# All Questions

114,971
questions

**5**

votes

**1**answer

111 views

### Trace inequality under consideration of definiteness

Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality
$$ \text{Tr} \...

**0**

votes

**0**answers

72 views

### Field theory, Abel-Ruffini theorem, technical question

Let me put the question first.
Let $F,K$ be subfields of $\mathbb{C}$. Suppose that $t,\rho\in \mathbb{C}$ are algebraic over $F$ and $\rho \in K$. If $F(t)\cap K\subset F$, is it true that $F(t,\rho)\...

**0**

votes

**0**answers

66 views

### A polynomial formed from the roots of another polynomial ad infinitum

Let $P(x)$ be a monic polynomial of degree $d$ with complex coefficients. Let $r_1(P),r_2(P),\dots, r_d(P)$ denote the set of roots, ordered so that $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...

**1**

vote

**0**answers

159 views

### On the error bound for the Prime Number Theorem for arithmetic progressions

Let $\chi$ be a Dirichlet character, $L(s,\chi)$ be the corresponding L-functions and $\Theta_{\chi}$ be the supremum of the real parts of the zeros of $L(s, \chi)$. Define $\pi(x; a, q)$ to be the ...

**10**

votes

**1**answer

199 views

### Good overviews on $\phi^{4}$-field theory?

I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about:
(1) What are the ...

**7**

votes

**1**answer

242 views

### Minimum cardinality of a cofinal collection of countable subsets of a set

Setup
Let $X$ be a set of cardinality $\kappa\geq \aleph_0$.
Edit:
Based on Todd Eisworth's suggestion:
What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that ...

**1**

vote

**0**answers

84 views

### Natural candidates for super-half-exponential which limit to half-exponential function from above

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However super-half-exponentials (functions whose composition grows ...

**0**

votes

**0**answers

51 views

### Lower bound for eigenvalue problem with single linear constraint

Consider the problem
$$\max_x x^\top B x$$
subject to $\|x\|=1$ and $b^\top x = a$, where $b$ is a unit vector but not necessarily an eigenvector of $b$. Suppose that $B$ is symmetric and positive ...

**2**

votes

**0**answers

36 views

### Directed graph minor theorems

In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that the directed graph minor theorem is true, using the definition
A directed graph is a minor of ...

**0**

votes

**1**answer

99 views

### Asymptotics for solution of transport equation and characteristics

Consider the transport equation $$u_t(t,x) + v(t,x) \cdot \nabla u(t,x) = 0.$$
Suppose that the solution of the characteristic equation
$$\dot X(t) = v(t,X(t)) $$
decays to zero as $t \to \infty$. ...

**12**

votes

**1**answer

210 views

### Rational homotopy invariance of algebraic $K$-theory

Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra
$$
K(...

**1**

vote

**0**answers

49 views

### n-dimensional polyhedron with special properties

I'd like to know if there exists a convex face transitive n-dimensional polyhedron with all dihedral angles equal to $\frac{2\pi}{3}$.
For n = 2,3,4 an example can be a regular hexagon, a rhombic ...

**1**

vote

**0**answers

42 views

### Uniform position for multiple components

(Modified from https://math.stackexchange.com/questions/3730261/uniform-position-theorem-for-reducible-varieties/3730457#3730457)
The uniform position theorem states (roughly) that a general ...

**2**

votes

**0**answers

85 views

### How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?
See also:
Malfatti circles

**6**

votes

**1**answer

607 views

### Why can't we embed Tarski's truth in PA?

I recently learned that ZFC can prove $Con(PA)$ because it can give a model of PA, but I'm not given the technical details. (My teacher thinks it is too obvious to even mention.)
What plagues me is ...

**1**

vote

**0**answers

66 views

### Braided category inside braided 2-category

Let $\mathcal{C}$ be a semistrict braided monoidal $2$-category in the sense of [BN] (so in particular a strict $2$-category). Let $\mathcal{C}_1$ be the category of $1$-morphisms (objects) and $2$-...

