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I'm trying to understand this paper on Maximum Entropy by Jaynes, and am stuck on something which should be rather simple.
We're attempting to maximize the entropy $-\sum_i p_i \ln(p_i)$ subject to the constraints $\langle f \rangle = \sum_i p_i f_i$ and $\sum_i p_i = 1$. Using Lagrange multipliers, we have:
$$L = -\su... |
I will give an answer to my own question by showing the inverse case (as claimed by user my2cts): Given that there is a 4-vector $j$ that satisfies $\partial_{\mu} j^{\mu}$, one can show that the integral of the component of $j$, which is perpendicular to a spacelike 3-dimensional hypersurface, performed over this surf... |
The Birkhoff–von Neumann theorem states that a doubly stochastic matrix (a matrix with non-negative entries in which rows and columns sum to 1) can be written as a convex combination of permutation matrices (0/1 matrices which contain precisely one 1 in each row and column). This immediately implies your result.
If you... |
What is Fluid Flow?
Fluid Flow is a part of fluid mechanics and deals with fluid dynamics. Fluids such as gases and liquids in motion are called fluid flow. It involves the motion of a fluid subjected to unbalanced forces. This motion continues as long as unbalanced forces are applied.
For example, if you are pouring w... |
The following is an elementary question about circuit complexity. It is different from the kind of thing I have seen discussed, so I would be interested in any work that has been done on this kind of question. My apologies if this is easy or well known. By a Boolean function, I mean a function from $2^n\rightarrow2^m$ ... |
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Sat, 08 Jun 2019
I have pondered category theory periodically for the past 35 years, but not often enough to really stay comfortable with it. Today I was pondering again. I wanted to prove that !!1×A \cong A!! and I was having trouble. I eventually realized my difficulty: my brain ... |
Summary
As orthocresol correctly notes, the presence of this kink is simply a feature of the $x$-shifted hyperbola that is the solution to the second-order kinetics problem, and the presence/characteristics of the kink will vary depending on the particular parameters of the problem.
However, this answer aims to demonst... |
In the paper ON A PAINLEVÉ-TYPE BOUNDARY-VALUE PROBLEM, the authors consider the BVP given by the ODE $$y''=y^2-x \tag{1} $$ with the boundary conditions $$\begin{align} y(0)&=0, \tag{2a} \\ y(x)& \sim \sqrt{x} \text{ as }x \to \infty \tag{2b}.\end{align}$$ They do so by studying the 1-parametric family of solutions $y... |
A particle is submitted to a time dependent force $$F(x,t)=\dfrac{k}{x^2}e^{-t/\tau}$$
Which is the Lagrangian of the particle?
I think that the force is derived from the potential $V$ and this potential has not explicit dependence of $\dot x$. So i can write
$$ \dfrac{d}{dt}\dfrac{\partial \mathcal L}{\partial \dot x}... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
Existence of stable and unstable periodic solutions for semilinear parabolic problems
1.
School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia
2.
Department of Mathematics, Yokohama National University, 156 Tokiwadai Hodogaya-ku - Yokohama
$\frac{\partial u}{\partial t} - \Delta u = g(x,u) ... |
Why and how is the Jordan Canonical form of a matrix in $M_3(\mathbb C)$ fully determined by its characteristic and minimal polynomials? And why does it fail for $n >3$?
Thanks.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only ta... |
Kumaran, V (1998)
Microscopic analysis of the coarsening of an interface in the spinodal decomposition of a binary fluid. In: Journal of Chemical Physics, 109 (8). pp. 3240-3244.
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Microscopic_analysis-61.pdf
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Abstract
The coarsening of a random int... |
Wu Ki Tung's writing is not too precise mathematically, it is true. "Group Theory in Physics" is the title, and the level of mathematical rigor is not the highest, thus this sloppy text, actually sloppy two-letter word: "
Since the basis elements of the Lie algebra are generators of infinitesimal rotations, it is quite... |
Let $x_1, x_2... x_n$ be a random sample from a distribution with pdf:
$$f(x;\mu,\sigma)=\frac1{\sigma}\exp\left({-\frac{x-\mu}{\sigma}}\right)\,,-\infty<\mu<\infty;\, \sigma>0;\, x\ge\mu$$
How do I find the MLE for the parameters if both parameters are unknown?
