# All Questions

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### Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture

Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
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### local systems with cyclic monodromy

In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows: Let $X$ be a smooth projective variety over some field $k$ of ...
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### Derivative of Set Functions

For any closed set $I\subseteq (0,1]$, I have a set function in the form of $$\theta(I)=\int_{q(I)}YdF(Y)$$ Where $q(I)$ if the image of I under the mapping $u \mapsto q(u)$. And $F(\cdot)$ is the ...
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### Derivative of diag (A) with respect to A [on hold]

Assume matrix A has no special structure, how to calculate ? Where means getting the element from the diagonal. I'm not sure whether $$\delta(A)$$ is a column vector consisting the diagonal element ...
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### Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular variety $X$, and let $X/G$ be the categorical quotient. I know that if $X$ has rational singularities, then so does $X/G$ ...
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### Hessian in local coordinates [migrated]

Let $f\colon M\to \mathbb{R}$ be a smooth function on a manifold $M$ with a critical point $p$. We define its Hessian at $p$ via $H(u, v)=(UVf)(p)$ where $u, v\in T_pM$ and $U$ and $V$ are vector ...
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### coends of stable infinity categories

Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote ...
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### Given an even function how to obtain the most close odd function and vise versa?

Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$? By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference ...
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### Positivity question on K3 surfaces

Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$. (Q1). do we have $L\cdot D\geq0$ ? If either one has positive self-intersection, the ...
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### Fourier Analysis in Kahane and Zelasko's characterization of maximal ideals in commutative Banach algebra

I've been told that Kahane and Zelasko (in A characterization of maximal ideals in commutative Banach algebra's Studia Math 29 1968) use a "non-trivial result" from Fourier analysis in their proof of ...
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### Exact Prime Counting Function [on hold]

There have been dozens of prime counting functions $\pi(n)$ that have been created over the years and yet the most common ones I see are either estimates (prime number theorem) or they require the ...
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### Which hyperbolic tilings are Cayley graphs?

I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here: Given a regular tiling of the hyperbolic plane is ...
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### Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS. It doesn't appear at this time. I can't find any reference to it either. Has anyone seen it? ...
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### Is this integration by parts legitimate?

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,$$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
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### A question about the inverse of a matrix [on hold]

and it's obvious that det A =w=1 or -1. My question is simple, but it seems to be chaos. I will be very happy if someone can replay.
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### reference request: Riesz/Newton potential and HLS inequality in L1.logL1

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,$$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
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### Bivariate Function Approximation

I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question: For any bivariate ...
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### What is a good poster for a math conference?

