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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 $\lambda$
... |
Projection Operators
We have already seen that if $V$ is a finite-dimensional nonzero vector space over the complex numbers, then $V$ has at least one eigenvalue, however, if $V$ is a finite-dimensional nonzero vector space over the real numbers, then any linear operator $T \in \mathcal L (V)$ need not have an eigenval... |
Even before quantization, charged bosonic fields exhibit a certain "self-interaction". The body of this post demonstrates this fact, and the last paragraph asks the question.
Notation/ Lagrangians
Let me first provide the respective Lagrangians and elucidate the notation.
I am talking about complex scalar QED with the ... |
I’m not sure to whom the image or the idea is due. Please comment if you have information. (See comments below for current information.)
The rules will naturally generalize those in Connect-Four. Namely, starting from an empty board, the players take turns placing their coins into the $\omega\times 4$ grid. When a coin... |
I came upon the term "implied state price density" in a couple of papers. As far as I understand the concept one basically tries to extract the "pricing density" from the market data.
For the sake of simplicity we assume a constant interst rate $r$ and also don't make any assumptions on the model used to evolve $S_t$.
... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
Definition 1:- Let $(X,d)$ be a metric space. $A$ be a subset of $X$. Then $\text{Boundary}(A)=\{x\in X:$open ball centered at $x$ intersects both $A$ and $A^c\}$ Definition 2:- $\overline A=A\cup A'$, $A'$ is the set of all limit points of $A$.Using these two definitions
My aim is to prove
If $A$ is closed iff $A$ con... |
I have derived a likelihood function for $\theta$ as follows:
$$L(\theta)=(2\pi\theta)^{-n/2} \exp\left(\frac{ns}{2\theta}\right)$$
Where $\theta$ is an unknown parameter, $n$ is the sample size, and $s$ is a summary of the data. I now am trying to show that
$s$ is a sufficient statistic for $\theta$.
In Wikipedia the ... |
Towards Understanding the Origin of Cosmic-Ray Electrons
We present the precision measurement of the electron flux with a particular emphasis on the behavior at high energies. The measurement is based on 28.1 million electron events collected by AMS from May 19, 2011 to November 12, 2017. This corresponds to a factor o... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
revised
It is not always (perhaps not even "usually") the case that "most" pairs of vertices are "much" less than the diameter apart. Based on meager experience (see below), it seems common that "most" pairs are "almost" the diameter apart, but relatively few are exactly that far apart. Asking that the average distance... |
I think my question is similar to this one, but different in that I consider a set of realisations, not only one. Sorry if this question is really easy, I'm just not sure how to go rigorously about it.
Say I have $N$ realisations from a multivariate normal distribution $\mathcal{N}(\mu,\Sigma)$. Intuitively, I would ex... |
I need to calculate the length of a curve $y=2\sqrt{x}$ from $x=0$ to $x=1$.
So I started by taking $\int\limits^1_0 \sqrt{1+\frac{1}{x}}\, \text{d}x$, and then doing substitution: $\left[u = 1+\frac{1}{x}, \text{d}u = \frac{-1}{x^2}\text{d}x \Rightarrow -\text{d}u = \frac{1}{x^2}\text{d}x \right]^1_0 = -\int\limits^1_... |
The Characteristic Polynomial of a Matrix
Recall from The Eigenvalues of a Matrix page that if $A$ is an $n \times n$ matrix, then the number $\lambda$ is said to be an eigenvalue of $A$ if there exists a nonzero vector $v$ such that $Av = \lambda v$. Furthermore, the vectors $v$ for which $Av = \lambda v$ are called t... |
The Open Mapping Theorem
Recall from the Open and Closed Mappings page that if $X$ and $Y$ are topological spaces then a function $f : X \to Y$ is said to be an open mapping if for every open set $U$ in $X$ the image, $f(U)$ is an open set in $T(X)$.
We are now ready to prove the very important Open Mapping theorem.
Th... |
A homework problem I recall from functional analysis was to prove that the weak closure of the unit sphere, $S$, in an
infinite-dimensional real normed vector space is the unit ball, $B$.
