## All Questions

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### objects which can’t be defined without making choices but which end up independent of the choice

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure …
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### When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity normal?

Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-s …
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### Lipschitz map of the circle onto a triangle

Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane. …
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### Karoubi versus Kasparov K-theory

I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$ with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ …
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### Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.

Let $M$ be a (discrete) monoid acting on a space $X.$ We may take the quotient of $X,$ by this action, $X/M$ that is the coequalizer of the action map $M \times X \to X$ against th …
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### Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ …
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### Need an explanation of this paragraph “Lebesgue Homoeomorphism”

I will just quote a part of one proof in "On uniformly regular topological measure spaces by Babiker: page 781" vol43 No4 Duke Math. J. 1976. Let $I$ be the unit interval endow …
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### exactness of sequence of groups

Hello, I have the question, which should has an easy answer, but I do not see that: To find a short exact sequence $0 \to A \to B \to C \to 0$ of abelian groups (where each homomo …
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### Determine the probability that two random vectors over a finite field are orthogonal

Hi all, Suppose that $\mathbf{f}=[f_1, f_2,\ldots,f_m]$ and $\mathbf{g}=[g_1,g_2,\ldots,g_m]$ are two $m$-dimensional vectors. All $f_i$'s are chosen uniformly randomly from a fin …
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### Joint distribution from multiple marginals

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial …
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### Permutations of $(Z/pZ)^*$

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$. Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of per …
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### Heisenberg Lie algebras

Dear forum, I would like to ask if $H(m)$ is the Heisenberg Lie algebra of dimension $2m+1$ and $M$ is an ideal of $H(m)$. Can we say that $M$ has a complement in $H(m)$?
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### What’s the definition of continuous of set-valued functions?

According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for ev …
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### Possible automorphism groups of a K3 surface

Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)? ...and does the answer change is I remove "polarized"? (polarized = equipped with an …
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### Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville: “The big experts in the field had already tried to make this approach w …
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### Degree of a finite locally free group scheme over a base scheme of characteristic p

Does a connected finite locally free group scheme G over a scheme S of characteristic p>0 has degree a power of p? I know that when S is the spectrum of a field k, it is true. So …
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### A question about “nice” functions

Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us c …
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### Centralizers of elementary abelian subgroups of $p$-groups

Let $P$ be a $p$-group. It is known that if $E$ is a maximal elementary abelian subgroup of rank 2 in $P$, then $C_P(E)/E$ is cyclic where $C_P(E)$ denotes the centralizer of $E$ i …
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### Graphs with vertex-separators of size a function of the diameter…

Hi there, I have a question somehow related to a previous question of mine http://mathoverflow.net/questions/131157/fundamental-cycle-separators-and-crossing-numbers. Consider a …
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### Octonions and the Fano plane.

Does the Fano plane mnemonic for octonion multiplication have any deeper meaning? http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg The symmetry group of the Fano …
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### Lang isogeny for group stacks

Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective c …
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### What does “Vertex Solution” mean?

Hello! I come across the word "vertex solution" in the context " We can also assume that x and y are vertex solutions,so that the sequence {x,y} remains in a finite set." Could a …
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### Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
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### regularity of eigenfunctions of Schrödinger Operator

Hello, I consider a compact and connected (smooth) Riemannian manofold $(M,g)$. I'm interested in the eigenfunctions of the Schrödinger Operator $L=-\Delta+ V$ acting on (smooth) …
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### Hyperbolic sets

I recently started reading about hyperbolic dynamics in the notes of L. Wen, http://www6.cityu.edu.hk/rcms/publications/ln5.pdf and in this (page 8) there is the following s …
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### How can I randomly draw an ensemble of unit vectors that sum to zero?

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question: Is there a …
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### Vector field pull back from embedding [closed]

Let $M$ and $N$ be finite dimensional smooth manifolds. A smooth map $f: M \to N$ is an embedding if and only if there is an open neighborhood $U$ of $f(M)$ in $N$ and a smooth ma …
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### Dual space of Bochner space: is there an easier proof to show they’re isometric?

It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$. If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are isometric? The proof that I know for …
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### Lawvere’s fixed point theorem and the Recursion Theorem

Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, \forall e\exists k[\Phi_e\text{ is total }\im …
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### Connection between subnet and superfilter.

Let's define a net and subnet in this way: A net is a any function of the form $n:(P,\le)\to X$ where $(P,\le)$ is a (preordered) directed set. A net $m:(P',\le)\to X$ is a subne …
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### What is barycentric simplicial subdivision? [closed]

In A Generalization of Brouwer's Fixed Point Theorem by Shizuo Kakutani, he defined $S^{(n)}$ be the $n$-th barycentric simplicial subdivision of $S$. In which $S$ is an $r$-dimens …
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### Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint

I am wondering what is known about optimization problems of the following type. Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalit …
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### Is a Cauchy principal value invariant under a “change of variables”?

Let $f \in C^{\gamma}_c(\mathbb{R}^n)$. Let $K:\mathbb{R}^n \backslash {\vec{0}} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties: 1) K smoot …
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### Question about the “nullstellensatz” for projective schemes

Yeasterday I asked a question on math stackexchange which simplifies to the following: Assume that $G$ is a graded ring and $A \subseteq G$ is a homogeneous radical ideal. …
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### DAG graph and topologic order question

I need to find the maximum number of topological sorts on Direct Acyclic Graph of N-order. I've checked by running Depth first search algorithm on various Direct Acyclic graphs, an …
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