# All Questions

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### All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open: Given ...
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### Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit ...
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### Extending inequality for $\ell^p$ with integer $p$ to real $p$

Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are two decreasing sequences of positive numbers such that $a_1<b_1$ and $$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ ...
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### Exact statistics in the Frobenius coin problem

The Frobenius coin problem guarantees that if $(a,b)=1$, then $$ax+by$$ does not represent exactly $(a-1)(b-1)/2$ numbers all below $ab-a-b$ if $x,y\geq0$ holds. I am confused by following argument. ...
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### Self avoiding walk problem?

As in the image we can see that there are black spots and moving from spot to another is 1 move. Can we create a function which will tell us the position after say 119 moves, 143 moves etc without ...
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### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
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### Reference request: Linear evolution equations of “hyperbolic type”

Does anyone have any accessable link to the following paper by Kato? Linear evolution equations of “hyperbolic type” Note: It is the first paper, not the sequel numbered by II. After several ...
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### spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
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### Smoothness of the fourth power of the geodesic distance in a Finsler geometry

The simplest form of Finsler metric is: $ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth ...
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### Genericity of irreducible automorphisms of free groups

I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic". I would like to ask that if for example : it is true that for any generating set $X$ of ...
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### Weighted maximal number of disjoint chains in the integer divisibility poset for $\{1,2,\ldots,n\}$

In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset ${\cal P}_n$ of the set of natural numbers $\{1,\ldots,n\}$ is considered. I have a somewhat dual ...
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### Size of KL-divergence neighbourhoods

I am new here. I was reading another post here and this got me wondering what can be said about the size of the following kl divergence neighborhoods. Consider these two kl-divergence neighbourhood ...
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### Distribution of $e^*f$, if $e$ is a complex Gaussian vector and $f$ is a unit norm complex vector