Looking back at what I turned in, I argued as follows:
Note that $S$ would be weakly dense in $B$ if, for any nonempty (relatively) ... |
$$\int f(x) dx \sim \sum_i^k f(x) \Delta x$$
$$
\begin{align*}
\Delta x & = \int \dot{x}(t) dt \\
& \sim \dot{x}(t) \Delta t
\end{align*}
$$
These are all special cases of "Deterministic Quadratures", where the integration step size $\Delta x$ is non-random ("deterministic") and we are summing up a bunch of quadrilater... |
Difference between revisions of "Degree of irreducible representation divides order of group"
(4 intermediate revisions by the same user not shown) Line 18: Line 18:
* [[Degree of irreducible representation divides index of center]]
* [[Degree of irreducible representation divides index of center]]
−
* [[Degree of irre... |
Box Topological Products of Topological Spaces
Recall that if $\{ X_i \}_{i \in I}$ is an arbitrary collection of topological spaces and $\displaystyle{\prod_{i \in I} X_i}$ is the Cartesian product of these spaces then we can define the product topology on $\displaystyle{\prod_{i \in I} X_i}$ to be the topology $\tau$... |
Recall from Substitution Rule the method of integration by substitution. When evaluating an integral such as
\[\int_2^3 x(x^2 - 4)^5 dx,\]
we substitute \(u = g(x) = x^2 - 4\). Then \(du = 2x \, dx\) or \(x \, dx = \frac{1}{2} du\) and the limits change to \(u = g(2) = 2^2 - 4 = 0\) and \(u = g(3) = 9 - 4 = 5\). Thus t... |
Answer
The circumference of the earth is 24,800 miles
Work Step by Step
We can convert the angle to radians: $\theta = 7^{\circ}12'$ $\theta = (7+\frac{12}{60})^{\circ}$ $\theta = (7.2^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 0.1257~rad$ We can find the earth's radius: $S = \theta ~r$ $r = \frac{S}{\theta}$ $r = \frac{4... |
Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions.
Integrals of Expon... |
I'm dealing with Fourier series and I'm trying to figure out $\log(1+e^x) - \frac{x}{2}$ is even??? I've tried the $f(-x) = f(x)$ method but it doesn't give me the equality. But I've plotted it, and it is even? :S
$$\begin{align} \ln(1+e^x)-\frac x2 &= \ln(1+e^x)-\ln(e^{\frac x2}) \\ &= \ln\left((1+e^x)e^{\frac {-x}2}\... |
I am seeking a closed form for the function $$f(x)=\,_3F_2\left(\tfrac12,\tfrac12,\tfrac12;\tfrac32,\tfrac32;x\right)$$
I expect there to be one, because of this post and Wolfram. The Wolfram link produces closed forms involving $\mathrm{Li}_2$ for any value of $x$ that I've tried so far, so I can only assume that a ge... |
I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
Guy, Unsolved Problems In Number Theory, F21, attributes to Bohuslav Divis the conjecture that in each real quadratic field there is an irrational with all partial quotients 1 or 2; more general... |
Bernoulli Differential Equations
We are now going to look at a method for solving another class of differential equations. Let $p(x)$ and $g(x)$ be continuous on an interval of interest, and consider the following non-linear differential equation:(1)
If either $n = 0$ or $n = 1$, the differential equation above is line... |
After reading this blog post, I learned the BSD conjectural formula for the coefficient of the leading term $a_0$ of the L-function of an elliptic curve $E$, namely$$a_0 \stackrel{?}{=} \frac{\Omega_E\cdot Reg_E \cdot \prod_p c_p \cdot \#Sha(E/\mathbb{Q})}{(\# E_{tors}(\mathbb{Q}))^2}$$All the terms are defined, if peo... |
This is an old revision of the document! Introduction
In that example we investigate a AlGaAs/GaAs quantum well in quantum mechanical point of view. The simulation involves band profile calculation in the hetero-structure, and compare its influence with different QM solvers.
