-1
votes
1answer
17 views

Discrete Maths Relations on the set {1,2,3,4}

I just want to make sure that I am doing these correctly. Here is what I have: Reflexive, symmetric, antisymmetric and transitive: And i have - {(1,1) (2,2) (3,3) (4,4)}. not Reflexive, not ...
0
votes
0answers
9 views

Is there any other method to solve this Cauchy problem for wave equation in 3D?

I want to solve this Cauchy problem for wave equation in 3D: \begin{align*} &u_{tt}=4\Delta_3 u(x,t), \qquad x=(x_1,x_2,x_3)\in\mathbb{R}^3, \quad t>0,\\ &u(x,0)=(x_2x_3)^2, \quad ...
1
vote
2answers
12 views

Choosing co-efficients from a restricted set to ensure a vector is orthogonal to another

Let $\vec{u} = (p_1u_1,...,p_nu_n) \in \mathbb{R}^n$ where $p_i \in \{-1,1\}$. Let $\vec{v} \in \mathbb{R}^n$. Is there a systematic way to choose $p_i$ such that $\vec{u} \cdot \vec{v} = 0$, or to ...
0
votes
0answers
14 views

troubles integrating by parts with $\int \sqrt{x^2+y^2}D_x\phi(x,y)$

I want to show that $f(x)=\|x\|^a$ is weakly differentiable in $B_1(0)\subset\mathbb R^2$ iff $a>-1$. Therefore I want to show that for all $\phi\in C_0^\infty$ we have $$ (*)\quad\int \phi D_i ...
1
vote
2answers
19 views

Divergence change of variables (to polar)

I would wish to simplify this integral $$\oint_{\partial D}(f\nabla f)\cdot\hat{n}\,ds$$ in terms of a line integral of $g$ on $[0,2\pi]$ where $g(\theta)=f(e^{i\theta})$. Background info: ...
2
votes
0answers
20 views

Using Menger's theorem to prove max-flow min-cut theorem

Is it possible to use Menger's theorem to prove the max-flow min-cut theorem for rational capacities?
0
votes
1answer
39 views

I need to solve these inequalities [on hold]

$ x - y - z > 0$ $ 2y - x - z > 0$ $ 4z - x - y > 0$ Any POSITIVE values of x, y and z that satisfy these inequalities. Is it possible? If it is what are the values?
0
votes
2answers
20 views

Euler Equation with a certain type of solution

I have the Euler equation in the following form $$x^2h''(x)+xh'(x)=b^2h(x)$$ with the condition $h(a)=0$. The general solution to this equation is $$h(x)=c_1x^{b}+c_2x^{-b}$$ Now, my question ...
0
votes
0answers
5 views

Singular matrix with homogeneous Neumann boundary conditions?

I'm numerically solving a PDE on a rectangular domain using a 9-point central difference scheme and purely homogeneous Neumann boundary conditions. Should the resulting matrix be singular? I thought ...
1
vote
2answers
26 views

Conditional Probability Problem About a Quiz in Which Every Question Has 4 Answers

Imagine that a quiz has some questions and every question has 1 correct answer between 4 choices. The probability that the student knows the answer of each question is $\frac{2}{3}$. What's the ...
0
votes
1answer
15 views

An open continuous image of a Baire space is a Baire space proof question

Let $(X,T)$ and $(Y,T_1)$ be topological spaces and $f: (X,T) \rightarrow (Y,T_1)$ be a continuous open mapping. If $(X,T)$ is a Baire space, prove that $(Y,T_1)$ is a Baire space. I start with: ...
1
vote
1answer
27 views

Why all irreducible representations appeear in the regular representation?

Let $G$ be a finite group and $R$ the regular representation. That is, as a vector space $R = F(G)$ is the free vector space with basis $G$. If the basis is $\{e_g : g \in G\}$ the action is defined ...
0
votes
1answer
17 views

reference request: lie algebra-lie group

I am looking for a reference where I can find a (relatively) elementary and self contained proof of the fact that all real, finite dimensional Lie algebras are the Lie algebra of some Lie group. ...
1
vote
1answer
31 views

Partial Fractions with Quadratic Factor

I understand that if we have a quadratic factor such as in $\frac{8}{(x^2 + 1)(2x-3)} $ and we want to decompose, we should have a linear factor above $ x^2 +1$. Is the reason behind this ...
2
votes
1answer
31 views

Order of a number

Does there always exist some integer $a$ such that $a^{p-1} \equiv 1 \pmod{p}$ where $p$ is a prime and $p-1 = \text{ord}_{p}(a)$? I was wondering about this and it is basically saying, "can the ...
0
votes
0answers
19 views

How to solve system of higher order linear homogeneous recurrences/difference equations with constant coefficients.

