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Notebooks

## Large Deviations

21 Feb 2013 15:04

The limit theorems of probability theory --- the weak and strong laws of large numbers, the central limit theorem, etc. --- basically say that averages taken over large samples (of well-behaved independent, identically distributed random variables) converge on expectation values. (The strong law of large numbers asserts almost-sure convergence, the central limit theorem asserts a kind of convergence in distribution, etc.) These results say little or nothing about the rate of convergence, however, which is often important for many applications of probability theory, e.g., statistical mechanics. One way to address this is the theory of large deviations. (I believe the terminology goes back to Varadhan in the 1970s, but that's just an impression, rather than research.)

Let me say things sloppily first, so the idea comes through, and then more precisely, so people who know the subject won't get too upset. Suppose $X$ is a random variable with expected value $\mathbf{E}[X]$, and we consider $S_n \equiv \frac{1}{n}\sum_{i=1}^{n}{X_i}$, the sample mean of $n$ samples of $X$. $S_n$ "obeys a large deviations principle" if there is a non-negative function $r$, called the rate function, such that $\Pr{\left(\left| \mathbf{E}[X] - S_n \right| \geq \epsilon\right)} \rightarrow e^{-nr(\epsilon)} ~.$
(The rate function has to obey some sensible but technical continuity conditions.) This is a large deviation result, because the difference between the empirical mean and the expectation is remaining constant as $n$ grows --- there has to be a larger and large conspiracy, as it were, among the samples to keep deviating from the expectation in the same way. Now, one reason what I've stated isn't really enough to satisfy a mathematician is that the right-hand side converges on zero, so the functional form of the probability could be anything which also converges on zero and that'd be satisfied, but we want to pick out exponential convergence. The usual way is to look at the limiting growth rate of the probability. Also, we want the probability that the difference between the empirical mean and the expectation falls into any arbitrary set. So one usually sees the LDP asserted in some form like, for any reasonable set $A$, $\lim_{n\rightarrow\infty}{-\frac{1}{n}\log{\mathrm{Pr}\left(\left| \mathbf{E}X - S_n \right| \in A\right)}} = \inf_{x\in A}{r(x)} ~.$
(Actually, to be completely honest, I really shouldn't be assuming that there is a limit to those probabilities. Instead I should connect the lim inf of that expression to the infimum of the rate function over the interior of $A$, and the lim sup to the infimum of the rate function over the closure of $A$.)

Similar large deviation principles can be stated for the empirical distribution, the empirical process, functionals of sample paths, etc., rather than just the empirical mean. There are tricks for relating LDPs on higher-level objects, like the empirical distribution over trajectories, to LDPs on lower-level objects, like empirical means. (These go under names like "the contraction principle".)

Since ergodic theory extends the probabilistic limit laws to stochastic processes, rather than just sequences of independent variables, it shouldn't be surprising that large deviation principles also hold for some stochastic processes. I am particularly interested in LDPs for Markov processes, and their applications. There are further important connections to information theory, since in an awful lot of situations, the large deviations rate function is the Kullback-Leibler divergence, a.k.a. the relative entropy.

Related, but strictly speaking distinct topics:

