6
votes
0answers
118 views
+300

graph product that commutes with automorphism, and semi direct

Is there a way to construct a "product" of graphs $G\rtimes H$ such that $Aut(G \rtimes H ) \simeq Aut(G) \rtimes Aut(H) $? A related topic is "Semidirect product" of graphs? but not quite ...
3
votes
1answer
53 views
+150

A few questions about KSVD algorithm (dictionary learning) in a paper

To learn more about dictionary learning, I am currently trying to understand the concept in detail and to do so, I've found the following paper quite informative: KSVD: an algorithm for designing ...
9
votes
0answers
72 views
+100

Why are diffeomorphism-invariant PDE not elliptic?

In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ...
0
votes
1answer
37 views
+50

Dixon's Factorization Method Modulo Question

Looking at Wikipedia's example for Dixon's Factorization Method, it shows the following. We will try to factor N = 84923 using bound B = 7. Our factor base is then P = {2, 3, 5, 7}. We then search ...
17
votes
2answers
272 views
+50

Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...
8
votes
0answers
160 views
+100

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a sense, unorganized (see Open problems in ...
1
vote
0answers
43 views
+50

Characterization of open maps in terms of nets

Here I asked about characterization of closed maps in terms of nets/sequences. I find this view illuminating, so I wanted to ask about open maps. A map $f: X \to Y$ is open if for each $x \in X$ and ...
3
votes
0answers
94 views
+50

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
1
vote
0answers
48 views
+50

Fast Hankel Transform

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
3
votes
2answers
102 views
+250

Connections on a manifold and principal connections on the frame bundle

Suppose $M$ is a manifold, and $E$ a vector bundle over $M$ equipped with a connection $\nabla $. If $F$ is the frame bundle of $E$, is there an explicit construnction of a connection on $F$ ...
0
votes
1answer
62 views
+50

When is the Laplace Beltrami Operator self-adjoint?

The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in $L^{2}$. I am wondering if this is also true for other Hilbert spaces $W^{k,2}$. If this is ...
1
vote
2answers
39 views
+50

How do correlation and causality effect this scenario?

I came up with a real world problem that I don't need to solve, but was intrigued by nonetheless. Imagine I was trying to figure out if a supermarket chain took credit as well as cash. My wife had ...
1
vote
2answers
78 views
+50

Let $a_{2n-1}=-1/\sqrt{n}$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges.

Let $a_{2n-1}=-1/\sqrt{n}$, $a_{2n}=1/\sqrt{n}+1/n$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges. What I have found so far is that $\prod_{k=2}^{2n} ...
10
votes
1answer
230 views
+50

Numbers having in decimal representation no common digits with all their proper divisors

Let us call a positive integer having in decimal representation no common digits with all its proper divisors "a good number". $54$ is a good number : $1,2,3,6,18,27$ $48$ is not a good number : ...
7
votes
0answers
60 views
+100

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\subsetneq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
2
votes
0answers
88 views
+50

The recurrence $a(n,k) = \sum_{0\leqslant j<n} a(n+j,k-1)$

I'm trying to find a closed form expression to the following recurrence relation: \begin{align} &a(0,k) = 0, \quad \forall k\geqslant 1; \\ &a(1,k) = 1, \quad \forall k\geqslant 1; \\ ...
3
votes
2answers
66 views
+50

Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\right ) \right \}$

Where $s\in \mathbb{C}$. I assume that this would be pretty easily handled by solving it by definition, but I haven't taken courses in complex analysis yet. Also, I can't think of any nice property of ...
2
votes
0answers
35 views
+50

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
2
votes
0answers
41 views
+50

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
2
votes
5answers
137 views
+200

Find the absolute maximum and minimum value of $f$

I have to find the absolute maximum and minimum value of $$f(x, y)=\sin x+\cos y$$ on the rectangle $[0, 2\pi] \times [0, 2\pi]$. I have done the following: $$\nabla f=(\cos x, -\sin y)$$ ...
3
votes
6answers
202 views
+200

The box has minimum surface area

Show that a rectangular prism (box) of given volume has minimum surface area if the box is a cube. Could you give me some hints what we are supposed to do?? $$$$ EDIT: Having found that for ...
5
votes
1answer
159 views
+500

Minimal “sumset basis” in the discrete linear space $F_2^n$

Let's $$ C\subseteq F^n_2, $$ $$ 2C=C+C=\{\bar\alpha+\bar\beta\ | \bar\alpha,\bar\beta\in C\}. $$ I need to find $C$ such that $2C=F_2^n$ and $|C|$ is minimal. I have found the following ...
3
votes
1answer
74 views
+50

Proving that certain incidence correspondence is a projective variety.

