6
votes
3answers
147 views
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 …
0
votes
2answers
19 views
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 …
0
votes
0answers
14 views
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. …
0
votes
0answers
10 views
Equivariant formality of a Lie group under conjugation by a maximal torus
Given an action of a group $G$ on a topological space $X$, the associated homotopy quotient is $$X_G := (EG \times X)/G,$$ where $EG$ is the total space of a universal principal $G …
1
vote
2answers
138 views
Real root of a cubic equation
I have a function f(x,n) can be expressed as a cubic function of x with coefficients that are functions of n. For example x^3 + (n-2)x^2 + (3n-6)x + n.
I want to prove that for e …
2
votes
1answer
57 views
How to cite a sequence from The On-Line Encyclopedia of Integer Sequences (OEIS)?
In my paper I want to provide a reference for a sequence (in this case - A001970) from The On-Line Encyclopedia of Integer Sequences (OEIS).
However, I couldn't find an official b …
0
votes
1answer
107 views
Hartogs Theorem and Canonical Bundles
Let $X$ be a normal complex affine algebraic variety. Suppose that $Y$ is an open subvariety of $X$, and that the codimension of $X\setminus Y$ in $X$ is at least $2$. One version …
19
votes
1answer
572 views
Yitang Zhang’s preprint on Landau-Siegel zeros
The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint on Landau-Siegel zeros? If this result is cor …
0
votes
0answers
8 views
Bounds on the crossing numbers of de Bruijn graphs and some incidence graphs
Hi there,
My question today is related to bounds for the crossing number $cr$ of the $k$-dimensional de Bruijn graph $B(t,k)$ on $t$ symbols (http://en.wikipedia.org/wiki/De_Bruij …
1
vote
0answers
51 views
Monoidal category
Let $(M,\otimes)$ be a small symmetric monoidal category.
Is it possible to choose in each isomorphy class $[A]$ a representative $A_0$ and for each $A\in M$ an isomorphism $\phi_A …
2
votes
0answers
50 views
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 $e_{i}^{2}=-1$ …
2
votes
1answer
49 views
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 …
2
votes
4answers
100 views
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 $E$ …
0
votes
0answers
35 views
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 …
0
votes
0answers
66 views
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 …
0
votes
1answer
63 views
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 …
1
vote
0answers
25 views
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 …
3
votes
0answers
101 views
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 …
0
votes
1answer
93 views
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 $H(m)$?
3
votes
3answers
186 views
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 …
3
votes
2answers
167 views
Algebraic closure of a polynomial ring
What could be conditions on $k\in\mathbb{C}[x,y,z]$ that would ensure that any polynomial $f\in\mathbb{C}[x,y,z]$ that is algebraically dependent of $k$ is indeed a polynomial in $ …
3
votes
1answer
90 views
The first eigenvalue of the Schrödinger operator is simple.
Hello,
let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\ …
3
votes
0answers
88 views
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 …
29
votes
1answer
2k views
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 …
1
vote
1answer
42 views
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 …
1
vote
1answer
142 views
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 …
0
votes
0answers
32 views
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 …
1
vote
0answers
14 views
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 …
12
votes
0answers
256 views
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 …
2
votes
0answers
91 views
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 …
0
votes
1answer
84 views
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 …
0
votes
0answers
23 views
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?
0
votes
1answer
51 views
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) …
3
votes
1answer
95 views
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 …
16
votes
2answers
416 views
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 …
-1
votes
2answers
190 views
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 …
0
votes
0answers
38 views
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 …
11
votes
1answer
232 views
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 …
1
vote
1answer
76 views
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 …
0
votes
0answers
90 views
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 …
2
votes
1answer
64 views
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 …
2
votes
1answer
197 views
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 …
2
votes
0answers
135 views
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. …
0
votes
1answer
88 views
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 …
3
votes
1answer
262 views
Set Theory exercise.
I find myself unable to solve question 24.1 of T. Jech's Set Theory:
If $\beta<\omega_1$ and if
$2^{\aleph_{\alpha}}\leq\aleph_{\alpha+\beta}$
for a stationary set of $\ …
2
votes
1answer
143 views
Closed geodesic loops around points in compact manifolds
Since in a compact Riemannian manifold $M$ the only totally convex subset is the whole manifold itself, see http://mathoverflow.net/questions/106169/closed-manifold-has-no-nontrivi …
0
votes
0answers
31 views
Proving a lower bound for the maximal eigen-value of a non-negative, irreducible, integer matrix
$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eig …
7
votes
0answers
129 views
A duality on partial permutations
A partial permutation matrix $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. S …
24
votes
13answers
2k views
Is there any proof that you feel you do not “understand”?
Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed …
13
votes
1answer
315 views
What are the main structure theorems on finitely generated commutative monoids?
I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I h …