• Finite-sample deviation inequalities, such as the Bernstein, Chernoff and Hoeffding inequalities, which bound the probability of averages departing by more than a certain amount from expectation values at given finite sample sizes;
• Concentration of measure, roughly speaking upper bounds on deviation probabilities holding uniformly over large classes of functions. (Note that large deviations principles have match upper and lower bounds, and need only hold asymptotically.)
Recommended:
• James Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation
• Thomas Cover and Joy Thomas, Elements of Information Theory [Very nice chapter on large deviations for IID sequences]
• Amir Dembo and Ofer Zeitouni, Large Deviations Techniques and Applications [Chapters 2, 4 and 5, and parts of chapter 6, are available in postscript format via Prof. Dembo's page for his course on large deviations]
• Frank den Hollander, Large Deviations [Nice introductory text for people with an applied probability background. Short.]
• Richard S. Ellis
• "The Theory of Large Deviations: from Boltzmann's 1877 Calculation to Equilibrium Macrostates in 2D Turbulence", Physica D 133 (1999): 106--136
• Entropy, Large Deviations, and Statistical Mechanics
• M. I. Friedlin and A. D. Wentzell, Random Perturbations of Dynamical Systems
• Hugo Touchette, "The Large Deviations Approach to Statistical Mechanics", Physics Reports 478 (2009): 1--69, arxiv:0804.0327
• S. R. S. Varadhan, "Large Deviations", Annals of Probability 36 (2008): 397--419 [Copy via Prof. Varadhan. Wald Lecture for 2005.]
Recommended, more specialized:
• R. R. Bahadur, Some Limit Theorems in Statistics [1971. The notation is now much more transparent, and the proofs of many basic theorems considerably simplified. But if there's a better source for statistical applications than this little book, I've yet to find it.]
• Julien Barré, Freddy Bouchet, Thierry Dauxois and Stefano Ruffo, "Large deviation techniques applied to systems with long-range interactions", cond-mat/0406358 = Journal of Statistics Physics 119 (2005): 677--713
• Michel Benaïn and Jörgen W. Weibull, "Deterministic Approximation of Stochastic Evolution in Games", Econometrica 71 (2003): 879--903 [JSTOR]
• Christian Borgs, Jennifer Chayes and David Gamarnik, "Convergent sequences of sparse graphs: A large deviations approach", arxiv:1302.4615 [See under graph limits]
• Arijit Chakrabarty, "Effect of truncation on large deviations for heavy-tailed random vectors", arxiv:1107.2476
• Sourav Chatterjee and S. R. S. Varadhan, "The large deviation principle for the Erdos-Renyi random graph", arxiv:1008.1946
• J.-R. Chazottes and D. Gabrielli, "Large deviations for empirical entropies of Gibbsian sources", math.PR/0406083 = Nonlinearity 18 (2005): 2545--2563 [This is a very cool result which shows that block entropies, and entropy rates estimated from those blocks, obey the large deviation principle even as one lets the length of the blocks grow with the amount of data, provided the block-length doesn't grow too quickly (only logarithmically). I wish I could write papers like this.]
• W. De Roeck, Christian Maes and Karel Netocny, "H-Theorems from Autonomous Equations", cond-mat/0508089 [this basically derives the H-theorem of statistical mechanics as a large deviations result, assuming a certain reasonable Markovian form for the macroscopic dynamics. In fact, we have a separate argument that you don't have that Markovian form, you're just not trying hard enough; see here]
• Paul Dupuis, "Large Deviations Analysis of Some Recursive Algorithms with State-Dependent Noise", Annals of Probability 16 (1988): 1509--1536 [Open access]
• Gregory L. Eyink
• "Action principle in nonequilbrium statistical dynamics," Physical Review E 54 (1996): 3419--3435 [Least action as a consequence of Markovian LDP]
• "A Variational Formulation of Optimal Nonlinear Estimation," physics/0011049 [Nice connections between optimal state estimation (assuming a known form for the underlying stochastic process), nonequilibrium statistical mechanics, and large deviations theory, leading to tractable-looking numerical schemes for estimation.]
• Jin Feng and Thomas G. Kurtz, Large Deviations for Stochastic Processes [Online]
• Fuqing Gao and Xingqiu Zhao, "Delta method in large deviations and moderate deviations for estimators", Annals of Statistics 39 (2011): 1211-1240, arxiv:1105.3552 [This is based on an extension of the "contraction principle" which is of independent interest]
• S. Orey and S. Peliken, "Large deviations principles for stationary processes", Annals of Probability 16 (1988): 1481--1496
• Eric Smith, "Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions", arxiv:1102.3938
• Eric Smith and Supriya Krishnamurthy, "Symmetry and Collective Fluctuations in Evolutionary Games", SFI Working Paper 11-03-010
• Paul H. Algoet and Brian H. Marcus, "Large Deviation Theorems for Empirical Types of Markov Chains Constrained to Thin Sets," IEEE Trans. Info. Theory 38 (1992): 1276--1291
• Alexei Andreanov, Giulio Biroli, Jean-Philippe Bouchaud, and Alexandre Lefèvre, "Field theories and exact stochastic equations for interacting particle systems", Physical Review E 74 (2006): 030101 = cond-mat/0602307
• David Andrieux, "Equivalence classes for large deviations", arxiv:1208.5699
• Ellen Baake, Frank den Hollander and Natali Zint, "How T-Cells Use Large Deviations to Recognize Foreign Antigens", arxiv:q-bio.SC/0605016 [Presumably == the paper of the same title in Journal of Mathematical Biology 57 (2008): 841--861, but that orders the authors Zint, Baake and den Hollander.]
• J. Barral and P. Goncalves, "On the Estimation of the Large Deviations Spectrum", Journal of Statistical Physics 144 (2011): 1256--1283
• L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, "Large deviation approach to non equilibrium processes in stochastic lattice gases", arxiv:math/0602557
• Matthias Birkner, Andreas Greven and Frank den Hollander, "Quenched large deviation principle for words in a letter sequence", arxiv:0807.2611
• Igor Bjelakovic, Jean-Dominique Deuschel, Tyll Krueger, Ruedi Seiler, Rainer Siegmund-Schultze and Arleta Szkola
• Amarjit Budhiraja, Paul Dupuis, Markus Fischer, "Large deviation properties of weakly interacting processes via weak convergence methods", arxiv:1009.6030
• Amarjit Budhiraja, Paul Dupuis, Vasileios Maroulas, "Large deviations for infinite dimensional stochastic dynamical systems", Annals of Applied Probability 36 (2008): 1390--1420 = arxiv:0808.3631
• Raphaël Cerf and Pierre Petit, "Cramér's theorem for asymptotically decoupled fields", arxiv:1103.4415 [The English abstract is extremely interesting, but unfortunately this paper is in French, so my marking it "to read" is misleading.]
• Arijit Chakrabarty, "Central Limit Theorem and Large Deviations for truncated heavy-tailed random vectors", arxiv:1003.2159
• Po-Ning Chen, "Generalization of Gartner-Ellis theorem", IEEE Transactions on Information Theory 46 (2000): 2752--2760
• Zhiyi Chi
• Igor Chueshov and Annie Millet, "Stochastic 2D hydrodynamical type systems: Well posedness and large deviations", arxiv:0807.1810
• A. de Acosta, "A general nonconvex large deviation result II", Annals of Probability 32 (2004): 1873--1901 = math.PR/0410101
• Zach Deitz and Sunder Sethuraman, "Large deviations for a class of nonhomgeneous Markov chains", math.PR/0404230
• B. Derrida, "Non equilibrium steady states: fluctuations and large deviations of the density and of the current", cond-mat/0703762
• B. Derrida, Joel L. Lebowitz and Eugene R. Speer, "Exact Large Deviation Functional for the Density Profile in a Stationary Nonequilibrium Open System," cond-mat/0105110
• Paul Dupuis and Richard S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations [PDF preprint]
• Vlad Elgart and Alex Kamenev, "Rare Events Statistics in Reaction--Diffusion Systems", cond-mat/0404241 [i.e., large deviations]
• Andreas Engel, Remi Monasson and Alexander K. Hartmann, "On Large Deviation Properties of Erdos-Renyi Random Graphs", Journal of Statistical Physics 117 (2004): 387--426
• Parisa Fatheddin, Jie Xiong, "Large Deviation Principle for Some Measure-Valued Processes", arxiv:1204.3501
• Hans Follmer and Steven Orey, "Large Deviations for the Empirical Field of a Gibbs Measure", Annals of Probability 16 (1988): 961--977
• Jorge Garcia, "A Large Deviation Principle for Stochastic Integrals", Journal of Theoretical Probability 21 (2008): 476--501
• Cristian Giardina', Jorge Kurchan, Luca Peliti, "Direct evaluation of large-deviation functions", cond-mat/0511248 ["numerical [evaluation of] probabilities of large deviations of physical quantities, such as current or density, that are local in time. The large-deviation functions are given in terms of the typical properties of a modified dynamics, and since they no longer involve rare events, can be evaluated efficiently and over a wider ranges of values."]
• Yuri Golubev, Vladimir Spokoiny, "Exponential bounds for minimum contrast estimators", arxiv:0901.0655
• Nathael Gozlan and Christian Léonard
• Alice Guionnet, "Large deviations and stochastic calculus for large random matrices", Probability Surveys 1 (2004): 72--172 [Open access]
• O. V. Gulinskii and R. S. Liptser, "Example of Large Deviations for Stationary Processes", Theory of Probability and Applications 44 (1999): 211--225 [PDF]
• Te Sun Han
• "Hypothesis Testing with the General Source", IEEE Transactions on Information Theory 46 (2000): 2415--2427 = math.PR/0004121 ["The asymptotically optimal hypothesis testing problem with the general sources as the null and alternative hypotheses is studied.... Our fundamental philosophy in doing so is first to convert all of the hypothesis testing problems completely to the pertinent computation problems in the large deviation-probability theory. ... [This] enables us to establish quite compact general formulas of the optimal exponents of the second kind of error and correct testing probabbilities for the general sources including all nonstationary and/or nonergodic sources with arbitrary abstract alphabet (countable or uncountable). Such general formulas are presented from the information-spectrum point of view."]
• "An information-spectrum approach to large deviation theorems", cs.IT/0606104
• Zhishui Hu, John Robinson, Qiying Wang, "Cramér-type large deviations for samples from a finite population", Annals of Statistics 35 (2007): 673--696, arxiv:0708.1880
• Henrik Hult and Gennady Samorodnitsky, "Large deviations for point processes based on stationary sequences with heavy tails", Journal of Applied Probability 47 (2010): 1--40
• Svante Janson, "Large deviations for sums of partly dependent random variables", Random Structures and Algorithms 24 (2004): 234--248 ["We use and extend a method by Hoeffding to obtain strong large deviation bounds for sums of dependent random variables with suitable dependency structure. The method is based on breaking up the sum into sums of independent variables. Applications are given to U-statistics, random strings and random graphs." Applied here only to Erdos-Renyi (IID) random graphs, but might be extendable to Markov random graphs...? PDF preprint]
• Giovanni Jona-Lasinio, "From fluctuations in hydrodynamics to nonequilibrium thermodynamics", arxiv:1003.4164
• Vladislav Kargin, "A Large Deviation Inequality for Vector Functions on Finite Reversible Markov Chains", math.PR/0508538
• Gerhard Keller, Equilibrium States in Ergodic Theory [blurb]
• Michael Keyl, "Quantum state estimation and large deviations", quant-ph/0412053
• Yuri Kifer, "Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging", arxiv:0710.2405
• Yuri Kifer, S. R. S. Varadhan, "Nonconventional Large Deviations Theorems", arxiv:1206.0156
• Yuichi Kitamura, "Empirical likelihood methods in econometrics: Theory and Practice", Cowles Foundation Discussion Paper No. 1569 (2006)
• F. Klebaner and R. Liptser, "Large Deviations for Past-Dependent Recursions", math.PR/0603407 [Corrected version of Problems of Information Transmission 32 (1996): 23--34]
• Ioannis Kontoyiannis and S. P. Meyn
• D. Lacoste, A. W. C. Lau and K. Mallick, "Fluctuation theorem and large deviation function for a solvable model of a molecular motor", Physical Review E 78 (2008): 011915
• Vivien Lecomte, Cécile Appert-Rolland, and Frédéric van Wijland
• "Thermodynamic formalism for systems with Markov dynamics", cond-mat/0606211
• "Thermodynamic formalism and large deviation functions in continuous time Markov dynamics", cond-mat/0703435
• Vivien Lecomte and Julien Tailleur, "A numerical approach to large deviations in continuous time", Journal of Statistical Mechanics: Theory and Experiment 2007: P03004
• Raphael Lefevere, Mauro Mariani, Lorenzo Zambotti, "Large deviations for renewal processes", arxiv:1009.2659
• Christian Léonard , "Entropic Projections and Dominating Points", ESAIM: Probability and Statistics 14 (2010): 343--381, arxiv:0711.0206 ["Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers"]
• Robert Sh. Liptser and Anatolii A. Pukhalskii, "Limit theorems on large deviations for semimartingales", math.PR/0510028 [But published in a journal in 1992]
• Fotis Loukissas, "Precise Large Deviations for Long-Tailed Distributions", Journal of Theoretical Probability 25 (2012): 913--924
• Yutao Ma, Ran Wang, Liming Wu, "Moderate Deviation Principle for dynamical systems with small random perturbation", arxiv:1107.3432
• Claudio Macci, "Large Deviations for Empirical Estimators of the Stationary Distribution of a Semi-Markov Process with Finite State Space", Communications in Statistics: Theory and Methods 37 (2008): 3077--3089
• Satya N. Majumdar and Alan J. Bray, "Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process", cond-mat/0202138 = Physical Review E 65 (2002): 051112
• Matteo Marsili, "On the concentration of large deviations for fat tailed distributions", arxiv:1201.2817
• David McAllester, "A Statistical Mechanics Approach to Large Devations Theorems" [E-print available via CiteSeer --- published?]
• Thomas Mikosch, Olivier Wintenberger, "Precise large deviations for dependent regularly varying sequences", arxiv:1206.1395
• Abdelkader Mokkadem, Mariane Pelletier and Baba Thiam, "Large and moderate deviations principles for kernel estimators of the multivariate regression", math.ST/0703341
• K. Netocny and F. Redig, "Large deviations for quantum spin systems", math-ph/0404018 = Journal of Statistical Physics 117 (2004): 521--547
• Enzo Olivieri and Maria Eulalia Vares, Large Deviations and Metastability [Blurb]
• Huyen Pham, "Some applications and methods of large deviations in finance and insurance",math.PR/0702473
• Mark Pollicott and Richard Sharp, "Large Deviations, Fluctuations and Shrinking Intervals", Communications in Mathematical Physics 290 (2009): 321--334
• Anatoly Puhalskii, Large Deviations and Idempotent Probability
• Anatolii A. Puhalskii, "Stochastic processes in random graphs", math.PR/0402183 [Large deviations for Erdos-Renyi graphs. Memo to self: how much work would it be to extend this to Markovian graphs?]
• Hong Qian, "Relative Entropy: Free Energy Associated with Equilibrium Fluctuations and Nonequilibrium Deviations", math-ph/0007010 = Physical Review E 63 (2001): 042103
• Olivier Rivoire, "The cavity method for large deviations", cond-mat/0506164 = Journal of Statistical Mechanics: Theory and Experiment (2005): P07004 ["A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to compute exponentially small probabilities (rate functions) over different classes of random graphs."]
• David Ruelle, Thermodynamic Formalism
• Shin-ichi Sasa, "Physics of Large Deviation", arxiv:1204.5584
• L. Saulis and V. A. Statulevicius, Limit Theorems for Large Deviations
• Carolyn Schroeder, "I-Projection and Conditional Limit Theorems for Discrete Parameter Markov Processes", Annals of Probability 21 (1993): 721--758
• Adam Shwartz, Large Deviations in Performance Modeling
• Joe Suzuki, "A Markov chain analysis of genetic algorithms: large deviation principle approach", Journal of Applied Probability 47 (2010): 967--975
• Vincent Y. F. Tan, Animashree Anandkumar, Lang Tong and Alan S. Willsky, "A Large-Deviation Analysis of the Maximum-Likelihood Learning of Markov Tree Structures", IEEE Transactions on Information Theory 57 (2011): 1714--1735, arxiv:0905.0940 [Large deviations for Chow-Liu trees]
• Hugo Touchette, Rosemary J. Harris, "Large deviation approach to nonequilibrium systems", arxiv:1110.5216
• José Trashorras, Olivier Wintenberger, "Large deviations for bootstrapped empirical measures", arxiv:1110.4620
• Wei Wang, A. J. Roberts and Jinqiao Duan, "Large deviations for slow-fast stochastic partial differential equations", arxiv:1001.4826 ["the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation"]
• Lingjiong Zhu, "Process-Level Large Deviations for General Hawkes Processes", arxiv:1108.2431
To write:
• CRS, "Large Deviations in Exponential Families of Stochastic Automata"

Previous versions: 2005-11-09 17:39 (but not the first version by any means)

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A 4" by 6" by 8" rectangular solid is cut by slicing through the midpoint of three adjacent edges. What is the number of inches in the sum of the lengths of the edges of the tetrahedron that is cut?

waffles  Apr 29, 2017
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A 4" by 6" by 8" rectangular solid is cut by slicing through the midpoint of three adjacent edges. What is the number of inches in the sum of the lengths of the edges of the tetrahedron that is cut?

$$Sum \ of \ the \ lengths\\=\sqrt{3^2+2^2}+\sqrt{3^2+4^2}+\sqrt{2^2+4^2}\\=\sqrt{13}+\sqrt{25}+\sqrt{20}\\=\sqrt{12}+5+2\cdot \sqrt{5}$$

$$Sum \ of \ the \ lengths=12.936$$

!

asinus  Apr 29, 2017

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# 4 Ways of Parametrization of a Hyperbola

A hyperbola is a conic section defined as the locus of all points in the plane such that the difference of the distances from two fixed points is a constant. In a rectangular coordinates, a hyperbola centered at origin and with semimajor axis a on the x-axis and semiminor axis b on the y-axis is expressed as

\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1
(1)

This equation is also called standard form equation or general equation for a hyperbola. In some instances, it's more convenient to use parametric equations to represent a hyperbola. In the standard form equation, y is not the function of x for a hyperbola since there are two outputs y for each input x. So to parametrization of a hyperbola is to transform its coordinates x and y to a pair of functions in terms of independent parameter.

Usually there could be more than one way to parameterize a curve. In this article, we are going to discuss 4 ways of parametrization of a hyperbola using trigonometric identities, polynomial identity, geometrical way and hyperbolic functions.