Band-structure calculation
The growth direct... |
I am tasked with finding the current
I through the following circuit at an array of frequencies. I have a solution however I am fairly new to AC systems and just want to make sure I am on the right track.
The values of \$ V_R, V_C, \$ and \$ V_L\$ were measured using an oscilloscope, and we can assume for the purpose o... |
I'm trying to understand Stern-Gerlach experiment on a computational level. Suppose we have a neutral particle with magnetic moment (e.g. a neutron), and apply an inhomogeneous magnetic field to it (let it change linearly with coordinate). As I understand, its Hamiltonian would look like:
$$\hat H=-\frac{\hbar^2}{2m}\n... |
Answer
$1$
Work Step by Step
$\tan45^{\circ}=\frac{\sin45^{\circ}}{\cos45^{\circ}}$ $\tan45^{\circ}=\frac{\frac{\sqrt2}{2}}{\frac{\sqrt2}{2}}$ $\tan45^{\circ}=1$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hours to send in a draft. An e... |
So, given some data,
Mathematica 10.2 can now attempt to figure out what probability distribution might have produced it. Cool! But suppose that, instead of having data, we have something that is in some ways better -- a formula. Let's call it $f$. We suspect -- perhaps because $f$ is non-negative over some domain and ... |
Let $(M,g)$ be a Riemannian manifold with Levi Civita connection $\nabla$. Then $\nabla$ satisfies a compatibility condition:
$(\nabla_ZX,Y)+(X,\nabla_ZY)=Z((X,Y))$ where $(\cdot,-)$ is a Hermitian pairing. In general, if we have a connection $\nabla$ on bundle $E$ (in our case $E=TM$) one can define the dual connectio... |
I am working with neural networks for a while right now and I've read that I could use a simple feedforward neural network to create a self driving car.
I was wondering how this possibly works because usually for any data that depends on time, I thought that I need to use a RNN.
I am working with Java so implementing a... |
The missing factor method is a particularly nice way to understand fraction division. It builds on what we know about multiplication and division, reinforcing that these operations have the same relationship whether the numbers are whole number, fractions, or anything else. It makes sense. But we’ve seen that it doesn’... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
Frequently we will want to estimate the empirical probability density function of real-world data and compare it to the theoretical density from one or more probability distributions. The following example shows the empirical and theoretical normal density for EUR/USD high-frequency tick data \(X\) (which has been tran... |
Focus Questions
The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions.
What is the unit circle and why is it important in trigonom... |
@Secret et al hows this for a video game? OE Cake! fluid dynamics simulator! have been looking for something like this for yrs! just discovered it wanna try it out! anyone heard of it? anyone else wanna do some serious research on it? think it could be used to experiment with solitons=D
OE-Cake, OE-CAKE! or OE Cake is ... |
Learning Objectives
Graph plane curves described by parametric equations by plotting points. Graph parametric equations.
It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at \(140\) feet per second and at an angle of app... |
Let me go a little further: I believe that there also exists a topologically mixing subshift on two symbols with no fully-supported invariant measure. I don't know of a reference for this result, but I think that a direct construction should be "not too hard", in the sense that a completely detailed proof would take up... |
Higher Order Homogenous Differential Equations - Constant Coefficients
Recall that if we have a second order linear homogenous differential equation with constant coefficients $a, b, c \in \mathbb{R}$, that is:(1)
We saw that the roots $r_1$ and $r_2$ of the characteristic equation $ar^2 + br + c = 0$ had importantance... |
Here is an archetypal vignette for Newtonian mechanics. Two compound atoms called $\mathbf{A}$ and $\mathbf{B}$ have an interaction with each other by swapping $\sf{X}$ another particle which is called the
exchange particle. The interaction is caused when $\mathbf{A}$ emits $\sf{X}$ at event $\mathbf{A}_{\it{i}}$ which... |
The cross product or vector product is a binary operation on two vectors in three-dimensional space (R3) and is denoted by the symbol
x. Two linearly independent vectors a and b, the cross product, a x b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them.
\[\LARGE A\tim... |
I am working on a physics reserach project for school and I have run into some troubles working Mathematica. I am a fairly inexperienced mathematica user so any help would very much be appreciated.
