# All Questions

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### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...
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16 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
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### Does null geodesic flow live on a natural compact bundle?

Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary). A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$. The geodesic flow can be seen as a ...
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### Prove that the Log-Euclidean distance is negative-definite

Let $\Bbb{S}_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices. The Log-Euclidean distance between two points of ...
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Let $f:{\mathbb R}\rightarrow {\mathbb R}_+$ be a density function with finite expectation. This is, $$\int_{\mathbb R}x f(x)dx<\infty.$$ Suppose that we want to integrate $I(a)=\int_a^{\infty} x ... 0answers 26 views ### Euler equation formula [on hold] when I am using Euler equation for Fourier transform integrals of type$ \int_{-\infty}^{\infty} dx f(x) exp[ikx] $I am getting following integrals:$\int_{-\infty}^{\infty} dx f(x) cos(kx)$(for ... 0answers 38 views ### Twisting by a multiplicative Character in Katz, Perversity and Exponential sums Let$C(x_1,\ldots,x_n)$be a nonsigular cubic form with integral coefficients. In his Proof that$C$fulfills the Hasse-Principle, if$n\geq 9$, Hooley used the following estimate that was provided ... 0answers 52 views ###$F[[T]] \times F[[1/T]]$fundamental domain, show compactness Let$p$be a prime number. What is the easiest way to see that$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$is compact? Here$\mathbb{F}_p[T, 1/T]$is embedded in ... 1answer 58 views ### Jacobson-Morozov theorem Jacobson-Morozov theorem for a semisimple algebraic group$G$(presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism$\phi$from$SL_2$... 1answer 39 views ### Comparison of Lp norm of matrix and its entry wise norm. [on hold] I need to know the relation between operator norm of a matrix i.e.$ \Vert A\Vert_p$for case of p=1 and 2 and its entry wise Frobenius norm$ \Vert A\Vert_F$. 0answers 45 views ### What is the symmetry of SU(3) - when seen as a manifold? [on hold] Simply asked: is it more correct to state that the symmetry of the SU(3) manifold is$Z_3$or$S_3$? Or neither of the two? SU(3) has a kind of threefold symmetry; but which one exactly? When ... 0answers 38 views ### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way Here is the claim: Given a polytope$K$in a unit ball in$\mathbb{R}^d$, there exists a universal constant$C(d)>0$depending only on$d$and a countable collection of convex polytopes ... 0answers 53 views ### How do i show that the fixed points of this dynamics$ x_{n+1}=x_{n}^2-x_{n-1}^2 $are stable? [on hold] Is there somone who can show me how do i show that the fixe point of this dynamics $$x_{n+1}=x_{n}^2-x_{n-1}^2$$ are stable ?$x_{0}+x_{1}>0 $,$x_{0}=0,x_{1}=\frac{1}{2}$*My attempt only I ... 0answers 39 views ### Normal Sub-groupoid [on hold] Is there a definition of normal groupoid? For normal sub-quasi-group I found two: The first one: a sub-quasi-group$H$is called normal if there exists a normal congruence$\theta$such that$H$... 0answers 30 views ### Expected size of determinant of$AA^T$for non-square random Toeplitz$A$If$A$is chosen uniformly at random over all possible$m$by$n$Toeplitz (0,1)-matrices, what is the expected size of the value of the determinant of$AA^T$? We can assume$m \leq n$and all ... 2answers 71 views ### Identities involving sums of Catalan numbers The$n$-th Catalan number is defined as$C_n:=\frac{1}{n+1}\binom{2n}{n}=\frac{1}{n}\binom{2n}{n+1}$. I have found the following two identities involving Catalan numbers, and my question is if ... 2answers 87 views ### Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem. My bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are ... 0answers 38 views ### TTF triples are recollements The notion of recollement $$\mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} ... 0answers 56 views ### n-recollements and perverse t-structures A recent preprint on arXiv brought my attention on the notion of n-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a ... 0answers 42 views ### Beauville's Integrable System with singular spectral curves Let us consider Beauville's Integrable System. So, we live on \mathbb{P}^1. There is the moduli space of matrices M_r(d)/\mathrm{PGL}(r) with polynomial entries of degree less than or equal d. ... 0answers 59 views ### Normal subgroupoid? [on hold] Is there a definition of normal groupoid? For normal sub-quasi-group I found two: The first one: a sub-quasi-group H is called normal if there exists a normal congruence \theta such that H ... 0answers 10 views ### K nearest neighbors estimation with a kernel If I have a bunch of data points x_1,\dots,x_n, I can build a density function f(x) based on these data points by defining f(x) = c/d_k(x) for an appropriate constant c, where d_k(x) is the ... 0answers 50 views ### All relations among degree n monomials in n variables In the course of my work, I have run into the problem of finding exactly all relations among degree n monomials in k[x_1,\dotsc,x_n], with specific interest in the case n=3 (e.g. x_1^2 x_2 ... 1answer 193 views ### Idea of using etale site I have just read an article which mentions that, when Grothendieck considered using etale morphism, he did borrow the idea from Riemann that multivalued function on an open subset of complex plane ... 1answer 25 views ### Convergence to equilibrium via gradient descent J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ... 0answers 54 views ### Proof of formula for dimension of moduli of stable vector bundles on smooth curves [migrated] Let C be a smooth curve of genus g \ge 2 over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on C of ... 