201203185433-00209.warc.gz

CC-MAIN-2020-50

http://www.rcsb.org/structure/3REZ

math

Quaternary organization of GPIb-IX complex and insights into Bernard-Soulier syndrome revealed by the structures of GPIb beta and a GPIb beta/GPIX chimeraMcEwan, P.A., Yang, W., Carr, K.H., Mo, X., Zheng, X., Li, R., Emsley, J. (2011) Blood 118: 5292-5301 - PubMed: 21908432 - DOI: 10.1182/blood-2011-05-356253 - Primary Citation of Related Structures: - PubMed Abstract: Platelet GPIb-IX receptor complex has 3 subunits GPIbα, GPIbβ, and GPIX, which assemble with a ratio of 1:2:1. Dysfunction in surface expression of the complex leads to Bernard-Soulier syndrome. We have crystallized the GPIbβ ectodomain (GPIbβ(E)) and de ... Platelet GPIb-IX receptor complex has 3 subunits GPIbα, GPIbβ, and GPIX, which assemble with a ratio of 1:2:1. Dysfunction in surface expression of the complex leads to Bernard-Soulier syndrome. We have crystallized the GPIbβ ectodomain (GPIbβ(E)) and determined the structure to show a single leucine-rich repeat with N- and C-terminal disulphide-bonded capping regions. The structure of a chimera of GPIbβ(E) and 3 loops (a,b,c) taken from the GPIX ectodomain sequence was also determined. The chimera (GPIbβ(Eabc)), but not GPIbβ(E), forms a tetramer in the crystal, showing a quaternary interface between GPIbβ and GPIX. Central to this interface is residue Tyr106 from GPIbβ, which inserts into a pocket generated by 2 loops (b,c) from GPIX. Mutagenesis studies confirmed this interface as a valid representation of interactions between GPIbβ and GPIX in the full-length complex. Eight GPIbβ missense mutations identified from patients with Bernard-Soulier syndrome were examined for changes to GPIb-IX complex surface expression. Two mutations, A108P and P74R, were found to maintain normal secretion/folding of GPIbβ(E) but were unable to support GPIX surface expression. The close structural proximity of these mutations to Tyr106 and the GPIbβ(E) interface with GPIX indicates they disrupt the quaternary organization of the GPIb-IX complex. Centre for Biomolecular Sciences, School of Pharmacy, University of Nottingham, United Kingdom.

s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376006.87/warc/CC-MAIN-20210307013626-20210307043626-00417.warc.gz

CC-MAIN-2021-10