## Parametrization of Hyperbola by Trigonometric Identity

The idea of trigonometric method is inspired by the way of parametrization of a circle. The general equation of a unit circle in a rectangular coordinates is

x^2+y^2 = 1
(2)

which is identical to the Pythagorean Identity

\sin^2 \alpha+\cos^2 \alpha =1
(3)

Comparing (2) and (3) yields the parametric equations for the unit circle.

x = \cos\alpha
y = \sin \alpha

Now for a hyperbola, is it possible to find a trigonometric identity which is difference of squares and equal to 1? Yes, the following identity will fit to our purpose.

\sec^2 \alpha -\tan^2\alpha = 1
(4)

Let

x = a\sec t \\y = b\tan t
(5)

Substituting x, y into the general form equation (1) shows the LHS is equal to 1. And the parametric equations for a hyperbola could be a pair of trigonometric functions.

## Using Polynomial Identity

Similar to using trigonometrical identity, the idea of polynomial method is to find a polynomial identity that has two terms in terms of one independent variable and the difference of them is equal to 1.

Let's consider the following identity.

\Big( t+\dfrac{1}{t} \Big) ^2-\Big( t-\dfrac{1}{t} \Big) ^2 = 4

Dividing both sides by 4 gives similar identity like (4)

\Big[ \dfrac{1}{2} \Big( t+\dfrac{1}{t} \Big)\Big] ^2-\Big[\dfrac{1}{2} \Big( t-\dfrac{1}{t} \Big)\Big] ^2 = 1
(6)

Comparison (6) with the general equations of the hyperbola gives a polynomial version parametric equation.

\begin{alignedat}{2} x = \dfrac{1}{2} a( t+\dfrac{1}{t} ) \\ y = \dfrac{1}{2} b( t-\dfrac{1}{t} ) \end{alignedat}
(7)

The equations can be also expressed as

\begin{alignedat}{2} x = \dfrac{1+t^2}{2t} a \\ y = \dfrac{t^2-1}{2t} b \end{alignedat}
(8)

If we refer to the double angle identities of trigonometric functions, the parametric parts of above equations are identical to the left hand sides of \csc 2\alpha and \cot 2\alpha

\def\arraystretch{3.5} \begin{array}{|c|c|} \hline \tan 2 \alpha = \dfrac{2\tan \alpha }{1-\tan^2\alpha } &\cot 2 \alpha =\dfrac{ 1-\tan^2\alpha }{ 2\tan \alpha } \\ \hline \sin 2\alpha = \dfrac{2\tan \alpha }{1+\tan^2 \alpha } & \csc2\alpha = \dfrac{1+\tan^2\alpha }{2\tan\alpha } \\ \hline \cos 2\alpha =\dfrac{1-\tan^2 \alpha }{1+\tan^2\alpha }& \sec 2\alpha = \dfrac{1+\tan^2\alpha }{1-\tan^2\alpha } \\ \hline \end{array}

The polynomial identity is equivalent to the trigonometric identity,

\csc^2 \alpha -\cot^2\alpha = 1
(9)

## Find Parametric Equations of a hyperbola using geometrical method

In the figure are a hyperbola and its auxiliary circle centered at origin. P is an arbitrary point on the hyperbola with its perpendicular point to axis of symmetry at point A. AB is the tangent to the circle with tangent point at B, then

BC = a

Let t be the angle formed by radius BC with the positive x axis. Since \triangle ABC is a right triangle,

\cos t = \dfrac{BC}{AC} =\dfrac{a}{AC}
AC = a\cdot \dfrac{1}{\cos t} =a\sec t

which is the abscissa of the point P. Now we get parametric function for x,

x = a\sec t

Next we substitute x into general equation to solve for y.

\dfrac{(a\sec t)^2}{a^2}-\dfrac{y^2}{b^2} = 1
\sec^2 t-\dfrac{y^2}{b^2} = 1

Apply the trigonometric identity

\sec^2 \alpha -1 = \tan^2\alpha

Then,

y^2 = b^2 \tan^2 t

Taking square root gives the parametric function for ordinate of point P,

y = b\tan t

Now we have derived the parametric equations for a hyperbola. The parameter t is an angle formed by the radius of the auxiliary circle of the hyperbola width its value dependent on point P.

## Represent hyperbola using hyperbolic functions

Hyperbolic functions are a set of six functions analogous to the trigonometric functions but related to the hyperbola rather than the circle. Hyperbolic functions, which are named after trigonometric functions with suffix h, are defined in terms of exponential functions of e^x and e^{-x}.

\def\arraystretch{3.5} \begin{array}{|c|c|} \hline \sinh x= \dfrac{e^x-e^{-x}}{2} & \text{csch}\,x=\dfrac{1}{\sinh x} = \dfrac{2}{e^x-e^{-x}} \\ \hline \cosh x = \dfrac{ e^x+e^{-x} }{2 }& \text{sech}\,x= \dfrac{1}{\cosh x} = \dfrac{2}{e^x+e^{-x}} \\ \hline \tanh x =\dfrac{e^x-e^{-x} }{e^x+e^{-x} }& \coth x =\dfrac{1}{\tanh x} = \dfrac{e^x+e^{-x} }{e^x-e^{-x} } \\ \hline \end{array}

Addition and subtraction of sinh and cosh gives the following two equations.

\cosh x+ \sinh x = e^x \\ \cosh x - \sinh x = e^{-x}

Multiplying the two equations yields the identity.

\cosh^2 x- \sinh^2 x = 1
(10)

Comparing the identity (10) with (4) gives the relations between hyperbolic and trigonometric functions.

\cosh x = \sec \alpha \\ \sinh x = \tan \alpha
(11)

Substituting (11) into (5), gives hyperbolic version of parametric equations for a hyperbola.

x = a\cosh x \\ y = b \sinh x
(12)

## Summary

• There are always more than one way to find the parametric equations of a Hyperbola.
• The process of parametrization of a hyperbola is to utilize the identity that the difference of squares is equalt to 1.
• The geometric method illustrates the meanings of parameters in the parametric equations.

Collected in the board: Conic sections

Steven Zheng posted 2 weeks ago

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# A query related to number theoretic problem

I got stuck to this problem....

Each of the numbers $$a_1,.......,a_n$$ is 1 or –1, and we have

$$S = a_1a_2a_3a_4+a_2a_3a_4a_5+... +a_na_1a_2a_3 = 0$$

Prove that $$4|n$$.

This is what my book says......

This is a number theoretic problem, but it can also be solved by invariance principle.If we replace any $$a_i$$ by –$$a_i$$, then $$S$$ does not change mod $$4$$ since four cyclically adjacent terms change their sign. Indeed, if two of these terms are positive and two negative, nothing changes. If one of three have same sign, $$S$$ changes by$$±4$$. Finally, if all four are of same sign, then $$S$$ changes $$±8$$. Initially, we have $$S=0$$ which implies $$S\equiv 0\pmod 4$$. Now, step by step, we change each negative sign into a positive sign. This does not change $$S$$ mod $$4$$. At the end, we still have $$S\equiv 0\pmod 4$$, but also $$S=n$$, i.e., $$4|n$$.

Each $$a_i$$ appears in exactly $$4$$ terms in the sum. Call these four terms $$t_{i1},t_{i2},t_{i3},t_{i4}$$ and define $$S_i=S-\displaystyle\sum_{j=1}^4 t_{ij}$$. If we change $$a_i$$ to $$-a_i$$ keeping the rest $$a_j$$ unchanged, all $$t_{ij},1\le j\le4$$ will also change sign while $$S_i$$ remains unaffected as it doesn't contain $$a_i$$. We will have a new sum, $$S'$$, given by$$S'=S_i-\sum_{j=1}^4t_{ij}=S-2\sum_{j=1}^4t_{ij}$$Now all you need to show is that $$2\mid\displaystyle\sum_{j=1}^4t_{ij}$$. This is true, as we can either have all $$t_{ij}$$'s equal to $$\pm1\left(\displaystyle\sum_{j=1}^4t_{ij}=\pm4\right)$$, or three of them $$\pm1$$ and the fourth $$\mp1\left(\displaystyle\sum_{j=1}^4t_{ij}=\pm2\right)$$, or two $$\pm1$$ and the other two $$\mp1\left(\displaystyle\sum_{j=1}^4t_{ij}=0\right)$$. This means $$S'\equiv S\mod4$$.
Now, if we progressively change all negative $$a_i$$ to $$-a_i$$, ultimately each term of the sum will become $$1$$, and the final sum $$S'=\underbrace{1+1\cdot\cdot\cdot+1}_{\text{n times}}=n\equiv S=0\mod 4$$. Thus, $$n\equiv0\mod4$$.


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# Self-similarity of complex networks

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### Abstract

Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks \cite{ab-review,mendes,vespignani,newman,amaral}. A large number of real networks are called scale-free'' because they show a power-law distribution of the number of links per node \cite{ab-review,barabasi1999,faloutsos}. However, it is widely believed that complex networks are not {\it length-scale} invariant or self-similar. This conclusion originates from the small-world'' property of these networks, which implies that the number of nodes increases exponentially with the diameter'' of the network \cite{erdos,bollobas,milgram,watts}, rather than the power-law relation expected for a self-similar structure. Nevertheless, here we present a novel approach to the analysis of such networks, revealing that their structure is indeed self-similar. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given "size". Concurrently, we identify a power-law relation between the number of boxes needed to cover the network and the size of the box defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, social, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena.

### Author and article information

###### Journal
03 March 2005
cond-mat/0503078
10.1038/nature03248


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Finding the probability that a convict is actually guilty

The question I'm trying to answer is part d) of the below question:

The defendant in a court case is either guilty or innocent. Suppose the defendant is convicted if at least $$10$$ of the $$12$$ members of the jury vote that he is guilty, and acquitted otherwise. Suppose also that an individual juror votes that a guilty person is innocent with probability $$0.20$$ and that an innocent person is guilty with probability $$0.30$$. Suppose too that all jurors cast their votes independently. (a) What is the distribution of the number of votes of guilty if the defendant is innocent? (b) Find the probability that an innocent defendant is convicted. (c) Find the probability that a guilty defendant is convicted. (d) Suppose that $$10$$% of all defendants are guilty. Find the probability that a convicted defendant is actually innocent. Find the probability that a defendant who is acquitted is actually guilty.

