Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2020-05-29T15:30:26Zhttps://mathoverflow.net/feedshttps://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/3616730Random walk on triangular latticestackoverloadhttps://mathoverflow.net/users/1588142020-05-29T15:30:06Z2020-05-29T15:30:06Z
<p>Recently, I have been solving the problem: is random walk on triangular lattice transient. Let's define <span class="math-container">$N$</span> to number of times visiting the origin, then the problem is to show, that <span class="math-container">$\mathbf{E}(N) = \infty$</span>. To calculate <span class="math-container">$\mathbf{E}(N)$</span> I need to calculate <span class="math-container">$\mathbf{P}(V_k = o)$</span>, where <span class="math-container">$V_i$</span> denotes the current position after <span class="math-container">$i$</span>-th step. Calculating <span class="math-container">$\mathbf{P}(V_k = o)$</span> is a bit tricky because I cannot come up with the proper invariant when it is the case (when it is regular integer grid, then the invariant is clear: equal number of horizontal and vertical steps). Any suggestions are appriciated.</p>
https://mathoverflow.net/q/3616720Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?Bobbrownhttps://mathoverflow.net/users/1187222020-05-29T15:25:12Z2020-05-29T15:25:12Z
<p>Let <span class="math-container">$(X,d)$</span> be an <span class="math-container">$n$</span>-dimensional <span class="math-container">$(n< \infty)$</span> complete geodesic metric space. Let <span class="math-container">$S$</span> be a sufficiently small metric <span class="math-container">$(n-1)$</span>-sphere in <span class="math-container">$X$</span>. Let <span class="math-container">$\epsilon>0$</span> be the radius of <span class="math-container">$S$</span>. Pick a point <span class="math-container">$p \in S$</span>. Find a metric <span class="math-container">$(n-1)$</span>-sphere <span class="math-container">$S'$</span> of radius <span class="math-container">$0<r < \epsilon/2$</span> around <span class="math-container">$p$</span>. Is the intersection <span class="math-container">$S \cap S'$</span> always an <span class="math-container">$(n-2)$</span>-metric sphere? </p>
https://mathoverflow.net/q/3616710Zero-dimensional functions in the planeD.S. Liphamhttps://mathoverflow.net/users/957182020-05-29T15:04:58Z2020-05-29T15:28:16Z
<p>Is the following true?</p>
<p><strong>Conjecture.</strong> Let <span class="math-container">$\varphi:C\to [0,\infty)$</span> be an upper semi-continuous function, where <span class="math-container">$C\subseteq \mathbb R$</span> is a Cantor set. Let <span class="math-container">$X$</span> be a zero-dimensional subset of the graph of <span class="math-container">$\varphi$</span>. Suppose <span class="math-container">$\langle c,\varphi(c)\rangle\in X$</span>, and <span class="math-container">$\varphi(c)>0$</span>. Then there exist <span class="math-container">$a,b\in \mathbb R$</span> with <span class="math-container">$a<c<b$</span>, and a continuous function <span class="math-container">$f:[a,b]\to (0,\infty)$</span> such that <span class="math-container">$f(c)<\varphi(c)$</span> and the graph of <span class="math-container">$f$</span> misses <span class="math-container">$X$</span>.</p>
<p><em>Upper semi-continuous</em> means that <span class="math-container">$\varphi^{-1}[0,t)$</span> is open in <span class="math-container">$C$</span> for every <span class="math-container">$t>0$</span>.</p>
<p><a href="https://i.stack.imgur.com/zlL6q.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zlL6q.jpg" alt="enter image description here"></a></p>
<p>Remarks:</p>
<p>(1) The assumption that <span class="math-container">$X$</span> is zero-dimensional is critical. Because there exist "Lelek functions" <span class="math-container">$\varphi$</span> such that <span class="math-container">$X=$</span> (entire graph of <span class="math-container">$\varphi$</span>) is a counterexample.</p>
<p>(2) If yes, then <span class="math-container">$\mathbb R\times(0,\infty)\setminus X$</span> is path-connected. As shown here: <a href="https://mathoverflow.net/questions/351807/is-the-complement-of-a-zero-dimensional-subset-of-the-plane-path-connected">Is the complement of a zero-dimensional subset of the plane path-connected?</a>, the complement of a zero-dimensional subset of the plane is not necessarily path-connected. But in my conjecture, the zero-dimensional set is particularly "nice" (it is contained in the graph of an upper semi-continuous function).</p>
<p>(3) If <span class="math-container">$\varphi$</span> is continuous, then the answer is yes. In this case <span class="math-container">$\varphi$</span> can be extended to a continuous function <span class="math-container">$\varphi':\mathbb R\to[0,\infty)$</span>. And then given a point of <span class="math-container">$X$</span> above the horizontal axis, we can just shift a piece of the graph of <span class="math-container">$\varphi'$</span> down by some small constant to obtain <span class="math-container">$f$</span>.</p>
<p>(4) Though not obvious, the conjecture implies negative answers to the questions here: <a href="https://mathoverflow.net/questions/352788/the-core-of-complete-erd%c5%91s-space">The "core" of complete Erdős space</a>. </p>
https://mathoverflow.net/q/3616700Fully faithful non-exact functor between abelian categoriesBas Winkelmanhttps://mathoverflow.net/users/1216602020-05-29T14:40:30Z2020-05-29T14:40:30Z
<p>I am looking for an example of two abelian categories <span class="math-container">$C$</span> and <span class="math-container">$D$</span>, and a functor <span class="math-container">$F:C \to D$</span> which is full and faithful but not exact.</p>
https://mathoverflow.net/q/3616680Checking axiom of Category $\mathcal{O}$CJShttps://mathoverflow.net/users/1356742020-05-29T14:27:45Z2020-05-29T14:40:05Z
<p>Let <span class="math-container">$K$</span> be a finite extension of <span class="math-container">$\mathbb{Q}_p$</span> and <span class="math-container">$G$</span> be a split connected reductive algebraic group over <span class="math-container">$K$</span> with Borel <span class="math-container">$B$</span>. We have the associated Lie algebras <span class="math-container">$\mathfrak{g}=$</span>Lie<span class="math-container">$(G)$</span> and <span class="math-container">$\mathfrak{b}=$</span>Lie<span class="math-container">$(B)$</span>. </p>
<p>Let <span class="math-container">$M$</span> be a <span class="math-container">$U(\mathfrak{g})$</span>-module with <span class="math-container">$N \subset M$</span> a finite dimensional <span class="math-container">$K$</span>-module, which is <span class="math-container">$B$</span>-invariant and generates <span class="math-container">$M$</span> as a <span class="math-container">$U(\mathfrak{g})$</span>-module. </p>
<p>I read that <span class="math-container">$M$</span> is then locally <span class="math-container">$\mathfrak{b}$</span>-finite, i.e <span class="math-container">$U(\mathfrak{b}) \cdot m \subset M$</span> is finite dimensional for all <span class="math-container">$m \in M$</span>, but I have trouble to see this. As <span class="math-container">$U(\mathfrak{b})$</span> seems so big for me, I cannot think of a finite basis for <span class="math-container">$U(\mathfrak{b}) \cdot m \subset M$</span> by knowing only <span class="math-container">$N$</span>.</p>
https://mathoverflow.net/q/3616641Some basic questions on quotient of group schemesDaebeom Choihttps://mathoverflow.net/users/956012020-05-29T13:24:59Z2020-05-29T13:24:59Z
<p>Let <span class="math-container">$S$</span> be a fixed base scheme and <span class="math-container">$G, H$</span> be group schemes over <span class="math-container">$S$</span>. Since I am mainly interested in commutative group schemes over fields, we may assume that <span class="math-container">$G,H$</span> are commutative and <span class="math-container">$S$</span> is a field if this helps.</p>
<p>(1) Let <span class="math-container">$f:G\to H$</span> be a morphism of group schemes. To define the cokernel of this map, we need to choose which topology to work with. Some people use the fppf topology (<a href="http://gerard.vdgeer.net/AV.pdf" rel="nofollow noreferrer">as in van der Geer & Moonen's book</a>) and other people use the fpqc topology (<a href="https://www.springer.com/gp/book/9780387963112" rel="nofollow noreferrer">as in Cornell-Silverman</a>). My question is: what is the difference of those two topologies in terms of group schemes? Is fppf quotient and fpqc quotient of group schemes different? Which topology do people prefer when they are working with group schemes?</p>
<p>(2) Let <span class="math-container">$H$</span> be a (normal) closed subgroup scheme of <span class="math-container">$G$</span>. I think there are at least three plausible definitions of the quotient <span class="math-container">$G/H$</span>:</p>
<ol>
<li><p>Categorical quotient: Since <span class="math-container">$H$</span> naturally acts on <span class="math-container">$G$</span>, we can think categorical quotient <span class="math-container">$G/H$</span> of the action <span class="math-container">$H\times G\to G$</span>.</p></li>
<li><p>Fppf/fpqc quotient: <span class="math-container">$G/H$</span> represents the quotient of <span class="math-container">$H\to G$</span> in the category of fppf/fpqc sheaves.</p></li>
<li><p>Naive quotient: A group scheme <span class="math-container">$G/H$</span> with a surjective (wrt fppf/fpqc topology) map <span class="math-container">$p:G\to G/H$</span> such that kernel of <span class="math-container">$p$</span> is the inclusion <span class="math-container">$H\to G$</span></p></li>
</ol>
<p>Are they equivalent in some good situations? In van der Geer & Moonen's book, it is proved that a fppf quotient is also a categorical quotient. But I cannot find proof nor prove other directions.</p>
<p>context of the question (2): Let <span class="math-container">$f:A\to B$</span> be an isogeny of abelian varieties with kernel <span class="math-container">$\ker f$</span>. Then we have the dual exact sequence <span class="math-container">$0\to \widehat{B}\to \widehat{A}\to \widehat{\ker f}\to 0$</span>. In <a href="https://www.jmilne.org/math/CourseNotes/av.html" rel="nofollow noreferrer">Milne's book on abelian variety</a>, to prove the dual exact sequence, consider <span class="math-container">$0\to \ker f\to A\to B\to 0$</span> as an exact sequence in the category of commutative group schemes over a field and use a long exact sequence with <span class="math-container">$\text{Hom}(-, \mathbb{G}_m)$</span>. To use the long exact sequence, we need to prove <span class="math-container">$B$</span> is <span class="math-container">$A/\ker f$</span> as a fppf/fpqc quotient (In fact I don't know which topology to work with. This is why I ask the question (1)...). However, I only know that <span class="math-container">$B$</span> is the `naive quotient (3)' <span class="math-container">$A/\ker f$</span>. </p>
<p>(3) Is the category of commutative group schemes over a field an abelian category? This statement is in <a href="https://www.jmilne.org/math/CourseNotes/av.html" rel="nofollow noreferrer">Milne's book on abelian variety</a>, but I cannot find proof. The main point is existence of cokernel, i.e. representability of fppf/fpqc quotient. However, I only know the following theorem in Cornell & Silverman,</p>
<p><strong>Theorem</strong>. Let <span class="math-container">$G$</span> be a finite type <span class="math-container">$S$</span>-group scheme and let <span class="math-container">$H$</span> be a closed subgroup scheme of <span class="math-container">$G$</span>. If <span class="math-container">$H$</span> is proper and flat over <span class="math-container">$S$</span> and if <span class="math-container">$G$</span> is quasi-projective over <span class="math-container">$S$</span>, then the quotient sheaf <span class="math-container">$G/H$</span> is representable.</p>
<p>and this is too weak to prove our statement.</p>
<p>Also one more quick question: do you know any good reference dealing with sufficiently general group schemes? I know <a href="https://link.springer.com/chapter/10.1007%2F978-1-4613-8655-1_3" rel="nofollow noreferrer">Shatz's paper in Cornell-Silverman</a>, <a href="https://link.springer.com/chapter/10.1007/978-1-4612-1974-3_5" rel="nofollow noreferrer">Tate's paper in Cornell-Silvermann-Stevens</a>, and <a href="https://www.math.uni-frankfurt.de/~stix/skripte/STIXfinflatGrpschemes20120918.pdf" rel="nofollow noreferrer">Stix's lecture note</a>, but they focus on finite flat group schemes. Also, I know some other articles & books which mainly focus on affine algebraic groups. Are there some more general references?</p>
<p>Thank you for reading my stupid questions.</p>
https://mathoverflow.net/q/3616611A generalization of integral Poincaré dualityMatthttps://mathoverflow.net/users/1031502020-05-29T12:33:19Z2020-05-29T12:33:19Z
<p>In <a href="https://core.ac.uk/download/pdf/82458627.pdf" rel="nofollow noreferrer">this paper</a>, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field <span class="math-container">$\mathbb{k}$</span>:</p>
<blockquote>
<p>An augmented differential graded algebra <span class="math-container">$R$</span> over <span class="math-container">$\mathbb{k}$</span> is <em>Gorenstein</em> if <span class="math-container">$\text{Ext}_R(\mathbb{k},R)$</span> is concentrated in a single degree and has <span class="math-container">$\mathbb{k}$</span>-dimension one.</p>
<p><span class="math-container">$X$</span> is <em>Gorenstein over <span class="math-container">$\mathbb{k}$</span></em> if the cochain algebra
<span class="math-container">$C^*(X,\mathbb{k})$</span> is Gorenstein.</p>
</blockquote>
<p>This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.</p>
<p>Does there exist a parallel notion of a Gorenstein space over <span class="math-container">$\mathbb{Z}$</span> which similarly generalizes Poincaré duality over <span class="math-container">$\mathbb{Z}$</span>?</p>
https://mathoverflow.net/q/361660-1Probability distribution of sum of squares related to order distribution?VS.https://mathoverflow.net/users/1365532020-05-29T12:22:41Z2020-05-29T12:22:41Z
<p>If we pick <span class="math-container">$k$</span> uniformly random integers <span class="math-container">$x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$</span> then what is the probability distribution of the quantity
<span class="math-container">$$\sum_{i=1}^{k-1}a_i(y_{i+1}-y_i)^2$$</span>
where <span class="math-container">$y_1\leq y_2\leq\dots\leq y_{k-1}\leq y_k$</span> holds and <span class="math-container">$a_i\in\mathbb Z_+$</span> with <span class="math-container">$y_{\sigma(i)}=x_i$</span> where <span class="math-container">$\sigma$</span> is a permutation in <span class="math-container">$S_n$</span>?</p>
<p>At least at <span class="math-container">$a_i=1$</span> do we know?</p>
https://mathoverflow.net/q/3616590Projection estimate for finite element basis?Yidong Luohttps://mathoverflow.net/users/1143342020-05-29T11:45:31Z2020-05-29T11:45:31Z
<p>Let <span class="math-container">$ \mathcal{T}_n $</span> be equidistant mesh of <span class="math-container">$ (0,1) $</span> with <span class="math-container">$ n+ 1 $</span> nodes, <span class="math-container">$ X_n $</span> be corresponding finite element space, <span class="math-container">$ P_n $</span> be the orthogonal projection mapping <span class="math-container">$ L^2(0,1) $</span> onto <span class="math-container">$ X_n $</span>. </p>
<p>We know from [Mats G. Larson, Fredrik Bengzon:The Finite Element Method: Theory, Implementation, and Applications] that
<span class="math-container">\begin{equation}
\Vert \Psi - P_n \Psi \Vert_{L^2} \leq \frac{C}{n^k} \Vert \Psi \Vert_{H^k} \quad C \ \ \textrm{is a constant which does not depend on} \ n
\end{equation}</span>
holds for <span class="math-container">$ k = 2 $</span>. <strong>Now can we determine that above estimate holds for arbitrary <span class="math-container">$ k \in \mathbb{N} $</span>?</strong> <strong>As to the more general case <span class="math-container">$ k \in \mathbb{R}_+ $</span>?</strong> Could anyone help provide some references?</p>
<p>(This question is proposed in MSE before.)</p>
https://mathoverflow.net/q/3616570Probability distribution of sum of squares of sum/difference of uniform random variablesVS.https://mathoverflow.net/users/1365532020-05-29T11:14:30Z2020-05-29T11:35:15Z
<p>If we pick <span class="math-container">$k$</span> uniformly random integers <span class="math-container">$x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$</span> then what is the probability distribution of the quantities
<span class="math-container">$$\sum_{\substack{i,j=1\\i\leq k}}^n(x_i-x_j)^2$$</span>
<span class="math-container">$$\sum_{\substack{i,j=1\\i\leq k}}^n(x_i+x_j)^2$$</span> when <span class="math-container">$k=t^\alpha$</span> at some <span class="math-container">$\alpha\in(0,\infty)$</span>?</p>
<p>For reference sum of squares of normally distributed variables is given by <a href="https://en.wikipedia.org/wiki/Chi-square_distribution" rel="nofollow noreferrer"><span class="math-container">$\chi^2$</span>-distribution</a> while sum of uniform random variables is given by <a href="https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution" rel="nofollow noreferrer">Irwin–Hall distribution</a>.</p>
https://mathoverflow.net/q/3616541How does the knot contact homology augmentation polynomial change under a surgery on the base manifold?wilsonwhttps://mathoverflow.net/users/392912020-05-29T10:32:38Z2020-05-29T13:40:34Z
<p>So for every knot <span class="math-container">$K \in S^3$</span>, there is a knot contact homology of <span class="math-container">$K$</span>, and we can find the augmentation variety for this homology. The defining polynomial is known as the augmentation polynomial <span class="math-container">$A(K)$</span>, which, as Aganagic <em>et al.</em> conjectured, corresponds to the classical A-polynomial when setting <span class="math-container">$Q=1$</span>.</p>
<p>Now my question is that, how does this <span class="math-container">$A(K)$</span> change when I do surgery on <span class="math-container">$S^3$</span>. As a concrete example, I want to find <span class="math-container">$K=\text{unknot}$</span> and the surgery is the contact surgery manifold obtained on a Legendrian unknots. It has Thurston-Bennequin number <span class="math-container">$\mathtt{tb} = -2$</span> and surgery coefficient <span class="math-container">$-1$</span>.</p>
<p>What I knew was that "contact" surgery with coefficient <span class="math-container">$-1$</span> is equivalent to attaching a Legendrian handle on the manifold, so in that case is simply a good old Legendrian surgery.</p>
<p>My thought was that this surgery changes the symplectization from simply <span class="math-container">$\mathbb R\times S^3$</span> to <span class="math-container">$\mathbb R\times M^3$</span>, but now the picture looks like this:</p>
<p><a href="https://i.stack.imgur.com/txQyH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/txQyH.png" alt="surgery effect"></a></p>
<p>So I think from Ekholm's talk I learned that I need to consider the closed Reeb orbits, but I am stuck. Ekholm in one of his lectures calculated the value of unknot in <span class="math-container">$\mathbb RP^3$</span>, but I do not understand how he arrived at his answer.</p>
<p>Is my above thinking correct? Or am I completely wrong? Thank you everyone!</p>
https://mathoverflow.net/q/3616523Constructive proof of the approximate Brouwer's Fixed Point Theorem for $\Delta^n$ludwigmachhttps://mathoverflow.net/users/1216592020-05-29T09:46:37Z2020-05-29T13:49:13Z
<p>The approximate Brouwer Fixed Point Theorem (<strong>aBFPT</strong>) for the standard <span class="math-container">$n$</span>-simplex is:</p>
<blockquote>
<p>Let <span class="math-container">$f$</span> be a uniformly continuous function from <span class="math-container">$\Delta^n$</span> into itself.
Then for each <span class="math-container">$\varepsilon>0$</span> there exists <span class="math-container">$x\in\Delta^n$</span> such that
<span class="math-container">$d(f(x),x)<\varepsilon$</span>.</p>
</blockquote>
<p>Does anyone know a reference for a full proof of <strong>aBFPT</strong> via Sperner's Lemma in <a href="https://www.springer.com/gp/book/9783642649059" rel="nofollow noreferrer">Bishop-style</a> constructive mathematics (<strong>BISH</strong>)?</p>
<p>A constructive proof of <strong>aBFPT</strong> via <a href="http://www.math.pitt.edu/~gartside/hex_Browuer.pdf" rel="nofollow noreferrer">Gale's</a> Hex Theorem is in Hendtlass's <a href="https://core.ac.uk/download/pdf/19417936.pdf" rel="nofollow noreferrer">PhD thesis</a>. A constructive proof for the <span class="math-container">$2$</span>-simplex is in <a href="https://www.sciencedirect.com/science/article/pii/S0304397511002787?via%3Dihub" rel="nofollow noreferrer">this paper</a> by van Dalen. That's all I could find.</p>
<p>If you don't know a reference but have a proof handy I'd very much like to see that as well.</p>
<p>Thank you in advance!</p>
<p>Note: The status of <strong>BFPT</strong> in constructive mathematics has been discussed in <a href="https://mathoverflow.net/questions/202811/does-the-brouwer-fixed-point-theorem-admit-a-constructive-proof">this MO post</a>. For quick orientation: <strong>aBFPT</strong> may be viewed as constituting the constructive core of the full classical <strong>BFPT</strong>. The latter implies <strong>LLPO</strong> and is thus inherently nonconstructive (indeed, the two are equivalent over <strong>BISH</strong> since both are equivalent to <strong>WKL</strong> in the presence of countable choice). Classically, <strong>BFPT</strong> can be retrieved from <strong>aBFPT</strong> by a simple application of the Bolzano-Weierstrass theorem, which is equivalent to <strong>LPO</strong> over <strong>BISH</strong> and thus constructively inadmissible. </p>
https://mathoverflow.net/q/3616511Is the set $\operatorname{Unif}(0,\frac{1}{n})$ for odd and even $n$ a 2-alternating capacity?Seyhmus Güngörenhttps://mathoverflow.net/users/363562020-05-29T09:39:29Z2020-05-29T10:51:31Z
<p>Let <span class="math-container">$\Omega$</span> be a complete metrizable space <span class="math-container">$\mathscr A$</span> its Borel <span class="math-container">$\sigma$</span>-algebra and <span class="math-container">$\mathscr M$</span> the set of all probability measures on <span class="math-container">$\Omega$</span>. Every non-empty subset <span class="math-container">$\mathscr P \subset \mathscr M$</span> defines an upper probability <span class="math-container">$$v(A)=\sup\{P(A)|P\in\mathscr{P}\},\quad A\in\mathscr A.$$</span></p>
<p>Then, a <span class="math-container">$2$</span>-alternating capacity is defined with the following conditions:</p>
<p><span class="math-container">$1.$</span> <span class="math-container">$v(\emptyset)=0$</span> and <span class="math-container">$v(\Omega)=1$</span></p>
<p><span class="math-container">$2.$</span> <span class="math-container">$A\subset B\Longrightarrow v(A)\leq v(B)$</span></p>
<p><span class="math-container">$3.$</span> <span class="math-container">$A_n\uparrow A\Longrightarrow v(A_n)\uparrow v(A)$</span></p>
<p><span class="math-container">$4.$</span> If <span class="math-container">$\mathscr P$</span> is weakly compact, then we also have <span class="math-container">$F_n\downarrow F,\,\, F_n\, \text{closed}\Longrightarrow v(F_n)\downarrow v(F)$</span> </p>
<p><span class="math-container">$5.$</span> <span class="math-container">$v(A\cup B)+v(A\cap B)\leq v(A)+v(B)$</span></p>
<p>That is also to say that the set <span class="math-container">$$P_v=\{P\in\mathscr M|P(A)\leq v(A)\quad\text{for all}\quad A\in\mathscr A\}$$</span> is not larger than the closed convex hull of the set <span class="math-container">$\mathscr P$</span> determining <span class="math-container">$v$</span>.</p>
<p><strong>Example</strong> <span class="math-container">$1:$</span> Let <span class="math-container">$\Omega=\{1,2,3\}$</span>, <span class="math-container">$P_0=\{\frac{1}{2},\frac{1}{2},0\}$</span>, <span class="math-container">$P_1=\{\frac{4}{6},\frac{1}{6},\frac{1}{6}\}$</span> and let <span class="math-container">$v$</span> be the upper probability determined by <span class="math-container">$\mathscr P=\{P_0,P_1\}$</span>. Then, <span class="math-container">$$P_v=\Bigg\{\frac{3+t}{6},\frac{3-t-s}{6},\frac{s}{6}\Bigg|0\leq s,t\leq 1\Bigg\}$$</span> whereas the convex closure of <span class="math-container">$P$</span> is the proper subset of <span class="math-container">$P_v$</span> determined by <span class="math-container">$s=t$</span>. In this example, <span class="math-container">$v$</span> is <span class="math-container">$2$</span>-alternating.</p>
<p><strong>Example</strong> <span class="math-container">$2:$</span> Let <span class="math-container">$\Omega=\{1,2,3,4\}$</span>, <span class="math-container">$P_0=\{\frac{5}{10},\frac{2}{10},\frac{2}{10},\frac{1}{10}\}$</span>, <span class="math-container">$P_1=\{\frac{6}{10},\frac{1}{10},\frac{1}{10},\frac{2}{10}\}$</span> and let <span class="math-container">$\mathscr P=\{P_0,P_1\}$</span>. Here <span class="math-container">$v$</span> is not <span class="math-container">$2$</span>-alternating. Let <span class="math-container">$A=\{1,2\}$</span> and <span class="math-container">$B=\{1,3\}$</span>. Then,</p>
<p><span class="math-container">$$v(A\cup B)+v(A\cap B)=\frac{15}{10}> v(A)+v(B)=\frac{14}{10}$$</span></p>
<blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$U^n=\operatorname{Unif}(0,\frac{1}{n})$</span> the uniform measure on unit interval. And <span class="math-container">$\mathscr{P}_0$</span> corresponds to those with odd <span class="math-container">$n$</span>, <span class="math-container">$\mathscr{P}_1$</span> contains exactly those with even <span class="math-container">$n$</span>. Is <span class="math-container">$\mathscr{P}_0$</span>, <span class="math-container">$\mathscr{P}_1$</span> or <span class="math-container">$\{\mathscr{P}_0,\mathscr{P}_1\}$</span> a <span class="math-container">$2$</span>-alternating capacity? If we change the definition with "odd and even <span class="math-container">$n$</span> less than <span class="math-container">$10^{10}$</span>", does <span class="math-container">$\mathscr{P}_0$</span>, <span class="math-container">$\mathscr{P}_1$</span> or <span class="math-container">$\{\mathscr{P}_0,\mathscr{P}_1\}$</span> become a <span class="math-container">$2$</span>-alternating capacity?</p>
</blockquote>
<p>The reference of this question is <a href="https://projecteuclid.org/download/pdf_1/euclid.aos/1176342363" rel="nofollow noreferrer">this paper</a>. Examples are from page <span class="math-container">$254$</span>.</p>
https://mathoverflow.net/q/3616490$V$-like actions of $V$Ville Salohttps://mathoverflow.net/users/1236342020-05-29T09:19:08Z2020-05-29T15:20:11Z
<p>This continues <a href="https://mathoverflow.net/questions/361556/is-there-a-prefix-continuous-bijection-between-finite-words-and-eventually-zero">my question about prefix-continuous bijections</a> (since the answer was "yes").</p>
<p>Notation and conventions: Let <span class="math-container">$A$</span> be a finite alphabet and <span class="math-container">$L \subset A^*$</span> a language. Let <span class="math-container">$G$</span> be a group. For words <span class="math-container">$u, w$</span>, juxtaposition <span class="math-container">$uw$</span> denotes word concatenation, and if we have an action <span class="math-container">$G \curvearrowright L$</span> and <span class="math-container">$g \in G, u \in L$</span> we write <span class="math-container">$gu$</span> for the image of <span class="math-container">$u$</span> in the action of <span class="math-container">$g$</span>; concatenation associates first.</p>
<p>An action <span class="math-container">$G \curvearrowright L$</span> is <strong>veelike</strong> (or <span class="math-container">$G$</span> acts veelike) if for all <span class="math-container">$g \in G$</span> there exists <span class="math-container">$n \in \mathbb{N}$</span> such that for all <span class="math-container">$u \in A^*$</span> with <span class="math-container">$|u| = n$</span> there exists <span class="math-container">$u' \in A^*$</span> such that <span class="math-container">$guv = u'v$</span> for all <span class="math-container">$uv \in L$</span>. Now we have the following simple observation.</p>
<blockquote>
<p>Thompson's <span class="math-container">$V$</span> acts veelike on the regular language defined by the regular expression <span class="math-container">$\emptyset + (0+1)^* 1$</span>.</p>
</blockquote>
<p>Proof. This is the regular language <span class="math-container">$L$</span> containing the empty word <span class="math-container">$\emptyset$</span> and all words ending in the symbol <span class="math-container">$1$</span>. To find the action, recall the defining action of <span class="math-container">$V$</span> on the boundary of the infinite binary tree. We can see the tree as <span class="math-container">$\{0,1\}^\omega$</span>, and elements of <span class="math-container">$V$</span> can be presented by picking two maximal prefix codes, bijecting them and then rewriting the prefix of the input word according to said bijection. Observe that the set
<span class="math-container">$$ X = \{x \in \{0,1\}^{\omega} \;|\; \exists m: \forall i \geq m: x_i = 0\} $$</span>
of infinite paths/words that are eventually zero is invariant under its action, and <span class="math-container">$V$</span> clearly acts faithfully on it (e.g. because it's a simple group and the action is nontrivial). There is an obvious bijection from <span class="math-container">$X$</span> to <span class="math-container">$L$</span>: just cut out the sequence after the last <span class="math-container">$1$</span>, if one exists, and map the all zero sequence to the empty word <span class="math-container">$\emptyset$</span>. The action of <span class="math-container">$V$</span> on <span class="math-container">$L$</span> is "veelike" in the obvious sense (same formula); that's basically the definition. Conjugating the action of <span class="math-container">$V$</span> to <span class="math-container">$L$</span>, it stays veelike since on all long enough words you do exactly the same rewrites as you would on infinite words beginning that way. Square.</p>
<p>(I have not written a more careful proof than that, it's natural so what could go wrong.)</p>
<p>My question is the following:</p>
<blockquote>
<p>On which alphabets <span class="math-container">$A$</span> and languages <span class="math-container">$L \subset A^*$</span> does Thompson's <span class="math-container">$V$</span> admit a veelike action?</p>
</blockquote>
<p>In particular:</p>
<blockquote>
<p>Does Thompson's <span class="math-container">$V$</span> act veelike on the set of binary words <span class="math-container">$\{0,1\}^*$</span>?</p>
</blockquote>
<p>One possible way to prove this would be to find a bijection <span class="math-container">$\phi : X \to \{0,1\}^*$</span> such that conjugating the action of <span class="math-container">$V$</span> from <span class="math-container">$X$</span> to <span class="math-container">$\{0,1\}^*$</span> gives you a veelike action.</p>
<blockquote>
<p>Does such a bijection exist?</p>
</blockquote>
<p>I claim that the bijection of @PierrePC to <a href="https://mathoverflow.net/questions/361556/is-there-a-prefix-continuous-bijection-between-finite-words-and-eventually-zero">my previous question</a> does not work (the answer is correct but I did not state all the necessary properties for the bijection). Namely, after conjugation the action indeed rewrites only prefixes, but you need to see the whole word to know how they are rewritten, i.e. the action is not continuous in the right sense.</p>
<p>More concretely, through this bijection the element that just flips the first bit acts
<span class="math-container">$$ 00000000000000001... \mapsto 10000000000000001... $$</span>
<span class="math-container">$$ 00000000100000001... \mapsto 10000000100000001... $$</span>
which are conjugated respectively to
<span class="math-container">$$ 00000000000000000 \mapsto 1000000000000000 $$</span>
<span class="math-container">$$ 0000000010000000 \mapsto 1000000010000000 $$</span>
which is highly veeunlikely because the length change depends on the <span class="math-container">$1$</span> arbitrarily far inside the word.</p>
https://mathoverflow.net/q/3616433Tilings of lattice polytopes by transformations of lattice polytopesDisplay namehttps://mathoverflow.net/users/1275212020-05-29T08:54:13Z2020-05-29T14:59:16Z
<p>A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice polytopes necessarily lattice polytopes?</p>
<p>If we allow dilations in the general sense to enter the "quasi club", then any figure may just be obtained from dilating a unit hypercube, so the result is obviously false. What if we restrict dilations to transformations induced by matrices with integer entries? This stronger question is true in 1 dimension since <span class="math-container">$[a,b]$</span> having integral endpoints implies <span class="math-container">$[ac,bc]$</span> has integral endpoints for all <span class="math-container">$c \in \mathbb{Z}.$</span></p>
<p>Let's be even more general. Fix a dimension <span class="math-container">$n,$</span> let <span class="math-container">$S$</span> be a set of affine transformations, and define <span class="math-container">$S$</span>-lattice polytopes as those in the image of <span class="math-container">$s: P \to P$</span> for some <span class="math-container">$s \in S$</span> where <span class="math-container">$P$</span> is the set of <span class="math-container">$n$</span> dimensional lattice polytopes. How large can <span class="math-container">$S$</span> be such that the statement "in a tiling of lattice polytopes by <span class="math-container">$S$</span>-lattice polytopes, all <span class="math-container">$S$</span>-lattice polytopes are lattice polytopes" is true?</p>
<p>For <span class="math-container">$n=1, S = \{x \to ax+b | a \in T, b \in \mathbb{R}\}$</span> where <span class="math-container">$T = \mathbb{Z}$</span> works. In fact, <span class="math-container">$T$</span> can be replaced by any extension <span class="math-container">$R \supseteq T$</span> such that <span class="math-container">$R$</span> is linearly independent over <span class="math-container">$\mathbb{Z}$</span> (defining an infinite set to be linearly independent iff every finite subset is), and this characterizes all maximal <span class="math-container">$S$</span> completely.</p>
https://mathoverflow.net/q/3616422Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional idealsuser43263https://mathoverflow.net/users/432632020-05-29T08:21:10Z2020-05-29T14:16:51Z
<p>This is a follow-up (but self-contained) question to my <a href="https://mathoverflow.net/questions/361520/what-is-the-state-of-the-art-for-solving-polynomials-systems-over-fields-that-ar">previous one</a>. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in general.<br>
I learned that the theory is more involved that I thought (I'm not working in algorithmic algebraic geometry, so I'm only familiar with the very basics, like Buchberger's algorithm, or the definition of the dimension of an ideal). Therefore it is necessary to ask a more specific questions that the previous, general one, which is more tailored to my needs. </p>
<p>My setup is the following:</p>
<p><em>Regarding complexity:</em> I'm interested in solving a large number of polynomial systems (on commodity hardware), on the order of <span class="math-container">$10^4$</span>. But each of the systems is of relatively small size - my baseline consists of least 6 different variable and 4 equations. If I could tackle this, I'd already be happy. Going further, I don't expect the systems to grow beyond about 20 different variables and 20 equations.<br>
So perhaps I don't actually need the fastest possible algorithm and can make do with simpler, older ones - but I will let you be the judge of that.</p>
<p><em>Regarding the polynomials:</em> There are no restrictions their coefficients, so, depending on the field I'm working in they can take any number.</p>
<p><em>Regarding the field:</em> Regarding the field I'm working in, my baseline is <span class="math-container">$\mathbb{R}$</span>, but I'd also be interested in <span class="math-container">$\mathbb{Q}$</span> and <span class="math-container">$\mathbb{Z}$</span>. If there are methods that are much easier for one field than another, than I will the choice of the field to study be influence by the time I need to invest to learn that method, i.e. the easiest one wins.</p>
<p><em>Regardin the dimension of the ideal spanned by the polynomials:</em> The ideal has dimension <span class="math-container">$2$</span> or <span class="math-container">$3$</span> over the complex numbers, in most cases I tested so far with the help of CAS.</p>
<p><em>What I'm looking for</em>: I'm interested is learning about methods (I'm happy with specific references) that tell me</p>
<p>1) <em>whether</em> the system has a solution at all or not. Working over, e.g., <span class="math-container">$\mathbb{C}$</span>, this would be easy (e.g. compute a Gröbner basis: If it contain the <span class="math-container">$1$</span>, if and only if the solution variety is empty). But this doesn't work unfortunately for non-algebraically closed fields. Given the answers from my previous question, I'm inclined to think that answer this question shouldn't be too hard (perhaps even trivial for the expert computational geometer, which I'm not unfortunately).</p>
<p>2) if it has an infinite number of solutions (if the variety is zero-dimensional, things are easy of course), I would like to pick out one single variable, say <span class="math-container">$n_0\in \{1,\ldots,n\}$</span>, project the solution variety <span class="math-container">$V(f_1,\ldots,f_s)\subseteq \mathbb{
R}^n$</span> (supposing we work over the field <span class="math-container">$\mathbb{R}$</span>) along this variable onto <span class="math-container">$\mathbb{R}$</span> to investigate whether there exists an interval <span class="math-container">$[-\alpha,\alpha]$</span> around <span class="math-container">$0$</span> which is contained in this projected set (I don't need to understand the projected set fully). That is what I menat by "rough shape" in the title.</p>
https://mathoverflow.net/q/3616360Ergodic action on unitary groupA beginner mathmaticianhttps://mathoverflow.net/users/1368602020-05-29T06:28:20Z2020-05-29T14:17:49Z
<p>Let <span class="math-container">$G$</span> be a locally compact Hausdorff group. Assume that <span class="math-container">$\theta:G\to U_d$</span> is a group homomorphism where <span class="math-container">$U_d$</span> is a finite dimensional unitary group. Consider a action of <span class="math-container">$G$</span> on <span class="math-container">$U_d$</span> by <span class="math-container">$g.u:=\theta(g)u,$</span> <span class="math-container">$u\in U_d.$</span> Consider <span class="math-container">$U=\overline{\theta(G)}.$</span> Is it true that <span class="math-container">$G$</span> acts ergodically on <span class="math-container">$U$</span> where <span class="math-container">$U$</span> is equipped with the invariant probability measure of <span class="math-container">$U_d$</span>?</p>
https://mathoverflow.net/q/3616200Two isomorphic reduced group $C^*$-algebrasmathbeginnerhttps://mathoverflow.net/users/1531962020-05-29T02:25:22Z2020-05-29T12:54:35Z
<p>Suppose that <span class="math-container">$C^*_r(G)\cong C^*_r(H)$</span>, can we conclude that <span class="math-container">$G\cong H$</span>?</p>
https://mathoverflow.net/q/3616180Correlation of stopping times for integral of Brownian motion incrementOOESCouplinghttps://mathoverflow.net/users/998632020-05-29T01:07:39Z2020-05-29T15:05:37Z
<p>Let <span class="math-container">$\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$</span>, where <span class="math-container">$(B_{s})_{s\geq 0}$</span> is a Brownian motion (starting at <span class="math-container">$B_{0}=0$</span>) and epsilon is small <span class="math-container">$0<\epsilon\ll 1 $</span>. Consider the stopping time</p>
<p><span class="math-container">$$T_{a}:=\inf\{t\geq \epsilon: \mu(t)\geq a \},\text{ for $a\geq 0$.}$$</span></p>
<p>For "good" deterministic choices <span class="math-container">$a<b<c<d<$</span> and <span class="math-container">$t_{1},t_{2}\geq 0$</span> we are studying the correlation of </p>
<p><span class="math-container">$$\{T_{b}-T_{a}\geq t_{1}\}\text{ and }\{T_{d}-T_{c}\geq t_{2}\}.$$</span></p>
<blockquote>
<p>Q: One approach is by studying a mixing condition: trying to estimate the difference</p>
</blockquote>
<p><span class="math-container">$$| P[\{T_{d}-T_{c}\geq t_{2}, T_{b}-T_{a}\geq t_{1}\}]-P[\{T_{d}-T_{c}\geq t_{2}\} ]\cdot P[ \{T_{b}-T_{a}\geq t_{1}\}]|$$</span>
when <span class="math-container">$[a,b]$</span> and <span class="math-container">$[c,d]$</span> are distant enough. In other words, we are trying to find <strong>whether</strong> there are any constraints on <span class="math-container">$a<b<c<d$</span> so that the above difference goes to zero as the distance between the two intervals <span class="math-container">$c-b$</span> increases.</p>
<p>If you think this question is ill-posed or too obvious for Mathoverflow, please put your hint in the comments and I will delete the question.</p>
<p><em>Approach</em></p>
<p>The process <span class="math-container">$\mu(x)$</span> is not Markov because of the <span class="math-container">$B_{s-\epsilon}$</span> term and so strong Markov property (SMP) doesn't apply. But in the spirit of (SMP), we introduce the event </p>
<p><span class="math-container">$$ E:=\{T_{c}> T_{b}+2\epsilon\}.$$</span></p>
<p>So just for the sake of understanding (even if we can't get a good control on <span class="math-container">$E^{c}$</span>), lets focus on the above difference with the first term intersected with <span class="math-container">$E$</span>:
<span class="math-container">$$| P[\{T_{d}-T_{c}\geq t_{2}, T_{b}-T_{a}\geq t_{1}\}\cap E]-P[\{T_{d}-T_{c}\geq t_{2}\} ]\cdot P[ \{T_{b}-T_{a}\geq t_{1}]|.$$</span></p>
<p>Equivalently, by Baye's law we are trying to estimate</p>
<p><span class="math-container">$$| P[\{T_{d}-T_{c}\geq t_{2}\}\cap E\mid \{T_{b}-T_{a}\geq t_{1}\}]-P[\{T_{d}-T_{c}\geq t_{2}\} ]|.$$</span></p>
<p>I will update as I find things. </p>
https://mathoverflow.net/q/3616041Reference request: The transform of a bounded random variable has a zero in the complex planeJohan Wästlundhttps://mathoverflow.net/users/143022020-05-28T20:34:29Z2020-05-29T12:46:14Z
<p>Together with coauthors I'm working on a paper where we use the following Proposition: </p>
<blockquote>
<p>If a real-valued random variable <span class="math-container">$X$</span> has bounded support, then except in the trivial case that <span class="math-container">$X$</span> has all its mass in a single point, its moment generating function <span class="math-container">$$ M(z) = E(e^{zX})$$</span> has a zero in the complex plane.</p>
</blockquote>
<p>Notice that the result is the same whether we are talking about the moment generating function, the characteristic function, the Laplace transform or the Fourier transform. Since the moments of <span class="math-container">$X$</span> grow at most exponentially, they are all entire functions and just rotations of each other.</p>
<p>It seems to us that the proposition must be both well-known and important, and we are baffled that we haven't been able to find it stated explicitly with a simple and self-contained proof.</p>
<blockquote>
<p>Is there a statement and simple proof of the proposition in the literature?</p>
</blockquote>
<p>The proposition is a consequence of the Hadamard factorization theorem: Since <span class="math-container">$M(z)$</span> is of order (at most) 1, it can be written as <span class="math-container">$e^{az+b}$</span> times a product involving its zeros. If there aren't any, we are left with <span class="math-container">$M(z) = e^{az}$</span>, and <span class="math-container">$X$</span> must have all its mass at <span class="math-container">$a$</span>. </p>
<p>But it can be proved with much easier complex analysis (see below), and this is why we're asking. </p>
<p>There is a lemma in William Feller's book <a href="http://www.ru.ac.bd/wp-content/uploads/sites/25/2019/03/101_06_Feller_An-Introduction-to-Probability-Theory-and-Its-Applications-Vol.-2.pdf" rel="nofollow noreferrer">An Introduction to Probability Theory and Its Applications</a> vol II p 525, that implies our proposition and that seems to have been distilled out of Harald Cramér's proof of the decomposition theorem conjectured by Paul Lévy. It states that if <span class="math-container">$\exp(c\cdot X^2)$</span> has finite expectation for some <span class="math-container">$c>0$</span> (a weaker condition than boundedness), then either <span class="math-container">$X$</span> is normal or its characteristic function has a zero. </p>
<p>But both Cramér's original paper <em>Über eine Eigenschaft der normalen Verteilungsfunktion</em> and Feller's book simply refer to the Hadamard factorization theorem. Feller even says about the lemma that "Unfortunately its proof depends on analytic function theory and is therefore not quite in line with our treatment...".</p>
<p>There is a very reasonable (and useful to us) proof of the same lemma in the recent paper <em>Three remarkable properties of the Normal distribution</em> by Eric Benhamou, Beatrice Guez and Nicolas Paris, so we're certainly not complaining, just wondering if something even simpler has been published. </p>
<p>To establish our Proposition, some easy parts of the proof of the Hadamard theorem will do: If <span class="math-container">$M(z)$</span> has no zeros, we can introduce the function <span class="math-container">$$ K(z) = \int_0^z \frac{M'(t)}{M(t)}\,dt,$$</span>
and we have <span class="math-container">$M(z) = e^{K(z)}$</span> throughout the complex plane (actually <span class="math-container">$K$</span> is known as the <em>cumulant generating function</em>). Assuming without loss of generality that <span class="math-container">$X$</span> is supported on <span class="math-container">$[-1,1]$</span>, we obtain
<span class="math-container">$$ \text{Re}(K(z)) \leq \left|z\right|, $$</span> which leads by the Borel-Carathéodory theorem to
<span class="math-container">$$ \left| K(z) \right| \leq 4\left| z \right|.$$</span>
This makes <span class="math-container">$K(z)/z$</span> a bounded entire function, and Liouville's theorem finishes the proof.</p>
<p>Spelling out the Borel-Carathéodory argument with a couple of more equations will reduce the whole thing to undergraduate complex analysis, and this is what we are thinking of doing. </p>
https://mathoverflow.net/q/3615885Spectral flow of Dirac operator twisted by instantonGorahttps://mathoverflow.net/users/1177232020-05-28T17:10:21Z2020-05-29T15:29:22Z
<p>Suppose <span class="math-container">$E$</span> is a <span class="math-container">$SU(2)$</span>-bundle over a closed three manifold <span class="math-container">$M$</span> and <span class="math-container">$S$</span> is the spinor bundle over <span class="math-container">$M$</span>. Also assume <span class="math-container">$D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$</span> is a family of Dirac operators twisted by a family of connections <span class="math-container">$A(t)$</span> on <span class="math-container">$E$</span>, <span class="math-container">$t\in[0,1]$</span>. </p>
<p>We have also a family of curl-div operators <span class="math-container">$L_{A(t)}:\Omega^0(E)\oplus\Omega^1(E)\to \Omega^0(E)\oplus\Omega^1(E)$</span> where
<span class="math-container">\begin{equation}
L_{A(t)}=
\begin{pmatrix}
0&d^*_{A(t)}\\
d_{A(t)}&*d_{A(t)}\\
\end{pmatrix}
\end{equation}</span>
My <strong>main question</strong> is:</p>
<p>Suppose <span class="math-container">$A(0)$</span> and <span class="math-container">$A(1)$</span> are two <strong>flat</strong> connections and <span class="math-container">$\mathbb A:=\{A(t):t\in[0,1]\}$</span> is an <strong>instanton</strong> on <span class="math-container">$M\times[0,1]$</span>.
Are there any relations between Spectral flow of <span class="math-container">$\{D_{A(t)}:t\in[0,1]\}$</span> and Spectral flow of <span class="math-container">$\{L_{A(t)}:t\in[0,1]\}$</span>?</p>
<p>Note: From the work of Atiyah-Patodi-Singer we could relate them with expressions involving eta invariants and some characteristic classes. Are they get simplified in our present situations? If the question above was not expressed properly, that's my fault. Please let me know, I would try to modify them.</p>
https://mathoverflow.net/q/3615728Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2Julien Marchéhttps://mathoverflow.net/users/145472020-05-28T13:39:53Z2020-05-29T14:06:59Z
<p>Fix <span class="math-container">$x,y,z\in \mathbb{C}^*$</span> and let <span class="math-container">$M=S^1\times S^1\times S^1$</span> with <span class="math-container">$\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$</span> mapping the three generators to diagonal matrices with entries <span class="math-container">$(x,x^{-1})$</span>, <span class="math-container">$(y,y^{-1})$</span>, <span class="math-container">$(z,z^{-1})$</span> respectively.
The question is can you construct a (smooth) 4-manifold <span class="math-container">$W$</span> bounding <span class="math-container">$M$</span> with a representation <span class="math-container">$\tilde{\rho}:\pi_1(W)\to \operatorname{SL}_2(\mathbb{C})$</span> extending <span class="math-container">$\rho$</span>?</p>
<p>Comments: </p>
<p>1) Such a 4-manifold exists by a non-constructive argument based on the fact that the map <span class="math-container">$H_3(\mathbb{C}^*,\mathbb{Z})\to H_3(\operatorname{SL}_2(\mathbb{C}),\mathbb{Z})$</span> induced by the inclusion of diagonal matrices in <span class="math-container">$\operatorname{SL}_2(\mathbb{C})$</span> is almost 0.</p>
<p>2) I have a homological argument saying that the extension <span class="math-container">$\tilde{\rho}$</span> cannot be abelian.</p>
<p>3) By standard arguments, the same manifold should work for <span class="math-container">$(x,y,z)$</span> in a Zariski open subset of <span class="math-container">$\mathbb{C}^*\times\mathbb{C}^*\times \mathbb{C}^*$</span>. </p>
<p>4) The general motivation is to illustrate topologically the low-dimensional homology of linear groups... </p>
https://mathoverflow.net/q/3615443Reference request: Variational techniques for complex "iterated" LagrangiansPaulShttps://mathoverflow.net/users/1587352020-05-28T03:15:58Z2020-05-29T11:50:15Z
<p>I am interested in solving variational problems of the form
<span class="math-container">$$
\min_u \int \Big\{L(x,y,u(x,y)) + \phi\Big(\int J(z,y,u(z,y))\,dz\Big)\Big\} p(x,y)\,dx\,dy.
$$</span>
for some known, smooth functions <span class="math-container">$L,J,\phi,p$</span>, and the minimization is with respect to functions <span class="math-container">$u:\mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}$</span>. We can assume e.g. <span class="math-container">$u$</span> is square integrable or even bounded; this part isn't crucial. Note that the derivatives of <span class="math-container">$u$</span> are not involved, which helps simplify things somewhat.</p>
<p>For example, a particular form that has come up is the following:
<span class="math-container">$$
\min_u \int \Big\{g(x) + \Big(\int h(z) u(z,y)\,dz\Big)^r\Big\} p(x,y)\, dx\,dy
$$</span></p>
<p>There is some structure here (e.g. the double integral involving two Lagrangian functions) that seems ripe for exploitation. But the second, inner integral seems to break the standard setup of the calculus of variations (what is the derivative of <span class="math-container">$\int J(z,y,u(z,y))\,dz$</span> with respect to <span class="math-container">$u$</span>?), but maybe I am missing something.</p>
<p>I am looking for any references covering these types of problems, or related ones. I would also be interested in (even more complicated) formulations involving derivatives of <span class="math-container">$u$</span>, but the simpler version seems difficult enough!</p>
<p><strong>EDIT:</strong> It seems that the most salient aspects of this problem can be exposed by considering the simpler form
<span class="math-container">$$
\min_u \int \phi\Big(\int J(z,y,u(z,y))\,dz\Big)\,dy.
$$</span>
From this angle, it is clear that <span class="math-container">$\phi$</span> is the complication (e.g. if <span class="math-container">$\phi$</span> were not involved, this would be a standard calculus of variations problem).</p>
https://mathoverflow.net/q/3611591Eigenspace of Gaussian Markov operatorH17https://mathoverflow.net/users/1340122020-05-23T15:29:34Z2020-05-29T13:02:55Z
<p>Consider the (one-dimensional) Gaussian distribution <span class="math-container">$Q := N(\nu,\tau^2)$</span> and the (Gaussian) Markov operator</p>
<p><span class="math-container">\begin{equation*}
\begin{array}{rccc}
R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) & \to & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) \\
& f & \mapsto & \int f(x)\, N(\cdot,\sigma^2)(\mathrm{d}x).
\end{array}
\end{equation*}</span></p>
<p>I am interested in the eigenspace <span class="math-container">$E_1 := \mathrm{kernel(I-R)},$</span> in particular in the dimension of <span class="math-container">$E_1.$</span></p>
<p>Obviously, the indicator function <span class="math-container">$\mathbb{1}_{\mathbb{R}}: x \mapsto 1$</span> and the identity <span class="math-container">$\mathrm{id}_{\mathbb{R}}: x \mapsto x$</span> are both eigenfunctions to the eigenvalue <span class="math-container">$1,$</span> that is, <span class="math-container">$\mathbb{1}_{\mathbb{R}},\ \mathrm{id}_{\mathbb{R}} \in E_1.$</span> </p>
<p>Are there more linearly independent eigenfunctions?</p>
https://mathoverflow.net/q/3586058Residually finite group with dense finite index subgroupsuser44172https://mathoverflow.net/users/441722020-04-26T15:28:31Z2020-05-29T14:14:18Z
<p>Let <span class="math-container">$G$</span> be a locally compact Hausdorff topological group whose underlying abstract group is residually finite. Let <span class="math-container">$H\subset G$</span> denote the intersection of all finite-index, <strong>closed</strong> subgroups. Is there an example of such a <span class="math-container">$G$</span> where <span class="math-container">$H$</span> is not trivial? Is there an example where <span class="math-container">$H=G$</span> (i.e., every finite-index subgroup is dense)? </p>
<p>My motivation for asking this question comes from the study of automorphism groups of connected, locally finite graphs. Therefore, the closer the example is to being of this type the better. E.g., an example which is an automorphism group of some structure on <span class="math-container">$\mathbb{N}$</span>, say a hypergraph structure, would be extremely interesting; an example where <span class="math-container">$G$</span> is separable would be more useful than an example with cardinality larger than the continuum, and so on.</p>
https://mathoverflow.net/q/3585420Lower bound for mutual inner products of N random unit vectors in $\mathbb{R}^n$, N > nuser27182https://mathoverflow.net/users/1022552020-04-25T20:31:43Z2020-05-29T12:01:00Z
<p>I have <span class="math-container">$N$</span> independent random unit vectors <span class="math-container">$\{v_i\}$</span> in <span class="math-container">$\mathbb{R}^n$</span>, where N > n. I need a concentration inequality of the form
<span class="math-container">$$\text{P}(|v_i \cdot v_j| > \epsilon \,\,\,\, \forall i, j = 1, \dots, N: i \neq j)\leq \psi(\epsilon)$$</span>
where hopefully <span class="math-container">$\psi(\epsilon)$</span> is something small.</p>
<p>I think that I can use Johnson-Lindenstrauss to do this for isotropic vectors (e.g. by choosing orthogonal basis for <span class="math-container">$\mathbb{R}^N$</span> and projecting into <span class="math-container">$\mathbb{R}^n$</span> with a random subgaussian matrix).</p>
<p>Are there results of this form that hold when the <span class="math-container">$\{v_i\}$</span> are not distributed isotropically, for instance Gaussian with covariance <span class="math-container">$\Sigma$</span>? For instance, when there is some weak correlation/dependence between the components of each of the <span class="math-container">$v$</span> --- maybe <span class="math-container">$|\Sigma_{ij}| \leq \alpha$</span> when <span class="math-container">$i\neq j$</span>? </p>
<p>(Any seemingly related results in this area are much appreciated!)</p>
https://mathoverflow.net/q/3545314Constant curvature difference surfacesNarasimhamhttps://mathoverflow.net/users/479732020-03-09T13:32:58Z2020-05-29T14:44:32Z
<p>Surfaces of Constant Mean Curvature CMC and constant Gauss Curvature Product <span class="math-container">$K$</span> are well-known in differential geometry of surfaces of revolution. Denote half sum and half difference curvatures as </p>
<p><span class="math-container">$$ k_1+ k_2 = 2 H_s; \, k_1- k_2 = 2 \, H_d. $$</span></p>
<p><strong><em>Significance of <span class="math-container">$H_d$</span></em></strong> </p>
<p>Now I assume that <span class="math-container">$H_d$</span> may as well be of equal interest with fundamental importance. The assumption is motivated by its representation in <strong><em>Mohr's Circle</em></strong> radius of curvature for stress, shell curvatures, moment of inertia (in Figure) among other such tensors. As is known, material stress failure theories in structural mechanics operate on stress/strain and other tensoral <em>differences</em> as <strong><em>shear</em></strong> entities. It occurs as an important curvature invariant in the equation of Mohr's Circle : </p>
<p><span class="math-container">$$\boxed{ (k_n-H_s)^2 +\tau_g^2= H_d^2} $$</span></p>
<p><a href="https://i.stack.imgur.com/KIPqn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KIPqn.png" alt="Mohr's Circle of Curvature"></a></p>
<p>I made a search in some textbooks that could be accessed. I was not lucky in finding references for the principal curvature difference profiles. Also the surface cannot be viewed as a particular case of CMC surfaces due to its sign change of curvature and so it is distinctly different. The calculation has a different expression and plots differently.</p>
<p>If <span class="math-container">$\phi$</span> is slope of tangent to meridian, primes on meridian arc</p>
<p><strong>Const <span class="math-container">$H=H_s$</span> Mean curvature Delaunay meridians</strong></p>
<p><span class="math-container">$$\phi^{'}+\frac{\cos \phi}{r} =2 H_s$$</span></p>
<p>With initial radius <span class="math-container">$r=r_1$</span> at <span class="math-container">$\phi=0$</span></p>
<p><span class="math-container">$$ \cos \phi = \frac{H_s (r^2-r_1^2)+r_1}{r}$$</span></p>
<p><span class="math-container">$$(-k_1,k_2)= (H_s(r^2+r_1^2)-r_1,\, H_s(r^2-r_1^2)+r_1)\,$$</span></p>
<p><strong>Const <span class="math-container">$H_d$</span> Difference curvature meridians</strong></p>
<p><strong><em>First Order ODE:</em></strong></p>
<p><span class="math-container">$$-\phi^{'}+\frac{\cos \phi}{r} = 2 H_d \tag1$$</span></p>
<p>First order ODE has the disadvantage of numerically computing indefinitely below points of singularity on <span class="math-container">$r=0$</span> axis so there appear multiple spindle meridian profiles below this symmetry axis.</p>
<p><strong><em>Second Order ODE:</em></strong></p>
<p><span class="math-container">$$\phi^{''}= 2 H_d \tan\phi\, (\phi^{'} +2 H_d) \tag2 $$</span></p>
<p><span class="math-container">$$ \cos \phi = \frac{r}{r_1}+2\, H_d\, r\, log \,\frac{r}{r_1}\tag3 $$</span></p>
<p><span class="math-container">$$(k_1,k_2)=(\frac{1}{r_1}+2H_d(1+log \,\frac{r}{r_1}),\frac{1}{r_1}+2H_d \,log \,\frac{r}{r_1} ) \tag4 $$</span></p>
<p><span class="math-container">$$ @\,r=0,\phi \rightarrow \pi/2$$</span></p>
<p>Second order ODE has the advantage of stopping computation at point of singularity so there appears no profile below symmetry line <span class="math-container">$r= 0.$</span></p>
<p>Shown below are profiles of constant difference <span class="math-container">$H_d$</span> of principal curvatures. </p>
<p>Three distinct shapes occur. <em>Progressive loops</em> <span class="math-container">$ H_d <-0.5$</span> ; <em>Ovaloids</em> between cylinder and sphere <span class="math-container">$ 0>H_d>-0.5$</span> and, <em>profiles with Inflection Point</em> for <span class="math-container">$ H_d >0 $</span> occurring at <span class="math-container">$ r=r_1e^{-(1+1/(2 r_1H_d))}. $</span></p>
<p>All profiles meet the axis of symmetry normally, however these are <strong><em>not</em></strong> umbilical points.</p>
<p><a href="https://i.stack.imgur.com/cxyYv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cxyYv.png" alt="Const<span class="math-container">$ H_d$</span> Unduloids"></a></p>
<p>Thanks in advance for your comments and for any references available on the topic.</p>
https://mathoverflow.net/q/20722417Vector field built from connection and metricMartin Hairerhttps://mathoverflow.net/users/385662015-05-21T16:43:48Z2020-05-29T11:02:28Z
<p>Consider a smooth finite-dimensional manifold $M$ with metric $g$ and connection $\nabla$. For some local coordinate system, denote by $g^{\alpha \beta}$ the inverse of the metric tensor and by $\Gamma^\alpha_{\beta\gamma}$ the Christoffel symbols for $\nabla$. The connection $\nabla$ is assumed to be torsion free, but it does in general not equal the Levi-Civita connection for $g$. Denote also by $R^\alpha_{\beta\gamma\eta}$ the curvature tensor built from $\nabla$. We can then define a vector field $h$ by
$$
h^\alpha = R^\alpha_{\beta\gamma\eta} g^{\gamma\zeta}\nabla_\zeta g^{\eta\beta}\;.
$$
Is there a "hands-on" geometric interpretation for $h$? (Does it maybe even have a name?)
In particular,
is there a geometric way of "seeing" when $h$ vanishes? (Of course there are the trivial cases where $\nabla$ is flat or Levi-Civita for $g$, but there must be others.)</p>
https://mathoverflow.net/q/1089658What is knot contact homology?Satoshi Nawatahttps://mathoverflow.net/users/176442012-10-05T22:43:20Z2020-05-29T11:12:26Z
<p>Recently, it was conjectured by the <a href="http://arxiv.org/abs/1204.4709">paper</a> of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed $A$-polynomial of a knot $K$ can be obtained by finding the difference equation of minimal order for the colored HOMFLY polynomials of the knot $K$. This conjecture seems to hold true for torus knots and twist knots. However, I do not understand what the knot contact homology is. </p>
<p>First of all, the knot contact homology describes knot invariants as invariants of the Legendrian submanifolds in the contact manifold. A knot is realized by an intersection of the cosphere bundle $ST^∗M$ of a 3-manifold $M$ with the unit conormal bundle $\Lambda_K$ where $ST^∗M$ admits a contact structure.</p>
<blockquote>
<p>1) Is there any way to visualize an intersection of $ST^∗M$ with $\Lambda_K$?</p>
</blockquote>
<p>The knot contact homology is constructed by the Legendrian differential graded algebra (DGA)</p>
<blockquote>
<p>2) Why do you need DGA to obtain homology theory invariant under Legendrian isotopy?</p>
</blockquote>
<p>From the combinatorial definition (Appendix B of the <a href="http://arxiv.org/abs/1205.1515">paper</a>), I cannot see the reason why this is isomorphic to Legendrian DGA. Although the differentials are determined by the braiding data of a knot, it seems to me that there is no contact structure involved.</p>
<blockquote>
<p>3) Could the isomorphism between the two DGA be explained in layman's terms?</p>
</blockquote>
<p>I do not understand what the augmentation polynomials of the knot contact homology are. </p>
<blockquote>
<p>4) Is there any relation between augmentation polynomials and Porincare-Chekanov polynomials?</p>
</blockquote>
<p>In addition, </p>
<blockquote>
<p>5) I would like to know if there is an explicit connection of knot contact homology to other knot homologies such as Khovanov-Rozansky and HOMFLY homology.</p>
</blockquote>
https://mathoverflow.net/q/29038What is Floer homology of a knot?Ilya Nikokoshevhttps://mathoverflow.net/users/652009-10-27T21:48:15Z2020-05-29T11:27:20Z
<p>I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology theory called Floer homology.</p>
<blockquote>
<p><strong>Question:</strong> what's the definition and properties of a Floer homology of a knot? How is it related to other knot homology theories? </p>
</blockquote>