Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2018-10-20T18:21:46Zhttps://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttps://mathoverflow.net/q/3133100Projecting a polyhedral cone onto its intersection with the infinity-norm ballmadison54https://mathoverflow.net/users/438392018-10-20T18:14:51Z2018-10-20T18:14:51Z
<p>For a point in a convex polyhedral cone, <span class="math-container">$x\in \mathcal{C}$</span>, is there an <em>efficient</em> algorithm to project <span class="math-container">$x$</span> onto the intersection with the <span class="math-container">$\infty$</span>-norm ballxx, <span class="math-container">$\mathcal{B}:= \{x \vert ||x||_{\infty} \leq 1\}$</span>?</p>
<p><span class="math-container">$$\textrm{argmin}_z ||x - z||_2^2 \; s.t. \; z \in \mathcal{C} \cap \mathcal{B}
$$</span></p>
<p><em>Efficient</em> here means an algorithm that exploits that <span class="math-container">$x\in \mathcal{C}$</span> to begin with and the special structure of <span class="math-container">$\mathcal{B}$</span> and is therefore more efficient than a generic algorithm projecting an arbitrary point onto the intersection of two polyhedral sets.</p>
<p><em>Note</em>: This is a more specific version of my <a href="https://mathoverflow.net/questions/297839/projecting-two-convex-polyhedra-onto-their-intersection">previous question</a>, now with additional assumptions on the polyhedral sets. The more general version doesn't seem to admit a more efficient solution.</p>
https://mathoverflow.net/q/3133090Definition of $C^{m,k}$-capacity of a pointRajesh Dachirajuhttps://mathoverflow.net/users/144142018-10-20T17:59:32Z2018-10-20T17:59:32Z
<p>I have come across the following notation and a new term <span class="math-container">$C^{m,k}$</span>-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.</p>
https://mathoverflow.net/q/3133070Laplace transform of a product of two functionsLFanalshttps://mathoverflow.net/users/1303262018-10-20T17:19:17Z2018-10-20T17:19:17Z
<p>I was asked to give this function <span class="math-container">$G′(t)=−G(t)(S+X(t))+S⋅H+R(t)/V$</span> in terms of <span class="math-container">$G(s)/R(s)$</span> and <span class="math-container">$G(s)/X(s)$</span> by using the Laplace transform. I know <span class="math-container">$G(0)=A$</span> and <span class="math-container">$X(0)=0$</span>, and all the other functions and constants are unknown to me.</p>
<p>When I transform this equation and I neglect the initial conditions, I get
<span class="math-container">$$sG(s)=−G(s)S−L((G(t)X(t))+R(s)/V$$</span></p>
<p>In order to solve this problem, firstly I set <span class="math-container">$X(t)=0$</span> and find <span class="math-container">$G(s)/R(s)=1/(V(s+S))$</span>.</p>
<p>But I am struggling to get <span class="math-container">$G(s)/X(s)$</span> because I would have to solve the Laplace transform of <span class="math-container">$G(t)X(t)$</span>. I know I could use the convolution of these to functions, but they are unknown to me, so it doesn't help.</p>
<p>Some people in my class say there is no solution for neither of the two transfer functions, while others don't agree.</p>
<p>Have I made any mistakes in my development? How would you solve this problem?</p>
https://mathoverflow.net/q/3133061Order relation between cohomology groupsKing Khanhttps://mathoverflow.net/users/1264072018-10-20T17:15:16Z2018-10-20T17:15:16Z
<p>We have <span class="math-container">$\mathbb{Q}$</span>-graded finite dimensional vector space <span class="math-container">$V=\bigoplus_{i=0}^{n}V_{i}$</span> and following cochain complex
<span class="math-container">$$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\xrightarrow[]{d_{n-3}} V_{n-2}\xrightarrow []{d_{n-2}}V_{n-1}\xrightarrow[]{d_{n-1}} V_{n}\rightarrow 0$$</span>
Furthermore <span class="math-container">$V=\bigoplus_{j=1}^{3}X_{j}$</span> and <span class="math-container">$V_{n}\cap X_{3}=\{0\}.$</span> The differentials have following properties:</p>
<p>1) the differentials <span class="math-container">$d_{i},\,\, 0\leq i\leq n-1$</span> acts on bases elements non-trivially.</p>
<p>2) <span class="math-container">$d_{i}(X_{j}\cap V_{i})\subseteq X_{j}\cap V_{i+1}\quad\mbox{for}\,\, j=1,2. $</span></p>
<p>The action of differentials on the bases elements of <span class="math-container">$X_{3}$</span> in the following way:</p>
<p><span class="math-container">$d_{n-1}(x_{3})=\alpha_{1}x_{1}+\alpha_{2}x_{2}$</span> where <span class="math-container">$\alpha_{1},\alpha_{2}\neq0$</span> and <span class="math-container">$x_{j}\in Bases(X_{j}).$</span></p>
<p><span class="math-container">$d_{i<n-1}(x_{3})=\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}$</span> where <span class="math-container">$\alpha_{1},\alpha_{2},\alpha_{3}\neq0$</span> and <span class="math-container">$x_{j}\in Bases(X_{j}).$</span></p>
<p>From property 2) we get subcochain complex:
<span class="math-container">$$0\rightarrow V_{0}\cap X_{1}\xrightarrow[]{d_{0}} V_{1}\cap X_{1}\xrightarrow[]{d_{1}}\ldots\xrightarrow[]{d_{n-3}} V_{n-2}\cap X_{1}\xrightarrow []{d_{n-2}}V_{n-1}\cap X_{1}\xrightarrow[]{d_{n-1}} V_{n}\cap X_{1}\rightarrow 0.$$</span>
Question 1) is it true <span class="math-container">$dim(Img(d_{i-1})\cap Ker(d_{i}\mid_{X_{1}}))=dim(Img(d_{i-1}\mid_{X_{1}}))?$</span></p>
<p>Question 2) is it true <span class="math-container">$dim(H^{i}(V);\mathbb{Q})\geq dim(H^{i}(V\cap X_{1});\mathbb{Q})?$</span></p>
https://mathoverflow.net/q/3133053What is a category of "Lepagean equivalent" or "variation problem"?user605302https://mathoverflow.net/users/1303252018-10-20T16:43:27Z2018-10-20T17:18:39Z
<p>I get to know about it form Mark Gotay's work <strong><em>An exterior differential system approach to the Cartan form</em></strong>, in that paper he defined the canonical Lepagean equivalent. The following is cited from it:</p>
<p>A <strong>canonical Lepagean equivalent</strong> of a variation problem <span class="math-container">$\left( M\stackrel{\pi}{\longrightarrow} X, \mathcal{I}, \mathcal{L} \right)$</span> is another variation problem <span class="math-container">$\left( W\stackrel{\rho}{\longrightarrow} X, \left\{0 \right\}, \Theta \right)$</span>, together with a surjective submersion <span class="math-container">$\nu:W \longrightarrow M$</span>, such that: (1) <span class="math-container">$\rho=\pi\circ\nu$</span>, and (2) if <span class="math-container">$\gamma \in \Gamma \left(\rho\right)$</span> satisfies <span class="math-container">$\nu\circ\gamma\in\Gamma \left(\pi,\mathcal{I}\right)$</span>, then</p>
