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6 votes
1 answer
725 views

Computer power in plane geometry

I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
5 votes
4 answers
2k views

Probability of random permutation having certain cycles

Are there any good references (either books or on-line) on the subject of the distribution of various cycle properties amongst permutations, particularly ones containing exact, closed-forms? For ...
3 votes
2 answers
640 views

Paper about Sasaki-Einstein manifolds

can you give me a good paper (in the sense of a simple introduction) about Sasaki-Einstein manifolds? Thank you and best regards Florian M.
1 vote
1 answer
99 views

Extension of the projective norm to a cross norm

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) ...
1 vote
0 answers
18 views

On an integral equation of Volterra type

Consider the following integral equation $$f(t) = A(t) + \int_0^t K(s,t)f'(s)ds,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$ where $A: \mathbb R_+\to [0,1]$ and $K: \{(s,t): 0\le s\le t\}\to[0,1]$ ...
3 votes
0 answers
69 views

Periodic objects in Frobenius categories

Let $A$ be a finite dimensional Gorenstein algebra and $C$ the stable module category of $A$. Question: Does there always exist an indecomposable periodic object $X$ in C, that is an object with $\...
0 votes
0 answers
6 views

A non-Kolmogorov system with Lebesgue spectrum: New examples?

It is known that a Kolmogorov system has Lebesgue spectrum, while not every system with Lebesgue spectrum is Kolmogorov. Some of the examples of the latter case are mentioned in Example 9.5.12 of the ...
1 vote
0 answers
18 views

Does $\mathscr{H}^{d-1} (A)<+\infty$ for $A\subset \mathbb R^d$ imply $A$ is (Borel) measurable?

I'm reading section 1.3.1 The quadratic case in $\mathbb{R}^{d}$ at page 17 from Santambrogio's Optimal Transport for Applied Mathematicians. The PDF is freely available from here. Let $\mu$ be a ...
0 votes
1 answer
167 views

Maximum number of colors for an optimal tiling which guarantees infinite paths

This question is a more applicable version of the question I've asked in mathexchange recently: What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square block ...
5 votes
2 answers
99 views
+100

Hamiltonian, energy, and conservation laws of nonlinear PDEs

In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
4 votes
1 answer
60 views

On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$ \bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all $\tau \...
2 votes
1 answer
64 views

Enlargement of filtration

Let $M_t$ be a continuous time real valued martingale, and $\mathcal F_t$ its natural filtration. Suppose that $\mathcal F_t \setminus \mathcal F_s$ is nonempty for all $t > s$. Let $\mathcal G$ be ...
0 votes
0 answers
41 views

Complete proof about Penrose tilings

It is well known that equivalence classes of Penrose tilings (say, by semikites and semidarts) are in bijection with binary sequences not containing 11 and modulo tail equivalence. However, I couldn't ...
2 votes
1 answer
139 views

Vanishing cycles exact sequence for degeneration of curves

Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$. Let $\eta\in D - \{0\}$ be a general point, and let ...
1 vote
1 answer
35 views

Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$

Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
13 votes
3 answers
1k views

Stable homology of arithmetic groups

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field. Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL_n(O_F)$, and as a result, ...
1 vote
1 answer
72 views

Infinitely many primes that split completely in an arithmetic progression

Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$. ...
0 votes
0 answers
22 views

Problems about Chern-Yamabe flow

I'm reading a paper about Chern-Yamabe flow. Chern-Yamabe flow which is about the Chern-Yamabe problem in Chern-Yamabe problem Here $\Delta$ is $\Delta=-\nabla^{\mu} \nabla_{\mu}$. The flow is: $$\...
15 votes
2 answers
3k views

Question about functional derivatives

This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
0 votes
1 answer
46 views

How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?

Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
1 vote
1 answer
44 views

Semi-orthogonal decomposition of Verra threefold

Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It ...
5 votes
1 answer
344 views

Polynomial size embeddings of toric varieties from polytopes?

Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan. $X_P$ is always projective, because the collection of characters corresponding to the points $\...
0 votes
0 answers
33 views

Asking for reference about a relation related to Fourier transform

Sorry for the not-perfect question. I am asking for a reference for the following relation: $$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$ Could ...
31 votes
3 answers
6k views

What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas? More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
10 votes
1 answer
164 views

Fixed-point free diffeomorphisms of surfaces fixing no homology classes

One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
0 votes
1 answer
52 views

Number of ways to write a finite set of cardinality n as the union of r distinct binary subsets

I want to know the number of ways to write a finite set of cardinality $n$ as the union of $r$ distinct two-element subsets. Is there a nice formula in binomial coefficients?
0 votes
0 answers
46 views

A Newton identity and the primes--the Faber partition polynomials and modular arithmetic

Dress and Siebeneicher in their tale of the Burnside family express an opinion (1.2) that, if I read it correctly, leads me to believe that the classic Faber partition polynomials which satisfy $\ln[A(...
1 vote
0 answers
61 views

Sudakov's lower bound type inequality for supremum of Chi-squared random variables

Let $\varepsilon$ be $n$-dimensional standard Gaussian veector, i.e., $\varepsilon \sim N_n(0, I_n)$. Let $\mathcal{P}$ be a subset of symmetric projection matrices in $\mathbb{R}^{n \times n}$ with $|...
2 votes
1 answer
88 views

Anti-Takagi: Given a Hermitian matrix $M$, is there a canonical form under $P \mapsto PMP^*$ where $PP^T = I$?

