# All Questions

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### Проверка статистических гипотез

Проверка гипотез В математической статистике рассматривается задача проверки гипотез: по выборке проверить гипотезу Н(о)-все наблюдения выборки имеют распределение Р(о) при альтернативе Н(1)-все ...
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### about the horizontal lift in a principal bundle

I'm currently studying Fibre Bundle by Nakahara's book, and I'm a bit confused about the following: Imagine we have a Principal Bundle $P(M,G)$ with open chart {$U_i$} and a local section ...
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### Advanced use of commutation matrices

I am aware of matrix operators vec and kronecker product, commutation matrices and various related identities like stated in ...
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### Can all the sporadic groups be expressed as permutation groups based on a single big cycle?

Working on M11, I came up with that it can be generated using the following permutations: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [[2, 0, 1, 7], [3, 4, 5, 6]] [[4, 0, 6, 7], [2, 3, 1, 5]] [[0, 7], [4, 6], ...
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### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$? For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...
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### What characterizations of relative information are known?

Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as relative information, relative entropy, information gain or Kullback–Leibler divergence is defined ...
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### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ? How can we prove the ...
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### Convex Optimization in an Ellipsoid

Suppose we want to minimize a linear objective inside an ellipsoid that is, $\min _x l^Tx$ such that $(x - \mu)^TA(x - \mu) \leq \beta ^2$. Here, A is PSD and $\mu$ is a fixed vector. Can this be ...
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### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...
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### all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...
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### 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 ...
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### P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow. Besides, I know that there ...
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### A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$ [on hold]

The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$. I am a prospective undergraduate mathematics student in Zimbabwe ...
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### Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated? In view of Ian Agol's answer, I ...
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### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} ...
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### F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...
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### Horn's spectrum problem with random Hermitian matrices

An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. ...
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(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ... 0answers 17 views ### Fractal Dimension Estimate from Hurst Paramater using Higuchi's Algorithm [on hold] I am using R Stats package to calculate the Hurst parameter of my Center of Pressure Data set. I am using Higuchi's Algorithm to calculate it. I went and read his paper and saw that "index H is ... 1answer 44 views ### Proving moduli of uniform continuity in RCA_0 Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$to prove various theorems of analysis for all continuous functions with a suitable modulus of uniform continuity. And he ... 0answers 15 views ### regularization and conversion of sgn(x) to difference of convex [on hold] sgn(x) or sign function has discontinuity in 0 which make it nonconvex function. however i have tried to represet sgn(x) as a limit for a sequence of converging function which are smooth and ... 2answers 96 views ### Simple question: different definitions of Bousfield localization I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let ... 0answers 54 views ### Strongly real elements of odd order in sporadic finite simple groups Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ... 3answers 209 views ### family of polynomials with square discriminant The title pretty much sums it up: do people know of nice parametrized families of polynomials (with integer coefficients) with square discriminant. I should say that one such family consists of ... 0answers 36 views ### Compute the index of the Dirac operator on$C_0(R^2)$to obtain Bott element in$K_0$I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case$G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ... 2answers 307 views ### Deep/precise relationship between two approaches to FLT for polynomials,$n = 3\$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...