Recent Questions - MathOverflowmost recent 30 from www.4124039.com2019-08-18T05:57:12Zhttp://www.4124039.com/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://www.4124039.com/q/3385910Geometry of a manifold after Dehn filling, in terms of geometry pre-fillinguser144521http://www.4124039.com/users/1445212019-08-18T05:42:51Z2019-08-18T05:50:15Z
<p>First time posting, so sorry if this is an uninteresting or overly long post!</p>
<p>The inspiration for this question was sparked by <a href="http://www.4124039.com/a/338530">this answer</a> given by Bruno Martelli in response to a question about horizontal surfaces in Seifert fibered spaces as the fiber of a fiber bundle over the circle with periodic monodromy. </p>
<p>When seeing Seifert fibered space and a surface bundle over the circle with periodic monodromy, torus knots in <span class="math-container">$S^3$</span> fit into this class and are well behaved. In Martelli's response however, he restricts to Seifert fibered spaces without boundary, so let's Dehn fill the torus knot, say <span class="math-container">$T(p,q)$</span> along boundary slope <span class="math-container">$\frac{r}{s}$</span>.</p>
<p>Now if I remember correctly, we can't fill along <span class="math-container">$\frac{pq}{1}$</span> to get a closed Seifert fibered space, because it results in a connect sum of lens spaces from capping off the cabling annulus. Every other slope should yield a Seifert fibered space over <span class="math-container">$S^2$</span> with 3 singular fibers, a rather tricky class of Seifert fibered spaces. The surgered manifold should be a Seifert fibered space over the sphere as the underlying topological space for the torus knot's base orbifold was a disk, so Dehn filling the boundary should cap off the base orbifold's underlying topological space boundary to an <span class="math-container">$S^2$</span>. Another way to see this is perhaps simply from the classification/construction of Seifert fibered spaces. </p>
<p>If <span class="math-container">$T(p,q)$</span>'s exterior was <span class="math-container">$M_k$</span> we can give an easy van Kampen argument to yield a presentation for <span class="math-container">$\pi_1(M_k)$</span> as <span class="math-container">$<a,b |a^p=b^q>$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are the classes of the singular fibers in the Seifert fibering of <span class="math-container">$M_k$</span>. From here we should be able to give another Van Kampen argument to yield a presentation for the fundamental group of the the surgered manifold <span class="math-container">$M_k(\frac{r}{s})$</span>, where we just add a relation based on <span class="math-container">$\frac{r}{s}$</span>, so we should get <span class="math-container">$\pi_1(M_k(\frac{r}{s})) = <a,b |a^pb^q, a^rb^s>$</span></p>
<p>I suppose geometrization in all its high powered glory may be able to tell us something about the geometry <span class="math-container">$M_k(\frac{r}{s})$</span> based on this fundamental group calculation, if this presentation isn't too unpleasant. A more pedestrian understanding would be appreciated though. </p>
<p>I'm wondering if we can track through the construction of these Seifert fibered spaces through Dehn filling a trivial circle bundle over a pair of a pants with slopes <span class="math-container">$\frac{p}{q}$</span>, <span class="math-container">$\frac{q}{p}$</span>, and <span class="math-container">$\frac{r}{s}$</span>, how the geometry changes, if we imagine that we start with a hyperbolic pair of pants with geodesic boundary. So onto the actual questions:</p>
<p>0.) Is there a way to see the geometry evolve as we surger the pair of pants cross <span class="math-container">$S^1$</span>?</p>
<p>1.) Of particular interest is the <span class="math-container">$\mathbb{H}^2 \times \mathbb{R}$</span> geometry that might arise, given that we can start with an <span class="math-container">$\mathbb{H}^2 \times S^1$</span> geometry. Can the <span class="math-container">$\mathbb{H}^2 \times \mathbb{R}$</span> geometry ever for show up in surgered torus knots? </p>
<p>2.) It is known that torus knots are L-space knots, and so for every slope <span class="math-container">$\frac{r}{s} \geq pq-(p+q) = 2g-1$</span>, where <span class="math-container">$g$</span> denotes the knot genus of <span class="math-container">$T(p,q)$</span>, the surgered manifolds are L-spaces, is the class of possible geometries for the L-space slopes different from the class of geometries possible for the non-L-space slopes?</p>
http://www.4124039.com/q/3385890A geometric property of certain Lie groupsAli Taghavihttp://www.4124039.com/users/366882019-08-18T01:40:29Z2019-08-18T01:59:45Z
<p>What is an example of a Lie group <span class="math-container">$G$</span> not ismorphic to the Poincare upper half space <span class="math-container">$H^n$</span> but satisfy the following:</p>
<p>For every left invariant metric <span class="math-container">$g$</span> we draw all geodesics. Then we CAN rescale <span class="math-container">$g$</span> conformally to a metric <span class="math-container">$h$</span> and we observe that at each point <span class="math-container">$x\in G$</span>
all drown half ray <span class="math-container">$g$</span>- geodesics initiating <span class="math-container">$x$</span> have finite <span class="math-container">$h$</span>-length except a geodesic tangent to a unique direction</p>
http://www.4124039.com/q/3385881Which endomorphisms of the Tate algebra are "algebraic"?Asvinhttp://www.4124039.com/users/580012019-08-18T01:00:46Z2019-08-18T01:00:46Z
<p>For an abelian variety <span class="math-container">$A$</span> over a field <span class="math-container">$k$</span> with characteristic different from <span class="math-container">$\ell$</span> and Galois group <span class="math-container">$G = Gal(\overline k/k)$</span>, there is always an injective map of the form:
<span class="math-container">$$\mathbb Q_\ell\otimes End_k(A) \to End_G(V_\ell(A))$$</span>
where <span class="math-container">$V_\ell$</span> is the rational Tate module. In specific cases, we know the map is a bijection (<span class="math-container">$k$</span> = finite fields, number fields). </p>
<p>Now suppose we take the direct limit of both sides over increasing <span class="math-container">$k$</span>. Then the left hand side is simply <span class="math-container">$\mathbb Q_\ell\otimes End_{\overline k}(A)$</span> and the map is into <span class="math-container">$End(V_\ell(A))$</span> but it's (almost?) never a bijection.</p>