<p><span class="math-container">$\gamma ^{*}\Theta=\left(\nu\circ\gamma\right)^{*}\mathcal{L}$</span></p>
<p><span class="math-container">$\Theta$</span> is a Lepagean equivalent of <span class="math-container">$\mathcal{L}$</span> if</p>
<p><span class="math-container">$\Theta\equiv\nu^{*}\mathcal{L} $</span> mod <span class="math-container">$\nu^{*}\mathcal{I}$</span></p>
<p>And in the paper he said that the assignment <span class="math-container">$\left( M\stackrel{\pi}{\longrightarrow} X, \mathcal{I}, \mathcal{L} \right) \rightsquigarrow \left( W\stackrel{\rho}{\longrightarrow} X, \left\{0 \right\}, \Theta_{\mathcal{}} \right)$</span> is an affine functor from the <strong>category of constrained variation problems</strong> to the <strong>category of free ones</strong>. This is a welcome feature of the canonical Lepagean equivalent, as classical Lepagean equivalents are not functorial in general (why?).</p>
<p>I'm interested in the category feature of such variation problems, what are the objects, what are the arrows, is the functor invertible, why are classical Lepagean equivalents not functorial in general, is there any specified research about it?</p>
https://mathoverflow.net/q/3133041Families of distributions with a certain symmetry property?Jameshttps://mathoverflow.net/users/457072018-10-20T16:09:33Z2018-10-20T16:17:05Z
<p>Consider the probability distribution <span class="math-container">$\mathcal{N}_n$</span> on <span class="math-container">$\mathbb{R}^n$</span> whose density is <span class="math-container">$$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$</span></p>
<p>This multivariate normal distribution has symmetry group <span class="math-container">$O(n)$</span>. But the family of these distributions has two additional symmetry properties:</p>
<p>(a) If <span class="math-container">$X$</span> has distribution <span class="math-container">$\mathcal{N}_n$</span>, and <span class="math-container">$\pi$</span> is an orthogonal projection onto an <span class="math-container">$k$</span> dimensional subspace, then <span class="math-container">$\pi \circ X$</span> has distribution <span class="math-container">$\mathcal{N}_k$</span>.</p>
<p>(b) If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent random variables with distribution <span class="math-container">$\mathcal{N}_k$</span> and <span class="math-container">$\mathcal{N}_j$</span>, then <span class="math-container">$X \oplus Y$</span> (the random variable on <span class="math-container">$\mathbb{R}^{k+j}$</span>) has probability distribution <span class="math-container">$\mathcal{N}_{k+j}$</span></p>
<p>My question: is <span class="math-container">$(\mathcal{N}_n : n \in \mathbb{N})$</span> the <em>only</em> family of probability distributions (up to rescaling) which has properties (a) and (b)?</p>
https://mathoverflow.net/q/313301-4Geometrical sum [on hold]Griezyhttps://mathoverflow.net/users/1303242018-10-20T15:27:55Z2018-10-20T15:31:21Z
<p>I'm doing some exercises from my book, and I'm stuck on this one task where you should count out the indicated sum of this:</p>
<p><span class="math-container">$$\sum_{i=1}^7 \left(\frac13\right)^i$$</span></p>
<p><a href="https://gyazo.com/cd978d87c56f76295fa6bb5c5cfbf315" rel="nofollow noreferrer">https://gyazo.com/cd978d87c56f76295fa6bb5c5cfbf315</a></p>
<p>I do not understand how to do it exactly, because of the 5 and 1/3. I suppose your meant to use the formula: <span class="math-container">$a\frac{1-r^2}{1-r}$</span> but I do not know how to implement it here. Some help would be appreciated.</p>
https://mathoverflow.net/q/3132991Gaussian curvature of conformal transformationsMathStudenthttps://mathoverflow.net/users/423262018-10-20T14:45:22Z2018-10-20T15:54:46Z
<p>Let <span class="math-container">$g$</span> be a smooth metric and <span class="math-container">$g'=e^{v}g$</span>, where <span class="math-container">$v$</span> is also a smooth function. Then it is well-known that</p>
<p><span class="math-container">$(*) -\Delta_g u +2k_g=2 k_{g'}e^{v}$</span>,</p>
<p>where <span class="math-container">$k_g$</span> and <span class="math-container">$k_{g'}$</span> are the Gaussian curvature of the metrics <span class="math-container">$g$</span>, and <span class="math-container">$g'$</span>, respectively. </p>
<p>Now let <span class="math-container">$\Omega$</span> be a bounded region in <span class="math-container">$R^2$</span> with <span class="math-container">$0 \in \Omega$</span>, and <span class="math-container">$g=e^{u}|dx|^2$</span> be conformal to the Euclidian metric in <span class="math-container">$\Omega \ \setminus 0$</span> satisfying</p>
<p><span class="math-container">$\Delta u+ k_{g} e^{u}=\alpha \delta(0)$</span>.</p>
<p>Let <span class="math-container">$g'=e^{w}e^{u}|dx|^2$</span>, where <span class="math-container">$w$</span> is a smooth function. I wonder how the equation (*) generalizes to this singular case, i.e. what is the equation satisfied by the Gaussian curvature of <span class="math-container">$g'$</span> in terms of <span class="math-container">$k_g$</span>? </p>
https://mathoverflow.net/q/3132981Generation of strictly contraction SemigroupsS. Chohttps://mathoverflow.net/users/1249042018-10-20T14:32:06Z2018-10-20T17:58:42Z
<p>Let <span class="math-container">$T(t)$</span> be a <span class="math-container">$C_0$</span>-semigroup on Banach space <span class="math-container">$X$</span>, and <span class="math-container">$A$</span> its generator. By Lumer-Philipps theorem we know that if <span class="math-container">$A$</span> is densely defined and m-dessipative operator then it generates a <span class="math-container">$C_0$</span>-semigroup of contractions, I.e., <span class="math-container">$$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$</span>
My question here: is there any theorem which gives conditions to generate a strictly contraction <span class="math-container">$C_0$</span>-semigroup? That is,
<span class="math-container">$$\|T(t)\| < 1, \quad \forall t > 0.$$</span>