I need to find the roots of the transcendental equation $$\zeta_n \tan(\zeta) - \sqrt{R^2-\zeta^2_n}=0$$ and then collect ... |
Electronic Communications in Probability Electron. Commun. Probab. Volume 22 (2017), paper no. 2, 6 pp. A heat flow approach to the Godbillon-Vey class Abstract
We give a heat flow derivation for the Godbillon Vey class. In particular we prove that if $(M,g)$ is a compact Riemannian manifold with a codimension 1 foliat... |
How I can evaluate the limit superior of a sequence?
How I can evaluate the limit superior of a sequence? I dont found in the documentation something related to this tool.
EDIT: the limit superior of a sequence $(x_n)$ is defined as
$$\limsup x_n=\lim_{n\to\infty} \sup \{x_k:k\ge n\} =\inf\{\sup \{x_k:k\ge n\}: n\in \m... |
Observation of New Properties of Secondary Cosmic Rays Lithium, Beryllium and Boron
Lithium, beryllium, and boron nuclei in cosmic rays are thought to be produced by the collisions of nuclei with the interstellar medium. They are called secondary cosmic rays. Precise knowledge of their spectra in the GV-TV rigidity reg... |
Continuity of global attractors for a class of non local evolution equations
1.
Instituto de Matemática e Estatística-Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, CEP 05508-090, São Paulo-SP, Brazil
2.
Unidade Acadêmica de Matemática e Estatística UAME/CCT/UFCG, Avenida Aprígio Veloso, 882, Bair... |
Towards Understanding the Origin of Cosmic-Ray Positrons
Studies of light cosmic ray antimatter species, such as positrons, antiprotons, and antideuterons, are crucial for the understanding of new phenomena in the cosmos, since the yield of these particles from cosmic ray collisions is small. Our data published in 2013... |
@user193319 I believe the natural extension to multigraphs is just ensuring that $\#(u,v) = \#(\sigma(u),\sigma(v))$ where $\# : V \times V \rightarrow \mathbb{N}$ counts the number of edges between $u$ and $v$ (which would be zero).
I have this exercise: Consider the ring $R$ of polynomials in $n$ variables with integ... |
Precision Measurement of Cosmic-Ray Nitrogen and its Primary and Secondary Components
Nitrogen nuclei in cosmic rays are thought to be produced both in astrophysical sources, mostly via the CNO cycle [H. A. Bethe, Phys. Rev. 55, 434 (1939)], and by the collisions of heavier nuclei with the interstellar medium. Therefor... |
Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.
I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’... |
1. Lines (definitions)
Everyone knows what a line is, but providing a rigorous definition proves to be a challenge.
Definition: Line
A
line with slope \(m\) through a point \(P = (a,b)\) is the set of all points \((x,y)\) such that
\[\dfrac{y-b}{x-a}= m.\]
2. The Slope Intercept Form of the equation of a Line
Given a p... |
Frequently we will want to estimate the empirical probability density function of real-world data and compare it to the theoretical density from one or more probability distributions. The following example shows the empirical and theoretical normal density for EUR/USD high-frequency tick data \(X\) (which has been tran... |
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recog... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
The Trace of a Square Matrix
Before we look at what the trace of a matrix is, let's first define what the main diagonal of a square matrix is.