I have not been able to find an answer to this question (maybe I am just bad at searching). For a (linear homogeneous) recurrence of order $k$ i.e $$x_n = a_{1}x_{n-1} + a_{2}x_{n-2} + \cdots + ...
1
vote
0answers
15 views

Rational Canonical Form for Fields of positive characteristic

Recently, I was working on the rational canonical forms of matrices and then one query popped up which i didn't manage to find in the literature (that I am aware of course). So, if $k$ is an infinite ...
0
votes
0answers
29 views

right-angled triangle problem

If the hypotenuse is 8 cm, one of the sides is X cm and the other 4 cm longer how do i find the two unknown sides? I started by applying the Pythagorean theorem like this $x^2+(4x)^2=8^2$ but i don't ...
-6
votes
2answers
81 views

How is this $\int dx=x+c$. [on hold]

$$\int dx=x+c$$ My teacher is keep telling that and not showing why is $x$ $$\int x^2dx=\frac{x^3}{3}+c$$ this is easy.
-2
votes
0answers
19 views

Please help me in this problem with some example. [on hold]

The closure of a subset of a metric space is the set of points whose distance from the set is ?
0
votes
1answer
17 views

Global Lipschitz implies bounded in coefficient

Consider $g:\mathbb{R}^2\to \mathbb{R}$ of the form $g(x,y)=p(x)q(y).$ Assume $g$ is uniformly Lipschitz in $x,y$ in the sense that there exists $K>0$ such that for any $(x_1,y_1),(x_2,y_2)\in ...
0
votes
0answers
6 views

Introductory text on partitions, matroids, geometric lattices

Can anyone recommend a text which explains matroids, lattices of subsets, and how they are related? Possibly motivated with examples from different applications or areas of math.
1
vote
1answer
28 views

Find the range of the function $f(x) = 4x + 8$ for the given domain $D = \{-5, -1, 0, 6, 10\}$

The question is to find the range of each function for the given domain $f(x)=4x+8$, $D=\{-5, -1, 0,6, 10\}$. Is the range just $R= \{-12,4,8,32,48\}$ or am I mistaken? Could you elaborate why my ...
0
votes
1answer
43 views

Question on limits

Let $$L=\lim_{x\to \infty} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}$$.where L is a finite real number. Then value of a,L for which this is true . Options were there answer is $a=2,L=1/64$. ...
1
vote
0answers
26 views

Proving a divisor is canonical

I am given hyperelliptical curve $C\dots y^2=f(x)$ of genus $2$ and divisor $D=(a,\sqrt{f(a)})+(a,-\sqrt{f(a)})$ which I must prove is canonical. I know $D$ is canonical if and only if $\deg D=2g-2=2$ ...
1
vote
0answers
30 views

Math notation to show two numbers in a range that added together get the max of the range [on hold]

I am completely new to math notations, it's been about 30 years since high school, and I am writing a research paper (completely on my own, not for a degree). I basically want to show that two real ...
0
votes
0answers
10 views

How to count all cycles (simple or not) in a directed complete graph?

I came up with an algorithm for counting cycles (simple or not) of length less or equal to n in a given directed complete graph Kn. I am looking for a more concise way of counting cycles but have not ...
0
votes
0answers
33 views

Help in a demonstration “left to the reader”

I'm reading Conway's complex analysis book by myself and on page 144 he state the following theorem: I need some help how to prove the part (b).
2
votes
1answer
38 views

If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges

I'm learning about Fourier series, specifically Cesàro summable sequences and series, and need help with the following problem: Show that if the series $\sum_{k=1}^{\infty} a_k$ is Cesàro ...
1
vote
1answer
43 views

Prove or disprove: $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(-\infty,\infty)$

I'm trying to find out if $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(\infty,\infty)$. Here's my attempt at an answer: If $ x\in(-\infty,0]:$ $$e^{-|x-n|}=e^{(-(x-n))}=e^{x-n}\le e^{-n}$$ ...
0
votes
0answers
20 views

Maxwell–Boltzmann distribution

Given the velocity dstribution for the velocity vector in natural units \begin{align} f(\vec{v}) = \left( \frac{m}{2 \pi T} \right)^{3/2} \cdot \exp\left({-\frac{m ~\vec{v} \cdot \vec{v}}{2 ...
0
votes
0answers
24 views

Cubic polynomial with 1 real root and 2 complex conjugated roots (real coefficients)

I am stuck on this problem about cubic polynomials. I rely on the Wikipedia page on the topic. Using wikipedia notations (chapter "General formula for roots") : For the case where $\Delta > 0$, ...
1
vote
0answers
16 views

Clamp angle between two vectors

I'm programming a snake-type game in 3 dimensions without grid constraints where the location of the new 'block' is determined by: [Normalized(lastBlockLocation - cursorLocation) * blockLength] + ...
0
votes
0answers
10 views

When is a stable domain in a minimal surface area minimizing?