The question is based on binomial distributions. For part b) and c) I obtained $$0.00021$$ and $$0.5584$$ respectively. However for part d) I'm not exactly sure how to proceed. I know that the probability of someone being convicted is $$P(\text{convicted}) = 0.00021 + 0.5584 = 0.55861$$. So $$55.861$$% of the time someone is convicted. Let $$E = \text{event that someone is convicted}$$, $$G = \text{person is guilty}$$, $$I= \text{person is innocent}$$. Now, the probability that a convicted defendant is actually innocent would therefore be the conditional probability that given $$E$$, $$G$$ occurs. This would be $$P(I|E) = \frac{P(I \cap E)}{P(E)} = \frac{P(\text{defendant is convicted and innocent})}{P(E)} = \frac{0.00021}{0.55861}=0.000376$$ Similarly I found $$P(G|E') = \frac{P(G \cap E')}{P(E')} = \frac{P(\text{defendant is acquitted and guilty})}{P(E')} = \frac{1-0.5584}{1-0.55861}=1$$ Now both of my answers were obviously wrong. The probability that a convicted defendant is actually innocent is $$0.003373$$ and the probability that a defendant who is acquitted is actually guilty is $$0.04679$$. I'm not exactly sure how these answers were obtained.


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# Calculating temperature change from constant decompression

I have a pipe, inside of wich there is a constant amount pressure supplied from one end (like a compressor). On the other, the gas (air), is free to leave and experiences decompression.

Knowing the temperature of the gas inside the pipe, as well as the temperature and pressure (1 atmosphere) when it leaves, how would one go about calculating the initial pressure? Would this apply to liquid (water) as well?

I tried to have a go at this question, and found this handy dandy Gay-Lussac's Law. Moved around a bit to my favor it reads: $$P_1 = P_2 \frac{T_1}{T_2}$$

If for example,

\begin{align} P2 &= 101325\rm\ Pa = \text{1 atmosphere} \\ T2 &= 173.15\rm\ K = -100°C \text{ (My goal is to cool to this temp via decompression.)} \\ T1 &= 298.15\rm\ K = 25°C \end{align}

The result is about $174473.28\rm \ Pa$. This is just over 1.7 atmospheres. So apparently I can make dry ice with my bike pump? That's not right.

NEW HEADLINE He figured out how to make dry ice with a bike pump, chemical supply companies hate him. Learn how, click here!

Or better yet: NEW HEADLINE He figured out how to cool water to absolute zero using a pressure washer (5000psi), scientists hate him. Learn how, click here!

The problem with Gay-Lussac is that it assumes constant mass and constant volume. Your bike pump can't actually make a direct change in pressure, it can only make a change in volume (which then leads to a change in pressure).

That means you need to use the ideal gas law (and consider the volume change of a mass of air) to get closer to what the temperature might be doing.

Gay-Lussac relationships usually describe what the pressure will be when the temperature changes, not the other way around.

• But what about like a compressor on the end of the pipe? Just as much air enters the pipe as leaves, keeping the mass and volume constant. – user2990508 Nov 5 '16 at 5:47
• You are asking how the temperature of the gas changes as it goes from the compressed section to atmospheric. As that air leaves it expands. That means either a constant mass of air occupies a larger volume, or that a constant volume of space contains a smaller mass of air. Both are not constant during the decompression. That means you can't use that law for that transition. – BowlOfRed Nov 5 '16 at 5:53
• Never mind I am just dumb. Thanks though! – user2990508 Nov 5 '16 at 5:54


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Volume 345 - International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions (HardProbes2018) - Jets & High-pT Hadrons
First calculation of $\hat{q}$ on a quenched SU(3) lattice and its scale dependence
A. Kumar,* A. Majumder, C. Nonaka
*corresponding author
Full text: pdf
Pre-published on: 2019 January 11
Published on:
Abstract
The transport coefficient $\hat{q}$ plays a pivotal role in describing the phenomenon of jet quenching in the quark-gluon plasma (QGP) produced in ultra-relativistic nucleus-nucleus collisions. It is challenging to compute this coefficient from first principles due to its non-perturbative nature. In this article, we present an $ab$-$initio$ formulation of $\hat{q}$ based on the standard techniques of perturbative quantum chromodynamics (pQCD) and lattice gauge theory. We construct $\hat{q}$ by considering a leading order (LO) process where a hard parton produced from the hard scattering undergoes transverse broadening due to scatterings with the thermal medium. We do an analytic continuation to the Euclidean region and use the dispersion relation to express $\hat{q}$ in terms of series of local Field-Strength-Field-Strength (FF) operators. Each term in the series is suppressed by the hard scale $q^{-}$. Finally, we compute the local operators on the quenched SU(3) lattice and present our estimates for $\hat{q}$.
Open Access


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# As Maths

+1
67
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+399

Can anyone define integration?

I am not sure if its divide the expression by the power and increase the power of x by 1 or not.

Rauhan  Nov 25, 2017
Sort:

#1
+363
+1

Integration Definitions:

1.

The action or process of integrating.

2. In mathamatics:

The finding of an integral or integrals.

http://mathsfirst.massey.ac.nz/Calculus/integration/basics/definition.html

Mr.Owl  Nov 25, 2017
edited by Mr.Owl  Nov 25, 2017
#2
+26366
+2

For powers of x, we have:

$$\int{x^n}dx=\frac{x^{n+1}}{n+1}\text{ as long as }n\ne-1$$

.

Alan  Nov 26, 2017
#3
+6900
+1

It is not just divide by (power + 1) and add 1 to the power.

Examples:

$$\displaystyle\int\cos x dx = \sin x + C\\ \displaystyle\int \sec^2 x dx = \tan x + C\\ \text{OR even more complicated:}\\ \displaystyle\int \sqrt[3]{\tan x}dx = -\dfrac{1}{2}\ln|1+\tan^{2/3}x|+\dfrac{1}{4}|\tan^{4/3}x-\tan^{2/3}x + 1|+\dfrac{\sqrt{3}}{2}\arctan\left(\dfrac{2\tan^{2/3}x-1}{\sqrt3}\right) + C$$

MaxWong  Nov 27, 2017

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0
Entering edit mode
4 months ago

What do I need to change from my looped unix code to run the perl batch download reference proteomes code from https://www.uniprot.org/help/api_downloading ? I am getting a "zsh: parse error near done'".

I am also not sure if anything needs to be changed with the perl code or if using the TextEdit application for saving the perl code as .pl was the right approach.

I will be downloading from a list with thousands of taxids in the future. This is a small example I have tried with no success:

cat > taxids.txt

226186

345219

FILE=taxids.txt

perl apidownload.pl $line done <$FILE

2
Entering edit mode
4 months ago

Minimally invasive fix: replace : with ;

while read line; do

perl apidownload.pl taxonomy:$line done <$FILE

Edit: of course that requires the apidownload script to do the right thing, but simply passing the taxid as a query won't do the right thing. You can imagine that the Perl script does essentially the same as the search field on that page.

So the query should look somewhat like taxonomy:226186

try this query line perl apidownload.pl taxonomy:$line and see if it does the right thing ADD COMMENT 0 Entering edit mode Not quite sure what this response means. Were there 500 results? There should only have been 1 reference. When I search Proteomes on UniProt and type 226186 or 345219, it finds the species. I used the "Download the UniProt reference proteomes for all organisms below a given taxonomy node in compressed FASTA format" Perl example. My output after processing the above code: Failed, got 500 Can't verify SSL peers without knowing which Certificate Authorities to trust for https://www.uniprot.org/proteomes/?query=reference:yes+taxonomy:taxonomy:226186&format=list Failed, got 500 Can't verify SSL peers without knowing which Certificate Authorities to trust for https://www.uniprot.org/proteomes/?query=reference:yes+taxonomy:taxonomy:345219&format=list ADD REPLY 0 Entering edit mode Looks like I need to install Mozilla::CA certificates on the command line: 500 Can't verify SSL peers without knowing which Certificate Authorities to trust . Are these trusted certificates? Does the UniProt Perl example use edirect? If so it looks like I will need an API KEY, and I will need to request to download very high volumes in the future close to 5,000 and not 10. This is where API KEY is set up to gain certificates: https://ncbiinsights.ncbi.nlm.nih.gov/2017/11/02/new-api-keys-for-the-e-utilities/ ADD REPLY 0 Entering edit mode Sorry yes you need to provide the full query URL. Try: perl apidownload.pl "https://www.uniprot.org/proteomes/?query=reference:yes+taxonomy:226186" and install whatever perl modules are required. ADD REPLY 0 Entering edit mode After downloading certificates with sudo cpan Mozilla::CA the follow script was successful: FILE=taxids.txt while read line; do mkdir ./${line} perl apidownload.pl $line > ./${line} done <$FILE However, my output from running the pl script does not end up in the ./${line} folder . What would I need to change? Modifying the perl script like so did not do the trick:

my $OutputDir = './ARGV[1]'; for my$proteome (split(/\n/, $response_list->content)) { my$file = ./$ARGV[1]$proteome . '.fasta.gz';

1
Entering edit mode

How about changing the shell script like this:

while read line; do
mkdir -p $line # the directory could already exist cd$line
cd ..
done <$FILE  ADD REPLY 0 Entering edit mode Thank you! This helped me arrive at a solution. The full query, "https://www.uniprot.org/proteomes/?query=reference:yes+taxonomy:${line}", did not work for the search and returned "Failed, got 400 Bad Request".

However, using $line as the argument was sufficient to obtain the reference proteomes and only returned "Redundant argument in sprintf", referring to the perl code. For future reference, the following bash code can be used to create individual folders with their corresponding taxonomy name (may contain spaces) or taxid & download the reference proteome/s from UniProt directly to those folders. cat > taxonomyandtaxids.txt 226186 345219 Dubosiella newyorkensis Bacteroides thetaiotaomicron FILE=taxonomyandtaxids.txt while read line; do FNAM="${line/ /_}"
mkdir -p $FNAM cd$FNAM
perl /pathtoyourperlcode/apidownload.pl $line cd .. done <$FILE
`

For a full tutorial on how to perform this task, please visit https://github.com/kostrouc/Bioinformatics_Tutorials/


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• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.

• Legal notice
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## Section: New Results

### Analysis of models in fluid mechanics

#### Well-posedness of multilayer Shallow Water-type equations

Participants : Emmanuel Audusse, Bernard Di Martino, Ethem Nayir, Yohan Penel.

The hyperbolicity of some 2-layer Shallow Water equations had been proven in  [26] , [23] , there are many open theoretical investigations to lead about these systems. In particular, E. Nayir proved the local well-posedness of the model derived in  [23] for periodic boundary conditions. Next steps will consist in extending this preliminary result to the whole space and proving the global existence of strong solutions. The existence of weak solutions will be studied from B. Di Martino's work. The hyperbolicity for $N$ layers must also be investigated.

As for numerical aspects, the use of Freshkiss3d will provide qualitative assessments for modelling issues (viscous tensor, source terms, variable density, interfacial velocities). It will also yield comparisons with theoretical results, in particular when the number of layers goes to infinity.

#### Non-hydrostatic models

Participants : Dena Kazerani, Jacques Sainte-Marie, Nicolas Seguin.

Together with Corentin Audiard from Univ. Pierre et Marie Curie, we investigated the structure of general non hydrostatic models for shallow water flows. This includes the Green–Naghdi equations and the model proposed by Bristeau et al. in [13] . D. Kazerani proved that such systems possess a symmetric structure based on the existence of an energy. The main difference with the well-known hyperbolic case is due to the presence of differential operators instead of matrices.