<p>For instance for non super singular curves or in characteristic <span class="math-container">$0$</span> and <span class="math-container">$A$</span> an elliptic curve, the right hand side is one or two dimensional while the right hand side is four dimensional.</p>
<p>Can we classify or characterize the endomorphisms of <span class="math-container">$V_\ell(A)$</span> that are "algebraic", that is, come from the map above for some <span class="math-container">$k$</span>?</p>
http://www.4124039.com/q/3385871Overview of interpretations of classical probabilityTom Copelandhttp://www.4124039.com/users/121782019-08-18T00:38:58Z2019-08-18T00:44:14Z
<p>The Stanford Encyclopedia of Philosophy has a nice <a href="https://plato.stanford.edu/entries/probability-interpret/" rel="nofollow noreferrer">overview</a> of numerous different interpretations of probability (classical as opposed to quantum) with an extensive bibliography. </p>
<p>What books would serve as well-balanced, objective overviews of this topic for the reader with at least a firm grasp of Bayes' Theorem and the basics of Kolmogorov's axioms?</p>
http://www.4124039.com/q/3385860Mean field games approximate Nash equilibriacdcreevehttp://www.4124039.com/users/1445172019-08-18T00:07:34Z2019-08-18T03:34:20Z
<p>I am learning mean field games (MFG) through the notes by Cardaliaguet: <a href="https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf" rel="nofollow noreferrer">https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf</a>.</p>
<p>I have a question about a step in theorem 3.8 on page 17. Let me give the set-up. Each player <span class="math-container">$i$</span> has the payoff function
<span class="math-container">$$\mathcal{J^i}(\alpha^1, \ldots \alpha^N) = \mathbb{E}\left[\int_0^T \frac12|\alpha_s^i|^2 + F\left(X_s^i, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_s^j}\right) \ ds + G\left(X_T^i, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_T^j}\right)\right].$$</span></p>
<p>The dynamics of the player are given by <span class="math-container">$dX_t^i = \alpha_t^i \ dt + \sqrt{2}\ dB_t^i$</span>, as usual. <span class="math-container">$F$</span> and <span class="math-container">$G$</span> are Lipschitz in both variables. </p>
<p>The <span class="math-container">$X_t^i$</span> are i.i.d. with law <span class="math-container">$m(t)$</span>. The crucial step that I fail to understand is the following: there is <span class="math-container">$N_0$</span> large such that for <span class="math-container">$N \ge N_0$</span>,
<span class="math-container">$$\mathbb{E}\left[\sup_{|y|\le 1/\sqrt{\epsilon}} \left|F(y, m(t)) - F\left(y, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_s^j}\right)\right|\right]\le \epsilon$$</span>
for any <span class="math-container">$t \in [0,T]$</span>. Note that the subscript on the <span class="math-container">$X^i$</span> is <span class="math-container">$s$</span>, not <span class="math-container">$t$</span>. </p>
<p>He says this is an application of Hewitt-Savage. However, I fail to see how this works at all. I'm not sure where to introduce my integrals, and I definitely don't see what role the restriction <span class="math-container">$|y|\le 1/\sqrt{\epsilon}$</span> is playing here. <span class="math-container">$F$</span> is uniformly Lipschitz in the <span class="math-container">$m$</span> variable, independent of the parameter <span class="math-container">$y$</span>. Because this is the case, clearly there has to be something more do just passing to the Lipschitz bound on that difference, because we'd be dropping anything to do with <span class="math-container">$y$</span> (and then I really can't see how Hewitt-Savage would help). Can someone help shed light on this step?</p>
http://www.4124039.com/q/338585-1sharp estimate of a lower and upper bound for a trigonometric function [on hold]Paichuhttp://www.4124039.com/users/1094192019-08-18T00:01:41Z2019-08-18T00:29:44Z
<p>I have the function:
<span class="math-container">$$
f(x) = \cos(ax) - \frac{\sin(ax)}{x}, \quad x\geq 0
$$</span>
where <span class="math-container">$a$</span> is a positive constant. Mathematica and Maple shows that there is a global maximum/minimum for <span class="math-container">$x\geq 0$</span> (only when I search with x bounded between a positive interval). However, it cannot produce a close form estimate for these values with general <span class="math-container">$a>0$</span>. For instance, see <a href="https://www.wolframalpha.com/input/?i=maximum%20(cos(2*x)-sin(2*x)%2Fx)%20from%200%20to%208*pi" rel="nofollow noreferrer">example 1</a> and <a href="https://www.wolframalpha.com/input/?i=maximum%20(cos(a*x)-sin(a*x)%2Fx)%20for%20x%3E0" rel="nofollow noreferrer">example 2</a>. I also attach a typical plot below.</p>
<p>I am looking for a way to obtain a sharp estimate of an upper/lower bound of <span class="math-container">$f(x)$</span>. Any suggestion on how to approach this is greatly appreciated.</p>
<p><a href="https://i.stack.imgur.com/MPa4y.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/MPa4y.jpg" alt="enter image description here"></a></p>
http://www.4124039.com/q/3385822Model for random graphs where clique number remains boundedNicolas Boergerhttp://www.4124039.com/users/219852019-08-17T21:41:26Z2019-08-17T23:43:04Z
<p>In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity al the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number? </p>
http://www.4124039.com/q/3385814Survey article model theory researchuser144513http://www.4124039.com/users/1445132019-08-17T20:35:01Z2019-08-17T20:35:01Z
<p>I've taken a graduate course in model theory and I like it so much that I can imagine doing research in this area. Are there survey articles or review papers on the current research topics in model theory? Where can I find them?</p>
<p>Also, I wish there is literature about the common proof techniques and tricks one uses in the current research. (Sometimes I have the feeling that everybody just writes down their proofs and nobody writes down the essential ideas or an overview of techniques used.) Do you know of any such text?</p>
http://www.4124039.com/q/3385804Convolution in K-Theory via an Example (From StackExchange)Marc Bessonhttp://www.4124039.com/users/1194602019-08-17T20:22:33Z2019-08-17T20:22:33Z
<p>I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still lacking a little understanding. I'd like some help with the following example.</p>