Definition: If $A$ is an square $n \times n$ matrix, then the Main Diagonal of $A$ consists of the entries $a_{11}, a_{22}, ..., a_{nn}$ (entries whose row number is the same as... |
Consider a repetitive chain of events noted by $\Psi = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2} , \sf{\Omega}_{3} \ \ldots \ \right)$ where $\sf{\Omega}_{1} = \sf{\Omega}_{2} = \sf{\Omega}_{3}$ etc. Earlier we gave an example of $\Psi$ as a simple movie loop called
The Almost-Dead March. That movie was boring so here i... |
I am attempting to calculate the functional derivative of a functional $$E[\rho] = \int G(\rho(\mathbf{r}),\nabla\rho(\mathbf{r}),\mathbf{r})d\mathbf{r},$$ where $$G(\rho(\mathbf{r}),\nabla\rho(\mathbf{r}),\mathbf{r})=\rho(\mathbf{r})^{4/3}\left(\alpha-\frac{(\nabla\rho(\mathbf{r})\cdot\nabla\rho(\mathbf{r}))^{3/4}}{13... |
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Now showing items 1-9 of 9
Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 ... |
New Reconstruction Method in the Electromagnetic Calorimeter (ECAL) Analysis
The key detector for measurements of electrons and positrons in AMS is the Electromagnetic Calorimeter, ECAL (see Figure 1). The ECAL consists of a multilayer sandwich of lead foils and ∼50,000 scintillating fibers with an active area of 648 ×... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
In this section, we show that every integer has a primitive root. To do this we need to introduce polynomial congruence.
Defintion: polynomial congruence
Let \(f(x)\) be a polynomial with integer coefficients. We say that an integer \(a\) is a root of \(f(x)\) modulo \(m\) if \(f(a)\equiv 0 (mod\ m)\).
Notice that \(x\... |
This is related to a question I answered earlier which raised a question in my mind.
My question is the following,
Suppose we have a vector space $\mathbb{V}$ with real coefficients.
Let $\textbf{T}$ be an operator on this space which has the following two properties:
$ \textbf{T}( \textit{u} + \textit{v})= \textbf{T}(... |
I am dealing with the decomposition of the representation $5\otimes5$ of $SU(5)$:
$$5\otimes5=15\oplus10 $$
demonstration:
$$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$
$$=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}\epsilon^{ijxyk}\epsilon_{xyklm}u^lv^m$$
where the term $\frac{1}{2}(u^iv^j+u^jv^i)$ has 1... |
The Class Equation for Groups Acting on a Set
Recall from The Orbit and Stabilizer of a Point in a Group Acting on a Set page that if $(G, \cdot)$ is a group acting on a (nonempty) set $A$, then for each $a \in A$ we defined the orbit of $a$ in $G$ to be the set:(1)
And for each $a \in A$ we defined the stabilizer of $... |
We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. Previously, der... |
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The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
Perhaps this is trivial but I would like to plot the following function:
\begin{equation} p(t)=e^{\left( -\frac{d}{1-c}\right)\left[ W_0\left[B(1+x/r)^{1/d}\right]-W_0[B] \right]} \end{equation} where $W_k$ is the Lambert-W function for the $k=0$ branch and \begin{equation} B=\frac{(1-c)r}{1-(1-c)r}e^{\frac{(1-c)r}{1-(... |
Mitra, J and Raychaudhuri, AK and Gayathri, N and Mukovskii, Ya M (2002)
Point-contact spectroscopy of single crystal $La_0_._75Sr_0_._2_5MnO_3$ and resistivity due to electron-phonon interaction. In: Physical Review B, 65 (14). pp. 140406-1.
PDF
Point-contact_spectroscopy.pdf
Restricted to Registered users only
Downlo... |
Is there a good characterization of the set $S$ of positive integers $n$ such that $\frac{1}{n}$ can be represented as a difference of Egyptian fractions with all denominators $< n$? For example, $44 \in S$ because $$ \dfrac{1}{44} = \left( \frac{1}{33} + \frac{1}{12}\right) - \frac{1}{11} $$
If I'm not mistaken, the f... |
Diagonal Matrices of Linear Operators Examples 2
Recall from the Diagonal Matrices of Linear Operators page that if $V$ is a finite-dimensional vector space and $T \in \mathcal L (V)$, then $T$ is said to be diagonalizable if there exists a basis $B_V$ such that $\mathcal M (T, B_V)$ is a diagonal matrix.
We saw that i... |
Dynamic programming algorithms are the bread and butter for structured prediction in NLP.
They are also quite a pain to debug, especially when implemented directly in your language of choice, and not in some high-level programming language for dynamic programming algorithms, such as Dyna.
In this post, I suggest a way,... |
Let's say we have two types of balls, black and white. There are $B$ black balls and $W$ white balls, s.t. $B + W = N$ where $N$ is the total number of balls.
We want to divide these $N$ balls evenly into $k$ groups, each with integer size $n = \frac{N}{k}$ balls per group. We further require that there are fewer than ... |
I have been asked this question by school kids, colleagues and family (usually less formally):
When ascending a flight of stairs, you exchange mechanical work to attain potential energy ($W_{ascend} = E_{pot} = m \cdot g \cdot h$).
However, when descending, you have to exert an equivalent force to stop yourself from ac... |
Arun from the National Public School inBangalore sent us this detailed solution:
It is clearly seen that the angle at $C$ is twice the angle at $D$.Here we have the fundamental theorem with regard to angles incircles -
" the angle subtended by an arc of a circle at the centre is doublethe angle subtended by it at any p... |
Inaccessible
Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy (although there are some weaker large cardinal notions, such as universe cardinals).
If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC, but this is not an equivalence, since the weaker notion of universe car... |
I am trying to estimate residential demand for electricity in a country where electricity is sold (to all households (HH)) at an increasing two-part tariff. By choosing marginal prices as my key independent variable (I am mostly interested in estimating the price elasticity of demand), I have come to understand that OL... |
№ 9
All Issues Volume 45, № 6, 1993
Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 731–743
This is a brief survey of the development and applications of the concept of the characteristic function for different classes of linear operators.
Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 744–752
The maximal commuting proper extensions of... |
In the beginning of chapter 3 on scattering theory in Weinberg's QFT book there is a use of the Cauchy residual theorem that I just cannot get.
First some notation, we are looking at states that are effectively non-interacting, and are considered to be a direct product of one-particle states described by their momenta ... |
To measure the escape velocity, if I use the equation -
$$\frac{-GMm}{r^2}=\frac{v.dv}{dr} $$
and I put my final distance to be $\infty$ , then I get the answer,
$$u = \sqrt\frac{2GM}{R} \;. \tag{1}$$ Which is quite obvious!
But, if I use the equations -
$$U_{\infty} - U_i = \frac{GMm}{R}$$
and
$$K_{\infty}+U_{\infty} ... |
The Feynman propagator for the free electron field is the Fourier transform w.r.t. $y$ of the time-ordered 2-point VEV $\left<0\right|\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]\left|0\right>$, taking $\hbar=c=1$,$$\mathcal{F}[\left<0\right|\mathcal{T}[\hat\psi(x)\hat\psi(x+y)]\left|0\right>](k)=
\frac{k\!\cdot\!\gamma+m_e}{... |
Note: According to OPs formulation in the bounty text this answer is aimed to give at least a glimpse of some aspects around the definition of set and set theories with focus on the axiom of choice. The top voted answers already contain the essential information.
On the definition of sets:
In modern times sets (and all... |
The XYZPipeJunction is removed using circular trig functions from Both.
Both = ContourPlot3D[ x^4 + y^4 + z^4 - (x^2 + y^2 + z^2)^2 + 3 (x^2 + y^2 + z^2) == 3, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]Dice = ContourPlot3D[ x^4 + y^4 + z^4 - (Cos[x] Cos[y] Cos[z])^2 + 3 (Cos[x] Cos[y] Cos[z]) == 3, {x, -2, 2}, {y, -2, 2}, {z,... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message stubner Joined: 14 Mar 2006 Posts: 7
Posted: Wed Apr 19, 2006 2:19 pm Post subject: absolute values Hi everybody,
it seems I ahven't used much absolute values lately since only yesterday I found that things like $|x|$ or $|o|$ look offbalance t... |
Orbital Graphs
Suppose that (as per usual) \Omega = \{1..n\} for some n \in \mathbb{N} and G \leq S_{\Omega}.
