3
votes
0answers
66 views

Is there a classification of 2d extended TQFTs with defects?

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly ...
2
votes
1answer
55 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...
2
votes
1answer
95 views

pseudovarieties and profinite group : do * and g() commute?

Let $V$ and $W$ be pseudovarieties of finite monoids. We denote with $gV$ the pseudovariety of categories generated by $V$, and by $V*W$ the semidirect product of pseudovarieties $V$ and $W$. Does ...
3
votes
1answer
116 views

intersection of holomorphic curve with hyperplane

Let $f : \mathbb{C} \rightarrow \mathbb{C}^n$, $n>1$ be an entire function. Assume for simplicity that $f(0)=0$. Let $B$ be the closed ball of centre $O$ and radius $R$. Is there an upper bound ...
7
votes
0answers
60 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
2
votes
1answer
173 views

Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...
19
votes
0answers
257 views

3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$. Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...
14
votes
1answer
500 views

Question on the irrationality of $e$

I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing ...
5
votes
1answer
162 views

Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...
7
votes
1answer
200 views

how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
0
votes
0answers
39 views

Characterization of the maximizer of a function based on a parameter's value

Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter. I have two optimization problems. ...
1
vote
0answers
55 views

Mixture with varying concentrations

Let $(\Omega ,\mathcal F, \mathbb P)$ be a probability space and suppose $$\mathbb P(X \in A) = H(A) = \prod _{i=1}^m H_i(A),\quad \forall A\in \mathcal F$$ be a distribution of a random vector $X = ...
3
votes
2answers
214 views

Proof that image of a polynomial map is a cone

Consider the nonlinear mapping $\phi: \mathbb R^{2 \times 2} \to \mathbb R^3$ given by $X \mapsto \begin{pmatrix} x_{11} x_{21} \\ x_{11} x_{22} + x_{21} x_{12} \\ x_{12}x_{22} \end{pmatrix}$. I ...
6
votes
1answer
185 views

For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...
3
votes
0answers
174 views

adding a boundary to the finite upper half-plane

Let $\Bbb{F}_q$ be a finite field, let $\delta \in \Bbb{F}_q$ be a non-square, let $\Bbb{F}_{q^2} = \Bbb{F}_q\big( \sqrt{\delta} \big)$ be the corresponding quadratic extension, and let ...
0
votes
0answers
41 views

Self adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
0
votes
0answers
20 views

Necessary Conditions for Saddle Value point [migrated]

This questions is from the Kuhn-Tucker paper "Nonlinear Programming" in Section 2 Lemma 1. I don't understand how those conditions are necessary for a saddle point. I always thought that a saddle ...
3
votes
2answers
228 views

Derivative of an eigenvector with respect to his own 3x3 real symmetric matrix

$\mathbf{C}$ is a real, positive-definitive 3x3 symmetric matrix (I am thinking about the right Cauchy-Green tensor in solid mechanics). We perform eigendecomposition and get: $$\mathbf{C} = ...
2
votes
2answers
74 views

Integrating a barycentric monomial over a simplex

Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...
5
votes
0answers
112 views

Universal anti-Horn classes?

Is there published work about universal anti-Horn classes? (See also related question Is there any research of universal algebras axiomatized by non-Horn clauses? asking about universal classes ...
-3
votes
0answers
103 views

meromorphic sections of line bundles over riemann surfaces [closed]

What is the obstruction on two holomorphic line bundles over a Riemann surface (with non-zero genus), which are associated to two divisors with the same degree, being isomorphic? In genus zero case, ...
20
votes
3answers
1k views

Does the Brouwer fixed point theorem admit a constructive proof?

Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
0
votes
0answers
56 views

C$^*$-algebras - compact operator of rank one [closed]

I am studying Murphy's book "C$^*$-algebras and operator theory". On section 2.4 he define this operator: If $x, y$ are elements of a Hilbert space $H$ we define the operator $x \otimes y$ on $H$ by ...
1
vote
0answers
39 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O] Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and ...
4
votes
1answer
93 views

Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS

Using Hardy-Littlewood-Sobolev inequality, we can prove that: $$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C ...
4
votes
2answers
268 views

Maps to the group completion

Let $M$ be an H-space, topological monoid (homotopy-commutative if necessary): What does the group comletion $\Omega BM$ represent in homotopy category? Is $[X,\Omega B M]$ always equal to the ...
3
votes
0answers
64 views

How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...
-3
votes
0answers
80 views

collapse of classical and intuitionistic negation [closed]

Can someone show me a sequent calculus deduction that establishes that adding a classical negation to intuitionistic logic will result in a collapse between the two negations (ie intuitionistic ...
-1
votes
0answers
23 views

