# All Questions

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### Equitably distributed curve on a sphere

Let $\gamma=\gamma(L)$ be a simple (non-self-intersecting) closed curve of length $L$ on the unit-radius sphere $S$. So if $L=2\pi$, $\gamma$ could be a great circle. I am seeking the most equitably ...
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### Are there any special properties of graph eigenvalues of perfect matchings? [on hold]

Say if the graph is just a perfect matching of its vertices OR if it is an union of a few perfect matchings OR its an union of a few perfect matchings and a part of another? Anything if one further ...
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### Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
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### When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ ...
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### Irreflexivity of relations on sets [on hold]

How can I know if the relations: xy >= 1 and x=y+1 or x=y-1 Are irreflexive on Z(all integers)? Thank you!
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### Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether ( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...
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### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to \mathbb{R}$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
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### Does this function have any exponential growth?

Has anyone seen any function of the following type? $$g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.$$ The question is whether for some constant ...
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### What is the characterization of a graph Laplacian? [on hold]

Given a matrix, what properties must it have so that its ensured that there exists a graph whose Laplacian it would be? (...may be you can consider weighted and unweighted cases separately...) And ...
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### the ratio between product of two trace functions maximization

Consider the following Optimization [\begin{array}{l} \mathop {\max }\limits_{\bf{X}} \,\frac{{trace\left( {{\bf{XA}}} \right)trace\left( {{\bf{XB}}} \right)}}{{trace\left( {{\bf{XC}}} ...
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### Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies Normality ...
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### Estimating the moments of a random variable

Suppose i wanted to estimate the expectation and variance of a random variable $X$. More over suppose i could write a variable $X$ as a sum of indicator random variables $X=\sum_{i=1}^{k} X_{i}$. Are ...
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Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon ... 1answer 98 views ### Resolvent of a triangular matrix Suppose$A$is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in$x$) matrix$(xI-A)^{-1}$? Of course,$(xI-A)^{-1}= N(x)/p_A(x)$, where$p_A$is the ... 0answers 40 views ### Behaviour of Markov type under uniform homeomorphism of spheres A metric space$(X,d_X)$has Markov type$p$(with$p \in [1,2]$), if, for every stationary Markov chain$\{Z_n\}_{n=0}^\infty$on$Y$(a finite space) and every mapping$f:Y \to X$, one has $$... 1answer 51 views ### Estimating mean and variance of a distribution based on error-prone estimates of its cdf Suppose I have some random variable X taking values in [a, b] with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ... 1answer 271 views ### Could there be an exact formula for the Ramsey numbers? Let R(k) denote the diagonal Ramsey number, i.e. the minimal n such that every red-blue colouring of the edges of K_n produces at least one monochromatic K_k. Is it possible that there ... 0answers 18 views ### Question on Stationary & Cointegration Test (Augmented Dickey Fuller & Engle Granger test) [on hold] I'm performing the stationary and cointegration test on stock prices. What I'm confused is 1) the difference between ADF stationary test and ADF cointegration test. 2) Also, in ADF stationary test, ... 0answers 27 views ### Mean time for the renewal process [on hold] The system is as below. Energy keeps coming at a node with a constant rate \rho. Node has files of size exponential(\lambda) to be transmitted. At time zero, say the energy at the node be zero. ... 0answers 79 views ### Degrees of multilinear polynomials satisfying some constraints [on hold] Let t<\sqrt{n}. \Bbb Z^t[x_1,\dots,x_n]=\{f\in\Bbb Z[x_1,\dots,x_n]: deg(f)\leq t and f is multilinear\}. Fix an ordering of S=\{0,1\}^n. If f\in\Bbb Z[x_1,\dots,x_n], let f(S) be ... 0answers 64 views ### Derivability of a function defined on the tangent bundle. Foundations of Finsler metrics My question is linked to the foundations of Finsler metrics (with weak derivability assumptions). Let M be a manifold of dimension n, and F is a function from the tangent bundle TM to ... 1answer 94 views ### ideal of maximal minors is cohen-macaulay? Let k be an algebraically closed field. Let A be an m \times n matrix with linear forms a_{ij} \in k[x_1, \ldots, x_p]_1 as entries. Let I be the ideal generated by the maximal minors of ... 1answer 121 views ### On the size of centralizers in a non-abelian finite simple group It is known that for a finite non-abelian simple group G we have |G|<|C_G(x)|^3 for some involution x. Is there a better bound for the order of centralizer of a nontrivial element of G (not ... 0answers 101 views ### Example of symplectic 4-manifolds with no Lefschetz fibration structure? I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ... 0answers 60 views ### Asymptotic analysis of a sum of complex summands using integral I'm trying to find the exact asymptotics of a sum:$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$as n\rightarrow\infty. Here x,y are complex numbers, |x|\leq1, ... 0answers 85 views ### “simulteneous eigenvectors” under the full set of weighted Laplacians on a g-fold product of the Poincare half plane This question is closely related to the following MO question Characterizing the real analytic Eisenstein series Let \mathfrak{h}=\{z=x+iy\in\mathbf{C}\} be the Poincare upper half plane endowed ... 1answer 103 views ### Absolutely continuous functions it is well known that if a function f:[0,T]\to\mathbb{R} satisfies the inequality$$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$for$s<t$and some$m\in L^1([0,T])$then$f$is absolutely ... 1answer 121 views ### abelian p- subgroups of E_6(q) Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power? 1answer 172 views ### Non-abelian freeness of SU_2 The distribution of the trace of a random element of$SU_2$is the Sato-Tate distribution. The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution. ... 1answer 85 views ### Dehn twist about an arbitrary curve I need to know if there is an algortihm to write down a Dehn twist about an arbitrary curve on an orientable surface$S$, as product of a set of generators of$MCG(S)$. Since we have the conjugacy ... 0answers 42 views ### finding points on elliptic curve over finite field [on hold] Find the points on the elliptic curve y^2 = x^3 + 2x + 2 in F17 (field of prime 17). Do I have to guess a first point and then use an algorithm to spit out all other points? 3answers 2k views ### A game of stones How I arrived at this question is a rather long story having to do with the honors calculus class I am teaching. At this point it's sheer curiosity on my part. Here is the game. ... 0answers 100 views ### Proper monomorphisms in complex analytic spaces In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ... 1answer 175 views ### Example of proof using the generic matrix There is a really nice proof of the Cayley-Hamilton Theorem using the generic matrix. I expose it briefly. One defines the generic matrix$G:=(X_{ij})_{ij} ...
Given some integers $n > \beta + \alpha, \beta > \alpha$ and $\alpha > 0$, is there a real number $\delta$ for which $\frac{n-\alpha}{n-\beta} \geq (\frac{\beta}{\alpha})^{\delta}$, where ...
Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one. For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...