For example, for <span class="math-container">$A=\Delta$</span> the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm <span class="math-container">$\|T(t)\|$</span> in this case? Or at least, Is <span class="math-container">$I-T(t)$</span> invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?</p>
https://mathoverflow.net/q/3132940Proof differential product correlated brownian motionsKeithhttps://mathoverflow.net/users/1303212018-10-20T13:52:03Z2018-10-20T13:52:03Z
<p>I was wondering how to prove/compute the differential of the product of two Brownian motions. I know how to do it in case they are independent as follows: </p>
<p>Suppose <span class="math-container">$dX_t= \mu_t dt +\sigma_t dW_t$</span> and <span class="math-container">$dY_t = v_t dt + \rho_t d \bar{W}_t$</span>, where <span class="math-container">$W_t$</span> and <span class="math-container">$\bar{W}_t$</span> are independent, i.e. <span class="math-container">$(dW_t d\bar{W}_t= 0)$</span></p>
<p>Then I can prove <span class="math-container">$d(X_t Y_t)$</span> with the following trick:</p>
<p>I start with decomposition:
<span class="math-container">$(X_t+Y_t)^2=X_t^2+Y_t^2+ 2X_t Y_t$</span>,</p>
<p>Which leads by differentiation to
<span class="math-container">$d(X_t Y_t)= \frac{1}{2}[d(\{X_t +Y_t\}^2) - d(X_t^2) - d(Y_t^2)]$</span></p>
<p>Then I apply Ito-lemma to all three parts separately as follows:
<span class="math-container">$d(X_t^2)=(2\mu_tXt+\sigma_t^2)dt+ 2\sigma_t X_tdW_t= 2X_t dX_t + \sigma_t^2dt$</span>
<span class="math-container">$d(Y_t^2)=(2v_t Y_t+\rho_t^2)dt+ 2\rho_t Y_t d\bar{W}_t= 2Y_t dY_t+\rho_t^2 dt$</span>
<span class="math-container">$d((X_t+Y_t)^2) = 2(X_t+Y_t)dX_t+ 2(X_t+Y_t)dY_t + (\sigma_t^2+\rho_t^2)dt$</span></p>
<p>Combined I eventually find:
<span class="math-container">$d(X_t Y_t) = Y_t dX_t + X_t dY_t$</span>. </p>
<p>Now I am wondering if it is possible to do the prove similarly but then for correlated brownian motions? I'm relatively new to stochastic processes.</p>
https://mathoverflow.net/q/3132930Two reasons why the Collatz conjecture could failDominic van der Zypenhttps://mathoverflow.net/users/86282018-10-20T13:48:07Z2018-10-20T17:12:18Z
<p>Let <span class="math-container">$\mathbb{N}$</span> denote the set of positive integers. The <em>Collatz function</em> <span class="math-container">$f:\mathbb{N}\to\mathbb{N}$</span> is given by <span class="math-container">$f(n) = n/2$</span> for <span class="math-container">$n$</span> even and <span class="math-container">$f(n) = 3n+1$</span> for <span class="math-container">$n$</span> odd. Given <span class="math-container">$k\in\mathbb{N}$</span> we associate to <span class="math-container">$k$</span> its <em>Collatz sequence</em> <span class="math-container">$(c^{(k)}_n)_{n\in\mathbb{N}}$</span> given inductively by <span class="math-container">$$c^{(k)}(1) = k\text{ and } c^{(k)}_{n+1} = f(c^{(k)}_n)\text{ for all } n\geq 1.$$</span>
One version of the <a href="https://en.wikipedia.org/wiki/Collatz_conjecture" rel="nofollow noreferrer">Collatz conjecture</a> states that <span class="math-container">$$1\in \text{im}(c^{(k)}) \text{ for all }k\in\mathbb{N}.$$</span>
Note that for all <span class="math-container">$k\in\mathbb{N}$</span> the sequence <span class="math-container">$c^{(k)}$</span> is either injective or eventually periodic. So any <span class="math-container">$c^{(k)}$</span> with <span class="math-container">$1\notin \text{im}(c^{(k)})$</span> would be either </p>
<p>(1) injective or </p>
<p>(2) eventually contain a period not containing <span class="math-container">$1$</span>. </p>
<p><strong>Question.</strong> Can the current state of research exclude one of the cases (1), (2)?</p>
https://mathoverflow.net/q/3132910Article request--Central trinomial coefficients and convolution type identitiesJacob.Z.Leehttps://mathoverflow.net/users/428162018-10-20T13:33:07Z2018-10-20T13:37:01Z
<p>R. Witula and D. Slota, Central trinomial coefficients and convolution type identities, Congressus Numerantium 201, 109-126 (2010). </p>
<p>I would like to be able to read the full text of the paper. But I have no access to the paper, so would you please mail me a PDF file or scanning copy of the paper. Thank you very much for your consideration.</p>
<p>My E-mail: jacob1234@163.com</p>
https://mathoverflow.net/q/313289-3Higher Derivative of Rational function [on hold]Jacob.Z.Leehttps://mathoverflow.net/users/428162018-10-20T12:51:36Z2018-10-20T12:51:36Z
<p>As is well known, every rational function <span class="math-container">$R(x)$</span> has a partial fraction decomposition. i.e.<span class="math-container">$$ R(x) = \frac{p(x)}{q(x)} = P(x) + \sum_{i=1}^m\sum_{r=1}^{j_i} \frac{A_{ir}}{(x-a_i)^r} + \sum_{i=1}^n\sum_{r=1}^{k_i} \frac{B_{ir}x+C_{ir}}{(x^2+b_ix+c_i)^r}.$$</span>
So, the difficulty of higher derivative of <span class="math-container">$R(x)$</span> mainly lies in the higher derivative of <span class="math-container">$\frac{Bx+C}{(x^2+bx+c)^r}$</span>.</p>
<p>I want to know the <span class="math-container">$n$</span>-th derivative of <span class="math-container">$\frac{Bx+C}{(x^2+bx+c)^r}$</span>. I will appreciate your help for any suggestion. Thank you very much.</p>
https://mathoverflow.net/q/3132833$P = [-°,Set]$ is a contravariant co/lax idempotent monad, whose multiplication is determined by the unitFosco Loregianhttps://mathoverflow.net/users/79522018-10-20T10:02:47Z2018-10-20T10:19:44Z
<p>A <em>unidetermined contramonad</em> is a 2-monad <span class="math-container">$T : {\cal C}\to \cal C$</span> such that</p>
<ol>
<li><span class="math-container">$T$</span> is contravariant, i.e. a contravariant endofunctor;</li>
<li>the multiplication <span class="math-container">$\mu_A : TTA \to TA$</span> is determined as <span class="math-container">$T\eta_A = T(A\to TA)$</span>.</li>
</ol>
<blockquote>
<p><strong>Q(-1)</strong>: Is this even a thing? Does this definition already exist under another name?</p>
</blockquote>
<p>An example of such monad is the presheaf construction <span class="math-container">$P : A\mapsto [A°,Set]$</span>; it has the Yoneda embeddings as units, and it is in fact possible to show that <span class="math-container">$T\eta_A$</span> acts as a multiplication, in view of the general fact that
<span class="math-container">$$
[[PA°,Set]°,Set] \underset{PP\eta}{\overset{P\eta P}\leftrightarrows} [PA°,Set]
$$</span>