Property of free submodules for a module over a PID [migrated]

This question was asked here and remains without solution. It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...
0
votes
0answers
11 views

A mix between the Horvitz-Thompson and ordinary estimator

I have two samples: unbiased $X$ with $N_1$ elements and biased $Y$ with $N_2$ elements from some distribution (let it be F = ChiDistribution(1) if needed, $N_1=N_2=50$). Elements of $Y$ are picked ...
-3
votes
0answers
112 views

Does the diffeomorphism group of an open manifold act naturally on the compactification of the manifold? [closed]

I need help regarding this question please: Does the diffeomorphism group of an open manifold act trnsitively/freely on the compactification of the manifold? any recommended references!
1
vote
1answer
148 views

Is there an analytic solution for this partial differential equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
-1
votes
0answers
67 views

Principal Z/nZ- bundles [closed]

I have two questions -Is it true that principal $\mathbb Z/n\mathbb Z$-Bundle over a curve $X$ can be viewed as $n$ unramified cover over $X$? -What proprieties should satisfy a vector bundle $V$ ...
1
vote
0answers
49 views

Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and ...
-4
votes
0answers
36 views

limit of this expression as n gotes to infinity [closed]

I am interested in finding the limit of this expression: Comb(2n,n) * (1/2)^n. Any idea on how to find it? Thanks in advance!
4
votes
1answer
131 views

On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer? Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence ...
3
votes
2answers
129 views

The Hadwiger number of $L(K_n)$

For $n\in\mathbb{N}$ we consider the set $\{1,\ldots,n\}$ and define the line graph $L(K_n)$ of the complete graph $K_n$ as follows: $V(L(K_n)) = \big\{\{a,b\}: a,b\in \{1,\ldots, n\}, a\neq b ...
0
votes
0answers
47 views

Opens of $\mathbb{C}$ ,Can be Critical Domain for Some Holomorphic Function [closed]

Consider that we have a holomorphic function on $U$ that is an open subset of $\mathbb{C}$. We say that $U$ is a critical domain for $f$ if we couldn't find an analytic extension of $f$ to an open of ...
4
votes
0answers
61 views

Topological systems of imprimitivity

Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense. Here is an attempt to define ...
10
votes
0answers
104 views

Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...
1
vote
1answer
43 views

Petrov classification/Weyl scalars

There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup: We want to show that Petrov type D (i.e. two principal null directions) ...
1
vote
1answer
112 views

Implication between Erdös-Faber-Lovasz conjecture and Hadwiger's conjecture?

The Erdös-Faber-Lovasz conjecture and Hadwiger's conjecture can be stated in a very similar form: Erdös-Faber-Lovasz conjecture: for all finite simple undirected graphs $G=(V,E)$ we have $\chi(G)\leq ...
10
votes
3answers
379 views

Mathematical difference between entropy and energy

I have a rather soft question. Let's assume that we consider the heat equation posed in $S^1$: $$ \partial_t u=\partial_x^2u. $$ It is well known that if we define the functionals $$ ...
1
vote
0answers
79 views

What does the Riemann–Stieltjes integral measure? [closed]

The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum: $$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$ where ...
0
votes
1answer
41 views

Countable, $T_1$, and not metacompact

Is there a countable space that is $T_1$ and not metacompact? (A space $(X,\tau)$ is not metacompact iff there is on open cover $\cal{U}_0$ such that for every open refinement $\cal V$ there is $x\in ...
-6
votes
0answers
135 views

Answer to “why is matrix called matrix and what does it have to do with the movie?” [closed]

In a first year linear algebra class, I was asked "why is matrix called matrix and what does it have to do with the movie?" by a student. The movie he was referring to was "The Matrix" from 1999. I ...
-5
votes
0answers
31 views

Equation proof possibly by induction discrete math [closed]

The question asks: Prove that for some b belonging to the natural numbers, sqrt(2)^n > n for every n >= b. Find such a b belonging to the natural numbers.
3
votes
0answers
95 views

classifying pairs of idempotent matrices

though classifying pairs of matrices up to simultaneous conjugation is known to be wild, it seems to me a folklore that classifying pairs of idempotent matrices (up to simultaneous conjugation) is ...
0
votes
0answers
16 views

Is trace of regular representation in Lie group a delta function? [migrated]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
-2
votes
1answer
124 views

Decomposition space of $\mathbb{C}$ by concentric circles [closed]

What are the topological properties of the quotient space $X$ obtained from $\mathbb{C}$ by identifying points of the same modulus? I.e., the space $X=\mathbb{C}/E$ where $E$ is the equivalence ...

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