This tag is used if a reference is needed in a paper or textbook on a specific result.

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3
votes
1answer
149 views

Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets

Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...
0
votes
1answer
176 views

Axiomatic explanation of why the volume of a parallelepiped is equal to the area of its base times its height [closed]

I asked this in MSE, it flashed and disappeared. Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope ...
5
votes
1answer
232 views

If a topological space has vanishing $n$th homology for every possible homology theory, does it have vanishing $n$th homotopy?

I don't have any strong preference as to whether or not the homology theories are required to be ordinary. Also, if this does not hold in general, does it hold for some nice category of spaces, like ...
5
votes
0answers
151 views

Does there exist a smooth version of Cohen's factorization theorem?

A famous factorization theorem of Cohen states that for any locally compact group $G$, $$L^1(G)=L^1(G)*L^1(G).$$ I want to know if analogous results exist for the class of smooth functions when $G$ is ...
4
votes
1answer
150 views

Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$

Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X ...
1
vote
0answers
51 views

Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
3
votes
1answer
85 views

Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...
1
vote
0answers
53 views

Convergence to equilibrium for time in-homogeneous diffusions

Consider the long time behavior for a time in-homogeneous diffusion such as $$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$ where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent ...
2
votes
0answers
111 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
8
votes
0answers
162 views

Simplices and cubes

Question: What is the first appearance in the literature of one of the following statements: The result of intersecting a simplex with a cell of the dual subdivision is a cube There ...
3
votes
1answer
233 views

Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.) Suppose we have an algebraic group $G$ ...
0
votes
0answers
62 views

The following ODE global existence theorem reference?

There is an ODE existence theorem of the form: Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function. Suppose that there is a constant $c$ such that if $y$ is a ...
6
votes
1answer
115 views

Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed: Suppose $P, ...
1
vote
0answers
134 views

Mathematical journals (maybe in the past) with regular competitions? [closed]

This question from SEM but no answer : In Yōko Ogawa's "The Housekeeper and the Professor," there is a character who regularly takes part in contests published in a fictitious mathematical journal ...
0
votes
1answer
92 views

Borel subsets of Polish groups

Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...
1
vote
0answers
37 views

Is there a polynomial time algorithm for Poly-trees (Oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
15
votes
1answer
944 views

The list of problems for Grothendieck's thesis

Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...
5
votes
1answer
181 views

Proof of a soft version of Moschovakis's lemma

The following fact, which I've heard being called "soft version of Moschovakis's lemma" (see top answer here) is the following: Under AD, if there is a surjection $\Bbb R\rightarrow\alpha$, then ...
1
vote
0answers
92 views

Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$

This question is related to my previous question (here). Let $P_\lambda$(q,t) be the Macdonald polynomials with partition $\lambda$. Let $\Lambda$ denote the ring of symmetric functions over the ...
-2
votes
0answers
79 views

Given 2 bounded power series, whether one can be written as a compound power series of the other one?

Let $S(x) = \sum\limits_{i = 0}^\infty {{a_i}{x^i}} ,F(x) = \sum\limits_{i = 0}^\infty {{b_i}{x^i}} $ be two real bounded power series for all positive real $x$, and we assume: $S(x),F(x) \in ...
3
votes
0answers
84 views

A “nice” Orthogonal Basis for Translation Invariant Symmetric Polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
1
vote
0answers
83 views

derivative of the adiabatic limit of the eta invariant

To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...
0
votes
0answers
21 views

Methods for RCPSP

I have an Resource Constrained Project Scheduling Problem (RCPSP) with and additional strict precedence graph $H$, where $(j, j') \in H$ means $j'$ should stay closely after $j$. Can you advise any ...
4
votes
1answer
201 views

Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
27
votes
4answers
877 views

Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix $$ \left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ...
2
votes
1answer
146 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
0
votes
1answer
47 views

Do the support sets of subspaces give the representable matroids?

