9
votes
1answer
756 views

Central extensions of group schemes

In the category of groups, it is elementary that all central extensions of a cyclic group are abelian. Is the same true, in the category of (finite?) group schemes over a field $k$, for central ...
1
vote
1answer
392 views

Fixed points of the Borel-Serre compactification

Let $\Gamma$ be an arithmetic group and $X$ its symmetric space. Borel-Serre constructed a space $\bar{X} \supset X$ such that $\bar{X}/\Gamma$ is a compactification of $X/\Gamma$ [Corners and ...
2
votes
1answer
416 views

Geometry Realization of Homology Class

Hello! My question is about the realization of homology class. The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a ...
0
votes
0answers
178 views

Gradient estimates for subsolutions of elliptic equations

Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and $\Delta u + \lambda u = 0,$ where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
7
votes
0answers
192 views

Binary Operation on a Cubic Surface

If $P$ and $Q$ are two sufficiently general points on a cubic surface, the line between them intersects the cubic surface at a unique third point, $f(P,Q)$. This gives a binary operation on (generic) ...
1
vote
1answer
82 views

Fixed submanifolds of the sphere at infinity of $\mathbb{H}^n$

Good afternoon, Take a submanifold $V$ of codimension $1$ of the sphere at infinity of $\mathbb{H}^n$ which is not the sphere at infinity of a totally geodesic hyperplane $\mathbb{H}^{n-1} \subset ...
2
votes
1answer
398 views

About Sobolev embedding theorem of the case $W^{s,2}$.

Let $s>n/2, \; f \in W^{s,2}(\Bbb R^n)$ . Then how can I show that there is an embedding into the space of uniformly bounded, continuous functions, that is, $$ |f(x)| \leqslant C\| f \|_{W^{s,2}}$$ ...
3
votes
1answer
794 views

Books on logic for someone aiming to go to grad school in the field?

I have taken two introductory courses on logic. One was an undergraduate level and the second one was at the graduate level. Both used a set of notes written by the instructor. I'm thinking about ...
5
votes
3answers
439 views

An isomorphism of 2-Schur modules

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings ...
18
votes
3answers
1k views

Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that $$\mathop{\rm area} \Sigma\ge ...
10
votes
1answer
864 views

Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic. The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...
0
votes
4answers
706 views

Realization of a bisimplicial set

Hello, I just need some clarification (or a good reference) for the definition of the realization of a bisimplicial set, this is what i have when $X$ is a bisimplicial set its realization is $\cup_n ...
-1
votes
1answer
165 views

How to define an “anisotropic vector” for a given object? [closed]

Dear experts, I am looking for a way to define an "alignment vector" (or anisotropy or orientation vector?) for a given geometrical object. I am not sure how to put this into correct technical terms, ...
0
votes
1answer
109 views

deformation of curves with three nodes

Let $X$ be a stable curve consisting of two components meeting at three points. Let $M$ be its versal deformation space. The locus in $M$ parametrizing singular curves is a divisor with three ...
2
votes
0answers
84 views

Ergodic actions with co-finite stabilizers

Let $G$ be a locally compact, second countable group acting on a standard probability space $(X,\nu)$, and let $\nu$ be $G$-invariant. Let $G_x = \{g \in G\,:\, gx=x\}$ denote the stabilizer of $x \in ...
1
vote
2answers
118 views

Singular points on the Hilbert scheme of a product

Let $X$ and $Y$ be smooth projective varieties, say over $\mathbb C$. Fixing a point $y\in Y$, we obtain a smooth, closed subvariety $X\times\{y\}$ of $X\times Y$, which in turn corresponds to a point ...
9
votes
1answer
755 views

Quasi-nilpotent trace class operators as limits of nilpotents

In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to: If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
6
votes
1answer
251 views

Characterization of $\kappa$-closed forcings

For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like. For a partial order $\mathbb{P}$, the ...
6
votes
1answer
255 views

Hamiltonian cycles in power-graphs

I've stumbled across a short note from 1993 where a nice question was asked: Suppose you take a graph with vertices $\{1,2,\ldots,n\}$ and connect $i,j$ by an edge if and only if $i+j$ is a $k$th ...
0
votes
0answers
408 views

Calculating entropy of adjacency matrix using eigenvalue decomposition?

How to calculate entropy using the eigenvalues when the eigenvalues are negative? Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
3
votes
1answer
314 views

Holonomy of a Kähler manifold

Hi, I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla ...
2
votes
1answer
209 views

zero locus of a sheaf homomorphism

The title says it all. I'm just starting to go through the existence-proof of the quot scheme (http://www.math.utah.edu/~bertram/courses/hilbert/ps/hilbert.ps). On page 7, almost at the bottom of ...
6
votes
3answers
781 views

Pullbacks of canonical divisors along branched maps

Let $f\colon X \to Y$ be a finite map of smooth surfaces. Let the divisor $D$ of $Y$ be the branch locus of $f$. We assume that $D$ is a union of nonsingular curves intersecting transversally with no ...
7
votes
1answer
339 views

Number of unique sortings of subset-sums

Take the set $A_n=\{a_1,...,a_n\}$. Let $S_n$ be the set of subset-sums of $A_n$. (The subset-sum of the empty set is assumed to be zero.) Assume that there are $2^n$ unique members of $S_n$. How many ...
2
votes
0answers
46 views

existence of Markov operators not generated by transition probability function

Transition probability functions can always be used to generate Markov operators, correct? So is it correct to say that a Markov process is a collection of Markov operators? On the other hand, are ...
4
votes
0answers
137 views

