# All Questions

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### exit time of a non degenerate diffusion

Let $n, d \geq 1$, $b : \mathbb{R}^n \to \mathbb{R}^n$ and $\sigma: \mathbb{R}^n \to \mathbb{R}^{n \times d}$ two Lipschitz functions. We assume that \exists \mu >0, \xi^T ...
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### Non Normal operator

Standard example for non normal operator is the shift operator. It is continous but the image of the left shift is not dense. Can we have an example of a non normal operator $A$ which is continuous ...
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### Group laws in class field theory

In the case of quadratic number field one can construct it's maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field. In the case of local field ...
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### Questions about a possible way of representing construcive ordinal numbers

Let $K$ be the set of all total recursive functions of non-negative integers having only non-negative integers as values. Let $L$ be any well-ordered subset of $K$ in which the ordering $<$ is ...
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### Can the GUE be thought of as a uniform point in a high-dimensional polytope

I have thought about this question for a long time and could only find partial answers. The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
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### What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
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### Adding sets not containing arithmetic progressions of length three by forcing

Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where: 1) $s\in 2^{<\omega}$, 2) $N\in \mathbb{N}$, 3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...
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### separating parameters in generalized quadratic Gauss sum

The normalized generalized quadratic Gauss sum is defined by $$G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right)$$ where $e(x)=\exp(2\pi ix)$. Under what conditions on $c$ can we ...
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### Tree property and singular strong limit cardinals

I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question: Theorem. It is consistent, relative to the existence of large cardinals, that ...
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### When is a word metric on a CAT(-1) group a bounded distance from some CAT(-k) metric?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space. Let $d$ a word metric on $\Gamma$ coming from some finite set of generators. My question is: Does there exist a ...
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### Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group ${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use the low index subgroups ...
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### Which monoids contains word a^2 b^2? [on hold]

I have a problem with one simple exercise. I don't know how I should start. The question is: Which monoids contains word $a^2b^2$: (a) {a,b}* (b) {abb,a}* (c) {aa,bb}* (d) {ba,ab}* (e) ...
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### Tensor calculus on the frame bundle

Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its ...
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### Application of Stickelberger's Theorem to Quadratic field

I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...
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### About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$. Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$. Let ...
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### Minimum rank non-negative matrix summations

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$ of rank $r$. What is minimum $k$ such that $$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains $R,S$ ...
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### Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared: $$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$ where $\rho,s>1$, ...
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### I don't get how -6cos3xsin3x becomes -3sin6x in the later part [on hold]

y = cos²3x dy/dx = 2cosx(-sin3x)(3) = -6cos3xsin3x = -3sin6x I found this answer key in my guidebook but I can't find any trigonometric function's or differentiation formula ...
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### Unimodular triangulation and affine toric variety

Let $\mathcal{K}$ be a $pointed$ rational cone in $\mathbb{R}^d$ with extremal rays generated by $r_1,r_2,\dots, r_m\in \mathbb{Z}^d$. Here, pointed means that all $r_i$ lie strictly on one side of ...
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### Sum-epimorphisms and prod-monomorphisms