I'm going to participate to a conference and they ask me to do a poster on my research. I've never made a poster for a conference/seen a poster session in a conference. So what is important? What do ...
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Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ... 0answers 75 views ### A section from subfactors to transitive groups A finite group-subgroup subfactor is a subfactor$(N \subset M)$isomorphic to$(R^G \subset R^H)$with$(H \subset G)$an inclusion of finite groups acting as outer automorphism on the hyperfinite ... 1answer 83 views ### Pullback of$L^p$functions via exponential map Let$M$be a complete Riemannian manifold, endowed with its exponential map$\exp: TM \longrightarrow M$. For any$C^k$- function$u, we get the Pullback $$\exp^* u = u \circ \exp$$ which is in ... 0answers 83 views ### Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion Is there a way to solve analytically the Fredholm integral equation of the second kind $$\int_0^{100} K(s, t) f(s) ds = \lambda f(t)$$ where the kernel has the piecewise 'linear' form \begin{align} ... 0answers 75 views ### 4-chromatic cubic graph that does not contain K4 [on hold] Is there a 4-chromatic cubic graph that does not contain the complete graph K4 as a subgraph? 0answers 76 views ### Wanted: a nontrivial weakly inadmissible Heegaard diagram This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram ... 2answers 193 views ### Understanding the definition of the quotient stack[X/G]$I'm trying to understand the definition of the quotient stack$[X/G]$as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles. Explicitly, let$G$be an affine smooth group ... 0answers 54 views ### Transmitting binary signals between vertices of the graph [on hold] We have some random undirected graph. Each edge at discrete moment of time can be activated or deactivated. For each vertex of graph we know$f: S \rightarrow S$.$S$- set of all possible states of ... 1answer 106 views ### Order of the zero of a meromorphic function under the action of$Gal(\mathbb{C},\mathbb{Q})$Let's take$X$a Riemann surface as an algebraic curve in$\mathbb{P}^n$. The group$Gal(\mathbb{C},\mathbb{Q})$(automorphisms of$\mathbb{C}$which act as the identity on$\mathbb{Q}$) acts on$X$... 0answers 52 views ### tangent developable surface in$\mathbb{R}^3$Let$C$be a regular curve embedded in$\mathbb{R}^3$(i.e. a real 1-dimensional manifold embedded in$\mathbb{R}^3$). Let$S$be the union of its affine tangent lines: $$S=\bigcup\limits_{p\in ... 1answer 102 views ### A question about the “size” of the neighbourhoods in which bundles are trivializable My question is about domains of trivializability of distributions on a smooth compact manifold. Assume that you have a sequence \{E^k\}_k of C^r distributions of rank n which is C^0 close to a ... 3answers 1k views ### Silly me & Van der Waerden conjecture So I walked into this very innocent-looking combinatorics problem, and quite soon I ended up with the problem to prove that any doubly stochastic n \times n matrix has a non-zero permanent. Now ... 1answer 90 views ### About generalized Minkowski inequality For which functions f:\mathbb{R}^+ \to \mathbb{R}^+ does the inequality f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ... 2answers 395 views ### An identity in an arbitrary commutative ring This fact might be either trivial, wrong, or well known. Let R be a commutative ring. Let u_1,\dots,u_{s-1},u_s\in R and m,M\in R. Let us assume that m,M satisfy$$(m-u_1) \dots ... 1answer 187 views ### Homotopy of localisations of colimits Let$X_k$be a family of spectra equipped with maps$f_k: X_k \to X_{k+1}$. If$Y$is a compact object (such as a sphere), then I can compute homotopy classes of maps from$Y$into the homotopy ... 0answers 158 views ### Blowdown and contraction I am sorry, my question is very naive. 2nd Edit: Let us suppose that$V$is a smooth complex projective variety, and$Y\subset V$is a smooth divisor and has an ample conormal line bundle. We would ... 1answer 179 views ### Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras? I am interested to identify (ideally classify) nilpotent Lie algebras that occur as nilradicals of parabolic subalgebras in (say) reductive Lie algebras. For example, all Heisenberg Lie algebras ... 1answer 212 views ### rationality question while dealing with an isogeny I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment ! So here is the situation. Let ... 0answers 153 views ### Connections between Riemann hypothesis for curves over finite fields and Ramanujan property for graphs this question relates to the beautiful construction of expander graphs using Cayley graphs of$PGL_2(\mathbb{F}_q)$, as exposited by Davidoff-Sarnak-Valette in their book, Elementary Number Theory, ... 1answer 123 views ### What do algebraic theories with strictly terminal trivial models look like? By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ... 0answers 149 views ### An optimization problem on the sphere Let$S$be a sphere centered at origin in$\Bbb R^{2n}$of radius$\sqrt{2n}$. Let$D$be a diagonal matrix. Let$U$be an orthogonal matrix. Let$r\in\Bbb Z_+$be a fixed integer. Let vector ... 1answer 99 views ### connections on principal bundles over$S^1$Suppose$G$is a compact connected Lie group and$P$is a$G$-bundle over$S^1$,$A$is a connection. Then we can choose a frame such that$A = a d\theta$where$a\in \mathfrak{g}$is constant. My ... 1answer 283 views ### Finite groups for which the element orders form an arithmetic progression Which are the finite groups$G$such that the element orders of$G$form an arithmetic progression? Several remarks:$S_3$,$A_4$and any$p$-group of exponent$p$satisfy this property. If$G$... 1answer 149 views ### Fiber products and torsors Suppose$G$is a finite linear group, and I have a$G$-torsor$Z \to X$. Suppose also I have a morphism$f : Y \to X$with some properties$P$. What should these properties$P$be in order to make the ... 0answers 25 views ### A combinatorial identity [migrated] Can anyone prove the following identity for me?$\sum_{i=0}^k \begin{pmatrix} n\\ i \end{pmatrix} \begin{pmatrix} -n\\ k-i \end{pmatrix}=0$for any positive integers$n,k$. I'm pretty sure this is ... 0answers 33 views ### About eigen value of a Sturm-Liouville equation [on hold] What is the distribution of eigenvalues of the following equatuion: $$y''+(\lambda r(x)-q(x))y=0$$ where$r(x)\$ has sigularity?

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