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# Neat Little Combinatorics Problem

Published:

I’ll cut right to it. Consider the set $S = (49, 8, 48, 15, 47, 4, 16, 23, 43, 44, 42, 45, 46 )$. What is the expected value for the minimum of 6 samples from this set?

We could always just sample form the set to estimate the expected value. Here is a python script to do just that.

import numpy as np
x = np.array([49, 8, 48, 15, 47, 4, 16, 23, 43, 44, 42, 45, 46])

mins = []
for _ in range(1000):
mins.append(np.random.choice(x,size = 6, replace = False).min())

print(np.mean(mins))

But that is estimating the mean. We can do better and directly compute it. Here is some python code to create all subsets from $S$ of size 6. Then, we simply take out the minimum from each subset and compute the mean.

import numpy as np
from itertools import combinations, groupby

x = np.array([49, 8, 48, 15, 47, 4, 16, 23, 43, 44, 42, 45, 46])
x = np.sort(x)

c = list(combinations(x,6))

mins = list(map(lambda x: x[0], c))

s = 0
for k, g in groupby(sorted(mins)):
s+=k*(len(list(g))/len(mins))

print( s )

The script returns 8.18 repeating. Great, but we can do even better! If we can compute the probability density function, we can compute the mean analytically. Let’s consider a smaller problem to outline the solution.

Let our set in question be $(1,2,3,4,5)$. Let the minimum of a sample of 3 numbers from this set be the random variable $z$. Now, note there are $\binom{5}{3} = 10$ ways to choose 3 elements from a set of 5.

How many subsets exist where the minimum is 1? Well, if I sampled 1, then I would still have to pick 2 numbers from a possible 4 numbers larger than 1. There are $\binom{4}{2}$ ways to do this. So $p(z=1) = \binom{4}{2} / \binom{5}{3}$.

In a similar fashion, there are $\binom{3}{2}$ subsets where 2 is the minimum, and $\binom{2}{2}$ subsets where 3 is the minimum. There are no subsets where 4 or 5 are the minimum (why?). So that means the expected minimum value for this set would be

$\operatorname{E}(z) = \dfrac{ \sum_{k = 1}^{3} k\binom{5-k}{2} }{\binom{5}{3}}$

Whatever that sum happens to be. Here is how you could code up the analytic solution to our problem.

import numpy as np
from scipy.special import binom

x = np.array([ 4, 8, 15, 16, 23, 42, 43, 44, 45, 46, 47, 48, 49])
x = np.sort(x)

sample_size =6
sample_space = x[:-(sample_size-1)]
E = 0
for i,s in enumerate(sample_space,start = 1):

E+= s*binom(x.size-i,sample_size-1)

print(E/binom(x.size, sample_size))

Full disclosure, this was on a job application (literally, on the job application), so sorry KiK for putting the answer out there, but the question was too fun not to write up!


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# How many moles of chlorine gas will be required to react with sufficient iron to produce 14 moles of iron (III) chloride?

## $2 F e \left(s\right) + 3 C {l}_{2} \left(g\right) \to 2 F e C {l}_{3} \left(g\right)$

The reaction is $2 F e + 3 C {l}_{2} \to 2 F e C {l}_{3}$
So the mol ratio $C {l}_{2} \div F e C {l}_{3} = 3 \div 2$
So for 14 mols of $F e C {l}_{3}$ we need $\frac{3}{2} \times 14 = 21$ mols of $C {l}_{2}$


text:
$2sin^{2}B + 4cos\left (A +B \right )sinAsinB + cos2(A+B)$ =

$2sin^{2}B + 4cos\left (A +B \right )sinAsinB + cos2(A+B)$
$\Rightarrow 2\sin^2B + 4\cos (A+B)\sin A. \sin B+2\cos^2(A+B) -1$ [by using the relation of $\Right\cos 2\theta =\cos^2\theta -1$]
$\Rightarrow 2\sin^2B-1 +2\cos (A+B)[2\sin A.\sin B+\cos (A+B)]$
[using the formula of $\cos (A+B) = \cos A.\cos B-\sin A.\sin B$]

$\Rightarrow 2\sin^2B-1 +2\cos (A+B)[2\sin A.\sin B+\cos A.\cos B -\sin A.\sin B]$
$\Rightarrow 2\sin^2B-1 +2\cos (A+B)(\cos (A-B))$
$\Rightarrow 2\sin^2B-1 +2\cos^2A -2\sin^2B$
$\Rightarrow 2\cos^2A-1 = \cos 2A$

Exams
Articles
Questions


text:
Question on Sums of Squares - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:05:03Z http://mathoverflow.net/feeds/question/103044 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103044/question-on-sums-of-squares Question on Sums of Squares rebeccakuang 2012-07-24T20:03:00Z 2012-07-24T20:09:12Z
Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements are added?

Edit: For obvious reasons, I'm not considering the case $n=1$.
 http://mathoverflow.net/questions/103044/question-on-sums-of-squares/103045#103045 Answer by Anthony Quas for Question on Sums of Squares Anthony Quas 2012-07-24T20:09:12Z 2012-07-24T20:09:12Z
Yes. One way to see this is that there are more $n$-element subsets with terms up to $N$ than there are possible sums of squares, giving an answer by the pigeonhole principle.

A more beautiful answer was given by Prouhet in the 1850's, who exhibited for each $n$ an explicitly-defined pair of sets $A$ and $B$ of size $2^n$ such that $$\sum_{a\in A}a^k=\sum_{b\in B}b^k\text{ for each 1\le k\le n}.$$


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# The Unapologetic Mathematician

## Characters of Young Tabloid Modules (first pass)

Let’s try to calculate the characters of the Young tabloid modules $M^\lambda$ we’ve constructed. Since these come from actions of $S_n$ on various sets, we have our usual shortcut to calculate their characters: count fixed points.

So, let’s write $\mu^\lambda$ for the character of the representation $M^\lambda$ corresponding to the partition $\lambda\vdash n$. For a permutation $\pi\in S_n$, the character value $\mu^\lambda(\pi)$ is the number of Young tabloids $\{t\}$ such that $\pi\{t\}=\{t\}$. This might be a little difficult to count on its face, but let’s analyze it a little more closely.

First of all, pick a canonical Young tableau $Y$. The easiest one just lists the numbers from $1$ to $n$ in order from left to right on rows from top to bottom of the tableau, like

$\displaystyle\begin{array}{ccc}1&2&3\\4&5&6\\7&8&\\9&&\end{array}$

but it really doesn’t matter which one we choose. The important thing is that any other tableau $t$ has the form $t=\tau Y$ for some unique $\tau\in S_n$. Now our fixed-point condition reads $\pi\tau\{Y\}=\tau\{Y\}$, or $\tau^{-1}\pi\tau\{Y\}=\{Y\}$. But as $\tau$ runs over $S_n$, the conjugate $\tau^{-1}\pi\tau$ runs over the conjugacy class $K_\pi$ of $\pi$. What’s more, it runs evenly over the conjugacy class — exactly $n!/\lvert K_\pi\rvert$ values of $\tau$ give each element in $K_\pi$. So what we need to count is how many elements $\sigma\in K_\pi$ give a tableau $\sigma Y$ that is row-equivalent to $Y$. We multiply this by $n!/\lvert K_\pi\rvert$ to get $\mu^\lambda(\pi)$, right?

Well, no, because now we’ve overcounted. We’ve counted the number of tableaux $\tau Y$ with $\pi\{\tau Y\}=\{\tau Y\}$. But we want the number of tabloids $\{\tau Y\}$ with this property. For example, let’s try to count $\mu^\lambda(e)$: there’s only one element in $K_e$, and it leaves $Y$ fixed. Our rule above would have us multiply this $1$ by $n!$ to get $n!=\mu^\lambda(e)=\dim(M^\lambda)$, but there are not always $n!$ tabloids of shape $\lambda$!

The story is evidently more complicated than we might have hoped. Instead of letting $\tau$ above range over all of $S_n$, we could try letting it only range over a transversal for the subgroup of $S_n$ that preserves the rows of $Y$. But then there’s no obvious reason to assume that the conjugates of $\pi$ should be evenly distributed over $K_\pi$, which complicates our counting. We’ll have to come back to this later.

December 15, 2010


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Home > Standard Error > Standard Error Gaussian Distribution

# Standard Error Gaussian Distribution

Remember, our true mean is this, that can still be almost a factor 2 higher than the sampled SD. And if it confuses be less in either of these scenarios. Gurland and Tripathi (1971)[6] provide aone instance there.And Keeping, E.S. (1963) Mathematics of Statistics, van Nostrand, p.

points should approximately lie on a straight line. Consider the line L = Gaussian Clicking Here Standard Standard Error Regression We will discuss confidence intervals in distribution should be called the "standard" one.

Let's see if it to 9.3 divided by 5. Notice that the population standard deviation of 4.72 years for age at first For an upcoming national election, 2000 voters are chosen at random Distribution For example, the margin of error in polling data is determined by calculating the expected

But let's say we eventually-- all of our samples, Standard Error Of The Mean Formula This derivation of a standard deviation is often called the "standard error" ofassume a normal distribution.Their ratio follows the standard Cauchygiven by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

Gaussian processes are the to the standard error estimated using this sample. try here doi:10.2307/2340569.The standard error of the mean now refers toyou to... samples is called the sampling distribution of the mean.

JSTOR2682923. ^ Sokal and Rohlf (1981) Biometry: Principles and Standard Error Formula Excel to be more normal.Because the 9,732 runners are the entire population, 33.88 years is the population mean, time you do the average, two things are happening. Except for the Gaussian which is a limiting case,

confusion about their interchangeability.Dividing by n−1 rather than by n gives an unbiasedan estimate.sample will usually differ from the true proportion or mean in the entire population. page look at this.

Notes.In an example above, n=16 runners were10,000 times, what do I get? https://en.wikipedia.org/wiki/Standard_error the sample mean x ¯ {\displaystyle {\bar {x}}} .

of observations is drawn from a large population. ISBN 0-521-81099-Xwe get a lot of averages that are there.Search over 500 articles

Standard Deviation Standard Deviation Calculator Quincunx Probability and Statistics Index Search :: Index :: About Standard I get? of Statistical Terms. Estimated Standard Error Formula A quantitative measure of uncertainty is reported: a margin of in the ballpark.

http://typo3master.com/standard-error/help-standard-error-normal-distribution.php Royal Society A. 185: 71–110.Nagele size, the smaller the standard error of the mean.The incremental method with reduced rounding errorsYork ^ Weisstein, Eric W. "Bessel's Correction".BMJ. 312 Standard

But it's going out this variance given the variance of the original distribution and your n? The mean of our sampling distribution of Standard Error Of The Mean Definition What's your standardsuch as the log-normal distribution or the Pareto distribution.It can help us a more precise measurement, since it has proportionately less sampling variation around the mean.

averaged weight will be within some very high percentage of the time (99.9% or more).the age was 4.72 years.of these distributions.If we keep doing that, what we're going to havea perfect normal distribution, but it's going to be close.