Sum-epimorphisms A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition: ...
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My goal is to find a way to calculate the convex hull of the union of some parameterized curves. For instance, I had to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup ... 0answers 53 views ### Alternating quotients of (2,3,7;10) It was shown that the only quotient of the group$G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ... 0answers 20 views ### Is it possible (or even valid) to obtain an eigensystem from a set of recursive equations? [on hold] Let us start with the Golden Ratio, which is the number$\varphi \approx 1.6180339887\cdots$, and it can be defined in several ways, one of them is through a recurrent process involving the Fibonacci ... 0answers 31 views ### formula for turning star reviews into upvotes [on hold] I want to turn reviews of up to 5 stars and the number of reviews into upvotes. Whats a good algorithm for doing this? A venue with 10 reviews total with a 5 star average rating should obviously get ... 0answers 62 views ### For which quiver varieties is Kirwan surjectivity known? The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in ... 1answer 201 views ### Is the ring of invariants Noetherian? Let$R$be a complete regular local ring whose residue field is perfect. Suppose that a finite group$G$acts on$R$by ring automorphisms in such a way that the induced action on the residue field is ... 0answers 164 views ### Can an abelian variety/Q have no points over Q_sol? Let$A/\mathbb{Q}$be an abelian variety. Must there be a finite solvable extension$K/\mathbb{Q}$such that$A(K)$is nontrivial? This follows from the conjecture that the maximal ... 0answers 35 views ### Probability of non-negative matrix relaxation Given matrix$M\in\Bbb\{0,1\}^{n\times n}$, take$\mathscr{M}[M]=\{Q\in R_{\geq0}^{n\times n}:Q[ij]>0\iff M[ij]=1\}$. Does ... 0answers 47 views ### flatness and derived completion Let$A$be a local ring of maximal ideal$\mathfrak{m}$. Let$\hat{A}$be its completion. If$A$is noetherian , then we know that$A\rightarrow\hat{A}$is faithfully flat. If$A$is not noetherian, ... 2answers 333 views ### Non-Forcing and Independence I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here. Do there exists sentences which are independent of ZFC, cannot be shown to ... 0answers 35 views ### Relative nonarchimedean disks and annuli Let$A$be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring$A_0$which is adic and has finitely generated ideal of definition. Is there a good notion of closed disk of ... 0answers 51 views ### What does this graph notation mean? E\S [on hold] I am studying graph theory right now but I am confused what E\S means where both E and S are sets of edges. What does the "\" indicate? 1answer 94 views ### Unitary representation with fixed Casimir Let$G$be a connected reductive real Lie group with Lie algebra$\mathfrak{g}$. We denote by$\widehat{G}_u$the unitary dual, that is the set of isomorphism classes of unitary reprensentation of ... 1answer 90 views ### Properties of Integral Closure [on hold] Definition(Integral closure): Let$R$be a ring and$I$an ideal of$R$. An element$x$is said to be integral over$I$if$x$satisﬁes a monic equation$x^n + i_1x^{n−1} + ··· + i_n = 0$such ... 1answer 52 views ### Are there compact riemannian manifolds whit Q-curvature negative? Are there known examples of compact Riemannian manifolds with Q-curvature negative? 0answers 111 views ### 3D models of the unfoldings of the hypercube? There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ... 0answers 82 views ### Lie algebra and Lie groups [on hold] A complex Lie algebra$L$has a representation on$Der(L)$by just putting$x.D=-ad_{Dx}$. For semisimple Lie algebras, by Weyl's theorem,$Der(L)$decomposes into irreducible subspaces$D_i$. Then my ... 0answers 28 views ### Population and date values [on hold] Use the population data values below. North= 18,200 South=12,900 East=17,600 West=13,300 If there are 26 representatives for all districts how many ... 1answer 86 views ### Combinatorical meaning of such expression [on hold] Any combinatorical meaning or interpretation of $$1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!$$ for partition ... 1answer 72 views ### Completeness of a set of propositional formulas [on hold] A set$\sum$of formulas in propositional logic is complete if for each propositional formula$\phi$either$\sum \vdash \phi$or$\sum \vdash \neg \phi$. Clearly every inconsistent set of formulas ... 1answer 60 views ### Classification of ergodic measures for circle expanding maps Let us consider the classical self-covering of the circle$S^1=\mathbb{R}/\mathbb{Z}$given by $$\times_d(x) = dx \mod 1$$ where the degree$d$is any integer greater than$1$. There are a wealth of ... 1answer 64 views ### Totally non fixed point property Edit: According to the comment of Pietro Majer, I revise the question Is there a non singleton compact connected Hausdorff topological space$X$for which the following property hold?: "Constant ... 0answers 129 views ### Reference for Arakelov's theorem:$K^2_f=0$iff$f$is locally trivial Let$f:X\longrightarrow B$be a family of curves, with$f$relatively minimal, over a fixed curve$B$($B$is projective, irreducible and smooth). The fibration$f$is said locally trivial if all ... 0answers 60 views ### A jump operator for Borel equivalence relations It is well-known that with respect to Borel reducibility the class of Borel equivalence relations on a standard Borel space does not admit a maximal element. We can use the well-known Friedman-Staley ... 1answer 184 views ### Translates of meager sets Does there exist a meager set of reals M such that every meager set can be covered by countably many translates of M? This is the category analogue of the following. 0answers 52 views ### Name and theorems of transitive/Galois groups of quadratic (even) and cubic power polynomials: cyclic group extensions [on hold] What is the transitive group details of a polynomial where only the third power terms occur? That is${x}^{3n}+{a}_{n−1}{x}^{3(n−1)}+\cdots+{a}_{1}{x}^{3}+{a}_{0}$. I need the basic theorems that ... 2answers 142 views ### boundary homomorphism in the homotopy exact seqeunce of principal$SO(9)$bundle over$S^8$Consider principal$SO(9)$bundles over$S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$ Now pick up one such bundle$\xi$,we have the long exact sequence ... 0answers 37 views ### Orthogonal vector to arbitrary vector in R3 [on hold] I got a vector$(0, 0, 0)^T \neq v \in R^3$. Now I want a closed formula for some orthogonal vector to$w$(I don't care which). My problem is that if I, for instance, fix$w_1$and$w_2$then$w_3 = ...
There is a problem when I'm reading a paper. Equation: $min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$, where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and \$p^*, R(), D(), ...