is an adjunction (<span class="math-container">$PP\eta\dashv P\eta P$</span>). (notation: whenever <span class="math-container">$X$</span> is a large category, <span class="math-container">$[X°,Set]$</span> is the category of <a href="https://ncatlab.org/nlab/show/small+presheaf" rel="nofollow noreferrer">small functors</a>).</p>
<blockquote>
<p><strong>Q0</strong>: Or is it (coming from the adjunction <span class="math-container">$P\eta\dashv \eta P$</span>)? I'm able to find two natural transformations:</p>
<ul>
<li><span class="math-container">$\alpha : F \Rightarrow P\eta(\eta P(F))$</span>, induced by the cowedge
<span class="math-container">$$ Fa\times A(-,a)\to F$$</span> (or, rather, induced by the action of <span class="math-container">$F$</span> on arrows): <span class="math-container">$\alpha$</span> mates to
<span class="math-container">$$ \tilde\alpha : Fa \to Set(A(-,a),F) $$</span>
(and <span class="math-container">$\tilde\alpha$</span> is invertible, for that matter: Yoneda lemma).</li>
<li><span class="math-container">$\beta : \Theta \Rightarrow \eta P(P\eta(\Theta))$</span>, induced by the action of <span class="math-container">$\Theta$</span> on morphisms: its components are
<span class="math-container">$$ PA(\hom(-,a),F)\to PA(\Theta F,\Theta(\hom(-a,)))$$</span>
who mate to a family of maps
<span class="math-container">$$ \tilde\beta : \Theta(F) \to PA(F, \Theta(\hom(-,a)))$$</span></li>
</ul>
<p>Now... who's the unit? Who's the counit?</p>
</blockquote>
<p><span class="math-container">$P$</span>, being a free cocompletion, is a <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad" rel="nofollow noreferrer">KZ-monad</a>. As soon as one wants to write down explicitly what this structure is, however, they have to face a few slight inaccuracies in how <span class="math-container">$P$</span> was defined;</p>
<ul>
<li><p>first of all it is not only contravariant, but also partially defined; it is a so-called <a href="https://ncatlab.org/nlab/show/relative+monad" rel="nofollow noreferrer">relative monad</a>, like a monad but not an endofunctor. In this particular case, <span class="math-container">$P : cat^\text{coop}\to Cat$</span> is a monad relative to <span class="math-container">$i^\text{coop} : cat^\text{coop} \to Cat^\text{coop}$</span>, the inclusion of small into locally small categories.</p>
<blockquote>
<p><strong>Q1</strong>: Am I wrong if I define a contravariant (total) monad to be a contravariant endofunctor <span class="math-container">$T : {\cal C}\to \cal C$</span> which is relative to <span class="math-container">$1^\text{op} : {\cal C}^\text{op} \to {\cal C}^\text{op}$</span>?</p>
</blockquote></li>
<li><p>second, it seems to satisfy mixed properties of a lax and a colax idempotent 2-monad: in particular, it seems to me that every <span class="math-container">$P$</span>-algebra <span class="math-container">$a :PA \to A$</span> is a left adjoint to the unit (so lax), and yet <span class="math-container">$\mu \dashv \eta P$</span> (so colax).</p>
<blockquote>
<p><strong>Q2</strong>: Am I committing a mistake? If not, does these mixed properties have to do with the fact that <span class="math-container">$P$</span> is contravariant? </p>
</blockquote></li>
<li><p>third it is "unidetermined", i.e. <span class="math-container">$\mu$</span> is determined by <span class="math-container">$\eta$</span>. </p>
<blockquote>
<p><strong>Q3</strong>: How does this affect the equations defining a KZ-monad, if at all?</p>
</blockquote></li>
</ul>
https://mathoverflow.net/q/3132742A question about distribution of fractional part of $2^k\alpha$Yu Dinghttps://mathoverflow.net/users/1303112018-10-20T04:37:15Z2018-10-20T10:23:07Z
<p>Let <span class="math-container">$\{x\}$</span> be the fractional part of <span class="math-container">$x$</span>, i.e. <span class="math-container">$\{x\}=x-[x]$</span>, where <span class="math-container">$[x]$</span> is the biggest integer <span class="math-container">$\leq x$</span>. </p>
<p>The question might be well known but I don't know where to look for: Assume <span class="math-container">$\alpha$</span> is an irrational number. Then does the sequence <span class="math-container">$\{\alpha\}$</span>,
<span class="math-container">$\{2\alpha\}$</span>, <span class="math-container">$\{4\alpha\}$</span>, <span class="math-container">$\{8\alpha\}$</span>, ... distribute uniformly on <span class="math-container">$[0, 1]$</span>? </p>
<p>In fact I was checking if the power series <span class="math-container">$z+z^2+z^4+z^8+...$</span> has bounded partial sum when <span class="math-container">$z=e^{2\pi i \alpha}$</span> and <span class="math-container">$\alpha$</span> is irrational. However even if we do have uniform distribution I am still not sure the partial sum is bounded.</p>
https://mathoverflow.net/q/3132714How does multiplication affect degrees?H A Helfgotthttps://mathoverflow.net/users/3982018-10-20T00:20:09Z2018-10-20T10:50:33Z
<p>Let <span class="math-container">$M(n) \sim \mathbb{A}^{n^2}$</span> be the space of <span class="math-container">$n$</span>-by-<span class="math-container">$n$</span> matrices, seen as an affine space over a field <span class="math-container">$K$</span>, and endowed with the usual matrix multiplication. Let <span class="math-container">$V$</span> and <span class="math-container">$W$</span> be subvarieties of <span class="math-container">$M(n)$</span>. The product <span class="math-container">$V\cdot W$</span> is a constructible set (by Chevalley's theorem); write <span class="math-container">$\overline{V\cdot W}$</span> for its Zariski closure.</p>
<p>Is it the case that <span class="math-container">$\deg(\overline{V\cdot W}) \leq \deg(V) \cdot \deg(W)$</span>?</p>
<hr>
<p>Note: this is a less ambitious version of a question I previously asked (<a href="https://mathoverflow.net/questions/312671/b%c3%a9zout-and-products-in-algebraic-groups">Bézout and products in algebraic groups</a>).</p>
https://mathoverflow.net/q/3132683When is $\Omega^1$ an equivalence?Marehttps://mathoverflow.net/users/619492018-10-19T23:12:21Z2018-10-20T13:00:08Z
<blockquote>
<p>Let <span class="math-container">$C$</span> be an abelian category with enough projectives and <span class="math-container">$\underline{C}$</span> the stable category of <span class="math-container">$C$</span> that is obtained by factoring out projective modules.