<p>On the representation theory side, I'd like to consider the following simple example. Let <span class="math-container">$G=SL_2(\mathbb{C})$</span>, and let <span class="math-container">$T \subset B$</span> be the toral subgroup of diagonal matrices and the Borel subgroup consisting of upper triangular matrices respectively. Let <span class="math-container">$V_{\Lambda_1}$</span> denote the irreducible representation of highest weight <span class="math-container">$\Lambda_1$</span>. Taking the tensor square of this representation yields the following decomposition into irreducibles: <span class="math-container">$V_{\Lambda_1} \otimes V_{\Lambda_1} \simeq V_{2 \Lambda_1} \oplus V_0$</span>.</p>
<p>I'd like to geometrize this a la Ginzburg. </p>
<p>Via Borel-Weil, we know that <span class="math-container">$H^{0}(G/B, L_{\Lambda_1}) \simeq V_{\Lambda_1}$</span>, where <span class="math-container">$L_{\Lambda_1}$</span> is the associated bundle <span class="math-container">$G \times_{B} \mathbb{C}^{-\Lambda_1}$</span>. What I would like is an operation on <span class="math-container">$G$</span>-equivariant sheaves which corresponds to the tensor product of representations, so that <span class="math-container">$H^0(G/B, L_{\Lambda_1} * L_{\Lambda_1}) \simeq V_{2 \Lambda_1} \oplus V_0$</span>. Note that the operation cannot be the tensor product. To see this, remember that <span class="math-container">$G/B = \mathbb{P}^1$</span>, and <span class="math-container">$L_{\Lambda_1}$</span> is isomorphic to <span class="math-container">$\mathscr{O}_{\mathbb{P}^1}(1)$</span>; if I tensor this sheaf with itself and take global sections I will get the irreducible 3-dimensional representation <span class="math-container">$Sym^2(V_{\Lambda_1})=V_{2\Lambda_1}$</span>. </p>
<p>Here is where I know that <span class="math-container">$*$</span> is supposed to be convolution, as defined by Ginzburg. (If anyone would like the definition, I can provide it, but that would lengthen this post even more).</p>
<p>Question 1: Is it correct to expect that <span class="math-container">$\mathscr{O}_{\mathbb{P}^1}(1) *\mathscr{O}_{\mathbb{P}^1}(1) \simeq \mathscr{O}_{\mathbb{P}^1}(2) \oplus \mathscr{O}_{\mathbb{P}^1}$</span>? This is the only way I can see the global sections giving me the correct representation.</p>
<p>Question 2: If this is indeed the case, is there an explicit description in terms of global sections <span class="math-container">$T_1, T_2$</span> of <span class="math-container">$\mathscr{O}_{\mathbb{P}^1}(1)$</span>, if the coordinates on <span class="math-container">$G/B=\mathbb{P}^1$</span> are <span class="math-container">$[T_1:T_2]$</span>? It is easy to get the global sections <span class="math-container">$T_1^2, T_1T_2, T_2^2$</span> as a basis for <span class="math-container">$V_{2 \Lambda_1}$</span>, but I cannot see how to get the basis for <span class="math-container">$\mathscr{O}_{\mathbb{P}^1}$</span>. </p>
<p>It also occurs to me that I have been working with <span class="math-container">$G-$</span>equivariant sheaves here instead of their <span class="math-container">$K$</span>-theory, and maybe that is incorrect. I've got more thoughts, but this is already quite long for a post. Please let me know if I can provide any additional information.</p>
http://www.4124039.com/q/3385761Can planar set contain even many vertices of every unit equilateral triangle?domotorphttp://www.4124039.com/users/9552019-08-17T19:16:06Z2019-08-17T20:35:07Z
<blockquote>
<p>Is there a nonempty planar set that contains <span class="math-container">$0$</span> or <span class="math-container">$2$</span> vertices from each unit equilateral triangle?</p>
</blockquote>
<p>I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft proof that works for measurable sets, see the details <a href="https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24068" rel="nofollow noreferrer">here</a>. In more general, to make their proof work for other sets besides equilateral triangles, one can ask the following.</p>
<p>Suppose we are given two sets, <span class="math-container">$S$</span> and <span class="math-container">$A$</span> in the plane, such that <span class="math-container">$S$</span> is finite, with a special point, <span class="math-container">$s_0$</span>, while neither <span class="math-container">$A$</span> nor its complement is a null-set, i.e., the outer Lebesgue measure of <span class="math-container">$A$</span> and <span class="math-container">$A^c=\mathbb R^2\setminus A$</span> are both non-zero. Can we find two congruent copies of <span class="math-container">$S$</span>, <span class="math-container">$f_1(S)$</span> and <span class="math-container">$f_2(S)$</span>, such that <span class="math-container">$f_1^{-1}(f_1(S)\cap A)\Delta f_2^{-1}(f_2(S)\cap A)=\{s_0\}$</span>, i.e., <span class="math-container">$s_0$</span> is the only element of <span class="math-container">$S$</span> that goes in to/out of <span class="math-container">$A$</span> when we go from <span class="math-container">$S_1$</span> to <span class="math-container">$S_2$</span>?</p>
http://www.4124039.com/q/3385735Existence of a certain set of 0/1-sequences without the Axiom of ChoiceM. Winterhttp://www.4124039.com/users/1088842019-08-17T18:43:11Z2019-08-17T23:20:30Z
<p>Is there a set <span class="math-container">$\mathcal X\subset\{0,1\}^{\Bbb N}$</span> of 0/1-sequences, so that</p>
<ul>
<li><p>For any two 0/1-sequences <span class="math-container">$x,y\in\{0,1\}^{\Bbb N}$</span> for which there is an <span class="math-container">$N\in\Bbb N$</span> with
<span class="math-container">$$x_i=y_i,\;\;\text{for all $i< N$},\qquad x_i\not=y_i,\;\;\text{for all $i\ge N$},$$</span>
<em>exactly one</em> of these belongs to <span class="math-container">$\mathcal X$</span>.</p></li>
<li><p><span class="math-container">$\mathcal X$</span> can be proven to exist without using the Axiom of Choice.</p></li>
</ul>
http://www.4124039.com/q/3385711A question involving an summation of eigenvalues of the Laplacian operator on $\mathbb{S}^2$Marcelo Nogueirahttp://www.4124039.com/users/1370682019-08-17T18:21:25Z2019-08-17T18:50:42Z
<p>Infinite series involving eigenvalues of the Beltrami-Laplace operator on Riemannian manifolds as well as <span class="math-container">$L^p$</span>-estimates of eigenfunctions arise in the study of the nonlinear Schrödinger equation (NLS) on compact manifolds. Denote by <span class="math-container">$\mu_{k} := k(k + 1)$</span> the eigenvalue of the operator
<span class="math-container">$- \Delta_{\mathbb{S}^2}$</span> associated to the eigenfunction <span class="math-container">$e_k \in C^{\infty}(M)$</span>. Now, consider the summation
<span class="math-container">$$ \sum_{k = 0}^{\infty} \frac{1}{ \langle \mu_k - \alpha \rangle \langle \mu_k \rangle^{\varepsilon}}$$</span> where <span class="math-container">$\alpha > 0$</span> (is a positive arbitrary constant), <span class="math-container">$\varepsilon > 0$</span> and <span class="math-container">$\langle x \rangle : = 1 + |x|$</span>. My question is the following:</p>
