# All Questions

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### Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces. Under these ...
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### Hall-Littlewood functions and functions on the nilpotent cone

The following observation between the spaces of global sections of line bundles on the nilpotent cone and the Hall-Littlewood polynomials is made in a recent physics preprint 1403.0585. Is this a ...
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### Sheaffication using a $\lambda$-transfinite colimit

I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway. I was reading this article ...
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k algebraically field, A k algebra and valuation ring of K (K field fraction of A) and we have the transcendence degree of K over k is one. i want to ask if A is noetherian ring?
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### Computation of symplectic quasi-state

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a ...
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### Intersecting Sets of Pythagorean Triples with Common Hypotenuses

For any $r\in\mathbb{N}$, let $A_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$. Given any $N\in\mathbb{N}$, does there exist $r,s$ ...
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### I need the isoperimetric proof in geometry method

I need the isoperimetric proof by Jakob Steiner in 1838. Where can I find it?
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### Entire solutions of finite difference equations

Let $(E):\ \sum_{k=0}^n P_k(z)g(z+k)=0$ with the $P_k\in\mathbb C[z]$ a finite differences equation in $\mathbb C$. Is it true that every any entire solution $g$ solution of (E) with finite ...
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### A Freiman-type of question

Recently I have bumped in the following question. This is not my field of research, but it looks to me very much related to Frieman-type of theorems. Let $n$ be a positive integer and let $S$ be a ...
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### Solvability and uniqueness of Fokker-Planck BVP

I have been searching for solvability for the following BVP, I believe that this is a Fokker Planck's equation but I can't find any comprehensible text on existence and uniqueness of the solutions of ...
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### Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...
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### Character theory ( representation of finite group ) [migrated]

Show that no simple group can have an irreducible character of degree 2
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For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$? Below are some of my ...
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### Can I always change the order of integration for an indefinite integral? [on hold]

The example I have is this: Does $\int$ dx $\int$ dy (x+y) = $\int$ dy $\int$ dx (x+y) ? I would instinctively have thought yes, of course, but working it out yields $c_1$ x + $c_2$ = $k_1$ y + ...
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### Estimation of $\sum_{n \leq x} \frac{k(n)}{n}$ , with $k(n)$ the squarefree kernel

I came across a poblem where they ask you to find an estimation of $\sum_{n \leq x} \frac{k(n)}{n}$, with $k(n) = \prod_{p \mid n} p$ the squarefree kernel of $n$, with an error term of $O(\sqrt{x})$. ...
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### Wave operator for focusing NLS

Consider the NLS equation $$\left\{ \begin{array}{rl} iu_t + \Delta u+u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\right.$$ where ...
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### The symmetric powers of the $O_{X}-$modules [on hold]

Let $C$ be a smooth projective curve of genus $g\geq 1$. The symmetric powers of the curve $C$ are $$Sym^{r}(C)=(C\times\dots\times C)/S_{r},$$ where $S_r$ denotes the symmetric group of degree $r$. ...
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### Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$. Is the expected time known ...
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### Reference Help: Matsuki duality Orbits

I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...
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### Help understanding a proof in Stallings' Triangle of Groups paper

I'm trying to understand the proof of theorem 1 in Stalling's Non-positively Curved Triangle of Groups. I have specific questions, but is there anywhere someone has written out the proof in more ...
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### Terminology for the equation $a=a+b$ in commutative semigroups

Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often ...
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### Reducing search space by probability

First question on this Stack, so I hope this is appropriate. This is not for homework, but a personal question related to a machine learning problem I am working on. However, I don't have enough ...
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### Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$ such that this sequence has uniform ...
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### Arithmetic progression and 3^m,3^{m+1} intervals

I'm trying to prove (or disprove) the following "conjecture".Given the following set of powers of two: $$A = \{ x \mid x = 2^n \text{ and } 2^{n-1} < 3^m < x < 3^{m+1} < 2^{n+1}\}$$ ...
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### moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc. Fix a non-negative integer $g$ and consider the space ...
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### Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$: S_2\times Q\rightarrow Q,\; ...
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### Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded？

I have the following question: Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...