When is the functor <span class="math-container">$\Omega^1 : \underline{C} \rightarrow \underline{C}$</span> an equivalence?</p>
</blockquote>
<p>We can assume <span class="math-container">$C$</span> is a module category of a ring <span class="math-container">$R$</span> in case that helps. For <span class="math-container">$R$</span> a finite dimensional algebra, this should be true iff <span class="math-container">$R$</span> is self-injective.</p>
<blockquote>
<p>Is there an explicit example of a ring <span class="math-container">$R$</span> which is not quasi-Frobenius and has the property that <span class="math-container">$\Omega^1$</span> is an equivalence on the stable module category?</p>
</blockquote>
https://mathoverflow.net/q/31325419References for Riemann surfacesseubhttps://mathoverflow.net/users/255902018-10-19T16:50:43Z2018-10-20T11:27:42Z
<p>I know this question has been asked before on MO and MSE (<a href="https://mathoverflow.net/questions/59605/reference-in-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/407004/good-book-for-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/1839673/books-on-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/200537/complex-analysis-book-with-a-view-toward-riemann-surfaces">here</a>) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one.</p>
<p>I am about to teach a first course on Riemann surfaces, and I am trying to get a fairly comprehensive view of the main references, as a support for both myself and students.</p>
<p>I compiled a list, here goes in alphabetical order. Of course, it is necessarily subjective. For more detailed entries, I made a bibliography using the bibtex entries from MathSciNet: <a href="https://www.brice.loustau.eu/teaching/RiemannSurfaces2018/References.pdf" rel="noreferrer">click here</a>.</p>
<ol>
<li>Bobenko. Introduction to compact Riemann surfaces. </li>
<li>de Saint-Gervais. Uniformisation des surfaces de Riemann: retour sur un théorème centenaire.</li>
<li>Donaldson. Riemann surfaces.</li>
<li>Farkas and Kra. Riemann surfaces.</li>
<li>Forster. Lectures on Riemann surfaces.</li>
<li>Griffiths. Introduction to algebraic curves.</li>
<li>Gunning. Lectures on Riemann surfaces.</li>
<li>Jost. Compact Riemann surfaces.</li>
<li>Kirwan. Complex algebraic curves.</li>
<li>McMullen. Complex analysis on Riemann surfaces.</li>
<li>McMullen. Riemann surfaces, dynamics and geometry.</li>
<li>Miranda. Algebraic curves and Riemann surfaces.</li>
<li>Narasimhan. Compact Riemann surfaces.</li>
<li>Narasimhan and Nievergelt. Complex analysis in one variable.</li>
<li>Springer. Introduction to Riemann surfaces.</li>
<li>Varolin. Riemann surfaces by way of complex analytic geometry.</li>
<li>Weyl. The concept of a Riemann surface.</li>
</ol>
<p>Having a good sense of what each of these books does, beyond a superficial first impression, is quite a colossal task (at least for me).</p>
<p>What I'm hoping is that if you know very well such or such reference in the list, you can give a short description of it: where it stands in the existing literature, what approach/viewpoint is adopted, what are its benefits and pitfalls. Of course, I am also happy to update the list with new references, especially if I missed some major ones.</p>
<p>As an example, for Forster's book (5.) I can just use the accepted answer <a href="https://math.stackexchange.com/questions/407004/good-book-for-riemann-surfaces">there</a>: According to <a href="https://math.stackexchange.com/users/71348/ted-shifrin">Ted Shifrin</a>:</p>
<blockquote>
<p>It is extremely well-written, but definitely more analytic in flavor.
In particular, it includes pretty much all the analysis to prove
finite-dimensionality of sheaf cohomology on a compact Riemann
surface. It also deals quite a bit with non-compact Riemann surfaces,
but does include standard material on Abel's Theorem, the Abel-Jacobi
map, etc.</p>
</blockquote>
https://mathoverflow.net/q/3132294When can a scheme be recovered from its descent groupoid?Gausslerhttps://mathoverflow.net/users/804672018-10-19T11:35:42Z2018-10-20T13:37:59Z
<p>Suppose that <span class="math-container">$ Y $</span> is a scheme and <span class="math-container">$ f\colon X\to Y $</span> a covering of <span class="math-container">$ Y $</span> in some Grothendieck topology on the category of schemes (i.e. if <span class="math-container">$\{ U_i\to Y\}$</span> is a covering in the topological sense, then <span class="math-container">$ X = \coprod U_i$</span>). Then we may consider the descent groupoid <span class="math-container">$\Gamma_0 = X$</span> and <span class="math-container">$ \Gamma_1 = X \times_{Y} X $</span> with source and target maps <span class="math-container">$ s,t\colon\Gamma_1 \rightrightarrows \Gamma_0$</span> given by the two projections (so for coverings in the topological sense, <span class="math-container">$ \Gamma_1 = \coprod U_i\cap U_j$</span>).</p>
<p>It is natural to expect from a covering that <span class="math-container">$ Y $</span> can be recovered from <span class="math-container">$\Gamma_0$</span> by gluing along the intersections, i.e. <span class="math-container">$ Y =\mathrm{Coeq}( \Gamma_1 \rightrightarrows \Gamma_0 ) $</span>. Equivalently, we would like the pullback square
<span class="math-container">$$
\require{AMScd}
\begin{CD}
X\times_{Y} X @>{t}>> X\\
@V{s}VV @VV{f}V \\
X @>>{f}> Y
\end{CD}
$$</span>
to be also a pushout square.</p>
<p>Do there exist sufficient conditions on a Grothendieck topology on schemes under which <span class="math-container">$Y$</span> is the pushout <span class="math-container">$$\displaystyle Y = \Gamma_0\coprod_{\Gamma_1} \Gamma_0 = X\coprod_{X\times_Y X} X$$</span> as above? Is this true for all the standard topologies (Zariski, etalé, fpqc, fppf)?</p>
https://mathoverflow.net/q/3131311how to calculate the following integral related to Chebyshev polynomialsJacob.Z.Leehttps://mathoverflow.net/users/428162018-10-18T11:10:34Z2018-10-20T13:17:52Z
<p>Chebyshev polynomials of the second kind <span class="math-container">$V_n(x)$</span> can be defined as
<span class="math-container">$$V_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$</span>
or through the recurrence relation
<span class="math-container">$$V_{n+1}=xV_n-V_{n-1}, V_0=1, V_1=x.$$</span>
First few low-order Chebyshev polynomials of the second
kind are as follows:
<span class="math-container">$$V_0=1, V_1=x,V_2=x^2-1,V_3=x^3-2x,$$</span>
<span class="math-container">$$V_4=x^4-3x^2+1,V_5=x^5-4x^3+3x.$$</span></p>
<p>I want to know how to calculate the following integral relate to Chebyshev polynomials:
<span class="math-container">$$\int_0^\pi (\frac{\sin nx}{\sin x})^m dx$$</span>
where <span class="math-container">$n,m\in \mathbb{Z}^+$</span>.</p>
<p>It is easy to see the following result:</p>
<p>For an even number <span class="math-container">$n \in \mathbb{Z}^+$</span> and and odd number <span class="math-container">$m \in \mathbb{N}$</span>, we have</p>
<p><span class="math-container">$$ \int_0^\pi (\frac {\sin nx}{\sin x})^{m} dx=0.$$</span></p>
<p>I conjectured that the result is a polynomial <span class="math-container">$P(n)$</span> of order <span class="math-container">$m−1$</span>. </p>
<p>I prefer to know the other two cases beside the above special case. Thank you.</p>
https://mathoverflow.net/q/3131286A combinatorial property of uncountable groups, IITaras Banakhhttps://mathoverflow.net/users/615362018-10-18T10:28:35Z2018-10-20T18:15:08Z
<blockquote>
<p><strong>Problem 1.</strong> Is it true that each uncountable group <span class="math-container">$G$</span> contains two subsets <span class="math-container">$A,B\subset G$</span> such that</p>
<p>1) for any <span class="math-container">$x,y\in G$</span> the intersection <span class="math-container">$xA\cap yB$</span> is finite and </p>
<p>2) for any function <span class="math-container">$\Phi:G\to 2^G$</span> assigning to each element <span class="math-container">$g\in G$</span> a finite subset <span class="math-container">$\Phi(g)\subset G$</span> there are two elements <span class="math-container">$x,y\in G$</span> and points <span class="math-container">$a\in A\setminus\Phi(x)$</span> and <span class="math-container">$b\in B\setminus \Phi(y)$</span> such that <span class="math-container">$xa=yb$</span>.</p>
</blockquote>
<p><strong>Remark.</strong> Such sets <span class="math-container">$A,B$</span> do exist if <span class="math-container">$G$</span> contains a subgroup <span class="math-container">$H$</span> that admits a homomorphism onto a group <span class="math-container">$\Gamma$</span> that contains an uncountable subset <span class="math-container">$U$</span> with infinite centralizer <span class="math-container">$C(U)=\bigcap_{u\in U}\{x\in\Gamma:xu=ux\}$</span>. This means that a counterexample if exists, should be very non-commutative, like a Jonsson group, constructed by <a href="https://www.sciencedirect.com/science/article/pii/S0049237X08713466" rel="nofollow noreferrer">Shelah</a>.
We recall that a group <span class="math-container">$G$</span> is <em>Jonsson</em> if it is uncountable but all proper subgroups of <span class="math-container">$G$</span> are countable.</p>
<blockquote>
<p><strong>Problem 2.</strong> What is the answer to the Problem for simple Jonsson groups (constructed by <a href="https://www.sciencedirect.com/science/article/pii/S0049237X08713466" rel="nofollow noreferrer">Shelah</a>)?</p>
</blockquote>
<p><strong>Comment.</strong> Problem 1 is a combinatorial reformulation of Question 2.2 from <a href="https://arxiv.org/pdf/1807.03028.pdf" rel="nofollow noreferrer">this survey of Protasov</a>. Question 2.2 asks if the countability of a group is equivalent to the normality of its finitary ballean. This question was also repeated (as Problem 12.6) in the paper <a href="https://arxiv.org/abs/1810.07979" rel="nofollow noreferrer">"The normality and bounded growth of balleans"</a> of Banakh and Protasov. </p>
https://mathoverflow.net/q/3131224Irrationality of the values of the prime zeta functionPierreTheFermentedhttps://mathoverflow.net/users/1037222018-10-18T09:03:04Z2018-10-20T14:24:09Z
<p><em>Preamble: I asked this question on <a href="https://math.stackexchange.com/questions/2820448/irrationality-of-the-values-of-the-prime-zeta-function">Math.SE</a> with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead.</em></p>
<hr>
<p>Since Apéry we know that <span class="math-container">$\zeta(3)$</span>, where <span class="math-container">$\zeta$</span> denotes the Riemann zeta function, is irrational. It is also well known that infinitely many values of the Riemann zeta function at odd positive integers are irrational. Moreover, various results by Zudilin have shown that certain subsets of zeta values at odd positive integers are irrational; for instance, at least one of <span class="math-container">$ζ(5), ζ(7), ζ(9)$</span>, or <span class="math-container">$ζ(11)$</span> is irrational.</p>
<p>Are there any similar results for <span class="math-container">$P(n)$</span>, where <span class="math-container">$P$</span> is the prime zeta function, i.e.,</p>
<p><span class="math-container">$$
{\displaystyle P(n)=\sum _{p\,\in \mathrm {\,primes} }{\frac {1}{p^{n}}}={\frac {1}{2^{n}}}+{\frac {1}{3^{n}}}+{\frac {1}{5^{n}}}+{\frac {1}{7^{n}}}+{\frac {1}{11^{n}}}+\cdots ?}
$$</span></p>
<p>A quick search on Wolfram Alpha reveals the following:</p>
<ul>
<li><a href="https://math.stackexchange.com/questions/2820448/irrationality-of-the-values-of-the-prime-zeta-function">is P(2) irrational?</a> - <em>unknown</em></li>
<li><a href="http://www.wolframalpha.com/input/?i=is%20P(3)%20irrational%3F" rel="nofollow noreferrer">is P(3) irrational?</a> - <em>unknown</em></li>
<li><span class="math-container">$\ldots$</span></li>
</ul>
<p>I was not able to find any papers or articles related to the irrationality of values of <span class="math-container">$P$</span> at positive integers. Have these been studied in a (more or less) serious manner, analogously to <span class="math-container">$\zeta$</span>? What are the current results?</p>
<hr>
<p>EDIT: In the comment section several links to Math.SE have been posted. These do not answer my question, since I am looking for <em>recent research</em> on the subject, not answers of the type "<strong>no</strong>" or "<strong>yes</strong>".</p>
https://mathoverflow.net/q/3130172Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEsZylhttps://mathoverflow.net/users/1234562018-10-16T23:06:33Z2018-10-20T14:28:35Z
<p>Where can I find a (readable and self-contained) proof of the following result?</p>
<blockquote>
<p>Let <span class="math-container">$\Omega$</span> be a Lipschitz domain of <span class="math-container">$\mathbb{R}^n$</span>, with <span class="math-container">$B(0,1) \subset \Omega$</span>. Let <span class="math-container">$u$</span> be the solution of <span class="math-container">$$-\mathrm{div}(A(x)\nabla u) = \delta_0,$$</span>
<span class="math-container">$$u|_{\partial \Omega} = 0,$$</span>
where <span class="math-container">$A$</span> is bounded, measurable and uniformly elliptic (<span class="math-container">$C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$</span>). Then on <span class="math-container">$B_{1/2}$</span>, we have