If we pick two distinct numbers \alpha, \beta \in \Omega, then the graph\Gamma_{(\alpha,\beta)} = \left<\Omega, \left(\alpha, \beta\right)^G\right> is the
Computing the orbital graph for a given pair (\alpha, \... |
trying to determine if the series is conditionally convergent or divergent. $$\sum_{n = 1}^\infty \frac{2^{n^{2}}}{n!}$$ with n! i tried the ratio test on the series $$\frac{2^{(n+1)^{2}}}{(n+1)!} * \frac{n!}{2^{n^{2}}} = \frac{2^{2n+1}}{(n+1)} $$ which is > 1 as $n\to \infty$ and is overall divergent ? not sure if I a... |
Can someone please advise how to compute the following (as my results go into thousands):
E.g. I have used
and the result (for T=12) = 580103.7261
Thanks
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Group Actions of a Group on a Set
Group Actions of a Group on a Set
Definition: Let $(G, \cdot)$ be a group and let $A$ be a (nonempty) set. A Left Group Action of the group $G$ on the set $A$ is a map $G \times A \to A$ denoted for all $g \in G$ and all $a \in A$ by $(g, a) \to ga$ that satisfies the following propert... |
Question
Four fair six-sided dice are rolled. The probability that the sum of the results being $22$ is $$\frac{X}{1296}.$$ What is the value of $X$?
My Approach
I simplified it to the equation of the form:
$x_{1}+x_{2}+x_{3}+x_{4}=22, 1\,\,\leq x_{i} \,\,\leq 6,\,\,1\,\,\leq i \,\,\leq 4 $
Solving this equation result... |
I'm trying to get the following system of equation to exact differential equation form:
$$ \left\{ \begin{array} \dot \dot x =y^2-x \\ \dot y = 2y \end{array} \right. $$
What I tried:
$$\frac{dx}{dy}=\frac{y^2-x}{2y} \quad \Rightarrow \quad 2ydx+(x-y^2)dy=0 $$
Next I tried to find an integration factor:
$$\frac{N_x-M_y... |
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Now showing items 1-9 of 9
Measurement of $J/\psi$ production as a function of event multiplicity in pp collisions at $\sqrt{s} = 13\,\mathrm{TeV}$ with ALICE
(Elsevier, 2017-11)
The availability at the LHC of the largest collision energy in pp collisions allows a significant advance in the measurement of $J/\ps... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
General Vector Notation
There are two common ways to denote a vector. The first way is with the use of an arrow such as $\vec{u}$. However, it is also common to use boldface such as $\mathbf{u}$. In the linear algebra section of MathOnline the former notation will be used much more frequently.
Component Form of a Vecto... |
I am studying entanglement entropy.
It's fullfilled for any local quantum system that the entanglement entropy of a region $A$ in a highly mixed state is extensvie,
$$ S_A \sim \frac{\text{Vol}(A)}{\epsilon^d} $$
where $\epsilon$ is the length between sites (or the UV cutoff for a regularized QFT). This is because
$$S_... |
Examples of Computing Jacobi Symbols
Recall from the Jacobi Symbols page that if $P, Q \in \mathbb{Z}$ and $Q$ has prime power factorization $Q = q_1^{e_1}q_2^{e_2}...q_k^{e_k}$ then the Jacobi symbol of $P$ over $Q$ is defined to be:(1)
where the terms in the product on the right are Legendre symbols. We proved some b... |
Is there an example of a commutative ring with an ideal that contains all the non-units?
I was trying to think of some subring of $\mathbb Q$, but I couldn't get it to work.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes ... |
For a basic intuition of the conditional probability formula, I always like using a two way table. Let's say there are 150 students in a yeargroup, of whom 80 are female and 70 male, each of whom must study exactly one language course. The two-way table of students taking different courses is:
| French German Italian |... |
Since $A$ is symmetric, there is an orthogonal matrix $D$ (i.e., $D^{-1} = D^T$) so that $A = D^T \Lambda D$, where $\Lambda$ is diagonal. Since
$$h^T Ah = A \Leftrightarrow (DhD^T)^T \Lambda (DhD^T) = \Lambda$$
Thus $$O(q) = D^T O_{\Lambda} D,$$
where $O_\Lambda = \{ h: h^T \Lambda h = \Lambda\}$. Thus to answer (1), ... |
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