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3
votes
1answer
103 views
+100

Orthogonality relation for integral with Bessel functions

I want to calculate the following integral $$\int_0^1 dx \,x \,\left(J_n'(x \,x_{nm})J'_n(x\,x_{nm'})x_{nm'} x_{nm}+\frac{n^2}{x^2}J_n(x \,x_{nm})J_n(x\, x_{nm'})\right)$$ where $x_{nm}$ is the $m$ th ...
3
votes
2answers
177 views
+50

Understanding matrix multiplication for visualizing what is happening under the hood

Take the case of this matrix multiplication: $$ A x= \begin{pmatrix} 1 & -1 & 2\\ 0 & -3 & 1\\ \end{pmatrix} \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} $$ The answer of which is $ ...
7
votes
1answer
132 views
+50

Do harmonic functions span the space of functions on manifolds?

If one considers the Laplace (or Helmholtz) equation in two dimensions, then through separation of variables in plane polar coordinates, the azimuthal dependence is seen to be of the form of a ...
2
votes
2answers
121 views
+100

Methods for estimating a 2D transformation without knowing the point matchings

$\mathbf{p}=[x_P,y_P]^T$ and $\mathbf{q}$ are 2D points. The transformation $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ maps $\mathbf{p}$ to $\mathbf{q}$ and has a vector of parameters $\boldsymbol{\theta}...
0
votes
1answer
65 views
+50

Probability of error for two independent process

Let assume that we have two possible event $a_1$ and $a_2$ that may happen with probability $p(a_1)$ and $p(a_2)$ with $p(a_1)+p(a_2)=1$. Now let us consider a detector which is affected by some noise ...
1
vote
0answers
66 views
+50

Recursive funtions (R.Soare)

I started learning Computability theory by myself and got stuck on this one problem from R.Soare's book "Recursively enumerable sets and degrees", problem 3.11. It doesn't seem to be too ...
1
vote
0answers
140 views
+50

Two ways of explaining the notation $\mathbb{R}^2$

I found two ways to explain the notation $\mathbb{R}^2$. First is by Cartesian product: $\mathbb{R} \times \mathbb{R}$. Secondly, regarding $2$ as any set with two elements. suppose it's $\{0,1\}$. ...
2
votes
0answers
216 views
+50

Poincaré-Hopf Theorem for manifolds with boundary in Milnor's book

I've just read Milnor's topology from the differentiable viewpoint, though I haven't been able to figure out the case of manifolds with boundary. Milnor says that the problem is that in this case the ...
1
vote
1answer
83 views
+50

Need help with gradient descent convergence calculation

We are given $\alpha=1-8\eta$, where $\eta$ is the step size. We are also given the following ($z$ is irrelevant, but basically it's the output of the algorithm at step $k$): We are asked what values ...
2
votes
0answers
63 views
+100

Proving a prior distribution defined by a trignometric basis on $L^2(\mathbb{T})$ is Hölder continous

Consider $L_2(\mathbb{T})$ with the basis $$\phi_{2k}(x)=\sqrt{2}\cos(2\pi k x)\\ \phi_{2k-1}(x)=\sqrt{2}\sin(2\pi k x)$$ for $k\in\mathbb{N}$. The functions $\phi_k$ belong to the domain $H^{2p}$ of ...
6
votes
0answers
153 views
+50

Compute exact integrals with quaternions

It's common knowledge that complex analysis is helpful in computing a bunch of exact real integrals. Is there any occurence of quaternions/quaternion formalism helping in the same way? If not, what ...
1
vote
0answers
108 views
+50

Compute summation of modules expression?

This question is an extension of the following question: How can I calculate summations of modulus expressions? In particular, what I want to look at is the sum $$\sum_{k=1}^n(pk\pmod{q})$$ where $p,q\...
0
votes
0answers
123 views
+50

Direct combinatorial proof of three successive entries with ratio $3: 4: 5$ in Pascal's triangle?