<span class="math-container">$$\frac{C_2}{|x|^{n-2}} \le u(x) \le \frac{C_1}{|x|^{n-2}}.$$</span></p>
</blockquote>
https://mathoverflow.net/q/3129796Adding constraints as penalty with $\| \cdot \|_0$ normTheWaveLadhttps://mathoverflow.net/users/735712018-10-16T14:28:29Z2018-10-20T13:36:32Z
<p>In the paper <a href="https://www.egr.msu.edu/~aviyente/elad06.pdf" rel="noreferrer"><em>Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries</em> (page 2)</a>, the authors rewrite the minimization problem </p>
<p><span class="math-container">\begin{align}
\min_{\alpha \in \mathbb R^k} \| \alpha \|_0 && s.t. && \|D \alpha - y \|_2^2 \leq T,
\end{align}</span></p>
<p>where <span class="math-container">$D \in \mathbb R^{n \times k}$</span> and <span class="math-container">$y \in \mathbb R^n$</span>, into</p>
<p><span class="math-container">\begin{align}
\min_{\alpha \in \mathbb R^k} \| D \alpha - y \|_2^2 + \mu \| \alpha \|_0
\end{align}</span></p>
<p>and state that </p>
<blockquote>
<p>for a proper choice of <span class="math-container">$\mu$</span>, the two problems are equivalent.</p>
</blockquote>
<p>There is no reference in the paper that explains why this would be true and in fact, I don't believe it is true, since <span class="math-container">$\| \cdot \|_0$</span> is not even convex. Does such a <span class="math-container">$\mu$</span> really exist?</p>
<p>I don't know if this is important, but the following assumptions were made: <span class="math-container">$y$</span> is a noisy version of <span class="math-container">$x$</span> with zero mean white noise of variance <span class="math-container">$\sigma^2$</span>, and that there exists <span class="math-container">$x$</span> so that <span class="math-container">$\| D \alpha - x \|_2 \leq \epsilon$</span> with <span class="math-container">$\| \alpha \|_0 \leq L \ll n$</span>. The <span class="math-container">$T$</span> from the first equation above is dictated by <span class="math-container">$\varepsilon$</span> and <span class="math-container">$\sigma$</span>.</p>
<p>I already asked two professors who also don't believe the statement is true. However, it has to come from somewhere. Does it maybe work with <span class="math-container">$\| \cdot \|_1$</span>? Or is this just an "analytic application" of a penalty method?</p>
https://mathoverflow.net/q/31210811Defining abstract varieties and their morphisms over a finitely generated subfield of the base fieldMikhail Borovoihttps://mathoverflow.net/users/41492018-10-05T14:47:38Z2018-10-20T11:20:04Z
<p>Let <span class="math-container">$k$</span> be an algebraically closed field.
By a finitely generated subfield of <span class="math-container">$k$</span> I mean a subfield <span class="math-container">$k_0\subset k$</span> that is finitely generated over the prime subfield of <span class="math-container">$k$</span> (that is, over <span class="math-container">$\mathbb Q$</span> or <span class="math-container">$\mathbb F_p$</span>).</p>
<p>I need a reference or a proof for the following (well-known? evident?) proposition:</p>
<blockquote>
<p><strong>Proposition.</strong>
<em>Let
<span class="math-container">$$f\colon X\to V$$</span>
be a morphism of <span class="math-container">$k$</span>-varieties.
Then the triple <span class="math-container">$(X,Y,f)$</span> can be defined over a finitely generated subfield of <span class="math-container">$k$</span>.
In other words, there exists a finitely generated subfield <span class="math-container">$k_0\subset k$</span>
and a morphism of <span class="math-container">$k_0$</span>-varieties
<span class="math-container">$$f_0\colon X_0\to Y_0$$</span>
such that
<span class="math-container">$(X_0,Y_0,f_0)\times_{k_0} k$</span> is isomorphic to <span class="math-container">$(X,Y,f)$</span>.</em></p>
</blockquote>
<p>A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for <em>abstract</em> varieties.</p>
https://mathoverflow.net/q/3120784Regular Borel measures and the measure of a singletonAndré Portohttps://mathoverflow.net/users/1007402018-10-05T03:56:21Z2018-10-20T15:50:57Z
<p>I'm studying this paper: <a href="http://matwbn.icm.edu.pl/ksiazki/sm/sm73/sm7313.pdf" rel="nofollow noreferrer">http://matwbn.icm.edu.pl/ksiazki/sm/sm73/sm7313.pdf</a></p>
<p>At the top of page 36, it states the following Proposition:</p>
<blockquote>
<p>Let <span class="math-container">$S$</span> be a compact and <span class="math-container">$\mu$</span> a regular Borel measure on <span class="math-container">$S$</span> with total variation 1. If for every partition of <span class="math-container">$S$</span> into two Borel subsets the measure of the smaller one is less than <span class="math-container">$1/3$</span>, then there is an <span class="math-container">$s_0\in S$</span> such that <span class="math-container">$\mu(\{s_0\})>2/3$</span>,.</p>
</blockquote>
<p>The author does not prove this Proposition. Maybe it's so obvious, but I simply have no idea of the proof.</p>
<p>I list below some approaches I've been trying with no success:</p>
<p><strong>1)</strong> I tried to do some kind of argument with ultrafilters. If we consider the family <span class="math-container">$\mathcal A$</span> of Borel sets with measure greater than <span class="math-container">$2/3$</span> it is a filter contained in the <span class="math-container">$\sigma$</span>-algebra of borelian sets. Moreover, <span class="math-container">$\mathcal A$</span> satisfies the following property:</p>
<blockquote>
<p>If <span class="math-container">$E$</span> is a Borel set of <span class="math-container">$S$</span> then either <span class="math-container">$E\in \mathcal A$</span> or <span class="math-container">$E^c\in \mathcal A$</span>.</p>
</blockquote>
<p>Therefore, if we pick an ultrafilter <span class="math-container">$U$</span> containing <span class="math-container">$\mathcal A$</span> then <span class="math-container">$U\cap\mathcal B_S=\mathcal A$</span>, where <span class="math-container">$\mathcal B_S$</span> denotes the family of borel sets of <span class="math-container">$S$</span>. The idea to do this was to conclude that <span class="math-container">$U$</span> cannot be a free ultrafilter, but I can't see any further argument.</p>
<p><strong>2)</strong> Define <span class="math-container">$ \lambda=\sup\{\mu(E): E\in\mathcal B_S \mbox{ and } \mu(E)\leq 2/3 \}.$</span></p>
<p>Let us see that <span class="math-container">$\lambda$</span> is assumed. Suppose the contrary and define
<span class="math-container">$$
\beta =\inf\{\mu(E): E\in\mathcal B_S \mbox{ and } \mu(E)\geq 1/3\}\geq2/3.