**0**

votes

**1**answer

96 views

### Faithful representation of group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...

**-1**

votes

**0**answers

46 views

### transform $ \phi '' + ( 1 +c^2/4 -|\phi |^2)\phi = 0 $ into $ \varphi '' + ( 1 - |\varphi |^2)\varphi = 0$

Assume that $\psi: \mathbb{R}\to\mathbb{C}$ is a solution of $\psi '' + i c\psi ' + (1-\vert\psi\vert^2)\psi = 0$, where $i^2 = -1$ and $c\in (0,\sqrt{2})$.
Applying the transformation $\Phi (\psi)=e^{...

**1**

vote

**2**answers

80 views

### Properties of the total variation norm on space of totally finite measure (from Bogachev)

Let $(X,d)$ be a metric space, $\mathcal{B}$ the Borel $\sigma$-algebra on $X$, and $\mathcal{M}(X)$ the space of totally finite measures on $\mathcal{B}$. Let $\|\mu\|_{TV}$ be the total variation ...

**2**

votes

**0**answers

148 views

### Infinitely many $n$ such that $\gcd(\lfloor n\sqrt{2}\rfloor, \lfloor n\sqrt{3}\rfloor)=m$

Is it true that for any positive integer $m$ there are infinitely many positive integers $n$ such that $\gcd(\lfloor n\sqrt{2}\rfloor, \lfloor n\sqrt{3}\rfloor)=m$?
$\lfloor x \rfloor$ is the floor ...

**2**

votes

**0**answers

173 views

### Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar.
Define an $\mathrm{SL}(2)$-action on $\...

**2**

votes

**0**answers

68 views

### Is there discrete Morse theory on acyclic categories?

Forman introduced discrete Morse theory on finite regular cell complexes. Minian introduced a version of discrete Morse theory for posets which generalizes Forman's original Morse theory https://arxiv....

**2**

votes

**0**answers

44 views

### On periods of symmetric algebras

Let $A$ be a symmetric finite dimensional algebra over a field of characteristic two (or even over the field with two elements) such that every simple $A$-module has the same period equal to $n$.
...

**10**

votes

**1**answer

307 views

### Approximating power series coefficients — Why does a clearly illegitimate method (sometimes) work so well?

For reasons that don't matter here,
I want to estimate the power series coefficients
$t_{ij}$ for the rational function
$$T(x,y)= {(1+x)(1+y)\over 1- x y(2+x+y+x y)}=\sum_{i,j} t_{ij}x^iy^j$$
Using a ...

**0**

votes

**0**answers

64 views

### Flat conical hypersurfaces in $\mathbb{R}^4$

What can be said about an isometric immersion $f:\mathbb{R}^3-\left\{0\right\}\to\mathbb{R}^4$ such that $\left\|f(x)\right\|=\left\|x\right\|$ for all $x\in\mathbb{R}^3-\left\{0\right\}$?

**1**

vote

**0**answers

62 views

+50

### Stochastic integral with respect to a random field

I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...

**6**

votes

**0**answers

200 views

### On a revised quantum Riemann hypothesis

This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the ...

**2**

votes

**1**answer

67 views

### why 1-species and 2-species of static Widom-Rowlinson model are equivalent?

In Elena Pulvirenti's slides she introduced a $\textbf{static Widom-Rowlinson model of one species}$. Consider $\Lambda\subset R^2$ with periodic boundary conditions, $\Lambda$ set of particle ...

**4**

votes

**1**answer

147 views

### Positive scalar curvature on the total space of a circle bundle

Let $(\Sigma_\gamma,g)$ be a closed and orientable Riemannian surface of genus $\gamma \geq 1$, $(M^3,\tilde{g})$ be a closed, connected and orientable Riemannian $3$-manifold, and $\pi : M \to \...

**0**

votes

**1**answer

49 views

### Fast way to generate random points in 2D according to a density function

I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...

**2**

votes

**0**answers

40 views

### Invertible bimodule for hereditary algebras

Let $A=kQ$ be a path algebra over a field $k$ for a finite acyclic quiver with enveloping algebra $A^e$.
Question: When is it true that $\tau_{A^e}(A) \cong A$ as a left and as a right $A$-modules? (...

**-2**

votes

**1**answer

69 views

### Permutation and combination [closed]

Problem Statement : Given input x,y,a,b where x is the number of 0's a binary string has , y is the number of 1's, a is the number of possible sub-sequence "01" while b the number of ...

