# All Questions

150,135
questions

0
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22
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### Can a non-separable C$^*$ algebra have separable GNS Hilbert space

Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if ...

0
votes

0
answers

23
views

### The Krull dimension of the tensor product of rings

The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...

0
votes

0
answers

22
views

### A function whose derivatives belong to $BMO(\mathbb{R}^n)$

I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...

3
votes

0
answers

173
views

### Math reviews in ZBMath that motivated you to read the paper

This is community wiki question.
I will be writing my first review for ZBMath. I would like to take some suggestion through examples.
In general, abstract is too small and introduction is too lengthy ...

1
vote

0
answers

19
views

### Two player games, Lipschitz isomorphisms, and a generalization of Gorelik principle

Let $E$ be a Banach space. It is known that for any weak neighborhood $U$ of $0$ in $E$ and $\tau>0$, there exists a weak$^*$-neighborhood $V=V(U,\tau)$ of $0$ in $E^*$ such that if $x^*\in V$ ...

2
votes

0
answers

21
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### Jacobi fields in singular metric on quotient space

Consider the square $\Omega = (0,\pi) \times (0,\pi/2) \ni (r,\theta)$ endowed with the Riemannian metric
\begin{equation}
f^2 \big(\mathrm{d} r^2 + \sin^2(r) \, \mathrm{d} \theta^2 \big),
\end{...

2
votes

0
answers

73
views

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

Is equation
$$
(x+1)y^2-xz^2=x^3+2x+2
$$
solvable in integers?
Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define ...

1
vote

0
answers

36
views

### MMP for surfaces over a curve where the geometric generic fiber is a rational curve

I am looking for an explaination or an reference for the following fact:
Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of ...

6
votes

2
answers

201
views

### What is the minimal density of a set A such that A+A = N?

Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$?
What I know:
...

2
votes

0
answers

47
views

### When does an iteration not add functions $\eta\to V$ at the final stage?

I am interested in better understanding the following property:
Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...

-4
votes

0
answers

23
views

### Calculation problem for Fixing small box in big box [closed]

I have received 26 small boxes in big box with following dementions.
Each small box: 32.5 X 25 X 6.5
Big box: 65 X 50 X 53
All small boxes are well accomudated in big box but when i do the mathemetic ...

1
vote

1
answer

68
views

### Simple proof for convexity of a real valued matrix function

I am looking for a simple and short proof showing that $X \to \|X X^\top\|_F^2$ is a convex function where $\|\cdot\|_F$ is the Frobenius norm. I have one proof by showing that the derivative is ...

2
votes

1
answer

148
views

### Inequality of inclusion-exclusion term

This question was initially posted on math.stackexchange.com but did not receive any answers for half a week.
While analyzing the properties of an algorithm I am working on (I'm a computer scientist), ...

2
votes

0
answers

41
views

### Modularity of the Drinfeld center of the category of G-graded vector spaces

Background: Let $G$ be a finite group, and $\mathrm{Vect}_G$ be the category of finite dimensional $G$-graded vector spaces over some algebraically closed field $k$ of char 0. It is well-known that $\...

-1
votes

0
answers

85
views

### Given a differential equation, can I find other differential equations that have the same solution? [closed]

For example, the function
$$x(t)=\frac{1}{2}e^{4t}-\frac{1}{2}e^{2t}+e^{-t}\tag{1}\label{eq1}$$
is the solution of the following equation:
$$
\begin{cases}
\ddot x-3 \dot x - 4x=3e^{2t},\\
x(0)=1,\\
\...

1
vote

0
answers

29
views

### Existence of a strongly regular vertex ordering on cubic graphs

Definition: Let $G=(V,E)$ be a cubic (i.e. $3$-regular) graph, and $<$ a total order on $V$. For $v\in V$ let $v^\downarrow$ denote the set of nodes $w\in V$ such that $w<v$, and let $\alpha(v) =...

3
votes

0
answers

42
views

### Are two homotopic principal bundles isomorphic?

Let $E_1 \to B$ and $E_2 \to B$ be two principal $G$-bundles, where $E_1$ and $E_2$ are two simply-connected manifolds and $G$ is a compact Lie group.
Suppose there exists a $G$-equivariant continuous ...

0
votes

1
answer

199
views

### How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...

-6
votes

0
answers

69
views

### Simple 4 - Color Theorem with Graph Theorem [closed]

There are four spaces. Each cell can contain red, yellow, green, and blue, and each adjacent cell cannot contain the same color.
ㅁㅁㅁㅁ
In this case, each cell can be expressed as A, B, C, and D as ...

3
votes

0
answers

65
views

### Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?

It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...

2
votes

0
answers

63
views

### Self adjoint operators from energy functionals

It is known that the equation
$$
\Delta f = 0
$$
on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...

0
votes

0
answers

38
views

### Meromorphic functions converging in measure

Let $f_1, f_2, \ldots$, and $g$ be measurable complex-valued functions on the open unit disk. We say that the sequence $f_1, f_2, \ldots$ converges in measure to $g$ if, for all $\epsilon, \mu >0$, ...

2
votes

1
answer

61
views

### Does every graph admit an embedding such that identically-colored edges do not cross?

Given a graph, is it always possible to color the edges of the graph using two colors such that there exists an embedding of the graph in the plane where only opposite-colored edges cross?
Simple ...

3
votes

2
answers

107
views

### Does $\mathsf{ZF}$ prove $\operatorname{Col}(\lambda,\kappa)$ preserves cardinals below $\lambda$?

Let $\lambda<\kappa$ be cardinals and consider the forcing $\operatorname{Col}(\lambda,\kappa)$ adding a generic surjection $\lambda\to\kappa$. More formally, $\operatorname{Col}(\lambda,\kappa)$ ...

10
votes

1
answer

496
views

### Recognizing free groups

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...

-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 ...

-6
votes

0
answers

64
views

### How exactly does curvature define the space geometry? [closed]

Example:
The definition of the local curvature $k(x)$ of the one-dimensional space,
defined by equation $y=f(x)$,
is given by the formula:
$$k(x) = \left| \frac{f''}{\sqrt{\left(1+{f'}^2 \right)^3}} ...

0
votes

1
answer

60
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

157
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

85
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

146
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}}\...

1
vote

0
answers

79
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

45
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

444
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 ...

9
votes

0
answers

120
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

32
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(...

-3
votes

0
answers

43
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

100
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

118
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

45
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 ...

11
votes

1
answer

473
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

83
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

14
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

146
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 ...

0
votes

1
answer

88
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 \...

3
votes

0
answers

41
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

48
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°)

1
vote

0
answers

66
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

120
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

315
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')\...