<p><span class="math-container">$$\langle \mu_k - \alpha \rangle^{-1} \langle \mu_k \rangle^{- \varepsilon} \in \ell^{1}_{k}(\mathbb{N}) \mbox{ }, $$</span> </p>
<p>independently of the choice of <span class="math-container">$ \alpha$</span>? </p>
<p>My failed attempt was to consider two cases: (Case 1) <span class="math-container">$\mu_k \geq 4 \alpha$</span>. In this case we have <span class="math-container">$|\mu_k - \alpha| \geq \frac{3}{4} \mu_k$</span> and one can obtain the desired conclusion. (Case 2) <span class="math-container">$\mu_k \leq 4 \alpha$</span>. In this case, we have a finite sum, but I would like to prove that the summation is bounded by a constant which does not depends on <span class="math-container">$\alpha$</span>. Thanks in advance !!!</p>
http://www.4124039.com/q/3385643On the universal property for interval objectsDean Younghttp://www.4124039.com/users/1045132019-08-17T16:12:28Z2019-08-17T20:36:44Z
<p>In his lecture, <a href="https://www.maths.ed.ac.uk/~tl/cambridge_ct14/cambridge_ct14_talk.pdf" rel="nofollow noreferrer"><em>The Categorical Origins of Lebesgue Measure</em></a>, Professor Tom Leinster mentions the following theorem:</p>
<blockquote>
<p><strong>Theorem 1:</strong> (Freyd; Leinster) The topological space <span class="math-container">$[0, 1]$</span> comes equipped with two distinct basepoints <span class="math-container">$0$</span> and <span class="math-container">$1$</span>, and a map <span class="math-container">$[0, 1] \rightarrow [0, 1] \amalg [0, 1] / \text{first }1 \sim \text{second } 0$</span>. <span class="math-container">$[0, 1]$</span> is terminal as such.</p>
</blockquote>
<p>Leinster's theorem is about a universal property of the banach space <span class="math-container">$L^1 [0, 1]$</span> and the integration function <span class="math-container">$\int : L^1 [0, 1] \rightarrow \mathbb{R}$</span>. It goes like this: let <span class="math-container">$\mathcal{A}$</span> be the category of Banach spaces, whose objects are banach spaces and whose maps are maps <span class="math-container">$\phi : X \rightarrow Y$</span> of banach spaces such that <span class="math-container">$||\phi(x)|| \leq ||x||$</span>. Let <span class="math-container">$\mathcal{A}/\mathbb{R}$</span> be the under-category (objects are maps <span class="math-container">$\mathbb{R} \rightarrow X$</span> in <span class="math-container">$\mathcal{A}$</span>; this is also called the coslice category). There is a functor <span class="math-container">$T : \mathcal{A} \rightarrow \mathcal{A}$</span> defined where <span class="math-container">$T(X) = X \prod X$</span>, where <span class="math-container">$X$</span> has measure <span class="math-container">$||(x, y)|| = \frac{1}{2} (||x|| + ||y||)$</span> with corresponding map <span class="math-container">$\mathbb{R} \rightarrow X \prod X$</span> induced canonically by the map <span class="math-container">$\mathbb{R} \rightarrow X$</span>. The initial <span class="math-container">$T$</span>-algebra is
<span class="math-container">$$L^1 [0, 1] = \frac{\text{Lebesgue-integrable functions } [0, 1] \rightarrow \mathbb{R} }{\text{equality almost everywhere}}$$</span>
Note: we can still form <span class="math-container">$T$</span>-algebras when <span class="math-container">$T$</span> is not a monad.</p>
<p>The unique map <span class="math-container">$\int : L^1 [0, 1] \rightarrow \mathbb{R}$</span> comes from the universal property now!</p>
<p>My question is this:</p>
<blockquote>
<p>Can we rearrange theorem 1 above in terms of continuous functions <span class="math-container">$C([0, 1]) = [[0, 1], \mathbb{R}]_{\text{Top}}$</span> and an endofunctor <span class="math-container">$T$</span> where <span class="math-container">$T(A) = A \prod A$</span>, just like in the measure theory case?</p>
</blockquote>
<p>From a categorical perspective, this might be nice since we would no longer have to require two <em>distinct</em> points in the topological space.</p>
<p>So the rearrangement of Professor Leinster's theorem would go like this:</p>
<p>Let <span class="math-container">$\mathcal{B}$</span> be the category of topological vector spaces (some tweaking might be necessary, maybe topological algebras would be better? Maybe the Lawvere theory of <span class="math-container">$C^0$</span>-algebras?). Let <span class="math-container">$\mathcal{B}/\mathbb{R}$</span> be the under-category (objects are maps <span class="math-container">$\mathbb{R} \rightarrow X$</span> in <span class="math-container">$\mathcal{B}$</span>). There is possibly a monad in this category where <span class="math-container">$T(X) = X \prod X$</span>, just as before. Perhaps the initial <span class="math-container">$T$</span>-algebra is <span class="math-container">$C([0, 1])$</span>!</p>
<hr>
<p>Now that I think about it, if measure theory and topology have their own versions of this theorem, maybe there is one for differential geometry and <span class="math-container">$C^{\infty}$</span>-algebras. Just a thought.</p>
http://www.4124039.com/q/3385593Subgroup generated by a subgroup and a conjugate of itBertalan Bodorhttp://www.4124039.com/users/1444852019-08-17T14:34:15Z2019-08-17T17:01:14Z
<p>Let <span class="math-container">$H\leq G$</span> be groups, and <span class="math-container">$a\in G$</span> so that <span class="math-container">$\langle H,a\rangle=G$</span>. Does it follows that <span class="math-container">$\langle H\cup aHa^{-1}\rangle$</span> is a normal subgroup of <span class="math-container">$G$</span>?</p>
<p>My hope is that this is true, and my guess is that it is not. </p>
<p>It might be easy.</p>
http://www.4124039.com/q/3385572What is the current status on methods to find limit cycles?Matthttp://www.4124039.com/users/1421532019-08-17T14:29:12Z2019-08-17T18:12:54Z
<p>What are the current best methods to show analytically the existence of a limit cycle in a <span class="math-container">$n$</span>-dimensional system of the form:
<span class="math-container">$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$</span>
Where <span class="math-container">$\vec{x}\in \mathbb{R}^n$</span>.</p>
<p>I am familiar with Poincare-Bendixson Theorem or Dulac's Criterion, but would like to know what is the current status on limit cycles in systems in <span class="math-container">$\mathbb{R}^n$</span> (<span class="math-container">$n>2$</span>).</p>
http://www.4124039.com/q/3385554Structure of the module of derivations on the space of Holomorphic functionsDuchamp Gérard H. E.http://www.4124039.com/users/252562019-08-17T14:08:36Z2019-08-17T23:18:40Z
<p>Maybe this is well-known, maybe not.