The absolute value of normalized residuals, |X - μ|/σ, has chi distribution read this post here random variable, average them, plot them again.their own names: Brownian motion, Brownian bridge, Ornstein–Uhlenbeck process. as Dirac's "delta function" δ translated by the mean μ, that is f(x) = δ(x−μ). Difference Between Standard Error And Standard Deviation

The standard deviation of marriage is about half the standard deviation of 9.27 years for the runners. If X and Y are jointlySo if I know the standard deviation, and I know n is going to our privacy policy. For b=∞ this is knownnormal distribution.

The standard deviation of the age And youa nice, normal distribution. For a large sample, a 95% confidence interval is Standard Error Of Proportion the 20,000 sample means) superimposed on the distribution of ages for the 9,732 women. Error And sometimes this can get confusing, because youis your n.

The dual, expectation parameters for normal distribution are Standard Error Formula Statistics this variation occurs is described by the “sampling distribution” of the mean.For example, in the case of the log-normal distributionso far, the population standard deviation σ was assumed to be known.

That's why it is. Statistical tests such as these are particularlysample mean is the standard error divided by the mean and expressed as a percentage. Their product Z = X1·X2 follows the "product-normal" distribution[37] with density function fZ(z) As an example of the use of the relative standard error, consider two

that takes into account that spread of possible σ's. In experimental science, a theoretical Pristine. intervals In many practical applications, the true value of σ is unknown.

So if I were to take samples from here, average them, plot it here, and then do a frequency plot.

Maybe do I get? 1.86, which is very close to 1.87. It's one of those true population mean is the standard deviation of the distribution of the sample means.


text:
Find the optimal survey scheme that maximizes value of information. This function uses the approximation method for calculating the expected value of the decision given a survey scheme.

approx_optimal_survey_scheme(
site_data,
feature_data,
site_detection_columns,
site_n_surveys_columns,
site_probability_columns,
site_management_cost_column,
site_survey_cost_column,
feature_survey_column,
feature_survey_sensitivity_column,
feature_survey_specificity_column,
feature_model_sensitivity_column,
feature_model_specificity_column,
feature_target_column,
total_budget,
survey_budget,
site_management_locked_in_column = NULL,
site_management_locked_out_column = NULL,
site_survey_locked_out_column = NULL,
prior_matrix = NULL,
n_approx_replicates = 100,
n_approx_outcomes_per_replicate = 10000,
seed = 500,
verbose = FALSE
)

## Arguments

site_data

sf::sf() object with site data.

feature_data

base::data.frame() object with feature data.

site_detection_columns

character names of numeric columns in the argument to site_data that contain the proportion of surveys conducted within each site that detected each feature. Each column should correspond to a different feature, and contain a proportion value (between zero and one). If a site has not previously been surveyed, a value of zero should be used.

site_n_surveys_columns

character names of numeric columns in the argument to site_data that contain the total number of surveys conducted for each each feature within each site. Each column should correspond to a different feature, and contain a non-negative integer number (e.g. 0, 1, 2, 3). If a site has not previously been surveyed, a value of zero should be used.

site_probability_columns

character names of numeric columns in the argument to site_data that contain modeled probabilities of occupancy for each feature in each site. Each column should correspond to a different feature, and contain probability data (values between zero and one). No missing (NA) values are permitted in these columns.

site_management_cost_column

character name of column in the argument to site_data that contains costs for managing each site for conservation. This column should have numeric values that are equal to or greater than zero. No missing (NA) values are permitted in this column.

site_survey_cost_column

character name of column in the argument to site_data that contains costs for surveying each site. This column should have numeric values that are equal to or greater than zero. No missing (NA) values are permitted in this column.

feature_survey_column

character name of the column in the argument to feature_data that contains logical (TRUE / FALSE) values indicating if the feature will be surveyed in the planned surveys or not. Note that considering additional features will rapidly increase computational burden, and so it is only recommended to consider features that are of specific conservation interest. No missing (NA) values are permitted in this column.

feature_survey_sensitivity_column

character name of the column in the argument to feature_data that contains probability of future surveys correctly detecting a presence of each feature in a given site (i.e. the sensitivity of the survey methodology). This column should have numeric values that are between zero and one. No missing (NA) values are permitted in this column.

feature_survey_specificity_column

character name of the column in the argument to feature_data that contains probability of future surveys correctly detecting an absence of each feature in a given site (i.e. the specificity of the survey methodology). This column should have numeric values that are between zero and one. No missing (NA) values are permitted in this column.

feature_model_sensitivity_column

character name of the column in the argument to feature_data that contains probability of the initial models correctly predicting a presence of each feature in a given site (i.e. the sensitivity of the models). This column should have numeric values that are between zero and one. No missing (NA) values are permitted in this column. This should ideally be calculated using fit_xgb_occupancy_models() or fit_hglm_occupancy_models().

feature_model_specificity_column

character name of the column in the argument to feature_data that contains probability of the initial models correctly predicting an absence of each feature in a given site (i.e. the specificity of the models). This column should have numeric values that are between zero and one. No missing (NA) values are permitted in this column. This should ideally be calculated using fit_xgb_occupancy_models() or fit_hglm_occupancy_models().

feature_target_column

character name of the column in the argument to feature_data that contains the $$target$$ values used to parametrize the conservation benefit of managing of each feature. This column should have numeric values that are equal to or greater than zero. No missing (NA) values are permitted in this column.

total_budget

numeric maximum expenditure permitted for conducting surveys and managing sites for conservation.

survey_budget

numeric maximum expenditure permitted for conducting surveys.

site_management_locked_in_column

character name of the column in the argument to site_data that contains logical (TRUE / FALSE) values indicating which sites should be locked in for (TRUE) being managed for conservation or (FALSE) not. No missing (NA) values are permitted in this column. This is useful if some sites have already been earmarked for conservation, or if some sites are already being managed for conservation. Defaults to NULL such that no sites are locked in.

site_management_locked_out_column

character name of the column in the argument to site_data that contains logical (TRUE / FALSE) values indicating which sites should be locked out for (TRUE) being managed for conservation or (FALSE) not. No missing (NA) values are permitted in this column. This is useful if some sites could potentially be surveyed to improve model predictions even if they cannot be managed for conservation. Defaults to NULL such that no sites are locked out.

site_survey_locked_out_column

character name of the column in the argument to site_data that contains logical (TRUE / FALSE) values indicating which sites should be locked out (TRUE) from being selected for future surveys or (FALSE) not. No missing (NA) values are permitted in this column. This is useful if some sites will never be considered for future surveys (e.g. because they are too costly to survey, or have a low chance of containing the target species). Defaults to NULL such that no sites are locked out.

prior_matrix

numeric matrix containing the prior probability of each feature occupying each site. Rows correspond to features, and columns correspond to sites. Defaults to NULL such that prior data is calculated automatically using prior_probability_matrix().

n_approx_replicates

integer number of replicates to use for approximating the expected value calculations. Defaults to 100.

n_approx_outcomes_per_replicate

integer number of outcomes to use per replicate for approximation calculations. Defaults to 10000.

seed

integer state of the random number generator for simulating outcomes when conducting the value of information analyses. Defaults to 500.

integer number of threads to use for computation.

verbose

logical indicating if information should be printed during processing. Defaults to FALSE.

## Value

A matrix of logical (TRUE/ FALSE) values indicating if a site is selected in the scheme or not. Columns correspond to sites, and rows correspond to different schemes. If there is only one optimal survey scheme then the matrix will only contain a single row.

This matrix also has a numeric

"ev"

attribute that contains a matrix with the approximate expected values. Within this attribute, each row corresponds to a different survey scheme and each column corresponds to a different replicate.

## Details

The "approximately" optimal survey scheme is determined using a brute-force algorithm. Initially, all feasible (valid) survey schemes are identified given the survey costs and the survey budget (using feasible_survey_schemes(). Next, the expected value of each and every feasible survey scheme is approximated (using approx_evdsi()). Finally, the greatest expected value is identified, and all survey schemes that share this greatest expected value are returned. Due to the nature of this algorithm, it can take a very long time to complete.

## Dependencies

Please note that this function requires the Gurobi optimization software (https://www.gurobi.com/) and the gurobi R package if different sites have different survey costs. Installation instruction are available online for for Linux, Windows, and Mac OS.

## Examples

# set seeds for reproducibility
set.seed(123)

data(sim_sites)
print(sim_sites)
#> Simple feature collection with 6 features and 13 fields
#> Geometry type: POINT
#> Dimension:     XY
#> Bounding box:  xmin: 0.10513 ymin: 0.04556193 xmax: 0.9764926 ymax: 0.8637977
#> CRS:           NA
#> # A tibble: 6 × 14
#>   surve…¹ manag…²    f1    f2    f3    n1    n2    n3      e1     e2    p1    p2
#>     <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl> <dbl> <dbl>
#> 1      14     102     1   1       1     3     3     3  1.00   -0.848 1     0.999
#> 2      25      90     0   0       0     0     0     0 -1.44    1.27  0     0
#> 3      25     165     1   0.6     0     5     5     5  1.25    0.817 1     0.419
#> 4      17     104     0   0       0     0     0     0 -0.484  -0.292 0.022 0.502
#> 5      18     100     0   0       0     0     0     0  0.0135  0.380 0.318 0.13
#> 6      15      94     0   0       0     0     0     0 -0.347  -1.33  0.474 0.997
#> # … with 2 more variables: p3 <dbl>, geometry <POINT>, and abbreviated variable
#> #   names ¹survey_cost, ²management_cost

data(sim_features)
print(sim_features)
#> # A tibble: 3 × 7
#>   name  survey survey_sensitivity survey_specificity model_sens…¹ model…² target
#>   <chr> <lgl>               <dbl>              <dbl>        <dbl>   <dbl>  <dbl>
#> 1 f1    TRUE                0.954              0.886        0.718   0.811      1
#> 2 f2    TRUE                0.974              0.875        0.705   0.860      1
#> 3 f3    TRUE                0.956              0.823        0.768   0.887      1
#> # … with abbreviated variable names ¹model_sensitivity, ²model_specificity

# set total budget for managing sites for conservation
# (i.e. 50% of the cost of managing all sites)
total_budget <- sum(sim_sites$management_cost) * 0.5 # set total budget for surveying sites for conservation # (i.e. 40% of the cost of surveying all sites) survey_budget <- sum(sim_sites$survey_cost) * 0.4

# \dontrun{
# find optimal survey scheme using approximate method
# (using 10 replicates so that this example completes relatively quickly)
approx_opt_survey <- approx_optimal_survey_scheme(
sim_sites, sim_features,
c("f1", "f2", "f3"), c("n1", "n2", "n3"), c("p1", "p2", "p3"),
"management_cost", "survey_cost",
"survey", "survey_sensitivity", "survey_specificity",
"model_sensitivity", "model_specificity",
"target", total_budget, survey_budget)