1answer 114 views ### Model over DVR for smooth projective curves Let C be a smooth, projective, geometrically irreducible curve of genus at least 2 over a complete discrete valued field F of characteristic zero (not necessarily algebraically closed). Let R ... 0answers 163 views ### Getting back into advanced mathematics [on hold] I hope that this is the correct forum to post my question on. I'm a 37 year old IT professional working in the Banking Sector who previously studied for a PhD in Computattional Fluid dynamics back in ... 0answers 37 views ### Relation between linear independence of lattice vectors and the toric variety defined by that lattice I have been reading some basic and elementary work on toric varieties, but even though people assured me that toric varieties are very well understood, several questions remained. Setup. Let ... 0answers 13 views ### lower incomplete gamma function Holomorphic extension [on hold] How to use repeated application of the recurrence relation for the lower incomplete gamma function to lead to the power series expansion? 0answers 45 views ### C^1 regularity of harmonic functions on Riemannian manifolds Consider a smooth, connected and complete Riemannian manifold M. It is well known that harmonic functions defined on some open subset of M are C^\infty. I'm interested in knowing whether there ... 0answers 36 views ### 3D matching modification Consider all instances of the 3D matching problem where all edges that intersect-intersect in "exactly" one vertex (1-edge intersection). Consider all instances of the 3D matching problem where ... 0answers 89 views ### Proof of Arnold-Liouville theorem in classical mechanics [on hold] I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ... 0answers 8 views ### The inter-request time distribution after aggregating some arrivals in the renewal process This is a follow-up question of the question "Aggregate arrivals from a Poisson Process" The inter-arrival time of a renewal process, t, conforms to a general distribution, denoted by PDF ... 0answers 28 views ### Algebraic independence in normed spaces A set of n points in \mathbb{R}^2 is algebraically independent over \mathbb{Q} if there is no polynomial dependency among the 2n coordinates. A result (Lemma 3.3) from "Globally linked pairs ... 1answer 86 views ### Expected size of determinant of AA^T for non-square random A If A is chosen uniformly at random over all possible m by n (0,1)-matrices, what is the expected size of the absolute value of the determinant of AA^T. We can assume m < n and all ... 0answers 152 views ### Question about Fermat's Last Theorem [on hold] Is there a way to prove that having x \gt 0, z \gt 0, n \gt 2 with x, z, n \in \mathbb{Z},$$ \sum_{k = 0}^{n - 1}{\binom{n}{k} x ^k} = z ^ n $$have no solution without using Fermat's Last ... 0answers 26 views ### A general method to integrate rational functions [on hold] \int\frac {x^3}{1+x^5} ATTEMPT: I did the following substitution: Let x=\frac{1}{t}. dx=\frac{-1}{t^2}dt. substituting back: I=\int\frac{-1}{1+t^5}dt which doesn't seems a simpler ... 0answers 16 views ### On important functions relflecting spectral properties of Jacobi operators [migrated] The spectral analysis of Jacobi (semi-infinite, tridiagonal) operators acting on \ell^{2}(\mathbb{N}) is deeply investigated. A crucial role is played by function m which is usually known as Weyl ... 0answers 60 views ### Volume of arithmetic quotients of symmetric spaces Now let \textbf{G} be some connected semisimple linear algebraic group over a number field F. Let G_{\infty} be \textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F). Let K_{\infty} be a maximal ... 1answer 80 views ### Differential Operators On A Curve And On Osculating Circle Given a 1D Riemannian manifold \Gamma embedded in 2D Euclidean space (e.g. a parametric curve on a plane \mathbb{R}^{2} ), and point x_{0}\in \Gamma, we denote S^{1}(x_{0}) the circle ... 2answers 105 views ### Subsets of \mathbb{N} whose lower density respects complements The lower density of A\subseteq\mathbb{N} is defined to be \lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}. We set$${\cal C} = \{A\subseteq \mathbb{N}: ... 4answers 542 views ### Algorithms for calculating R(5,5) and R(6,6) Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said: Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ... 0answers 31 views ### Logic resolution and logic consequence [on hold] Which of this are false? a) If some formula H results from premises D, then H could be derived from D with using resolution (reapetedly) rule. b) If some formula H results from premises D, then we ... 3answers 470 views ### Determining if a matrix is orthogonal Let g be an element of$GL_n(\mathbb C)$. We know that there are orthogonal groups$O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$for any$\beta$, invertible symmetric matrix. Though these ... 0answers 95 views ### If C is a closed, unbounded subset of Ord, are the sets generated by the new class of ordinals Ord restricted to C, a Universe? Given any universe, and some closed unbounded subset of Ord, is it possible toy generate a universe in the following way: Restrict Ord to a target club. Then generate all look the sets necessary to ... 2answers 647 views ### History of Geometric Analogies in Number Theory My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to ... 0answers 24 views ### Fast Algorithm to compute the Discrete Fourier Transform with a constraint on the summation index I really appreciate if anyone can help me with this problem. Problem: Let$W_n=e^{\frac{2\pi i}{N}}$which is the$N$th root of unity. The backward Discrete Fourier Transform of a complex vector ... 1answer 74 views ### A question about the Vandermonde determinant We know that the Vandermonde determinant of order$n$is the determinant defined as follows:$$\begin{vmatrix} 1&x_1&x_1^2&\dots&x_1^{n-1}\\ ... 0answers 36 views ### Mutual Information: How these two equations are equal? [on hold] I'm a biologist trying to apply the Mutual Information (MI) to some RNA secondary structure. I know that there exists two MI equation that, mathematically, are equal:$I(X,Y) = \sum_{x,y} p(x,y) ...

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