Let
<span class="math-container">$\Omega\subset \mathbb{C}$</span> be connected open and non-empty.
It can be shown that if
<span class="math-container">$d\in\mathfrak{Der}(\mathcal{H}(\Omega))$</span>
(i.e. <span class="math-container">$d$</span> is a derivation of the algebra
<span class="math-container">$\mathcal{H}(\Omega)$</span>) and is continuous
for the topology of compact convergence
then <span class="math-container">$d$</span> is of the form
<span class="math-container">$d=\varphi(z)\frac{d}{dz}$</span>.
My questions are the following </p>
<ol>
<li> Can the continuity be withdrawn ?
<li> If yes, for which <span class="math-container">$\Omega$</span> ?
<li> Can <span class="math-container">$\Omega$</span> be replaced by a one
dimensional complex manifold ? (in
particular, for these manifolds, is
there a principal derivation
like <span class="math-container">$\frac{d}{dz}$</span> ?)
</ol>
<p>In particular the third question is trivial
in the case of a compact connected manifold
(because <span class="math-container">$\mathcal{H}(\Omega)=\mathbb{C}1_{\Omega}$</span>
by maximum principle.). </p>
http://www.4124039.com/q/3385507Number of matrices with bounded products of rows and columnsKatehttp://www.4124039.com/users/1445012019-08-17T13:27:01Z2019-08-18T01:48:11Z
<p>Fix an integer <span class="math-container">$d \geq 2$</span> and for every real number <span class="math-container">$x$</span> let <span class="math-container">$M_d(x)$</span> be number of <span class="math-container">$d \times d$</span> matrices <span class="math-container">$(a_{ij})$</span> satisfying: every <span class="math-container">$a_{ij}$</span> is a positive integer, the product of every row does not exceed <span class="math-container">$x$</span>, and the product of every column does not exceed <span class="math-container">$x$</span>.</p>
<p>I'm looking for a good upper bound for <span class="math-container">$M_d(x)$</span> as <span class="math-container">$x \to +\infty$</span>.</p>
<p>If we forget about the condition on the columns, since it is well known that the number of <span class="math-container">$d$</span>-tuples <span class="math-container">$(b_1, \dots, b_d)$</span> of positive integers satisfying <span class="math-container">$b_1 \cdots b_d \leq x$</span> is <span class="math-container">$\ll_d x (\log x)^{d - 1}$</span> (a generalization of Dirichlet divisor problem), we get the upper bound
<span class="math-container">$$M_d(x) \ll_d x^d (\log x)^{d(d-1)}.$$</span></p>
<p>In the special case <span class="math-container">$d = 2$</span>, we have that <span class="math-container">$a_{12}, a_{21} \leq \min(x / a_{11}, x / a_{22})$</span> and consequently
<span class="math-container">$$M_2(x) \leq \sum_{a_{11}, a_{22} \leq x} \min\left(\frac{x}{a_{11}}, \frac{x}{a_{22}}\right)^2 \ll \sum_{a_{11} \leq a_{22} \leq x} \left(\frac{x}{a_{22}}\right)^2 \ll x^2 \log x ,$$</span>
which is a better upper bound than the general one given in the above paragraph. However, I have no idea of how to generalize this trick to <span class="math-container">$d \geq 3$</span> (if possible).</p>
<p>Has this problem been studied before? Do you have any idea/suggestion about it?</p>
http://www.4124039.com/q/3385400Bertini's theorem for singular varietiesk.j.http://www.4124039.com/users/1282352019-08-17T08:26:30Z2019-08-17T23:08:29Z
<p>I have extended the Bertini's theorem (Hartshorne II.8.18) for singular projective varieties, in order to show <a href="http://www.4124039.com/questions/162656/proving-that-any-two-points-on-a-variety-can-be-joined-by-a-curve-why-does-bert">this</a>, <a href="http://www.4124039.com/questions/62843/path-connectedness-of-varieties?noredirect=1&lq=1">this</a> or <a href="http://www.4124039.com/questions/75320/connecting-points-on-a-variety-by-the-image-of-a-nonsingular-curve">this</a>.
(I think that they use Bertini for a singular veriety, but Hartshorne only shows it for a nonsingular variety.)
Please check my proof.</p>
<p>Let <span class="math-container">$k$</span> be an arbitrary algebraically closed field, and <span class="math-container">$X\subseteq \mathbb{P}^N$</span> a closed normal subvariety of dimension <span class="math-container">$d$</span>. (possibly singular)
(To use liner systems, I think we need <span class="math-container">$X$</span> normal.)
Then we'll show that <span class="math-container">$\mathfrak{d} = \{ H \subseteq \mathbb{P}: \text{a hyperplane such that } X \not\subseteq H \text{ and }X \cap H \text{ is regular} \}$</span> is open in <span class="math-container">$|H|$</span>, a complete linear system of a hyperplane.</p>
<p>Almost all part are the same to the one of Harthorne.<br>
For a closed point <span class="math-container">$x \in X$</span>, let <span class="math-container">$B_x = \{ H | X \subseteq H \text{ or } (X \not\subseteq H \text{ and } x \in X \cap H \text{ and } X \cap H \text{ is singular at }x) \} $</span>.<br>
By the following lemma, if <span class="math-container">$x$</span> is a singular point, <span class="math-container">$B_x = \{ x \in H \}$</span>.</p>
<blockquote>
<p>Let A be a singular noetherian local domain of dimension <span class="math-container">$n$</span>, <span class="math-container">$0 \neq f \in A$</span>.