Find the smallest value of $n$ such that row $n$ of Pascal's triangle contains three successive entries with the ratio $3:4:5$. I was able to solve this with algebraic manipulation. We have$${{\binom{...
5
votes
0answers
144 views
+50

Central Limit Theorem & Random Walks

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\{ X_i \}_{i=1}^\infty$ be $\operatorname{i.i.d}$. Assume that $P(X_1 =1) = P(X_1 =-1) = \frac{1}{2} $ and define $S_n$ by $$\begin{cases} ...
0
votes
0answers
66 views
+50

quadratic programming with a fourth power term

I've the following modified Quadratic Program (QP) with an additional fourth power term. Suppose $X\in\mathbb{R}^{k \times n}$ $$ X^{\text{opt}} = \arg\min_{X \geq \mathbf{0}} \,\,\,\,tr(X^{\intercal}...
0
votes
2answers
130 views
+100

Moscow papyrus area of hemisphere

I found this interesting text on wikipedia that explains how egyptians calculated the area of (supposedly) hemisphere, referred to as "basket". It is contained in the Moscow Papyrus. Here is the text: ...
14
votes
2answers
392 views
+100

Find all functions $f:\mathbb{R}^+\to \mathbb{R}$ such that $xf(xf(x)-4)-1=4x$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}$ such that for all $x\in\mathbb{R}^+$ the following is valid: $$xf\big(xf(x)-4\big)-1=4x$$ All I could do is: $f(x)> {4\over x}$ for all $x$ so $f(...
3
votes
0answers
195 views
+50

Finding An Unknown Summand In A Summation

For simplicity, let $$p(k, v)=\frac{1}{2}\big(k-v+1\big)\big(k+v\big)\Gamma(v)-\frac{(kv+k+v)\Gamma(k+2)}{v(v+1)(v+2)\Gamma(k-v+1)},$$ and let $$q(k, v)=\sum_{n=v}^{k-1}{\frac{(k^2+k-n^2-n)}{2}\bigg(n{...
6
votes
1answer
188 views
+100

Recursive formula for a tree problem

Question: A binary is defined as a tree in which 1 vertex is the root, and any other vertex has 2 or 0 children. A vertex with 0 children is called a node, and a vertex with 2 children is called an ...
2
votes
0answers
96 views
+150

Embeddings of several Lie groups and their geometry

The question concerns some problems about the Lie groups and representations. And the geometry embedding of the several Lie groups. We start from a fixed common special unitary group SU(2), with the ...
2
votes
0answers
114 views
+50

Size constraints on reduced solutions to $x^p\equiv 1\mod q$

Are there any size constraints on the reduced solutions $x$, $(0<x<q)$, to $x^p\equiv 1\mod q$, for $p$ and $q$ specific primes? (Considering the primitive $p$-th roots of unity modulo $q$). I ...
3
votes
4answers
95 views
+50

Product rule for independent functions

Suppose we have two independent functions $f(u)$ and $g(v)$ then (i) :- $$\frac {d \;(f(u)\cdot g(v))}{dx} = \frac {d (f(u)\cdot g(v))}{du}\cdot\frac{du}{dx} $$ Since $g(v)$ doesn't change with ...
3
votes
0answers
50 views
+100

Characteristics of equations of the form $u_{xy}=f(u_x,u_y,u)$

In the usual treatment of hyperbolic differential equations, it is always assumed that there are two families of characteristics. That is, if the equation $L[u]-f(u_x,u_y,u)=au_{xx}+2bu_{xy}+cu_{yy}-f(...
5
votes
1answer
114 views
+150

If $\rho(z)<1$ then exists some $N$ such that $\lVert R(z)^N \rVert <3/4$ for all $z\in U$

Let $\rho(z)$ denote the spectral radius of $R(z)$, where $R:U \to \text{Hom}(\mathcal{L},\mathcal{L})$ and $U$ is a closed subset of $\mathbb{C}$, here $\mathcal{L}$ can be understood as a Banach ...
1
vote
1answer
64 views
+50

Is the following “generalized version of distributive law of sets” true?

Consider a collection of sets $\mathcal{T}_{i, s_i} \subseteq \mathbb{R}^n$ with $i \in \{0,\ldots,k\} =: \mathcal{I}$ and $s_i \in \mathcal{S}_i$, where $\mathcal{I}$ is a finite set and $\mathcal{S}...
14
votes
1answer
303 views
+200

Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

While reading about the square peg problem, I found this paper of Jerrard, where he described that for the spiral $$r = k\theta \quad 2\pi \leq \theta \leq 4\pi $$ if we join the endpoints, you can ...
3
votes
0answers
205 views
+300

Proof involving weak convergence: where to us compactness

I have the following claim to prove as homework: Consider a continuous random variable $W$ with PDF $f_W(\cdot)$ and probability measure $\mathbb{P}_W$. Let $B$ be a compact subset of $\mathbb{R}\...
1
vote
1answer
52 views
+50

A naive question about the Godement resolution

So, I have read several accounts on Godement resolution, including an exposition by Godement himself in "Topologie algébrique et théorie des faisceaux". And there is still one thing that ...
4
votes
0answers
93 views
+50

Forward characterization of measurable functions?