Differences of Numbers of Helicity States in 4-dimensional Strings

The question whether the states in $D=2m + 2$ dimensional string theory, which carry a representation of $SO(2m)$, span spaces which carry representations of $SO(2m+1)$ seems hopelessly complicated. ...
7
votes
2answers
1k views

Uniformly Convex spaces

My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill). I'm interested in constructing a uniformly convex Banach space with conditional structure ...
4
votes
1answer
163 views

Question about unusual highest weight modules for $U_q(sl(2))$

Background Let $U_q(sl(2))$ be the quantum group associated with $sl(2)$ i.e. the associative algebra with 1 over $Q(q)$ generated by $x^+,x^-,K,K^{-1}$ with relations $$KK^{-1}=K^{-1}K=1$$ ...
21
votes
2answers
680 views

Random permutations of Z_n

In http://www.springerlink.com/content/y19u81675243r237/fulltext.pdf, the author states the following without proof (equation 3.1): "Consider a random permutation $\pi$ of $\mathbb{Z}_n$. What is the ...
4
votes
2answers
224 views

hodographic transformation

Let $\phi(x,t)$ be smooth function. Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$. Let $u$, $v$ satisfies following equations: 1- $$u_\eta- v_\zeta= 0$$ ...
4
votes
1answer
1k views

Integration on high dimensional sphere

Hi, I need to integrate a function on an n-dimensional sphere surface. One way is to use the triangle function like: http://en.wikipedia.org/wiki/N-sphere#Spherical_volume_element, however, it is too ...
4
votes
1answer
276 views

Is the chromatic number of the real plane invariant under the norm?

Recall that chromatic number of $\mathbb{R}^2$ is the least $n$ such that there exists a function $f$ from $\mathbb{R}^2$ into a set of colors ${C_1,\ldots,C_n}$ with $f(x)\neq f(y)$ for ...
1
vote
2answers
393 views

Calculus of Binary Relations

I was reading "Origins of the Calculus of Binary Relations" by Vaughan Pratt where he says "it consists of two components, a logical or static component and a relative or dynamic component" but it ...
1
vote
0answers
212 views

canonical model of a reducible curve

Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...
0
votes
1answer
405 views

Cartan decomposition of a unitary group?

For local fields $F$, we consider two case 1) $E$=quadratic extension of $F$ , 2) $E = F \times F$. Let V be a 2-dim hermition space over E. In 1) case, by Cartan decompostion $U(2)$ can be ...
2
votes
1answer
155 views

Maximization of specific Likelihood function

N coins have probability $p_n = e^{-t_n/s}$ of heads, $t_n$ being specific for each coin. Coins 1 to m came up heads and m+1 to N came up tails. Now I'm trying to estimate $s$ using the Maximum ...
5
votes
1answer
256 views

Convex PBW bases

Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, ...
2
votes
2answers
345 views

non-Identity operator on a separable Hilbert space

Suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in ...
4
votes
4answers
444 views

Coproduct on coordinate ring of finite algebraic group

I'm reading Mukai's book "An introduction to invariants and moduli", and I am having trouble understanding one of his examples. It is example 3.49 on page 101. The setup is as follows. Let $G$ be a ...
5
votes
1answer
1k views

binary code with constant hamming distance

I want as many 80-bits words as possible with the constraint that the hamming distance between any couple of words is exactly 40. How many can I generate? Is there a generic formula telling me how ...
8
votes
4answers
1k views

Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?

Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more ...
2
votes
3answers
502 views

Notion of internality in model theory

Good evening, Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ? Thank you
6
votes
2answers
605 views

Which semigroups can be linearly ordered?

As usual I consider a semigroup to be a structure $(A, +)$ such that $+$ is an associative binary function over the set $A$. The notion of linearly-ordered semigroup corresponds to structures of the ...
6
votes
3answers
2k views

Maximal ideal in polynomial ring

Is it true that the intersection of a maximal ideal in $A[x]$ with $A$ is a maximal ideal in $A$? Let's say A is Noetherian. I would be surprised if it isn't true but somehow I can't seem to show it. ...
0
votes
1answer
193 views

Embedding a semigroup into a divisible semigroup

The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
6
votes
3answers
266 views

Minimum separating subdivision in Plane

Hi I was thinking about the following problem: Given a planar Graph embedded in the plane and a set of points $P$ contained in the faces (no face contains more than one point) I want to determine ...
1
vote
1answer
103 views

examples of space of direction at a point in an infinite dim Alexandrov space compact

The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.
3
votes
1answer
161 views

Optimizing a stochastic “flip and prune” procedure for selecting a subset of coins

I place some number of coins, $(c_1, ..., c_N) \in C$ on a table, where each coin is originally tails up. Let's call the "tails" state $0$ and the "heads" state $1$. I then perform the following ...
2
votes
0answers
143 views

Mappings between Banach spaces

What is the definition of an analytic mapping between two Banach spaces? This is a problem I ran into when solving an integral equation. One of the related coefficients is represented as a functional ...
4
votes
3answers
317 views

closed meagre sets

A closed meagre subset of $[0,1]$ is either countable or homeomorphic to the Cantor set: either way it is $0$-dimensional. Q.1. Is every closed meagre subset of an $n$-dimensional locally compact ...

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