curious little things aggregator
http://rybu.org/aggregator
curious little things - aggregated feedsenMath Overflow Recent Questions: Reference request: explicit equivariant localization formula on toric complete intersections
https://mathoverflow.net/questions/306062/reference-request-explicit-equivariant-localization-formula-on-toric-complete-i
<p>This post is about an equivariant integration formula in a famous paper <a href="https://arxiv.org/pdf/alg-geom/9701016.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/alg-geom/9701016.pdf</a> by Alexander Givental, where the author presented the formula without proof or reference. It turns equivariant localization into calculation of residues I am trying to combine it with global residue theorem to derive some vanishing result of some invariant.</p>
<p>$$
\int_X f(p,\lambda)=\sum_{\alpha}Res_{\alpha}\frac{f(p,\lambda)dp_1\wedge\cdots\wedge dp_k}{u_1(p,\lambda)\cdots u_n(p,\lambda)}
$$
where $X$ is a toric symplectic variety, $p_k$ forms base of $H^2(X)$ (the Picard group) and $u_i(p,\lambda)=\sum_ip_im_{ij}-\lambda_j$, where $m_{ij}$ is a integer matrix describing how $p_i$ span all the invariant divisors and $\lambda_i$ is the ordinary equivariant index for torus action. The RHS sums over fixed point or pole specified by some of the $u_i=0$.</p>
<p>This paper is hard to read because I am not an expert on symplectic geometry but I really need to re-derive this explicit formula. Maybe some experts could kindly provide me some hints or references and any other discussion is welcomed. Thanks in advance.</p>Sun, 15 Jul 2018 06:51:48 -0600Math Overflow Recent Questions: Testing whether a quiver algebra is cellular with a computer
https://mathoverflow.net/questions/306061/testing-whether-a-quiver-algebra-is-cellular-with-a-computer
<p>With a friend I made a program in the GAP-package QPA to check whether a given finite dimensional quiver algebra is quasi-hereditary. It is very slow since it has to go through all permutations of points in the quiver but in principle it works by using just linear algebra.
Now a similar class as quasi-hereditary algebras are cellular algebras:
<a href="https://en.wikipedia.org/wiki/Cellular_algebra" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Cellular_algebra</a>. A cellular algebra is quasi-hereditary iff it has finite global dimension.</p>
<blockquote>
<p>Question: Is it possible to have a programm in QPA that checks whether a given finite dimensional quiver algebra is cellular (by using given commands that preferably only use linear algebra)? </p>
</blockquote>
<p>I have no real experience with cellular algebras and the definitions make it look like the answer is no. But maybe there is an equivalent definition of cellular algebras that makes such a QPA-program easy?</p>Sun, 15 Jul 2018 06:25:55 -0600Math Overflow Recent Questions: representation of prime numbers
https://mathoverflow.net/questions/306060/representation-of-prime-numbers
<p>We know that primes of the form $p=4k+1$ can be written as sum of two squares, i.e., $p=x^2+y^2$ (uniquely iff $0<x<y$). However, this expression holds for composite numbers that all of the prime factors are of the form $4k+1$, or if they have prime factors of the form $4k+3$, these factors are of even power.</p>
<p>I am looking for some expression to represent prime numbers of the form $p=4k+1$ such that it does not hold for composite numbers? Is it possible to have such expression?</p>
<p>P.S.: If there are some conditions (not necessarily expressing as polynomials) that determine such prime numbers would be desirable.</p>
<p>Thank you.</p>Sun, 15 Jul 2018 06:20:40 -0600Math Overflow Recent Questions: Surjectivity of norm map on subspaces of finite fields
https://mathoverflow.net/questions/306058/surjectivity-of-norm-map-on-subspaces-of-finite-fields
<p>It is basic that the norm map $N:\mathbf{F}_{q^n}^* \to \mathbf{F}_q^*$ is surjective for finite fields. In fact $N(x) = x^{(q^n-1)/(q-1)}$. How well does this simple fact extend to subspaces?</p>
<p>A basic example is an intermediate extension $\mathbf{F}_{q^d}$. On $\mathbf{F}_{q^d}^*$ we have
$$N(x) = \left(x^{(q^d-1)/(q-1)}\right)^{(q^n-1)/(q^d-1)} = \left(x^{(q^d-1)/(q-1)}\right)^{n/d}$$
since the term in the brackets is in $\mathbf{F}_q^*$ and $(q^n-1)/(q^d-1) \equiv n/d \pmod {q-1}$. So $N$ is surjective on $\mathbf{F}_{q^d}^*$ if and only if $(n/d, q-1) = 1$. In particular $N$ fails to be surjective on a subspace of dimension $n/2$ whenever $n$ is even and $(n/2, q-1) > 1$.</p>