I tried using the usual MLE with likelihood function:
$$L... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
Spherical coordinates
Spherical coordinates can be a little challenging to understand at first. Spherical coordinates determine the position of a point in three-dimensional space based on the distance $\rho$ from the origin and two angles $\theta$ and $\phi$. If one is familiar with polar coordinates, then the angle $\... |
Is my logic right?
Suppose there is a particle $p$ that can either decay into $ \{$a spin-1 and a spin-0 particle$\}$ or two spin-0 particles, then the lowest possible spin of $p$ is 2. This is because we need the spin to be even and large enough to accommodate the spin-1 product.
ADDED: $p$ is such that $p\to \pi^-+\r... |
With the set of parameters available to you, you cannot do this.
If you have the
actual track instead of the desired track, you will be able to calculate the wind.
The simplest way to do this is using vector math.
There are three vectors to consider:
ground speed vector $\vec{V_{gs}}$ air speed vector $\vec{V_{as}} $ w... |
I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$.
For literature, I'm referring mostly to the original Schoof (1985) and Gregg Musiker ("Schoofs Algorithm for Counting Points on $E(F_q)$... |
Given $A_n\rightarrow \infty$ almost surely (a.s). Show that $\forall\ N > 0$, $P\left\{A_n<N\ \text{infinitely often}\right\} = 0$.
My thought: By the sake of contradiction, assume there exists an $\infty> N >0$ such that $P\left\{A_n<N\ \text{infinitely often}\right\} > 0$. This is equivalent to: $\lim_{n\rightarrow ... |
I am finding it very difficult to find resources giving a overview of techniques that can be used to approximate a closed curve. For example, the wikipedia article on curve fitting has further links to linear/polynomial regression techniques that I am familiar with but their section for closed curve fitting is very bri... |
This question arises when looking at a certain constant associated to (a certain Banach algebra built out of) a given compact group, and specializing to the case of finite groups, in order to try and do calculations for toy examples. It feels like the answer should be (more) obvious to those who play around with finite... |
Adamson, P. and Ader, C. and Andrews, M. and Anfimov, N. and Anghel, I. and Arms, K. and Arrieta-Diaz, E. and Aurisano, A. and Ayres, D. and Backhouse, C. and Baird, M. and Bambah, B. A. and Bays, K. and Bernstein, R. and Betancourt, M. and Bhatnagar, V. and Bhuyan, B. and Bian, J. and Biery, K. and Bocean, V. and Boge... |
Definition:Isomorphism (Abstract Algebra)/Group Isomorphism Contents Definition
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a (group) homomorphism.
If $G$ is isomorphic to $H$, then the notation $G \cong H$ can be used (although notation varies). Also known as Isomorphism as defined ... |
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Sat, 04 Jan 2014
There is a famous mistake of Augustin-Louis Cauchy, in which he is supposed to have "proved" a theorem that is false. I have seen this cited many times, often in very serious scholarly literature, and as often as not Cauchy's purported error is completely misunders... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
Difference between revisions of "Meissel-Lehmer method"
(3 intermediate revisions by 3 users not shown)
(No difference)
Latest revision as of 23:39, 28 February 2012
The
Meissel-Lehmer method is a combinatorial method for computing [math]\pi(x)[/math] in time/space [math]x^{2/3+o(1)}[/math]. It is analysed at LMO???
De... |
At one time, calories in foods were measured with a bomb calorimeter. A weighed amount of the food would be placed in the calorimeter and the system was then sealed and filled with oxygen. An electric spark ignited the food-oxygen mixture. The amount of heat released when the food burned would given an idea of the food... |
I'm told to use Gauss's Theorem to compute the flux of a field $\vec F = <x,y^2,y+z>$ along the boundary of the cylindrical solid $x^2+y^2 \le 4$ below $z=8$ and above $z=x$.
