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5
votes
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
3answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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 ...

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