The Takagi decomposition provides a canonical form for a complex symmetric matrix $S$ under $U \mapsto USU^T$ where $UU^* = I$. Question: Is there an anti-Takagi decomposition? I.e. Is there a ...
7 votes
2 answers
492 views

Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates?

In the quadratic case, it does. Given an irreducible quadratic polynomial $f(x)=ax^2+bx+c$, the discriminant of the quadratic number field $\frac{\mathbb{Q}[x]}{f(x)}$ is $\operatorname{sqf}(d)$ or $4\...
2 votes
0 answers
40 views

Orthonormal eigenspinors of the gauge-covariant dirac operator on $\mathbb{R}^4$, with extra conditions are possible?

Let $G$ be a simple Lie group, and $V$ a representation. Consider $\mathbb{R}^4$ with its flat Euclidean metric. Let $P$ be the trivial $G$-bundle on $\mathbb R^4$ equipped with some (non-trivial) ...
0 votes
0 answers
22 views

Question regarding properties of map which produces measures that are invariant to orthogonal rotation

Let $\mathcal{M}_1$ denote the set of probability measures on the unit ball in $\mathbb{R}^d$ (which comes with its Borel $\sigma$-field). Denote by $\sigma$ the uniform measure on the orthogonal ...
1 vote
0 answers
46 views

Some $p$-adic congruences involving permutations

Motivated by my study of determinants and permanents, here I present several conjectures on $p$-adic congruences involving permutations. As usual, we let $S_n$ be the symmetric group consisting of all ...
2 votes
1 answer
171 views

Game on groups (generalization of spinning switches puzzle)

Alice and Bob are playing a game as follows: Initially There're two subgroups $A,B$ of Sym(n) known to both Alice and Bob There're $n$ slots $S_1, \cdots, S_n$ and $n$ boxes $B_1, \cdots, B_n$. ...
7 votes
3 answers
711 views

Is there a "geometric" language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions. Here, ...
6 votes
0 answers
84 views

The distribution of certain Galois groups

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
2 votes
1 answer
71 views

Does pointwise convergence yield the convergence under Skorokhod topology?

Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
31 votes
1 answer
2k views

Connes's absolute geometry and Lurie's spectral algebraic geometry

Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
0 votes
0 answers
49 views

Automorphism group and independent sets in vertex-transitive graph

Suppose a graph $G$ is vertex-transitive. Then, is there any relation, or better how can the automorphism group $\operatorname{Aut}(G)$ aid in the computation of independent sets of $G$. I hope the ...
2 votes
1 answer
59 views

A compact family of holomorphic functions and their corresponding compact ranges?

Let $C = [0,1]^n \subset \mathbb{R}^n$ be the closed unit cube. For some open set $V \subset \mathbb{C}^n$ such that $C \subset V$, denote by $\mathcal{F}$ some compact family of holomorphic functions ...
3 votes
0 answers
88 views

Faithfulness of parabolic induction

I've only recently begun to study the representation theory of $p$-adic groups, so the following question might be quite silly. Let $F$ be a non-archimedean local field of residue characteristic $p$, $...
12 votes
5 answers
5k views

Number of spanning forests in a graph

Hello, I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels. Q1: I am ...
1 vote
0 answers
28 views

How can I create a cover for H's weight space?

$$ \mathcal{B}:=\left\{\mathrm{B}_{\boldsymbol{w}} \mid \mathrm{B}: \mathbb{R}^{d_{1}} \rightarrow \mathbb{R}^{q}, \operatorname{Lip}\left(\mathrm{B}_{\boldsymbol{w}}\right) \leq L_{B} \&\|\...
8 votes
1 answer
456 views

Existence of a vector field with a finite number of limit cycles.

The following question is related to the Seifert conjecture. Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
10 votes
0 answers
182 views

Surprisingly only real points on intersection of certains quadrics

Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by \begin{align} X_e &= 0\\ X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\ X_g &...
4 votes
1 answer
231 views

Quadratic extensions of cyclotomic numbers by absolute values of elements

Summary I was wondering whether there is an explicit description of the extension $\mathbb{Q}(\zeta_n, |z|)/\mathbb{Q}$ obtained from a cyclotomic field $\mathbb{Q}(\zeta_n)$ by adjoining any finite ...
3 votes
0 answers
137 views

Bailey's lemma in number theory

A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by $$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$ or equivalently $$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
2 votes
0 answers
93 views

Compactness of a nonlinear operator

Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$ ...
1 vote
0 answers
47 views

Ramsey-theoretic properties of Erdős cardinals

The $\beta$-Erdős cardinal is defined as the least ordinal $\eta$ such that $\eta \to (\beta)^{\lt \omega}_2$, that is for every function $f: [\eta]^{\lt \omega} \to 2$, there is an $f$-homogeneous ...
5 votes
1 answer
174 views

Points of differentiability of squared distance from a point in metric spaces

I posted this same question on MSE with no answer. Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...

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