I know by Gauss's Theorem that:
Net Flux = $\iint_{\partial D} \vec F \cdot \vec ndS = \iiint_D \nabla \cdot \vec FdV$
This computation is pretty... |
The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the "
critical value." If the ... |
One way you can try to see it is through an analogy with the Laplace transform...
$$e^{-at}\stackrel{\mathcal{L}}\longleftrightarrow \frac{1}{s+a}$$
Where $a\in\mathbb{C}$. If $\Re\{a\}>0$, the exponential decays and the system is stable. If $\Re\{a\}=0$, there is no damping and the system is marginally stable. $\Re\{a... |
From Wikipedia's Multiple Comparison
For hypothesis testing, the problem of multiple comparisons (also known as the multiple testing problem) results from the increase in type I error that occurs when statistical tests are used repeatedly. If
n independent comparisonsare performed, the experiment-wide significance leve... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
I found the following exercise in Introduction to Metric and Topological Spaces by Sutherland (Chapter 10 Question 20).
Prove that the topology on a space X is discrete iff the diagonal $\Delta=\{ (x,x) \mid x\in X\}$ is open in the topological product $X \times X$.
I believe I could prove in the $implies$ direction. I... |
Like JDługosz wrote, what will cause problems in the scenario you describe isn't so much your orbit as the fact that you are
within the gas giant's atmosphere.
I'm going to use Jupiter here to have some specific gas giant to use for examples. Feel free to look up the relevant data for any other gas giant, or come up wi... |
Let M be a deterministic Turing machine wich has the properties:
1) $\forall x,y \in \Sigma^* : t_M(xy) \ge t_M(x) + t_M(y)$
2) $\forall a \in \Sigma: t_M(a) \ge 1$ (Also 2) should be obvious for every DTM).
Then it follows that for all $x \in \Sigma^* : t_M(x) \ge |x| $. The graph $G_M$ induced by the transition funct... |
Tel No. +86-21-58386189
· A) Calculate the magnetic field within a wire carrying a current I uniformly distributed throughout the wire. B) A circular hole is now drilled through the wire, offset from the wire's center.
14 The Magnetic Field in Various Situations. 14–1 The vector potential. ... A rotating charged cylind... |
I am told that the MV CLT can be proved using the Cramér–Wold device.
The theorem is as follows (from Flury's "A First Course in Multivariate Statistics") Suppose $\bf{X}_1, \bf{X}_2, \ldots$, $\bf{X}_n$ are independent, identically, distributed, p-variate random vector, with mean vectors $\bf{µ}=E[\bf{X}_i]$ and covar... |
Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules:
$$\begin{array}{rcl}i:M&\longrightarrow&S\otimes_RM\\[.05in]m&\longmapsto&1\otimes m\end{array}$$
where the $S$-module $S\otimes_RM$ obtained by extension of scalars is considered an $R$-module via restri... |
Fisher information for sample $x$ in experiment $(\Omega, \mathcal{F}, P_\theta)$ is defined as $$Var \left[\nabla_{\theta}\ell(\theta, x) \right] = \mathbb{E}\left[[\nabla_{\theta} \ell(\theta, x)] [\nabla_{\theta}\ell(\theta, x)]^T\right] $$ where $\ell(\theta, x) = \log(f(x|\theta)$.
I do not understand how this def... |
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Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.
bidibangboom Posts: 34 Joined: May 10th, 2019, 6:38 pm
triple poster
Code: Select all
#C A... |
I need a data structure for storing a number $n$ of elements, each of whom is associated with some different time $t_i$. $n$ varies and while it has a theoretical upper limit, this is many orders of magnitude larger than what is typically used.
Through my application I can ensure that:
Inserted elements are always newe... |
Vanishing points are an important concept in 3D vision. Many papers related to them are using a notation called Gaussian sphere representation, which I find hard to understand at the beginning. This document will summarize what vanishing points (and their “Gaussian sphere representation”) are, how to represent them, wh... |
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hep-ph
Change to browse by: Bookmark(what is this?) General Relativity and Quantum Cosmology Title: Conical singularities and the Vainshtein screening in full GLPV theories
(Submitted on 21 Dec 2015 (v1), last revised 2 Mar 2016 (this version, v2))
Abstract: In Gleyzes-Langlois-Piazza-Vernizzi (... |
Pitch and Frequency
Sound is vibration transmitted through a medium, i.e. a solid, liquid or gas. A high pitch sound corresponds to a high frequency sound wave and a low pitch sound corresponds to a low frequency sound wave. The frequency is most often measured in
hertz(Hz). One hertz means that an event repeats once p... |
Research Open Access Published: Perturbational blowup solutions to the compressible Euler equations with damping SpringerPlus volume 5, Article number: 196 (2016) Article metrics
472 Accesses
Abstract Background
The
N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental eq... |
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Fri, 26 Apr 2019
What is the shed in “watershed”? Is it a garden shed? No.
I guessed that it meant a piece of land that sheds water into some stream or river. Wrong!
The Big Dictionary says that this shed is:
This meaning of “shed” fell out of use after the end of ... |
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Wed, 18 Sep 2019
Suppose you have a bottle that contains !!N!! whole pills. Each day you select a pill at random from the bottle. If it is a whole pill you eat half and put the other half back in the bottle. If it is a half pill, you eat it. How many half-pills can you expect to ha... |
Some mathematical elements change their style depending on the context, whether they are in line with the text or in an equation-type environment. This article explains how to manually adjust the display style.
Let's see an example
Depending on the value of $x$ the equation \( f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \) m... |
I am attempting to derive equations 2 and 6 from Xiao et al. paper "Valley contrasting physics in graphene" (Link to paper). The Hamiltonian for graphene with a staggered sublattice potential (in other words, a potential energy difference between the A and B sublattices) is given by: $$H = \frac{\sqrt{3}}{2}at(q_x\tau_... |
The argument that the paper seems to be making appears strange to me.
According to the paper, the goal of CV is to estimate $\alpha_2$, the expected predictive performance of the model on new data, given that the model was trained on the observed dataset $S$. When we conduct $k$-fold CV, we obtain an estimate $\hat A$ ... |
There is (at least) one way to prove unambiguity of a grammar $G = (N,T,\delta,S)$ for language $L$. It consists of two steps:
Prove $L \subseteq \mathcal{L}(G)$. Prove $[z^n]S_G(z) = |L_n|$.
The first step is pretty clear: show that the grammar generates (at least) the words you want, that is correctness.
The second s... |
Taking the derivative of $\ln(y) = b_0 + b_1 \cdot x$ with respect to $x$, you get
$$\frac{1}{y} \cdot \frac{dy}{dx}=b_1.$$ You can think of $$\frac{dy}{dx} \approx \frac{\Delta y}{\Delta x},$$ the change in $y$ for a small change in $x$. This means that
$$b_1=\frac{\Delta y}{y} \cdot \frac{1}{\Delta x}=\frac{\frac{\De... |
How would I solve the following.
An algorithm that is $O(n^2)$ takes 10 seconds to execute on a particular computer when n=100, how long would you expect to take it when n=500?
Can anyone help me answer dis.
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of com... |
I was playing with the idea of performing integration on a sound signal, since I've never heard of where it'd be used.
Is it used somewhere?
Here's something:
http://pcfarina.eng.unipr.it/Differentiation-Integration.htm
Signal Processing Stack Exchange is a question and answer site for practitioners of the art and scie... |
Edit: So, my original question (stated below) was to find an error in my "proof" that immediate parabolic basins for rational maps are always simply connected. Since I have not received any answers as of yet I would ask alternatively if someone could point out an explicit example of a rational map with a parabolic fixe... |
I'm trying to calculate the momentum distribution of a 1D system of non-interacting identical fermions in a harmonic trap.
Given Feynman's answer (from his Statistical Mechanics book) for the position density matrix of a single trapped particle at $T>0$, $ \rho_1 (x, x'; \beta) = \sqrt{\cfrac{m \omega}{2 \pi \hbar \sin... |
A model for this situation is to put 61000 ($n$) balls into an urn, of which 23000 ($n_1$) are labeled "A". 15000 ($k$) of these are drawn randomly without replacement. Of these, $m$ are found to be labeled "A". What is the chance that $m \ge 10000$?
The total number of possible samples equals the number of $k$-element... |
W.I.P.: Work in progress
Following p. 370 of Cramer's 1946
Mathematical Methods of Statistics, define $$\Xi_n = n(1 - \Phi(Z_n)) \,. $$ Here $\Phi$ is the cumulative distribution function of the standard normal distribution, $\mathscr{N}(0,1)$. As a consequence of its definition, we are guaranteed that $0\le \Xi_n \le ... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
It looks like you're new here. If you want to get involved, click one of these buttons!
Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
CHAPTER 21-SURFACE AREA, VOLUME AND CAPACITY Question 1.
Find the volume and the total surface area of a cuboid, whose:
(i) Length = 15cm, breadth = 10cm and height = 8cm.
Solution:-
Volume of a cuboid \(=\text {Length} \times \text { Breadth } \times \text { Height }=15 \times 10 \times 8=1200 \mathrm{cm}^{3}\)
Total ... |
Definition:Real Interval/Unit Interval/Closed Definition
The closed interval from $0$ to $1$ is denoted $\mathbb I$ (or a variant) by some authors:
$\mathbb I := \left [{0 \,.\,.\, 1} \right] = \left\{{x \in \R: 0 \le x \le 1}\right\}$
This is often referred to as the
closed unit interval. Also denoted as
Sources which... |
Definition:Semigroup Endomorphism Definition
Let $\left({S, \circ}\right)$ be a semigroups.
Let $\phi: S \to S$ be a (semigroup) homomorphism from $S$ to itself.
Then $\phi$ is a semigroup endomorphism. Also see
The word
endomorphism derives from the Greek morphe ( ) meaning μορφή formor structure, with the prefix endo... |
Knowledge of the specific fact that $(\sin x)' = \cos x$ actually predates the general knowledge of calculus and derivatives. It was known in the following form: that for very small $\Delta x$, when you increase $x$ to $x + \Delta x$, the increase in value of the sine, from $\sin x$ to $\sin (x + \Delta x)$, is proport... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
I am working on "Elementary Number Theory" By Underwood Dudley and this is problem 13 in Section 4.
The question is "What can the last digit of a fourth power be?" I got the correct answer but I'm wondering if there is another more elegant way to do it. I'm also wondering if my argument is solid or if I just got the ri... |
As with the review of Derivatives, it would be challenging to include a full review of Integrals. In this review, we try to include the most common integrals and rules used in STAT 414. There are many helpful websites as texts out there to help you review. We have provided links to Khan Academy for you to take a look a... |
Strategy
I would like to apply rational decision theory to the analysis, because that is one well-established way to attain rigor in solving a statistical decision problem. In trying to do so, one difficulty emerges as special: the alteration of SB’s consciousness.
Rational decision theory has no mechanism to handle al... |
Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction
Department of Mathematics, IME-USP, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil
$δ^{\prime}$
$i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log|}}u|2 = 0,(x,t) \in \math... |
There is a language: $L=\{w: w\in \{a,b\}^*, |w|_a\ \equiv |w|_b \equiv 0$ $(mod$ $5) \}$. My idea for DFA is - we count number of $a$$\pmod{5}$ and separately we count number of $b$$\pmod{5}$. So we end up with $5*5=25$ states, because each state have to keep track for both number of $a$ and $b$$\pmod{5}$. Is there an... |
I'm currently doing an exercise and faced with the following question:
We have the true form: $y_i=\beta_0 +\beta_1 d_i +u_i $
Where $d_i$ is a dummy variable. We have measured $d_i$ with measurement error such that 10% of those for whom $d_i=1$ have been recorded to have $d_i=0$ and similarly 10% of those for whom $d_... |
This work is based on Zengyi Dou's MS thesis. The oil well placement problem
The oil well placement problem is vital part of secondary oil production. Since the calculation of the net present value (NPV) of an investment depends on the solution of expensive partial differential equations that require tremendous computa... |
A message to the aliens, part 23/23 (wat)
Earlier articles:IntroductionCommon featuresPage 1 (numerals)Page 2 (arithmetic)Page 3 (exponents)Page 4 (algebra)Page 5 (geometry)Page 6 (chemistry)Page 7 (mass)Page 8 (time and space)Page 9 (physical units)Page 10 (temperature)Page 11 (solar system)Page 12 (Earth-Moon system)... |
This is a question about generalizing trace defined for positive operators to trace class operators, in the terminology below, in the context of functional analysis.
Let $T \in B(H)$ (i.e. $T$ is a bounded linear operator on a Hilbert space $H$). Say $T$ is positive (denoted $T \geq 0$) if $T$ is self-adjoint and $\lan... |
I have seen that if $G$ is a finite group and $H$ is a proper subgroup of $G$ with finite index then $ G \neq \bigcup\limits_{g \in G} gHg^{-1}$. Does this remain true for the infinite case also?
Not in general. Every matrix in $\text{GL}_2(\mathbf C)$ is conjugate to an invertible upper triangular matrix (use eigenvec... |
My book states that if we perturb a given Hamiltonian for the Schrödinger Equation
$$ H = \frac{p^2}{2m} +V(x) $$
to
$$ H' = \frac{p^2}{2m} + V(x) + \frac{\lambda p}{m} $$
then we can rewrite the perturbed Hamiltonian in the form
$$ H' = \frac{(p+\lambda)^2}{2m} + V(x) - \frac{\lambda^2}{2m} = \frac{p'^2}{2m} + V(x) - ... |
Permanent of an $m \times n$-matrix $A = \left\Vert a_{ij} \right\Vert$
The function $$ \mathrm{per}(A) = \sum_\sigma a_{1\sigma(1)}\cdots a_{m\sigma(m)} $$
where $a_{ij}$ are elements from a commutative ring and summation is over all one-to-one mappings $\sigma$ from $\{1,\ldots,m\}$ into $\{1,\ldots,n\}$. If $m=n$, t... |
Here is how I like to understand the Weil pairing.
The dual of an abelian variety $A$ is the scheme $\hat{A} = \mathrm{Hom}(A, B\mathbf{G}_m)$. Here $B\mathbf{G}_m$ is the stack of line bundles and $\mathrm{Hom}$ refers to homomorphisms of group stacks. One therefore has a perfect pairing
$A \times \hat{A} \rightarrow ... |
Question in brief
Let $a$ and $b$ be unit vectors in $\mathbb{R}^d$. Let $f$ be the $1-step$ transition function of a random walk on the $d$ dimensional unit sphere. I am interested in evaluating $\mathrm{E}[\exp(a^T f(b))]$ and one of things I want to know is any transition function/distribution exist where this quant... |
From Ravenel's article "Localization and Periodicity in Homotopy Theory":
Two spectra $E$ and $F$ are said to be
Bousfield equivalentwhen they give the same localization functor, or equivalently when $E_\ast (X)=0$ iff $F_\ast (X)=0$. The equivalence class of $E$ is denoted by $\langle E \rangle$. There is a partial or... |
I am reading the paper by Griffin and Brown (2010) where at one step in their MCMC procedure they need to sample from the following conditional posterior:
$$ p(\lambda|\gamma, \Psi)\propto \pi(\lambda)\frac{1}{(2\gamma^2)^{p\lambda}(\Gamma(\lambda))^p}\left(\prod_{i=1}^p\Psi_i\right)^\lambda $$
They say that $\lambda$
... |
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Tue, 08 Aug 2017
I should have written about this sooner, by now it has been so long that I have forgotten most of the details.
I first encountered Paul Erdős in the middle 1980s at a talk by János Pach about almost-universal graphs. Consider graphs with a countably infinite set of... |
I have heard vague information about point groups and symmetry before but I am unsure how they actually help me predicting physical properties of molecules. How do I assign point groups and what good does it do me?