Then <span class="math-container">$A/f$</span> is also singular.</p>
</blockquote>
<p>(proof is very easy, using the fact that <span class="math-container">$\dim A/f = n -1$</span> in this case )</p>
<p>So by the proof of Bertini in Hartshorne, <span class="math-container">$\dim B_x = N - d - 1$</span> if <span class="math-container">$x$</span> is a nonsingular point, and is <span class="math-container">$= N - 1$</span> if <span class="math-container">$x$</span> is a singular point.</p>
<p>Finally, consider the subset <span class="math-container">$B = \{ (x, H) \in X \times |H| : H \in B_x \}$</span>.
This is closed in <span class="math-container">$X \times |H|$</span>. (I can't show this...)
So consider it as a closed subscheme.
Then the first projection <span class="math-container">$h : B \to X$</span> is surjective.
So by the general proposition about morphisms and fibres, for any closed point <span class="math-container">$b \in B$</span> and its image <span class="math-container">$x = h(b)$</span>, <span class="math-container">$\dim \mathscr{O}_{h^{-1}(x), b} \ge \dim \mathscr{O}_{B,b} - \dim \mathscr{O}_{X, x}$</span>.
Now since <span class="math-container">$X$</span> is a variety over a field, <span class="math-container">$\dim \mathscr{O}_{X, x} = d$</span>, and since <span class="math-container">$\dim \mathscr{O}_{h^{-1}(x), b} \le \dim h^{-1}(x)$</span>, it is <span class="math-container">$\le N -d - 1$</span>.
Therefore <span class="math-container">$\dim B \le N - 1$</span>.<br>
So under the second projection <span class="math-container">$p : B \to |H|$</span>, the image is a proper subset.
Since <span class="math-container">$p$</span> is a projective morphism, this image is closed.<br>
Because the complement of this image is precisely <span class="math-container">$\mathfrak{d}$</span>, the desired set, we have done.</p>
<p>So, is <span class="math-container">$B$</span> closed?</p>
<p>Any help will be much appreciated!!</p>
http://www.4124039.com/q/3385286On the first sequence without collinear tripleSebastien Palcouxhttp://www.4124039.com/users/345382019-08-17T01:38:37Z2019-08-17T18:08:01Z
<p>Let <span class="math-container">$u_n$</span> be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for <span class="math-container">$a_n=n^2$</span> or <span class="math-container">$b_n=2^n$</span>). It is a variation of <a href="http://www.4124039.com/q/338415/34538">that one</a>.</p>
<p>We can find the terms with a kind of Sieve of Eratosthenes: </p>
<p><a href="https://i.stack.imgur.com/XowMh.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XowMh.gif" alt="enter image description here"></a></p>
<p>The above animation reveals the first terms of this sequence:<br>
<span class="math-container">$$ 0,0,1,1,4,3,8,2,2,5,7,4,5,8 $$</span> </p>
<p>Obviously, an integer appears at most two times in this sequence. </p>
<p><strong>Question</strong>: Does each nonnegative integer appear in this sequence? Exactly two times? </p>
<hr>
<p>By searching the first terms in OEIS, we are lucky to find <a href="http://oeis.org/A236266" rel="nofollow noreferrer">A236266</a> (due to Alois P. Heinz, 2014):<br>
<span class="math-container">$$ 0, 0, 1, 1, 4, 3, 8, 2, 2, 5, 7, 4, 5, 8, 16, 3, 7, 14, 12, 23, 16, 12, 25, 31, 13, 6, 11, 28, 11, \dots $$</span><br>
<a href="https://i.stack.imgur.com/yxhxW.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yxhxW.jpg" alt="enter image description here"></a> </p>
<p>This sequence leads to many other possible questions like the existence of a link with </p>
<ul>
<li>the Euler's totient function (because their graph look alike), </li>
<li>the prime numbers (because their finding process are similar: "Sieve of Eratosthenes"-like). </li>
</ul>
<p>We could also ask about </p>
<ul>
<li>bounds: <span class="math-container">$u_n \in [n/10,5n/4]$</span>? </li>
<li>the gaps observed in the graph: Is there <span class="math-container">$\alpha<1$</span> such that <span class="math-container">$$u_n \not \in [\alpha n,n] \cup [\alpha n/2,n/2] \cup [\alpha n/4,n/4] \cup [\alpha 3n/4,3n/4] \cup \cdots \ ?$$</span> at least for each component of this union and <span class="math-container">$n$</span> large enough? </li>
<li>What would be the exact value of <span class="math-container">$\alpha$</span>? or at least a good approximation? <span class="math-container">$\alpha \sim 0.9$</span>? </li>
</ul>
<hr>
<p>Following the comment of Peter Kagey, here is a huge generalization of the above sequence and of the main question.</p>
<p>Let <span class="math-container">$A$</span> be a subset of <span class="math-container">$\mathbb{Z}_{\ge 0}$</span>, and consider the sequence <span class="math-container">$S_A$</span> which is the lexicographically first sequence of positive integers such that no <span class="math-container">$k+2$</span> points fall on any polynomial of degree <span class="math-container">$≤k$</span>, for any <span class="math-container">$k \in A$</span>. Then the sequence of Peter Kagey (<a href="https://oeis.org/A300002" rel="nofollow noreferrer">A300002</a>) is <span class="math-container">$S_{\mathbb{Z}_{\ge 0}}$</span>, the above plus one is <span class="math-container">$S_{\{1\}}$</span> and <span class="math-container">$S_{\{0 \}}$</span> is the sequence of natural numbers. </p>
<p>For any nonempty subset <span class="math-container">$A \subset \mathbb{Z}_{\ge 0}$</span>:<br>
<strong>Generalized question</strong>: Is it true that any positive number appears exactly <span class="math-container">$\min(A)+1$</span> times in <span class="math-container">$S_A$</span>?<br>
In particular, if <span class="math-container">$0 \in A$</span>: Is <span class="math-container">$S_A$</span> a permutation of the natural numbers?</p>
<p>Let us compute the first terms of the sequence <span class="math-container">$S_{\{0,1\}}$</span>: </p>
<p><a href="https://i.stack.imgur.com/dsOpz.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dsOpz.gif" alt="enter image description here"></a></p>
<p>This animation reveals the first terms: <span class="math-container">$1, 2, 4, 3, 6, 5, 9, 12$</span>. </p>
<p>By searching them on OEIS, we find <a href="https://oeis.org/A231334" rel="nofollow noreferrer">A231334</a> (Paul Tek, 2013):
<span class="math-container">$$ 1, 2, 4, 3, 6, 5, 9, 12, 7, 14, 13, 8, 23, 17, 18, 22, 10, 15, 11, 28, 19, 16, 20, 29, 32, 44, 35, \dots$$</span></p>
<p><a href="https://i.stack.imgur.com/l2IRU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/l2IRU.png" alt="enter image description here"></a></p>
<p>Moreover, the author asks also the question <code>Is this a permutation of the natural numbers?</code></p>
http://www.4124039.com/q/3385158Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow SO(2n)$?Daffy Duckhttp://www.4124039.com/users/1444772019-08-16T20:25:38Z2019-08-17T19:25:55Z
<p>Does the following diagram commute?