Fact:   Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid. Not sure you ...
10
votes
1answer
339 views

Origin of group theory problem (bound on number of Sylow subgroups)

This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...
4
votes
0answers
45 views

Structure of Lagrangian algebras in the center of a Fusion algebra

(1) Let $\mathcal F$ be a spherical fusion tensor category. Then Müger showed that $R=\bigoplus_{H\in\mathrm{Irr}(\mathcal F)} H\boxtimes H^\mathrm{op}$ canonically has the structure of a Frobenius ...
6
votes
1answer
88 views

Reference for a PL flat torus embedding in $\mathbb{R}^3$

A piecewise linear flat torus embedded in $\mathbb{R}^3$ is shown at http://www.mathcurve.com/polyedres/toreplat/toreplat.shtml. It is flat in the sense that the angle defect at the vertices is zero. ...
1
vote
1answer
87 views

Conditions for the consistency of a system of affine polynomials

Let $f_1, f_2,\ldots,f_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective ...
4
votes
3answers
604 views

Looking for techniques of How to measure the Similarity/Dissimilarity between two images?

I would like to compute the similarity/dissimilarity between two images L and R. I know one way which is : computing the histogram of blocks of each image, and then using Bhattacharyya measure I ...
0
votes
1answer
89 views

Graph automorphisms that preserve independent sets [closed]

Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets. We call a graph automorphism $f:V \to V$ of $G$ good if it is non-trivial and ...
3
votes
1answer
97 views

Generalisation of “tangent space” to not-necessarily connected sets

I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following. Definition (Blob) Let $S\subset \mathbb{R}^n$ be a set, and $p \in S$. The ...
1
vote
0answers
86 views

Vector valued Sobolev spaces

My question is in reference to this question previously asked here. As asked there, consider a function $f \in H^s_x(L^2_y) \cap L^2_x(H^s_y)$. In the notation of Lions and Magenes (Chapter 4, Vol 2), ...
4
votes
0answers
174 views

Primitive Closure Arithmetic

I am researching a system that was designed by myself and what I call PCA (Primitive Closure Arithmetic), because it looks like PRA. The differences are: - PRA uses recursive definition with a ...
6
votes
1answer
118 views

Homology class of a Lagrangian Klein bottle

Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb ...
0
votes
0answers
69 views

Finding a set of generators of an ideal with certain property in $k[x_1, …, x_n]$

I am interested in the following problem, and I would appreciate any comments, inputs, answers, references! Let $k$ be a field. For each $1 \leq j \leq n$, let $$ I_j = (x_j, u_{2}^{(j)}, ..., ...
1
vote
0answers
90 views

cohomlogy of Diagonal ring

Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let ...
18
votes
1answer
926 views

Anti-Mandelbrot set

I clearly remember seeing a paper where the dynamic of the anti-conformal map $f(z)=\overline{z}^2+c$ was studied (the bar means complex conjugation). There was a picture of the analog of the ...
14
votes
1answer
244 views

Geometric Mean of $L(1,\chi)$ for quadratic Dirichlet characters

Let $S = \{D_1, D_2, D_3, \ldots \}$ be the set of all prime discriminants (or positive prime discriminants) of quadratic number fields. For such a discriminant let $\chi_j(n) = (\frac{D_j}n)$ be ...
1
vote
0answers
104 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
8
votes
0answers
361 views

Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question. In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...
2
votes
0answers
103 views

Nonnegative integers represented by $\prod_{i=1}^m \sum_{j=1}^n a_{i,j} x_j $, where the $a_{i,j}$ are positive integers

Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the ...
-1
votes
0answers
88 views

Sheaf of rank 1 on smooth projective variety

Let $X$ be a smooth projective variety over $\mathbb{C}$ and let $F$ be a torsion free sheaf of rank $1$ on $X$. Then why $F\cong I\otimes L$ (in a unique way), where $I$ an ideal sheaf of finite ...
-1
votes
0answers
30 views

Reference for Convergence/Collapse of Riemannian manifolds

Can someone please tell me what are the good references for studying convergence/collapse of Riemannian manifolds? I am aware of Petersen's book ; there is a chapter on Convergence. Thanks.
1
vote
2answers
160 views

What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?

Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...
47
votes
3answers
4k views

Is this differential identity known?

Recently I discovered the differential identity $$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$ valid for any odd natural number $k$; for ...
28
votes
5answers
1k views

The unpublished papers in reference to the published papers

Sometimes it happens that a published paper refers to an unpublished paper for a result used. In this case, if we want to check this result by ourselves, we need to access to this unpublished paper. ...
1
vote
0answers
59 views

The dimension of the space of automorphic forms with multiplier system

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...