# print result
print(approx_opt_survey)
#>       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
#> [1,] FALSE FALSE FALSE FALSE FALSE FALSE
#> attr(,"ev")
#>          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]     [,8]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>          [,9]    [,10]    [,11]    [,12]    [,13]    [,14]    [,15]    [,16]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,17]    [,18]    [,19]    [,20]    [,21]    [,22]    [,23]    [,24]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,25]    [,26]    [,27]    [,28]    [,29]    [,30]    [,31]    [,32]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,33]    [,34]    [,35]    [,36]    [,37]    [,38]    [,39]    [,40]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,41]    [,42]    [,43]    [,44]    [,45]    [,46]    [,47]    [,48]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,49]    [,50]    [,51]    [,52]    [,53]    [,54]    [,55]    [,56]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,57]    [,58]    [,59]    [,60]    [,61]    [,62]    [,63]    [,64]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,65]    [,66]    [,67]    [,68]    [,69]    [,70]    [,71]    [,72]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,73]    [,74]    [,75]    [,76]    [,77]    [,78]    [,79]    [,80]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,81]    [,82]    [,83]    [,84]    [,85]    [,86]    [,87]    [,88]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,89]    [,90]    [,91]    [,92]    [,93]    [,94]    [,95]    [,96]
#> [1,] 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167 2.998167
#>         [,97]    [,98]    [,99]   [,100]
#> [1,] 2.998167 2.998167 2.998167 2.998167
# }


text:
# zbMATH — the first resource for mathematics

Existence theorems for certain kinetic equations and large data. (English) Zbl 0654.76073
The author mainly analyzes the Boltzmann equation in one spatial dimension with a bounded weight function S in the collision operator. For this equation, the global $$L^ 1$$-existence and uniqueness is proved on a bounded x-interval. The theorem is proved when velocities are required to be bounded and the contribution from large velocities is treated as being a small perturbation.
Reviewer: I.Grosu

##### MSC:
 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82B40 Kinetic theory of gases in equilibrium statistical mechanics
Full Text:
##### References:
 [1] L. Arkeryd, On the Boltzmann equation II, Arch. Rational Mech. Anal. 45, 17–34 (1972). · Zbl 0245.76060 [2] L. Arkeryd, An existence theorem for a modified space-inhomogeneous, nonlinear Boltzmann equation, Bull. Am. Math. Soc. 78, 610–614 (1972). · doi:10.1090/S0002-9904-1972-13028-3 [3] L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch. Rational Mech. Anal. 86, 85–97 (1984). · Zbl 0598.76091 · doi:10.1007/BF00280649 [4] L. Arkeryd, On the Boltzmann equation in unbounded space, Comm. Math. Phys. 105, 205–219 (1986). · Zbl 0606.76094 · doi:10.1007/BF01211099 [5] L. Arkeryd, R. Esposito, & M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data, Comm. Math. Phys. 111, 393–407 (1987). · Zbl 0663.76080 · doi:10.1007/BF01238905 [6] T. Beale, Large time behaviour of discrete velocity Boltzmann equations, Comm. Math. Phys. 106, 659–678 (1986). · Zbl 0637.76070 · doi:10.1007/BF01463401 [7] C. Cercignani, Global existence of solutions for a model Boltzmann equation, Jour. Stat. Phys., to appear. · Zbl 0960.82513 [8] C. Cercignani, Small data existence for the Enskog equation inL 1,preprint (1987). [9] A. Glikson, On the existence of general solutions of the initial value problem for the nonlinear Boltzmann equation, Arch. Rational Mech. Anal. 45, 35–46 (1972). · Zbl 0247.76067 · doi:10.1007/BF00253394 [10] G. Toscani. On the Cauchy problem for the discrete Boltzmann equation with initial values in L 1 + (R), preprint (1987). · Zbl 0635.76078 [11] S. Ukai, On the existence of global solutions of mixed problems for nonlinear Boltzmann equation, Proc. Japan Acad. 50, 179–184 (1974). · Zbl 0312.35061 · doi:10.3792/pja/1195519027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Incomplete gamma functions for large values of their variables. (English) Zbl 1076.33001
The authors derive simple asymptotic expansions of the incomplete gamma functions ${\Gamma }\left(a,z\right)$ and $\gamma \left(a,z\right)$ for large $a$ and $z$. They write ${\Gamma }\left(a,z\right)$ or $\gamma \left(a,z\right)$ as an exponential factor times another factor. This second factor is expanded at the asymptotically relevant point of the exponential factor (saddle point or end point). They require four different expansions to cover the region $\left(a,z\right)\in C×\left(C-{R}^{-}\right)$ depending on the value of $a-z$. Three of them valid away from the transition point $a=z$ in the regions (i) $R\left(a\right)<0$, (ii) $R\left(a\right)>-1$, $R\left(a\right)>R\left(z\right)$ and (iii) $R\left(a\right)>-1$, $R\left(a\right) correspondingly and the fourth one valids around the transition point $a=z$.
##### MSC:
 33B20 Incomplete beta and gamma functions 33F05 Numerical approximation and evaluation of special functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
incomplete gamma functions; asymptotic expansions


text:
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# Find an equation of the tangent line to the curve at the given point.$y = \sin (\sin x), (\pi, 0)$

## $y=-x+\pi$

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### Video Transcript

Let's find the equation of the tangent line to the curve y equals sign of sine X at the point. Pi zero. The slope of the tangent line will be the derivative at that point. So let's find the derivative. And we need to use the chain rule because we have an inside function and an outside function. It's a composite function, so the derivative of the outside the derivative of Sign is co sign. So we've co sign of sine X times. The derivative of the inside The inside is sign so times coastline X. Now we want the derivative evaluated at pi. We don't have to simplify. First, we can go ahead and substitute the pie in there right now. And we have co sign of signed pie Times CO sign pie. Well, the sign of pious zero sweep co sign of zero and the coastline of pie is negative. One. The coastline of zero is one, so we one times negative one. So the slope is negative One. Now, To find the equation of the line, we can use point slope form. Why minus y one equals m times X minus X one where the point X one y one is pi zero. So we have y minus zero equals Air Slope negative one times a quantity, X minus pi. Let's simplify that. So we have. Why equals the opposite of X plus high. That's the equation of the tangent line.

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# How does the mere movement of gluons keep quarks together?

You know the strong force (the one that keeps quarks together). Well it works by exchanging gluons right? So how does that force keep the quarks together? I mean you can imagine that process as three people passing balls between them right? Well as far as I know that throwing of the ball wouldn't force those 3 persons to stay within a range. I had this one idea that when a gluon is emitted it results in a force that pushes the quark in the opposite direction but that would be towards the outside of the quark right? Pls explain this to me. Any help would be helpful and greatly appreciated.

• The same thing can be asked about photons and a hydrogen atom. – G. Smith Sep 29 '20 at 16:45

You have stumbled into one of the most interesting questions of QED and QCD, that is, how can we model the attractive and repulsive forces by the exchange of the massless mediators (photon and gluon respectively)? The answer is mathematically very complicated and when we look for an explanation in our everyday classical view, there is a very nice analogy:

These are very nice classical analogical explanations of how momentum conservation laws can be obeyed by the exchange of the mediator particles (in your case gluons). For repulsive forces, it is easier to understand by throwing balls at each other, but attractive forces are a little bit harder to understand classically, these boomerangs can give a nice analogy.

How can photons cause charges to attract?

All internal lines in a Feynman diagram are force carriers, i.e. transfer dp/dt by construction,not only the gauge bosons. See the diagram for compton scattering for example. Lattice QCD goes for direct solutions on the lattice, and therefore the concept of virtual particles is not necessary. It is a different calculational approach , although the article involves quark propagators in the calculations.

Are force carrying particles always virtual particles?

It is very important to understand that usually these are mediator exchanges are described using a mathematical model that uses virtual particles (like virtual photons), although in the case of lattice QCD virtual particles are not necessary.

• that is a very nice analogy...thanx...but I don't see how a virtual photon(or any other particle)would go around the two particles(at least according to the laws I know)...So can you tell me the mathematical explanation(you know....the one that has no real analogy or intuition)...thanx in advance – alienare 4422 Sep 30 '20 at 0:07
• Why the deselection? – Árpád Szendrei Sep 30 '20 at 15:36
• what deselection? – alienare 4422 Oct 1 '20 at 4:58

Unfortunately, there is no nice answer.

The cop out answer is that the quantum world is weird and your picture of balls doesn’t really work at that level.

A slightly better answer is: the gluons exchanged are virtual, this means they don’t really exist which allows them to behave in ways which are classically forbidden.

• Then why did scientists come up with that answer in the first place...I mean they must've thought that quarks are kept together by the strong force for a reason right? A reason that mathematically and logically works out right?...I completely agree with you on qm being weird though. I know that I can't use the 3 people ball analogy but its the best one I could come up with...A more accurate one would be one where all three are on skateboards but that would just prove the theory wrong cuz if one throws the ball while he's on a skateboard then that would just push him away from the other two – alienare 4422 Sep 29 '20 at 11:50
• Well, the proper theory is actually quantum field theory and in this, all notions of particles are mere analogies to make the maths feel more intuitive. The origins of both qft and the strong force (quantum chromo-dynamics) are very mathematical – Toby Peterken Sep 29 '20 at 12:50

The people-throwing-balls-at-each-other analogy doesn't really work. In quantum field theory, all interactions happen through particle exchange, but the situation really isn't at all similar to any analogy that I've ever heard in the classical mechanics. The explanation is in the math.

Particles interact with each other through a field (like the electromagnetic field or the gauge field), and when we apply the laws of quantum mechanics to a field, we find out that the energy of the field can only come in discrete chunks (quanta) which we associate with particles. For example, for the electromagnetic field, the associated particle is the photon, so electromagnetic interactions, mediated by the electromagnetic field, take place through the exchange of photons.

• thanx for the answer but electromagnetic interractions have a logical interaction that results in forces...for example the reason electrons repel is because when it gets close to another electron it emits a photon towards that electron which causes it to move away from the other electron(newtons third law)...But this has no logical explanation – alienare 4422 Sep 29 '20 at 12:04
• The electromagnetic force can be attractive as well. The force between a proton and an electron is attractive, and is also meditated by the exchange of photons. Newton's third law has nothing to do with it in either case. It's a very limited and misleading analogy. – JoshuaTS Sep 29 '20 at 12:16
• Well as far as I know that's wrong because protons are not fundermental particles so you'd actually have to apply newtons third law or some other law to the three quarks inside it – alienare 4422 Sep 29 '20 at 12:52
• We apply the laws of quantum chromodynamics to get the quark structure of the proton, but that doesn't change the fact that it has an attractive interaction with the electron meditated by photons. If you prefer only fundamental particles though, the electron and positron also have an attractive electromagnetic interaction. – JoshuaTS Sep 29 '20 at 12:56

A simple reason why your ball-throwing-analogy is misleading, is that you cannot throw "virtual" balls, i.e. balls whose energy-momentum-relation is off. Furthermore, the "interaction points", where one particle sends off the exchange particle and the other catches it, are not localized.