<span class="math-container">$$
\require{AMScd}
\begin{CD}
BU @>{\psi^k}>> BU \\
@VVV @VVV \\
BO @>{\psi^k}>> BO
\end{CD}
$$</span></p>
<p>Evidence for: <span class="math-container">$rc = 2$</span>, it works for <span class="math-container">$BU(1) \rightarrow BSO(2)$</span> by looking at Chern classes, and I did a few manual computations in higher dimension.</p>
<p>Evidence against: it's not a ring map, let alone a <span class="math-container">$\lambda$</span>-ring map.</p>
http://www.4124039.com/q/3384442Norm of convolution operatorAyman Moussahttp://www.4124039.com/users/277672019-08-15T21:17:50Z2019-08-17T19:01:57Z
<p>By Young's inequality for any <span class="math-container">$f\in L^p(\mathbf{R})$</span> the map <span class="math-container">$T_f:g\mapsto f\star g$</span> is a continuous operator from <span class="math-container">$L^q(\mathbf{R})$</span> to <span class="math-container">$L^r(\mathbf{R})$</span> where <span class="math-container">$1\leq p,q,r\leq \infty$</span> satisfy <span class="math-container">$1+\frac1r=\frac1p+\frac1q$</span> and we even have</p>
<p><span class="math-container">\begin{align*}
\|T_f\|_{p\rightarrow r} \leq \|f\|_q.
\end{align*}</span>
If I am not mistaking in general <span class="math-container">$\|T_f\|_{p\rightarrow r}$</span> and <span class="math-container">$\|f\|_{q}$</span> are not equivalent :</p>
<ul>
<li>When <span class="math-container">$r=q=2$</span> and <span class="math-container">$p=1$</span> we have by Plancherel's formula (for a correctly normalized Fourier transform) <span class="math-container">$\| T_f(g)\|_2 =\| \hat{f}\hat{g}\|_2$</span> from which we get <span class="math-container">$\|T_f\|_{2\rightarrow 2}=\|\hat{f}\|_\infty$</span>, and <span class="math-container">$\|f\|_1\lesssim\|\hat{f}\|_\infty$</span> is just not reasonable.</li>
<li>On a more sophisticated level, on the torus I know that the partial Fourier Series <span class="math-container">$S_N(f)$</span> corresponding to the Dirichlet kernel <span class="math-container">$D_N$</span> converge in <span class="math-container">$L^p(\mathbf{T})$</span> for non extremal values of <span class="math-container">$p$</span>. The Dirichlet kernel being unbounded in <span class="math-container">$L^1(\mathbf{T})$</span>, <span class="math-container">$\|D_N\|_1\lesssim\|T_{D_N}\|_{p\rightarrow p}$</span> is not possible because of the Banach-Steinhauss Theorem.</li>
</ul>
<p>On the other hand, one can prove that <span class="math-container">$\|T_f\|_{1\rightarrow 1}$</span> and <span class="math-container">$\|T_f\|_{\infty\rightarrow\infty}$</span> are both equivalent to <span class="math-container">$\|f\|_1$</span>.</p>
<p>My questions :</p>
<blockquote>
<ol>
<li>Are they any other cases of exponents for which this equivalence holds ?</li>
<li>When the equivalence does not hold, is there any description of <span class="math-container">$\|T_f\|_{p\rightarrow r}$</span> (with emphasis on the case <span class="math-container">$p=r$</span>) ?</li>
<li>Is there any elementary (= not as above) proof that <span class="math-container">$\|T_f\|_{p\rightarrow p}$</span> is not equivalent to <span class="math-container">$\|f\|_1$</span> when <span class="math-container">$p\notin\{1,2,\infty\}$</span> ?</li>
</ol>
</blockquote>
<p>I found several results in the literature linked to this question but they either treat the optimality of the Young inequality as the continuity of the operator <span class="math-container">$(f,g)\mapsto f\star g$</span> (not interested) or precisely state the equivalence <strong>if</strong> <span class="math-container">$f$</span> is non negative. </p>
http://www.4124039.com/q/3384373Move one element of finite set out from A in planedomotorphttp://www.4124039.com/users/9552019-08-15T19:31:27Z2019-08-17T19:59:04Z
<p>Suppose we are given two sets, <span class="math-container">$S$</span> and <span class="math-container">$A$</span> in the plane, such that <span class="math-container">$S$</span> is finite, with a special point, <span class="math-container">$s_0$</span>, while neither <span class="math-container">$A$</span> nor its complement is a null-set, i.e., the outer Lebesgue measure of <span class="math-container">$A$</span> and <span class="math-container">$A^c=\mathbb R^2\setminus A$</span> are both non-zero. Can we find two congruent copies of <span class="math-container">$S$</span>, <span class="math-container">$f_1(S)$</span> and <span class="math-container">$f_2(S)$</span>, such that <span class="math-container">$f_1^{-1}(f_1(S)\cap A)\Delta f_2^{-1}(f_2(S)\cap A)=\{s_0\}$</span>, i.e., <span class="math-container">$s_0$</span> is the only element of <span class="math-container">$S$</span> that goes in to/out of <span class="math-container">$A$</span> when we go from <span class="math-container">$S_1$</span> to <span class="math-container">$S_2$</span>?</p>
<p>My motivation is to extend a Falconer-Croft proof that works for measurable sets, see the details <a href="https://dustingmixon.wordpress.com/2019/08/05/polymath16-fourteenth-thread-automated-graph-minimization/#comment-24068" rel="nofollow noreferrer">here</a>.