$$</span>
Since <span class="math-container">$\lambda$</span> is not assumed, <span class="math-container">$\beta$</span> is not assumed. For each <span class="math-container">$n\in \mathbb N$</span>, pick <span class="math-container">$E_n\in\mathcal B_S$</span> such that
<span class="math-container">$$
\beta< \mu(E_n)< \beta+\frac{1}{n}.
$$</span>
If <span class="math-container">$m>n$</span> then <span class="math-container">$\mu(E_m\cap E_n)\leq \beta + 1/m$</span> and let us see that <span class="math-container">$$\mu(E_m\cap E_n) \geq \beta.$$</span>
Indeed, if <span class="math-container">$\mu(E_m\cap E_n)< \beta$</span>, then <span class="math-container">$\mu(E_m\cap E_n)\leq 1 - \beta$</span>, and consequently,
<span class="math-container">$$
2\beta < \mu(E_m)+\mu(E_n) = \mu(E_m\cap E_n) + \mu(E_m \cup E_n)
\leq 1 - \beta + 1,
$$</span>
and it follows that <span class="math-container">$\beta< 2/3$</span>, a contradiction.</p>
<p>We proved above that, if <span class="math-container">$\mu(E_n) \leq \beta + 1/n$</span> and <span class="math-container">$\mu(E_m) \leq \beta+ 1/m$</span> and <span class="math-container">$m>n$</span>, then
<span class="math-container">$$
\beta \leq \mu(E_m\cap E_n) \leq \beta + 1/m.
$$</span></p>
<p>Then, by induction, we prove that <span class="math-container">$$\beta \leq \mu\left(\bigcap^n_{j=1} E_j\right)\leq \beta + 1/n,\ \forall n\in\mathbb N.$$</span>
and consequently, <span class="math-container">$\mu(\bigcap^\infty_{j=1} E_j)=\beta$</span>. Therefore, <span class="math-container">$\beta$</span> is assumed and so <span class="math-container">$\lambda$</span> is assumed.</p>
https://mathoverflow.net/q/3048873Interpolation inequality related to the 5/3-Laplace operatorGabe Khttps://mathoverflow.net/users/1252752018-07-12T20:08:56Z2018-10-20T14:17:32Z
<p>I'm having trouble with an estimate that would be helpful in information geometry.</p>
<p>The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a compact manifold without boundary with $Vol(X) = 1$. We also assume that $g$ satisfies $\int_X g^{10} dx =1.$
I want to obtain a positive lower bound on the following functional:
$$DF(g) := \frac{ \| \nabla g \|_{L^{5/3}(X)}}{ 1 - \int_X g dx }$$</p>
<p>This bound will depend on $X$, but I would ideally want it to be done in any dimension. In the 1-dimension case, this can be done, and it feels like something that should follow from a Sobolev embedding inequality, but I'm not seeing how to do it.</p>
<p>Does anyone have any advice on an interpolation inequality or something similar which would be able to help with this?</p>
<p>Edit: An earlier version of this question had a related functional, but it turned out to not have the desired property. With extra calculations, I've been able to simple it to a form that seems more likely to be true.</p>
https://mathoverflow.net/q/302934-4Why sheaves are important and why do we care about them? [closed]Lolmanhttps://mathoverflow.net/users/567132018-06-16T10:59:28Z2018-10-20T10:15:09Z
<p>Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$</p>
<p>For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(C,J)\subseteq Fun(C^{op},Set).$$</p>
<p>Part of the beauty/usefulness of this procedure is that the resulting objects are topoi and working in them is "easy".</p>
<p>The first question is: why do we prefer $Set$ above all else? Why $Set$ seems to be the center of this construction?
Usually when consider functions/morphisms/functors targeting some structure, the set/category of functions/functors inherits this structure. ("Usually" our structure comes from products and limits so "usually" it works. I never worked with Hopf algebras to not use the usually.)
My guess is that it seems we want to "pullback" $Set$'s structure, that is a topos. Is this all there is to it?</p>
<p>The second question is why we do it for other categories as well? We consider sheaves of groups or rings. We will never get a topos, but we are nevertheless interested. And it seems that some people are interested even in sheaves over other less concrete categories. (I don't have an explicit example.)</p>
<p>So the main question is: why do sheaves always seem to pop out? Why it seems that sheaves contain interesting information?</p>
<p>[EDIT]</p>
<p>It seems my question is ambiguous and noons gets it...
A group us not a sheaf, is a category. A ring is not a sheaf, is a category. A metric space is not a sheaf is a category. A posed is not a sheaf is a category ( in this case both enriched and not).
When we consider many concepts in math close to these concepts it seems that sheaves comes out pretty often. I do not know about stochastic processes, but I wouldn't except them to be far away just because today none treats them as sheaves.
Most base theories are categories, because we need indexes. Most interesting construction on these theories are categories of sheaves. The question was why it is so.</p>
https://mathoverflow.net/q/3027652How to use these higher symmetries and conservation laws?W. muhttps://mathoverflow.net/users/985432018-06-14T09:47:27Z2018-10-20T13:35:13Z
<p>For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation.</p>
<p>However, unlike the classical symmetries (point symmetries), the higher symmetries (or Lie-Backlund symmetries; such as KdV hierarchy) seem useless, or there are something I am unfamilar. </p>
<p>Similar case are the conservation laws. For KdV equation, we have infinite many $\int udx$, $\int u^2dx$, $\int \frac12u_x^2-u^3dx$, $\cdots$. But it seems that only the first few conservation laws are useful.</p>
<p>I know some people treat the existence of infinite symmetries or conservation laws as a criterion whether the equation is integrable, but I don't see the real application.</p>
<p>The question is : how to utilize these infinite symmetries and conservation laws? </p>
https://mathoverflow.net/q/757778Elementary proof of the equidistribution theoremuser8761468https://mathoverflow.net/users/179032011-09-18T18:43:19Z2018-10-20T13:26:11Z
<p>I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is uniformly distributed on the unit interval. With "elementary" I mean that it does not make use of complex analysis in particular the <a href="http://mathworld.wolfram.com/WeylsCriterion.html" rel="noreferrer">Weyl's criterion</a>.</p>
<p>Thank you very much.</p>