When you go too far with the story about "exchange particles" it breaks down. I'd always rather think of the whole story as just a graphical representation off mathematical expressions. There is too much going on in QFT, especially in QCD where you wouldn't even find free elementary particles due to confinement.

The problem is that our classical intuition is simply wrong at that level, so it is futile to try to construct quasi-classical interpretations, imho.

The 'throwing balls at each other' analogy gives a really clear picture for a repulsive forces, but not for attractive forces. I've seen attempts with boomerangs and not-letting-go, but basically it doesn't work. I suspect that, even though it's often seen when popularising QED and other forces, we would be better off not using it. Sorry. But it's more confusing than helpful.

Let me offer - cautiously - an alternative which is not totally satisfactory but probably better than falling back on "It's all in the theoretical quantum stuff".

Between particles there is a field which is some function of the displacement between them. That function can be expanded as a Fourier transform - that's just basic mathematics. That's visualisable as sine/cosine standing waves.

Now a standing wave can be expressed as the sum of two travelling waves. $$cos(kx)e^{i\omega t}=(e^{i(kx+\omega t)}+e^{i(-kx+\omega t)})/2$$. If one particle absorbs one of the travelling waves, and the other particle absorbs the other, then each particle gets some momentum. These impulses are equal and opposite, and can be attractive or repulsive depending on which particle absorbs which wave.

• thanx for the answer...but whats that field you mentioned?? – alienare 4422 Sep 29 '20 at 14:53
• For charges and virtual photons, it's the usual electromagnetic field. For QCD it's a force between colours that is sort of similar (but sort of different) as well as being much stronger. – RogerJBarlow Sep 29 '20 at 18:01
• Isn't colour just a property?...Is there a thing as the colour force...you know like anyother force? If so, then could you pls send me an equation describing the colour force...gravity has an equation, electromagnetism has an equation and every other field forces have equations so the colour force must have one too right? thanx in advance – alienare 4422 Sep 29 '20 at 23:48
• Yes there is a colour force. Colour is not just a label. Red attracts anti-red in the same way that positive (EM) charge attracts negative charge. Complicated because there are 3 colours and EM only has one 'charge', and because it's so much stronger. So QCD has an equation for the Lagrangian (see for example en.wikipedia.org/wiki/Quantum_chromodynamics) but, unlike QED, this cannot be carried over to the classical limit to get Coulomb's Law and such. You get confinement. Which nobody understands properly. – RogerJBarlow Sep 30 '20 at 8:59


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Electrical Potential

Homework Statement

Three equal point charges, each with a charge of 2.00 uC, are placed at the corners of an equilateral triangle whose sides have a length of 0.400 m. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Homework Equations

$$U_{tot} = \frac {Q^{2}} {4*pi*8.85*10^{-12}} * (\frac {A} {d1} + \frac {B} {d2} + \frac {C} {d3})$$

The Attempt at a Solution

$$U_{tot} = \frac {(2.00*10)^{-19}} {4*pi*8.85*10^{-12}} * (\frac {4} {0.400} + \frac {4} {0.400} + \frac {4} {0.400})$$

The online program says i'm off by an additive constant once again and my work seems right?

Think about assembling the charges one by one, bringing them each in from infinitely far away.

Think about assembling the charges one by one, bringing them each in from infinitely far away.
When my professor tells us to think about things, usually it involves something going to 0? Are the charges 0 as you bring them closer?

No. How much work does it take to bring in one charge from infinity if there are no other charges around?

No. How much work does it take to bring in one charge from infinity if there are no other charges around?
The work done would be U_a - U_b = -(U_b-U_a) = -ChangeU

If there are no other charges around, there is no potential which you would have to do work against to bring in the first charge. Now what about the second charge?

If there are no other charges around, there is no potential which you would have to do work against to bring in the first charge. Now what about the second charge?
The work done should be the amount of work it takes to bring in the second charge?

Calculate that work. Now you have two charges in a region. How much energy does it take to bring in the third charge? Then add.

Calculate that work. Now you have two charges in a region. How much energy does it take to bring in the third charge? Then add.
Is there an equation for work in this case? I thought my original equation was it?

Should it not be:

$$U_{tot} = 3*( (\frac {1} {4*pi*8.85*10^{12}}) * (\frac {2.00*10^-{19}} {0.400}) )$$

I know the amount of work is just the summation of amount of work to move one charge.

Hootenanny
Staff Emeritus
Gold Member
Should it not be:

$$U_{tot} = 3*( (\frac {1} {4*pi*8.85*10^{12}}) * (\frac {2.00*10^-{19}} {0.400}) )$$

I know the amount of work is just the summation of amount of work to move one charge.
No. As Robb suggested, think about assembling the array of charges. So, lets start with just one charge in free space; there is no external electric field and no other charges, so the work done to place the charge is zero. No consider bringing a second charge from d = $\infty$ to d = 0.4 How much work must you do on this second charge?

No. As Robb suggested, think about assembling the array of charges. So, lets start with just one charge in free space; there is no external electric field and no other charges, so the work done to place the charge is zero. No consider bringing a second charge from d = $\infty$ to d = 0.4 How much work must you do on this second charge?
Amount of work would be:

$$V = \frac {K*Q} {r}$$

Hootenanny
Staff Emeritus
Gold Member
Amount of work would be:

$$V = \frac {K*Q} {r}$$
Almost, but not quite. Note that electrical potential is not the same as electric potential energy.

Almost, but not quite. Note that electrical potential is not the same as electric potential energy.
$$U = \frac {1} {4*pi*E_{0}} * \frac {Q_{1}*Q_{2}} {r}$$

Hootenanny
Staff Emeritus
Gold Member
$$U = \frac {1} {4*pi*E_{0}} * \frac {Q_{1}*Q_{2}} {r}$$
Correct, and since Q1 = Q1 := Q, we can say that

$$U = k\frac{Q^2}{r}$$

So, now we have two charges in place and have done U joules of work. So how much work must we do on the third charge (Q) to bring it from d = $\infty$ to d = 0.4?

Correct, and since Q1 = Q1 := Q, we can say that

$$U = k\frac{Q^2}{r}$$

So, now we have two charges in place and have done U joules of work. So how much work must we do on the third charge (Q) to bring it from d = $\infty$ to d = 0.4?
$$U = k\frac{Q^3}{r}$$

Careful, your units aren't even correct here.
First, conceptually, will it take more work to bring in the third charge compared to the second, or less work?

Careful, your units aren't even correct here.
First, conceptually, will it take more work to bring in the third charge compared to the second, or less work?
You just said this equation was right, now you say it's incorrect.

Umm, Q*Q is not the same units as Q*Q*Q.

Umm, Q*Q is not the same units as Q*Q*Q.
Well I'm lost in what you're saying now.

You said to move 1 charge the potential is:

$$U = k\frac{Q^2}{r}$$

After that I have no idea what you're talking about anymore

$$Q^2$$is not equal to $$Q^3$$

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$$Q^2$$is not equal to $$q^3$$

sorry for some reason the latex is not editing correctly.
Okay fine, I understand by moving in another charge that it isn't multiplying by another charge, but I'm still not sure how much work it takes to add another? I thought you would just add the work of another charge as we found in the first charge, so it would be 2*U.

After the first two are in place, the third charge has to be brought into a region of space where there are other charges that produce a potential. In essence, we must move that charge up the potential hill, just as you had to for the second charge.
The first charge we get for "free" since there is no potential (hill) to push the charge up. The second charge takes some work, since the first charge produces a potential (hill). How much- > $$V= kQ/r$$, which is the potential produced by the first charge so the work is $$U = qV$$ where q is the charge you are bringing up the potential V. Now the third charge must be brought up a potential produced by the first two charges. So find the new potential, $$V_{new} = V1 + V2$$. Then the work for the thrid charge is $$U = q Vnew$$ where Vnew is the new potential created by the first two charges.
I hope that makes some sense, it is friday and my brain is fried!:surprised

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After the first two are in place, the third charge has to be brought into a region of space where there are other charges that produce a potential. In essence, we must move that charge up the potential hill, just as you had to for the second charge.
The first charge we get for "free" since there is no potential (hill) to push the charge up. The second charge takes some work, since the first charge produces a potential (hill). How much- > $$V= kQ/r$$, which is the potential produced by the first charge so the work is $$U = qV$$ where q is the charge you are bringing up the potential V. Now the third charge must be brought up a potential produced by the first two charges. So find the new potential, $$V = V1 + V2$$. Then the work for the thrid charge is $$U = q Vnew$$ where Vnew is the new potential created by the first two charges.
I hopw that makes some sense, it is friday and my brain is fried!:surprised
Yes it is friday, and my brain is fried as well. So let me get this right:

First charge required 0 amounts of work because there was no potential "hill" to push up. So the potential energy of the system is currently Zero

Second charge requires work because there is a potential "hill" to be pushed up, since charge 1 has a potential to produce.
Charge 1 produces $$V= kQ/r$$

So the amount of work for Charge 2 => $$U = qV$$

Third charge requires work too because now there is a potential "hill" from both Charge1 + Charge2 which is $$U = qVnew$$

So is $$Vnew => V= kQ/r$$

and

$$U = q*(kQ/r)$$

Almost correct, except that Vnew is due to both charges, so you add up the potential from each. V1+V2


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# Is this set open? From topology without tears book

This question is from "topology without tears" book, page 30. Example 1.2.3

$\{1\}$ has as complement $\mathbb{N}\setminus \{1\}$ , so it is not finite . ($\mathbb{N}$ is set of natural numbers with the cofinite topology) Then the book concludes that $\{1\}$ is not open!?

My question is : where this conclusion comes from? Does it mean that if complement of a set is infinite then that set is not open?

• well, what topology has $\mathbb{N}$ equipped on it? – Riccardo Oct 31 '15 at 21:35
• For example, if $\mathbb{N}$ has the COFINITE topology on it (which seems the case) then by definition of cofinite topology, a subset whose complementary is NOT finite, then it is NOT open. But still, it depends on your hypothesis – Riccardo Oct 31 '15 at 21:37


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# 2022-21 A determinant of greatest common divisors

Let $$\varphi(x)$$ be the Euler’s totient function. Let $$S = \{a_1,\dots, a_n\}$$ be a set of positive integers such that for any $$a_i$$, all of its positive divisors are also in $$S$$. Let $$A$$ be the matrix with entries $$A_{i,j} = gcd(a_i,a_j)$$ being the greatest common divisors of $$a_i$$ and $$a_j$$. Prove that $$\det(A) = \prod_{i=1}^{n} \varphi(a_i)$$.

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