A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...
-4
votes
0answers
21 views

Parallelogram law in Hilbertspaces [on hold]

I am having trouble solving the following exercise: (a) Let $(V, ||\cdot||)$ a normed vector space over the field $\mathbb R$. Show that, for all $x, y \in V$ the equation $$ ||x+y||^2 + ...
1
vote
0answers
9 views

While finding an optimal strategy for a mixed nash equilibrium, why do we not consider strategies which are never a best response?

"A strategy cannot be plausibly chosen by a rational player if and only if it is never a best response." I understand the logic behind neglecting the strategies that are strictly dominated. But why ...
0
votes
1answer
25 views

prove $P_{2n}(0)=(-1)^n\frac{(2n)!}{2^{2n}(n!)^2}$ of Legendre Polynomials.

Use: $P_n(x)=\frac{1}{2^n}\sum_{m=0}^M(-1)^m\frac{(2n-2m)!}{m!(n-m)!(n-2m)!}x^{n-2m}$ where M=n/2 if even, (n-1)/2 if n odd, to prove $P_{2n}(0)=(-1)^n\frac{(2n)!}{2^{2n}(n!)^2}$ and $P_{2n+1}(0)=0$. ...
-13
votes
0answers
38 views

Function help needed ASAP [on hold]

How can I find the value of (x) for which the function of a curve f(x) is not defined??? Eg. F(x)= 3x^2 - 4x -24 =0???
0
votes
0answers
12 views

Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. ...
4
votes
3answers
79 views

Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular

Problem: Let $A$ and $B$ be real orthogonal matrices, $n$x$n$, where $n$ is an odd number. Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular. What have I done so far: ...
0
votes
0answers
20 views

Factor fields, is this reasoning correct?

Let H = $\mathbb{Z}_3[x]/<x^4+2x^3+2x^2+x+1>.$ Find the smallest field F such that $H \subset F. $ We can factor $x^4+2x^3+2x^2+x+1$ as $(x^2+x+2)^2$, so it is not irreducible. However, ...
0
votes
0answers
5 views

Using partition of unit to extend paths on vector bundles in time dependent sections.

Let $\pi: A\longrightarrow M$ be a vector bundle and $a: I\longrightarrow A$ be a path where $I=[0, 1]$. I would like to show there are time dependent section $\alpha: I\times M\longrightarrow A$ ...
0
votes
0answers
7 views

Rate of convergence for quadratures

How can I find the observed (practical) order of convergence of a quadrature? I remember the formula $\frac{|x_{n+1}-x_{n}|}{|x_{n}-x_{n-1}|^q}$ but does this work here aswell? This formula gives me ...
2
votes
2answers
19 views

Eigenvalue of a matrix and a polynomial of that matrix

Let $A$ be a $n \times n$ matrix over $F$, and let $c_1, ... c_n$ be its eigenvalues. Show that for every polynomial $g(x) \in F[x]$, the eigenvalues of $g(A)$ are $g(c_1), ... , g(c_n)$. I think by ...
0
votes
0answers
17 views

Poisson Distribution to give more accurate probabilty

I'm making a betting prediction app for football matches using data such as the average goals scored and average goals conceded for a given team. I then use Poisson distribution to give me a ...
0
votes
0answers
8 views

Minimal time to steady-state flow (1st Stokes Problem)

How to obtain a minimal time to steady-state flow? This is related to the Stokes 1st Problem. $$\frac{u}{U}=0.05=1 - \operatorname{erf}(\eta)$$ $$\eta = \frac{y}{2\sqrt{vt}} \approx 1.4$$ ...
1
vote
0answers
12 views

Distribution of a transform of bivariate to univariate random variable?

Suppose we have two random variables $$R\sim U[1-\varepsilon,1]\;\;\;\;\; \Theta\sim U[0,2\pi],$$ and a third random variable $$X=g(R,\Theta)=R\cos\Theta.$$ What is the density $p_X(x)$? The ...
0
votes
1answer
20 views

General solution of a linearized PDE of second order

$$\frac{\partial u}{\partial t}=\Delta u+\gamma f(u,v)\text{ for }x\in\Omega, t\geq 0$$ $$ \frac{\partial v}{\partial t}=d\Delta v +\gamma g(u,v)\text{ for }x\in\Omega,t\geq 0 $$ with ...
2
votes
4answers
60 views

Evaluate the algebraic limit $\lim_{n\to\infty} \frac{2n}{\sqrt{n+2}(\sqrt{n+1} + \sqrt{n-1})} $

$$\lim_{n\to\infty} \frac{2n}{\sqrt{n+2}(\sqrt{n+1} + \sqrt{n-1})} $$ I know that its value is somewhere in $(0, \infty)$ but I have no idea how to find the exact value. Update: I've had a ...
2
votes
0answers
17 views

“Neural Networks Learn in Continuous space” - What does this statement mean?

I was reading the paper on "Recurrent neural network based language model" by Tomas Mikolov et. al and I came across this statement "Neural Networks Learn in Continuous space". Please help me in ...

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