**2**

votes

**1**answer

70 views

### Fourier transform of a function of bounded variation

I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...

**1**

vote

**0**answers

37 views

### Integrability of Fourier transform of truncated fractional power

Is the Fourier transform of the function $f$ which agrees with $1_{[-1.1]}|x|^\alpha$ on $[-1,1]$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(\...

**11**

votes

**1**answer

366 views

### When does an open manifold admit two linearly independent vector fields?

$\DeclareMathOperator{\span}{span}$
$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\...

**0**

votes

**0**answers

27 views

### Conjugate point to spacelike hypersurface

Suppose you have a smooth spacelike hypersurface $\Sigma$ in some spacetime (four-dimensional Lorentzian manifold). Let $\gamma$ be a timelike geodesic meeting $\Sigma$ orthogonally and let $p$ be a ...

**6**

votes

**2**answers

478 views

### “Well-known fact” that every irreducible 3-manifold with non-empty boundary has an incompressible surface

I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one?
Also, it would be great if someone could provide me with a ...

**-5**

votes

**0**answers

53 views

### why can't i integration and differentiation? [closed]

Please kindly help solve this question for me. I need it quite urgent

**3**

votes

**0**answers

162 views

+100

### A density criterion and a submersion map of a Hodge bundle

In Voisin's excellent book 《Hodge theory and complex algebraic geometry II》5.3.4 - a density criterion, there is a important theorem:
Let $X$ be a compact Kähler manifold, $\pi:\mathcal X \rightarrow ...

**0**

votes

**1**answer

45 views

### Maximum number of edges in “square” hypergraph

For any set $X$ and any cardinal $\kappa$, let $[X]^\kappa$ denote the subsets of $X$ having cardinality $\kappa$.
A linear hypergraph is a hypergraph such that for all $e\neq e_1 \in E$ we have $|e\...

**9**

votes

**3**answers

805 views

### Are “large enough” finite etale covers arithmetic?

Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...

**5**

votes

**1**answer

154 views

### Dense generator whose closure under finite colimits takes several steps to form?

Let $\mathcal C$ be a locally finitely presentable category, and let $\mathcal C_0 \subseteq \mathcal C$ be a dense generator of finitely-presentable objects. Then
Every object $C \in \mathcal C$ is ...

**1**

vote

**1**answer

133 views

### If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?

Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...

**2**

votes

**0**answers

73 views

### Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem

I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \...

**1**

vote

**1**answer

136 views

### Existence of entire function that yields periodicity

I have the following question:
Does there exist an entire function $f(z)$ where $z=x+iy$ such that
$$g(x,y) =e^{-2\pi y^2}f(z)$$
is periodic in both $x$ and $y$ direction, i.e. $$\forall x,y: g(1,y)=g(...

**2**

votes

**0**answers

30 views

### On maximum principle of spectral fractional Laplacian

Suppose $(-\Delta)^s u=g \geq 0$ in $\Omega$ and $u=0$ in $ \partial \Omega.$ Also suppose $u$ is $C^{2}$ non-negative and $(-\Delta)^s u=0$ in $\Omega \setminus B$ and $u\leq a$ on $\partial B $ ...

**0**

votes

**0**answers

99 views

### Scheme-theoretic image of the inverse image of a morphism of schemes

Let $f:X \to Y$ be a finite, surjective morphisms between noetherian, integral varieties (over $\mathbb{C}$). I am looking for conditions on $f$ under which I can say that for any closed subscheme $Z \...

**0**

votes

**1**answer

60 views

### Rings or algebras with many nilpotent elements and efficient computation

Crossposted from quantum.SE
where comment appears to suggest that solving modulo 2 might
be possible.
Searching the web for '"quantum computer" nilpotent'
returns many results, so maybe the ...

**8**

votes

**2**answers

332 views

### Is it consistent to have a function that is sensitive to subset relation from the power set of a set to that set?

Is it consistent with $ZF$ to have a set $S$ and a function $F: P(S) \to S$ such that:
$\forall X,Y \in P(S): X \subsetneq Y \implies F(X) \neq F(Y)$

**0**

votes

**0**answers

29 views

### Boundary regularity of rectifiable multiplicity 1 hypercurrents

Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...