<p>As a sort of converse note that if $(n,q-1)=1$ then $N$ is surjective on every one-dimensional subspace.</p>
<blockquote>
<p>Is it true that if $V \leq \mathbf{F}_{q^n}$ is a $\mathbf{F}_q$-rational subspace of dimension $>n/2$ then $N$ is surjective on $V$?</p>
</blockquote>
<p>Equivalently, if $\dim_{\mathbf{F}_q} V > n/2$, can we always find $x^{q-1} \in V$?</p>Sun, 15 Jul 2018 05:52:57 -0600Math Overflow Recent Questions: irreducible component of support of function
https://mathoverflow.net/questions/306054/irreducible-component-of-support-of-function
<p>Vakil gives two equivalent definitions of associated points in his "Rising Sea":</p>
<ol>
<li>a prime ideal $p$ of a ring $A$ is called an associated prime for module $M$ if it is the annihilator of an element $ m \in M$, i.e. $p = \mathrm{Ann}(m) $</li>
<li>a point p in $\mathrm{Spec} A$ is called an associated point for module $M$ if there is an element $ m \in M$ such that $p$ is a generic point of $\mathrm{Supp }
\text{ }m$</li>
</ol>
<p>Where $A$ is a Noetherian, $M$ is finite generated over A.</p>
<p>Vakil asks readers to check this equivalence, and he gives a hint:</p>
<blockquote>
<p>if $p$ is an associated point, then there is an element $m$ with Support $\bar{p}$</p>
</blockquote>
<p>If I prove this then I finish the exercise, but I can't.</p>Sun, 15 Jul 2018 02:49:54 -0600Math Overflow Recent Questions: What is the expected value of n? [on hold]
https://mathoverflow.net/questions/306052/what-is-the-expected-value-of-n
<p>• A special die is designed with n faces, enumerated 1 through n with all faces being equally likely.
If X is the observed number when this die is thrown, what is the expected value of X? [C]<br>
(a) $\frac n2+1$<br>
(b) $\frac n2$<br>
(c) $\frac{n+1}2$<br>
(d) $\frac{n−1}2$<br>
Why is the answer c correct? I know that it is a discrete uniform distribution.</p>Sun, 15 Jul 2018 01:52:57 -0600Math Overflow Recent Questions: Reference request: Bipartite symmetric graphs are hamiltonian
https://mathoverflow.net/questions/306051/reference-request-bipartite-symmetric-graphs-are-hamiltonian
<p>Does anyone know whether bipartite symmetric graphs are hamiltonian?
I'm not sure whether anyone have proved it before, but a nonhamiltonian symmetric bipartite graph would lead to a counterexample to the Lovasz conjecture. I would appreciate any references or ideas.</p>Sun, 15 Jul 2018 01:23:14 -0600Math Overflow Recent Questions: Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?
https://mathoverflow.net/questions/306049/does-there-exist-a-lebesgue-nonmeasurable-set-e-in-mathbbr-satisfies-that
<p>Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$. </p>
<p>Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows:
$$\mathcal{B}_0=\{A\in \mathcal{B}_{\mathbb{R}}:\mu_L(E)=0\ \text{or}\ +\infty\}.$$
I want to prove that the family of all locally measurable sets of the measure space $(\mathbb{R},\mathcal{B}_0,\mu_L|_{\mathcal{B}_0})$, that is, $$\{E\subset \mathbb{R}:E\cap A\in \mathcal{B}_0\ \text{for all $A\in \mathcal{B}_0$ such that $\mu_L(A)<\infty$}\}$$
is not the family of all subsets of $\mathbb{R}$.</p>
<p>So I want to ask whether there exists a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that
$E\cap A$ is a Borel null set for every Borel null set $A$.</p>Sun, 15 Jul 2018 00:42:14 -0600Math Overflow Recent Questions: General solution for first-order difference equation
https://mathoverflow.net/questions/306048/general-solution-for-first-order-difference-equation
<p>I have the following first-order difference equation</p>
<p>$$y_{t} = \frac{x_{t}}{1-\rho L} + \epsilon_{t}$$</p>
<p>where $L$ denotes the backshift operator, i.e., $L(x_{t}) = x_{t-1}$. I can obtain a solution to this difference equation heuristically, but I am wondering if there is a general procedure. A solution (the solution?) is </p>
<p>$$y_{t} = \sum_{i = 0}^{\infty} \rho^{i}X_{t-i} + \epsilon_{t}$$</p>
<p>I looked in Goldberger's text and couldn't find it there. Any reference is appreciated also. Thanks.</p>Sat, 14 Jul 2018 22:48:31 -0600Math Overflow Recent Questions: How to visualize a Witt vector?
https://mathoverflow.net/questions/306046/how-to-visualize-a-witt-vector
<p>As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.</p>Sat, 14 Jul 2018 22:24:36 -0600