Let $M$ be the projective space of nonzero $m\times n$ matrices up to scalars (in $\mathbb{K}$). In Joe Harris' Algebraic Geometry: A first course, in order to find the dimension of $M_{k}=\{A\in ...
2
votes
0answers
36 views
+100

Mean value theorem for random variables

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$f(X+Y)=f(X)+f^\prime(X+\theta Y)(Y-X)$$ for real valued random variables $X$ and $Y$ and ...
3
votes
0answers
22 views
+50

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
1
vote
0answers
35 views
+100

Serge Lang ANT Theorem 7 p. 148

Let $\mathfrak c, \mathfrak f$ be admissible cycles for an extension of number fields $K/k$ with $\mathfrak f$ dividing $\mathfrak c$. If you're not familiar with the terminology, a cycle $\mathfrak ...
2
votes
0answers
80 views
+50

The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don't fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: ...
1
vote
1answer
78 views
+50

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...
5
votes
1answer
117 views
+300

Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
6
votes
0answers
103 views
+50

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
0
votes
0answers
30 views
+50

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
1
vote
0answers
64 views
+50

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (s-t)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
3
votes
0answers
35 views
+50

Symplectic reversing diffeomorphisms

Let $(M,\omega)$ be a compact symplectic manifold. Is there a diffeomorphism $f$ on M with $f^{*}\omega =-\omega$?
3
votes
0answers
49 views
+100

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
2
votes
1answer
119 views
+50

Finf f such that $F \circ F$ is a primitive of f

Find all primitivable functions $f:\mathbb{R} \to \mathbb{R}$ that admits a primitive $F:\mathbb{R} \to \mathbb{R}$ for which $F\circ F$ is a primitive of $f$. From $(F \circ F)'=f$ we get that ...
3
votes
1answer
41 views
+50

Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at ...
20
votes
1answer
295 views
+100

Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In a gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
2
votes
1answer
77 views
+50

Sum of i.i.d. random variables and finding an upper bound

Problem: Suppose that $(X_i)_{i\in\mathbb{N}^+}$ is a sequence of i.i.d. random variables. For some $n\in\mathbb{N}^+$, let $S_n=\sum_{i=1}^n X_i$. Furthermore, let $a$ be a positive constant, and ...
0
votes
1answer
50 views
+50

The closure of an open set in $\mathbb{R}^n$ is a manifold

I want to solve the following exercise from M. Spivak's Calculus on Manifolds (p. 114): (a) Let $A \subseteq \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n-1)$-dimensional manifold. ...
6
votes
1answer
49 views
+100

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
0
votes
0answers
61 views
+100

Behavior of a Solution to Heat equation Compactly Supported in Time

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
3
votes
1answer
54 views
+50

Behaviour of solutions to ODE near singular points

I am having trouble understand how to classify what happens to solutions of ODE near singular points. For example, I have a question that is about the ODE; $$(x^2-36)y''+(6-x)y'+(x^2+12x+36)y=0$$ ...
4
votes
1answer
78 views
+100

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
2
votes
0answers
35 views
+50

Classification of $3$-pointed rational curves

I tried to prove that $\mathbb P^1 \setminus \{0,1,\infty\}$ is the fine moduli space for the moduli problem, which assigns to a scheme $S$ the set of (isomorphim classes of) $4$-pointed rational ...
0
votes
1answer
43 views
+50

Can the following inequality be directly infered?

If we have a condition as follows $$\log(1+\mathbf{h}_2^* \mathbf{\Sigma} \mathbf{h}_2) \leq \log(1+\mathbf{h}_1^* \mathbf{\Sigma} \mathbf{h}_1)$$ where $\Sigma$ is positive semi definite matrix ...