Please answer at a low level.
Chemistry Stack Exchange is a question and answer site for scientists, acad... |
Let consider a steady state CSTR in which occurs the following reaction:
$$\ce{aA + bB \leftrightarrow cC}$$
Notations:
$F_i$ is the molar flow of $i$, $r$ is the rate of reaction, $V$ is the volume, $X$ is the conversion at the end of the reactor, $\nu_i$ is the stoichiometric coefficient of $i$ and $\xi$ is the exten... |
I'm told that this is true, but I can't imagine why. It seems like the fact that there is less air would make the engines less efficient... But that probably just shows how little I know about jet engines.
For a quick explanation, you need to know that
Thrust is the difference between the entry impulse of the air enter... |
When I went for interview at Imperial College, my interviewer (Dr. P. Kelly) asked me quite a nice question, which (for some unearthly reason) I decided to follow up on afterwards.
The question was a simple one: given a string (e.g.) "examplesgnome" consisting of two substrings (in this case, "examples" and "gnome"), h... |
Math/Display
< Math >
Contents Display Math
The famous result (once more) is given by \startformula c^2 = a^2 + b^2. \stopformula
This, when typeset, produces the following:
Numbering Formulae
The famous result (once more) is given by \placeformula \startformula c^2 = a^2 + b^2. \stopformula
This, when typeset, produce... |
Although András' comment already answers the question, I think it is worthwile to give a few more details explicitely here, in order to point out that the analyticity is in fact a consequence of the
resolvent identity only and has nothing to do with the operator whose resolvent we consider:
Let $X$ denote a complex Ban... |
I'll try with some calculations : please, check it and the formulae used ...
A solid ball with a mass $m$ of $1$ kg falls (with the usual approxiamtions : no drag, etc.) with an acceleration $a$ that is about $10 \ m/sec^2$.
This means that falling from a tower $80$ meters heigh, it will touch ground after $4$ sec, wit... |
Dynamics of { $\lambda tanh(e^z): \lambda \in R$\ ${ 0 }$ }
1.
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India, India
$\mathcal{M} = { f_{\lambda}(z) = \lambda f(z) : f(z) = \tanh(e^{z}) \mbox{for} z \in \mathbb{C} \mbox{and} \lambda \in \mathbb{R} \setminus \{ 0 \} }$
is st... |
Elementary discrete dynamical systems biology problems Problem 1
The mass of a fish is increasing by 100 grams per year.
Set up a dynamical system model that describes the evolution of the fish's mass. Be sure to define your notation, including the meaning of any variable for time. If the mass of a fish in year 0 is 45... |
I'm 27 and since I was about 15 I had the same doubt you do. Only a couple of years ago I realized why momentum is always conserved in a collision, whereas the same is not enforced for energy. (They must have told me this at some point -- or points --, but I guess sometimes I just don't pay much attention)
First of all... |
The method by which we calculate wavefunctionals is actually remarkably similar to the methods by which we calculate the position space representations of states in standard quantum mechanics. The details of this will depend heavily on your exact theory, but for simplicity I will assume we are working with a free scala... |
Despite my utmost effort the book has errors: typos, grammatical and substantive. This page is for reporting substantive errors not typos. I am however interested in knowing about all errors so please email me at rvc@petercorke.com and provide enough information to locate the error, and tell me what you think the error... |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\m... |
Let $\mu$ be a regular Borel Measure on $\mathbb{R}$ and suppose that there exists a $C > 0$ such that for every $x \in \mathbb{R}$ and $r > 0$\begin{align*}\mu((x - r,x+r)) \leq Cr.\end{align*}Let
m be the Lebesgue measure on $\mathbb{R}$. I have shown that $\mu << m$ ($\mu$ is absolutely continuous wrt m). I now want... |
Image: Probability distribution for the sum of two six-sided dice
A bar chart illustrating the probability distribution for a random variable $X$ that is given by the sum of the result of rolling two six-sided dice.
The probability distribution is \begin{gather*} P(x) = \begin{cases} \frac{1}{36} & \text{if $x \in \{2,... |
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