Let $e$ be a complex Gaussian vector where its elements are of zero mean and variance equals to $\sigma^2$. In addition, we define $f$ as a complex random unit norm vector uniformly distributed. Note ...
Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot ... 1answer 37 views ### Reference: First publishing of Mallivain Derivative as First Variation Who first showed the Malliavin derivative to be expressible in terms of the first Variation of the process it was deriving? 0answers 19 views ### Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions Let \mathfrak{g} be a complex simple Lie algebra with maximal torus \mathfrak{h}, Weyl group W. The adjoint representation \operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)} extends ... 11answers 24k views ### Sum of 'the first k' binomial coefficients for fixed n I am interested in the function \sum_{i=0}^{k} {N \choose i} for fixed N and 0 \leq k \leq N . Obviously it equals 1 for k = 0 and 2^{N} for k = N, but are there any other notable ... 1answer 158 views ### realizing uniform boundedness of Galois representations associated to elliptic curves This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear). Suppose I have an ... 1answer 41 views ### Supercommutator of exterior multiplication operators and their adjoints Let \mathfrak{h} be a complex Hilbert space and consider Grassmann algebra \mathcal{F}=\bigwedge\mathfrak{h} with its induced inner product. For \omega\in\mathcal{F} we also consider the ... 3answers 205 views ### Classification of open subset of \mathbb{R}^{3} [on hold] There is a theorem which gives a classification of connected open sets of \mathbb{R}^{2}. Unfortunately, I don't remember the correct statement, but it looks like this Theorem ? Let ... 1answer 338 views ### optimization of inverse matrix with constraint on matrix elements everyone! I have this optimization problem with constraint. D and T are symmetric matrices, where T is known and D is the unknown parameter. x and v are two known p-dimensional vectors. The ... 1answer 448 views ### Grothendieck's paper on principal bundles, reduction to a torus step In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma ... 5answers 3k views ### Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA? After having read Gunnar Carlsson's Topology and Data I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge ... 1answer 93 views ### How many points are in such set with the same norm-2 Let L=[a,b]\cap\mathbb{N} with a,b\in\mathbb{N}, let D\in\mathbb{N}, and let C=L^D. Then I would like to know how many points are there in C with the same given norm-2 d. I.e., I'm looking ... 1answer 216 views ### Nice sign-expansions of special surreal numbers What is the "right" surreal generalization of the fact that a real number r is rational if and only if its sign-expansion is eventually periodic? I can think of more than one natural way to ... 0answers 16 views ### Calculation of minimal right \operatorname{add}(M)-approximations given a finite dimensional quiver algebra A and a generator M with \operatorname{Ext}^1(M,M)=0. By Wakamatsus lemma, for any A-module N there exists a surjective A-linear map f\colon M_1 ... 0answers 73 views ### A family of maximal ideals Let m_i , i \in I, be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist j \in I such that \cap_{i\not= j} ... 0answers 176 views ### How close to being well-orderable does this make my powerset? Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set k satisfying k\times k \simeq k, and consider its powerset X = 2^k. I have a technical condition ... 1answer 92 views ### On linear integer inequalities with infinitely many solutions Suppose that a linear system of inequalities Ax \le b, where A\in Z^{m\times n} and b\in Z^m have integral coefficients, has an infinite number of integral solutions x. Can one conclude that ... 2answers 185 views ### p-torsion of an abelian variety of p-rank 0 Let k be an algebraically closed field of characteristic p > 0 and let A be an abelian variety over k such that A[p](k) = 0, i.e., such that A has p-rank 0. If I am not mistaken, ... 1answer 95 views ### Schauder estimate for the heat equation on compact manifolds I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here. Let M be a compact manifold without boundary. Consider Lu:=\partial_tu-\Delta u. Let f\in ... 0answers 83 views ### Is stable map space \overline{M_{0,n}}(\mathbb{P}^n,d) is irreducible for all n,d? I read a paper "Notes On Stable Maps And Quantum Cohomology, W.Fulton and R.Pandharipande". And I think that \overline{M_{0,n}}(\mathbb{P}^n,d) is irreducible. But I cannot find an exact statement ... 1answer 103 views ### Inequality for the maximum of Gaussian variables Let X=(X_1,\dots,X_n) and Y=(Y_1,\dots,Y_n) be centered Gaussian vectors with variance matrix \Gamma_X and \Gamma_Y. We assume that the matrix \Gamma_Y-\Gamma_X is positive definite. Is it ... 0answers 52 views ### Universal Witt vectors in full complete closed p-adic space omega? Is there a p-adic mathematical structure that incorporates the advantages of both universal Witt vectors (not p-typical-limited; implementing Frobenius and Verschiebung operations) and permitting ... 0answers 39 views ### How to calculate product of the coefficients of a polynomial? [on hold] I have a recurrence equation that results in a polynomial. The quantity I am interested in is the product of the coefficients of the terms in the polynomial. For example, if f_2(x)=a0+a1x+a2x^2, I ... 0answers 43 views ### Composition of derivations is zero on commutative ring? [on hold] Let Z be a commutative domain char Z>n>1.If d_{1},..,d_{n} are such derivations of Z that the composition d_{1}...d_{n} is a derivation,then d_{i}=0 for some 1\leq i \leq n. Can ... 0answers 24 views ### For a \sigma-finite measure is u_*(E)=lim_iu_*(E_i)? [on hold] let u be a \sigma-finite measure on a \sigma-ring S, let u_* be the inner measure induced by u and denote H(S) as hereditary \sigma-ring generated by S. {E_i} is an increasing ... 0answers 244 views +100 ### “The” natural double complex associated to a principal G-bundle? Let \pi: P \to M be a principal G-bundle. We have the associated adjoint bundle ad(P)= P \times_{ad} \mathfrak g whose sections correspond to infinitesimal guage trasformations. Consider the ... 0answers 41 views ### What are the explicit expressions of quantum Casimir elements for U_q(sl_3) and U_q(sl_4)? What are the explicit expressions of quantum Casimir elements for U_q(sl_3) and U_q(sl_4) in terms of E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}? Any help will be greatly appreciated! 3answers 491 views ### An inequality improvement on AMM 11145 I have asked the same question in math.stackexchange, I am reposting it here, looking for answers: How to show that for a_1,a_2,\cdots,a_n >0 real numbers and for n \ge 3: ... 1answer 247 views ### Improper integral \int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx with -a and b positive Is the following function real analytic in t>0:$$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$where -a and b are positive, and c\not=a? ... 2answers 437 views ### Uniform proof that a finite (irreducible real) reflection group is determined by its degrees? Given a finite (irreducible real) group G generated by reflections acting on euclidean n-space, it was shown by Chevalley in the 1950s that the algebra of invariants of G in the associated ... 0answers 128 views ### Time Hierarchy Theorem and P vs NP One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ... 1answer 312 views ### Homogeneous metrics of given volume element, on the n-sphere. Let \omega be an n-form on S^n, nowhere vanishing. Is there a Riemannian metric g on S^n, so that its volume form is \omega, and (S^n,g) is homogeneous? Is it unique, and if not, what ... 0answers 83 views ### A tricky 2d integral [on hold] I tried to calculate such integral:$$ \int d^2q \frac{\bf{q+q_2}}{(\mathbf{q}^2+m^2)((\mathbf{q-q_1})^2)^{1-i\eta}(\mathbf{q+q_2})^2}  where $q$,$q_1$ and $q_2$ are two dimensional vectors. Can ...
There is a classical result commonly attributed to W. Sierpiński that reads as follows: Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ ...