2
votes
0answers
47 views

calculating the Hilbert polynomial of a scheme given its primary decomposition

Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by $$ I_X =I_1 \cap \ldots \cap I_2 $$ I was ...
0
votes
0answers
408 views

A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
13
votes
2answers
595 views

If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...
4
votes
2answers
264 views

Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...". C. F. Gauss, Disquisitiones ...
-3
votes
1answer
118 views

Derivatives of infinite order [on hold]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature? For example, can one make sense of $$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 ...
4
votes
1answer
149 views

forcing square with small conditions

In the paper, Large cardinals and definable counterexamples to the continuum hypothesis, Foreman and Magidor mention a way to force $\square_{\omega_1}$ with countable conditions. (This is used in ...
3
votes
0answers
42 views

Progressively measurable vs adapted

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
1
vote
0answers
40 views

Complexity of mappings (forms) in R. Thom's “Structural stability and morphogenesis”

In his "Structural stability and morphogenesis", R. Thom (especially in the chapter about dynamics of forms) among other things speculates about a notion of complexity of a "form" (mapping between ...
-3
votes
0answers
28 views

Convergence of independent RVs to a constant [on hold]

Let $(X_n)$ be a sequence of independent RVs that converges in probability to some RV $X$. Show that $X$ is constant a.e. My attempt I think I should use Kolmogorov 0-1 theorem but I don't know how. ...
10
votes
0answers
308 views

Erdos multiplication problem revisited

The well-known problem is acquiring a cardinality of the set of distinct numbers in the multiplication table n x m. The very problem has been discussed in-depth and, as such, I require no further ...
1
vote
0answers
45 views

polynomial 0,1 integer programming

IS there a mathematical optimization branch that explicitly tries to optimize this (type) problem? $$\eqalign{ & \min \cr & \sum\limits_{i = 1}^N {(J*s[i] + {J_1}*s[i]*s[i + 1] + ...
0
votes
0answers
61 views

Generic path in the space of vector fields on the orientable surface

Who knows, does a theorem of such form exist: Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that 1) all fields in the family except finite number ...
2
votes
0answers
238 views

Rational structures on the flag variety over a finite field

Some Notions A variety over a field is defined to be a scheme of finite type over this field. An $\mathbb{F}_q$-rational structure of an $\bar{\mathbb{F}}_q$-variety $V$ is a $\mathbb{F}_q$-variety ...
2
votes
1answer
173 views

Tools for Removing Radicals from Equations

I am currently doing some investigations on Sylvester's 4 Point Problem Probability of 4 Points being in Convex Configuration and repeatedly face the problem of solving equations between sums of ...
6
votes
1answer
143 views

Integrals of representations over geodesics

Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...
0
votes
0answers
13 views

Approximation Hardness difference

What is the difference between a $n^{\epsilon}$ and $n^{1-\epsilon}$ bound on hardness of approximation? To be more specific, approximating the chromatic number is both $n^{\epsilon}$ and ...
2
votes
1answer
79 views

Mumford-Ramanujam examples in characteristic p

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...
1
vote
1answer
99 views

Perfectness of the Jacobian of a curve

Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer ...
2
votes
1answer
68 views

How the equality in the first case is equivalent to the inequality in the last case?

The motivation to this question can be found in: About equivalent statements of the Birch and Swinnerton-Dyer Conjecture My question is about the last equivalences: $\mathrm{ord}_{s=1} L(E/K,s) = ...
0
votes
0answers
126 views

Alternative definition of a category [on hold]

It is usual to define a (small) category $C$ as consisting, among other things, as: a set of objects $|C|$; and, for each pair of objects $X$ and $Y$, a set of morphims $C(X, Y)$. Now, since the ...
0
votes
0answers
88 views

book suggestion in fourier analysis and differential topology [on hold]

I am at present following Vinberg and Onischik's Lie Groups and Algebraic Groups and I find it a fantastic book with theory developed through a series of problems left to the reader. I would like to ...
0
votes
0answers
19 views

Explicit construction of transition semigroup from generator for completely independent spin system (Feller process)

As an example of how to obtain the transition semigroup from the probability generator for a Feller process, I am looking at the easiest spin system, namely with all sites independent. Notationwise, I ...
1
vote
1answer
122 views
+50

A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...
-1
votes
0answers
75 views

Help with extension of f (mentioned below) to f: Zp -> Zp ,continuous function [on hold]

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
4
votes
1answer
166 views

Constructing a “geometric” model structure on Cat by localizing the “categorical” model structure

Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the ...
1
vote
2answers
69 views

Criterion for deloopable based map

Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq ...
0
votes
3answers
207 views

question about the induced homomorphism of etale fundamental groups

Background/Setup For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
4
votes
2answers
140 views

Borel Sets in Sacks Generic Extension

Let $\mathbb{S}$ denote Sacks forcing. This is forcing with perfect trees or equivalently forcing with uncountable Borel subsets of ${}^\omega 2$ with the relation $\subseteq$. Let $G \subseteq ...
0
votes
0answers
91 views

When is the earliest large prime gap also the latest large prime gap?

Suppose one finds the earliest prime gap of at least a certain size $g$, so that $p_{n+1}-p_n=g$ and $n$ is the smallest index for which the gap is as big as $g$. Now consider the relative size of ...
2
votes
2answers
122 views

Expectation of a generalization of Dirichlet distribution

For the standard Dirichlet, the expectation of $X_i$ is $\alpha_i/\alpha_0$, where $\alpha_0 = \sum_i \alpha_i$ [http://en.wikipedia.org/wiki/Dirichlet_distribution]. I am considering the following ...
6
votes
2answers
154 views

elementwise functions of positive definite matrix

The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...
-1
votes
0answers
24 views

Expressing the superposition of two cubic bezier splines as a cubic bezier spline? [on hold]

To be clear, I'm not a mathematician, but a programmer. I'm trying to find an algorithm, and I'll try to be clear about the problem and its constraints. Please let me know if I need to provide more ...
2
votes
1answer
85 views

Symmetric invariants of a Schur Module

Let $V\cong\mathbb C^n$ be a complex vector space of dimension $n$. Let $\lambda\in\mathbb Z^r$ be a generalized integer partition $\lambda_1\ge\cdots\ge\lambda_r$ with $r\le n$. Denote by $\mathbb ...
3
votes
3answers
274 views

How to find an integer set, s.t. the sums of at most 3 elements are all distinct?

How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different. Example with $|A|=3$: Out of the set $A ...
1
vote
0answers
65 views

why is this result about Gaussian analytic functions equivalent to the Crofton formula

I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function. Definition A Gaussian analytic function ...
2
votes
1answer
163 views

Inverse of a Borel surjection

Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}_Y$. ...
1
vote
0answers
37 views

On $C_0(\Omega)$-module maps from $L_\infty(\Omega,\mu)$ to $L_q(\Omega,\nu)$

Let $\Omega$ be a locally compact space, and $\mu,\nu\in C_0(\Omega)^*$. By $H_{p,q}^{B}$ (resp. $H_{p,q}^{C}$) we denote the Banach space of continuous $B(\Omega)$-module (resp. $C_0(\Omega)$-module) ...
0
votes
0answers
9 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [migrated]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
5
votes
1answer
363 views

Is there a mistake in Vapnik's “Basic Lemma”?

I have a concern about the "Basic Lemma" which Valdimir Vapnik states and proves in his 1998 book Statistical Learning Theory (ch. 14.3, pp. 574–76): It seems like a certain coefficient should have ...
12
votes
1answer
272 views

Why are Witten-Reshetikhin-Turaev invariants expected to be integral?

A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ ...
-4
votes
0answers
190 views

Associated prime ideals and local cohomology [on hold]

Let $M$ be an $R$-module such that $\operatorname{Ass}(M/N)$ is a finite set for any submodule $N$ of $M$. Show that 1. $\operatorname{Ass}(M/r M)=\operatorname{Ass}(M/r^n M)$ for each natural $n$; 2. ...
1
vote
1answer
94 views

Vertex transitive and edge transitive and line graph

How can we find the proof of the following statement: An undirected graph is edge transitive if and only if its line graph is vertex transitive.
-3
votes
0answers
65 views

my question is about matrix [on hold]

let A be an n*n matrix with real entries which of the following is correct? 1.if A^2=0, then A is diagonalisable over complex numbers. 2.if A^2=I, then A is diagonalisable over real numbers 3.if ...
1
vote
0answers
90 views

Calogero-Moser eigenfunction

The folllowing function \begin{equation} J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} ...
3
votes
1answer
219 views

Generating finite groups

Let $G$ be a finite group possessing a generating set of order $n \in \mathbb{N}$. Let $H \leq G$ and $x_1, \dots, x_n \in G$ for which $\langle H, x_1, \dots, x_n \rangle = G$. Must there be $h_1, ...
6
votes
0answers
134 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
0
votes
0answers
56 views

How $f$ is approximated, in the $L^{p}-$ norm, by a function $f+h$ whose Fourier transform is constant in some nbhd of the point?

Fact.Suppose $f\in L^{1}(\mathbb R), x_{0}\in \mathbb R,$ and $\epsilon >0.$ Then there exists $h\in L^{1}(\mathbb R)$ with $\|h\|_{L^{1}}< \epsilon,$ such that $$\hat{h}(x)= ...
5
votes
1answer
309 views

Is the Manickam-Miklós-Singhi Conjecture solved?

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...
-2
votes
0answers
46 views

Differential equations [closed]

For k > 0,x = x(t),y = y(t) Solve this system: \begin{array}{l} x\frac{{{d^2}x}}{{d{t^2}}} = k\frac{{dy}}{{dt}}\\ x\frac{{{d^2}y}}{{d{t^2}}} = - k\frac{{dx}}{{dt}} \end{array}
2
votes
0answers
170 views

p-adic etale cohomology

Let $X$ be a smooth projective scheme over $\mathbb{Z}_p$, with special fiber $X_s$ over $\mathbb{F}_p$, generic fiber $X_{\eta}$ over $\mathbb{Q}_p$, and geometric generic fiber $\bar{X_{\eta}}$ over ...

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