In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images: $f$ is continuous if the ...
-1
votes
0answers
61 views
+50

How to most compactly define addition, a norm, and a metric in this space?

I have a vector space with some unusual equivalences between vectors, and I'm struggling to define a norm and metric which will preserve those equivalences in a simple way. The way I need to represent ...
2
votes
0answers
56 views
+50

Covering of a polytope by balls with midpoints at the vertices

I have a polytope $P=\operatorname{conv}(v_1,\ldots,v_m)\subset \mathbb{R}^n$ and a ball $B_r(x)$ (with center $x\in \mathbb{R}^n$ and radius $r>0$), such that $P\subset B_r(x)$. Is the statement $...
5
votes
0answers
78 views
+100

Help showing $y \cdot f(x+1) \leq \frac{C_0(p) \cdot y \cdot f(y)}{2} + \frac{x \cdot f(x)}{2}$

I was reading through this paper and noticed a seemingly simple algebraic lemma (Lemma 2 on page 6) for which a proof is not given. I cannot seem to prove the fact myself, so I was hoping to turn to ...
4
votes
0answers
77 views
+50

Number of real roots of polynomial and intermediate value theorem

Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a polynomial function with real coefficients satisfying \begin{align} p(x_1)<0, p(x_2)>0, p(x_3)<0,\ldots \text{(sign flips in alternating manner)} \...
4
votes
1answer
126 views
+200

Conjecture about divergent periodic summations of odd functions

$$f(x)=\lim_{m\to\infty}\frac{\sum_{n=-m}^m(x-n)^{1/3}}{\sum_{n=-m}^m(1-n)^{1/3}}\stackrel{?}{=}x$$ I had a 'proof' but I made the simple mistake of assuming 2 limits could be swapped, so I only ...
2
votes
0answers
99 views
+50

Green's Theorem Concepts: Circulation in R2

I am trying understand how circulation density arises in Green's theorem and I'd like to know if my line of thinking is on the right track. Here it goes :). Idea We know that if we have a vector field ...
6
votes
0answers
82 views
+50

What's the proper way to take a Groebner basis with respect to a quotient polynomial ring?

Suppose I have the quotient ring $R=\mathbb{Q}[x,y]/I$ for some ideal $I$, and I want to find a Groebner basis for another ideal $J\subseteq R$. When computing the basis, does it make a difference if ...
3
votes
0answers
62 views
+100

Separation of variables of 2nd order PDE yields 1st order ODEs

Consider the telegraph or Klein-Gordon equation on a rectangle*, $$ \begin{align} \left(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\right)\psi(x,y)=\gamma^2\psi(x,y), \end{align} $$...
0
votes
1answer
40 views
+100

Support of Gaussian measures on Hilbert spaces

I'm reading Kuo's lectures notes, Gaussian measures on Banach spaces, and I'm having trouble justifying a few lines in a proof. $\mu$ and $\nu$ are Gaussian measures on a Hilbert space $H$, $\mu = [0, ...
0
votes
1answer
148 views
+50

Prove or disprove: $1+\frac{\ln^c(ax)-\ln^ca}{\ln^c(2a)-\ln^ca}-x\leq 0$, where $1<x$, $a>2$, and $c$ is chosen so that $f'(2)=0$

Define: $$f(x)=1+\frac{\ln^{c}\left(ax\right)-\ln^{c}\left(a\right)}{\ln^{c}\left(2a\right)-\ln^{c}\left(a\right)}-x$$ where $1< x$ and $a>2$. Assume further that the parameter $c$ is chosen so ...
11
votes
2answers
353 views
+100

Guessing number of colors of beads in an urn

Motivation from cocktail bar Every time when I order the cocktail “Latex and Prejudice” (“Латекс и предубеждение”) in the Tesla bar in Saint Petersburg (Russia) the barkeeper selects by random a small ...
2
votes
1answer
44 views
+50

showing a ring is a principal ideal domain

Suppose $R$ is a principal ideal domain. Let $S$ be a multiplicatively closed subset of $R$ not containing $0$. Show that $S^{-1}R$, the localization of $R$ by $S$, is a principal ideal domain. I ...