# All Questions

150,265
questions

-3
votes

0
answers

42
views

### convergence of a numerical series using information about an entire series [migrated]

I'm on a problem that seems simple but turns out to be a bit twisted.
Let be $\sum_{n\epsilon N }^{}{u_nz^n}$ a power series with radius of convergence ρ = 1. Which of the following statements are ...

0
votes

1
answer

82
views

### An example of module which is square-free, CS, NOT C3, and NOT nonsingular

Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every ...

2
votes

1
answer

181
views

### Expected norm of a product of Gaussian matrices

Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?
$$E[\|C_n\|_F^2]=d^{n+1}$$
This ...

2
votes

0
answers

116
views

### Do the nearby cycle and Beilinson's vanishing cycle functors commute?

Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...

2
votes

1
answer

157
views

### Is this theorem true in the case of a general measure space?

I'd would like to confirm if the following proposition is indeed true in the case of an arbitrary measure space.
Theorem: Let $(X,\Sigma,\mu)$ be a measure space and $\{f_n\}_{n\in\mathbb{N}}\...

2
votes

0
answers

112
views

### Existence of a hyper plane

I am very new to algebraic geometry, and self-studying varieties. I have the following question.
Suppose $Y$ is a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P$ be a ...

-1
votes

0
answers

52
views

### Prove that this equation for natural m and n doesnt have an answer [closed]

$19^(19)=m^3 + n^4$
from $19^(19)$ i mean 19 to the power of 19
i've tried m and n for mod k, k=1,2,...,11 but i haven't reached a solution

6
votes

2
answers

656
views

### A conceptual proof that bounded index subgroups of a bounded torsion abelian group contain bounded index complemented subgroups

Call an abelian group $G = (G,+)$ $m$-torsion for some natural number $m$ if one has $m \cdot x = 0$ for all $x \in G$. A subgroup $H$ of $G$ is said to be complemented if one can write $G = H \oplus ...

12
votes

1
answer

338
views

### How exactly are realizability and the Curry-Howard correspondence related?

Consider, on the one hand:
the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...

0
votes

0
answers

41
views

### The discrete orthogonal polynomials

I want a document or something that explains the following proposition:
The discrete orthogonal polynomials are the polynomial solutions of the given diference equation:
$$
\sigma(x)\Delta\nabla P_n(...

-4
votes

0
answers

53
views

### Power summing function [closed]

f(x,p)=sum(n=1,n<=x,n^p) where p and x are integers. f(x,1)=(x^2+x)/2, and f(x,2)=x(x+1)(2x+1)/6, but what is f(x,p), where p is VERY BIG?

2
votes

1
answer

139
views

### On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...

1
vote

0
answers

146
views

### Deformations over $A_{\inf}$

Setup:
Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
Let $\mathcal{X}$ be a flat, projective $\...

2
votes

0
answers

50
views

### Can the Weyl algebra be free over its invariant subalgebra?

Let $k$ be an algebraically closed field of zero characteristic, let $P_n$ denote the polynomial algebra in $n$ indeterminates, and let $G$ be a finite group of linear automorphisms. Then, by ...

14
votes

1
answer

643
views

### Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...

-1
votes

1
answer

92
views

### How to prove that increasing the number of constant symbols of a first-order logic by the number of formulas keeps the number of formulas the same [closed]

Let $S$ be a set of theory symbols for a first-order logic, and let $C$ be a set of constant symbols in $S$ such that $|C| = |L(S)|$, where $L(S)$ is the set of all formulas generated by $S$ in the ...

-1
votes

0
answers

24
views

### Vertex expansion or vertex isoperimetric number of the cartesian product of cycles [closed]

Vertex expansion or vertex isoperimetric number of the cartesian product of cycles when all cycles are the same, C_m.
Exact value or any upper or lower bound?

2
votes

0
answers

176
views

### Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?

In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any ...

1
vote

1
answer

119
views

### Is this constraint convex?

I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} \exp \...

4
votes

0
answers

55
views

### Pfaffian elements and anomalies

If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...

-1
votes

0
answers

50
views

### Measuring to exact decimal places with ruler and compass exclusively [closed]

Do you know any way to construct a segment given its length in decimals, using only a ruler and compass, in an exact way?
For example:
a) 0.54896753
b) 12 decimals of acos(20°)

2
votes

0
answers

81
views

### Orthogonal representation of free products of two groups

Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...

0
votes

0
answers

134
views

### Research directions related to the Hilbert-Smith conjecture

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...

6
votes

1
answer

328
views

### When are the chirp signals orthogonal?

Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...

0
votes

0
answers

130
views

### Proof that a specific cubic equation has three real roots [migrated]

I'm trying to prove that the cubic equation
$a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$
has three real roots. The coefficients are
$a_3 = - 1 - \sigma - \tau - \chi$
$a_2 = -2 (\sigma +...

0
votes

0
answers

34
views

### Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$

Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...

2
votes

0
answers

24
views

### Is anything known about the equivariant homotopy theory of surfaces with the action of a finite subgroup of the mapping class group?

The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...

-1
votes

0
answers

43
views

### Measurable Function and Inverse Maps [migrated]

Every text I read about random variables starts by introducing the concept of measurable functions. It goes something like this:
Suppose you have 2 measurable spaces $(\Omega, \Gamma)$ and $(\Omega', \...

4
votes

0
answers

233
views

### Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?

I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...

0
votes

0
answers

47
views

### Primal optimal attained implies dual optimal attained

Given some optimization problem $\min_{x \in S \subset \mathbb{R}^n} f_0(x)$ $\text{s.t.}$ $f_i(x) \leq 0, 1\leq i\leq m$. We can find the dual problem $\max_{\lambda\in\mathbb{R}^m} g(\lambda)$ $\...

3
votes

0
answers

34
views

### When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?

Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...

2
votes

0
answers

53
views

### Exponential of Liouville Numbers

By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that
Any Liouville number is a $U$-number.
$\log \alpha$ is either an $S$- or a $T$-...

2
votes

0
answers

64
views

### Complemented C* Algebras

let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?

2
votes

0
answers

67
views

### How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...

5
votes

1
answer

254
views

### Does the oriental inject into the cube?

For every $n \geq 0$ there is an inclusion of the ordered set $\{0<1<\dots<n\}$ into the product
$\{0<1\}^{\times n}$ sending $i$ to the increasing sequence $(0 < \dots<0<1<\...

3
votes

0
answers

62
views

### Can the set of parafinite congruences be descriptive-set-theoretically complicated?

Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...

3
votes

0
answers

138
views

### Factorization of symmetric polynomials

Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The ...

4
votes

2
answers

301
views

### Probabilty measures that are both discrete and continuous

Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...

1
vote

1
answer

161
views

### Do balls in expander graphs have small expansion?

Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...

1
vote

1
answer

99
views

### Metropolis-Hastings kernel in measure theory

I'm facing difficulties in formulating the Metropolis-Hastings kernel for a specific problem where I need to sample from a probability distribution involving both discrete and continuous degrees of ...

1
vote

0
answers

63
views

### Select random point on elliptic curve

If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...

0
votes

0
answers

106
views

### Is the BGQ spectral sequence functorial with respect to morphisms of finite Tor-dimension?

It is well known that the BGQ (Brown-Gersten-Quillen) spectral sequence for the G-theory of a Noetherian scheme of finite Krull-dimension is contravariant with respect to flat morphisms.
My question ...

0
votes

0
answers

105
views

### When is a functor of chain complexes triangulated?

Let $\textsf{A}, \textsf{B}$ be abelian categories.
Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...

4
votes

0
answers

78
views

### Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{n,2}$ . $n$ even. Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold....

2
votes

1
answer

186
views

### When is a (co)edge trivial in graph cohomology?

Let $G$ be a connected graph and let $e$ be an edge in this graph. I would like to know if there are necessary and sufficient questions so that $e^{\vee}=0$ in $H^1(G)$? The question must be easy to ...

2
votes

0
answers

130
views

### A set inequality problem [migrated]

There is two different sets called set $a$ and $b$.Let $t$ be a positive integer,and put $t$ objects in another set called set $c$ ,and label the $t$ objects $c^1$,$c^2$...$c^t$.
Next,you put the ...

1
vote

0
answers

120
views

### There exists noncommutative geometric invariant theory?

In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$...

0
votes

0
answers

26
views

### Enumeration of flat integral $K_4$

Question:
What is known about the enumeration of all $(a,b,c,d,e,f)\in\mathbb{N}^6_+: \\ \quad\operatorname{GCD}(a,b,c,d,e,f)=1\ \\ \land\ \exists \lbrace x_1,x_2,x_3,x_4\rbrace\subset\mathbb{E}^2:\ \...

4
votes

0
answers

139
views

### Large sets of nearly orthogonal integer vectors

This question is motivated by the Question 5 from the 2017 Asia Pacific Mathematical Olympiad. To paraphrase, the question asks what is the largest cardinality of a set $S \subset \mathbb{Z}^n$ such ...

2
votes

1
answer

93
views

### A problem about the existence of increasing coloring groups

Got stuck on this one for months.
Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...