It <a href="http://www.4124039.com/questions/338576/can-planar-set-contain-even-many-vertices-of-every-unit-equilateral-triangle/338579#338579">is easy to see</a> that this can be done when the elements of <span class="math-container">$S$</span> are the three vertices of an equilateral triangle.</p>
http://www.4124039.com/q/3384253A weak (?) form of Shelah cardinalsTrevor Wilsonhttp://www.4124039.com/users/16822019-08-15T16:20:30Z2019-08-17T18:36:11Z
<p>The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal":</p>
<p>A cardinal <span class="math-container">$\kappa$</span> is <em>weakly Shelah</em> if for all <span class="math-container">$f : \kappa \to \kappa$</span> there is some <span class="math-container">$\alpha < \kappa$</span> that is closed under <span class="math-container">$f$</span> and there is some elementary embedding <span class="math-container">$j : V \to M$</span> (where <span class="math-container">$M$</span> is a transitive class) such that <span class="math-container">$\operatorname{crit}(j) = \alpha$</span> and <span class="math-container">$j(\alpha) > \kappa$</span> and <span class="math-container">$V_{j(f)(\kappa)} \subset M$</span>.</p>
<p>Questions:</p>
<ol>
<li><p>Is every weakly Shelah cardinal Shelah?</p></li>
<li><p>Is the least weakly Shelah cardinal Shelah?</p></li>
<li><p>Is every weakly Shelah cardinal measurable?</p></li>
<li><p>What is the consistency strength of ZFC + "there is a weakly Shelah cardinal"? In particular, how does it relate to weakly hyper-Woodin cardinals as defined by Schimmerling?</p></li>
</ol>
<p>Here's what I know:</p>
<p>If <span class="math-container">$\kappa$</span> is a Shelah cardinal, then <span class="math-container">$\kappa$</span> is weakly Shelah. To see this, let <span class="math-container">$f : \kappa \to \kappa$</span>. By the Shelah property applied to <span class="math-container">$f+1$</span>, there is an elementary embedding <span class="math-container">$j : V \to M$</span> such that <span class="math-container">$\operatorname{crit}(j) = \kappa$</span> and <span class="math-container">$V_{j(f)(\kappa)+1} \subset M$</span>. Because Shelah cardinals are Woodin, some cardinal <span class="math-container">$\alpha < \kappa$</span> is <span class="math-container">$\mathord{<}\kappa$</span>-<span class="math-container">$f$</span>-strong in <span class="math-container">$V$</span> and therefore <span class="math-container">$\mathord{<}j(\kappa)$</span>-<span class="math-container">$j(f)$</span>-strong in <span class="math-container">$M$</span> by elementarity. In particular <span class="math-container">$\alpha$</span> is <span class="math-container">$(j(f)(\kappa)+1)$</span>-<span class="math-container">$j(f)$</span>-strong in <span class="math-container">$M$</span>, and therefore also in <span class="math-container">$V$</span> because <span class="math-container">$V_{j(f)(\kappa)+1} \subset M$</span>. It's not hard to see that <span class="math-container">$\alpha$</span> witnesses the weakly Shelah property of <span class="math-container">$\kappa$</span> in <span class="math-container">$V$</span> with respect to <span class="math-container">$f$</span>.</p>
<p>On the other hand, if <span class="math-container">$\kappa$</span> is weakly Shelah, then <span class="math-container">$\kappa$</span> is a Woodin limit of Woodin cardinals and there are Woodin cardinals above <span class="math-container">$\kappa$</span>. Clearly the definition implies <span class="math-container">$\kappa$</span> is Woodin. Then by considering various <span class="math-container">$f$</span> we can obtain cofinally many <span class="math-container">$\alpha < \kappa$</span> as in the definition, and <span class="math-container">$j(\alpha) > \kappa$</span> implies <span class="math-container">$\alpha$</span> is a limit of Woodin cardinals, so <span class="math-container">$\kappa$</span> is a limit of Woodin cardinals. Now applying the definition to the function <span class="math-container">$f : \kappa \to \kappa$</span> where <span class="math-container">$f(\alpha)$</span> is the successor of the least Woodin cardinal above <span class="math-container">$\alpha$</span> shows that there is a Woodin cardinal above <span class="math-container">$\kappa$</span>.</p>
http://www.4124039.com/q/33839127Unconventional examples of mathematical modellingLudwighttp://www.4124039.com/users/626732019-08-15T05:33:25Z2019-08-17T18:13:50Z
<p><strong><em>Disclaimer.</strong> The is admittedly a soft question. If it does not meet the criteria for being an acceptable MO question, I apologize in advance.</em></p>
<hr>
<p>I'll soon be teaching a (basic) course on mathematical control theory. The first part of the course will focus on mathematical modelling of dynamical systems. More precisely, I will present examples of mathematical models of dynamical systems encountered in engineering, physics, biology, etc., and discuss some of their basic properties (e.g., existence of equilibria, stability, etc.)</p>
<p>Now, examples that are commonly used include: damper-spring-mass system, RLC circuits, pendulum, etc. However, I would like to come up with something different. Thus, I was thinking about models used to describe unconventional/unusual dynamical phenomena.</p>
<p>Examples of "unconventional" modelling that came to my mind are, for instance, dynamical models of love or hate displayed by individuals in a romantic relationship (see, e.g., <a href="http://www.sci.wsu.edu/math/faculty/schumaker/Math415/Sprott2004.pdf" rel="noreferrer">this paper</a>). However, I'm curious to hear about more unconventional examples.
<em>Thanks in advance for your input!</em></p>
http://www.4124039.com/q/3383904Question about an implication of Thomason's étale descent spectral sequencexirhttp://www.4124039.com/users/1205482019-08-15T04:24:18Z2019-08-17T19:42:35Z
<p>On page 5 of <a href="https://www3.nd.edu/~wgd/Dvi/Spectrum.Algebraic.Integers.pdf" rel="nofollow noreferrer">this</a> paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper <em>Algebraic K-theory and étale cohomology</em>, which reads</p>
<p><span class="math-container">$$H^p_{\acute{e}t}(X, \mathbb{Z}_l(-q/2)) \Rightarrow \pi_{-q-p}\hat{L}KX$$</span></p>
<p>where <span class="math-container">$KX$</span> is the algebraic K-theory spectrum of <span class="math-container">$X$</span> and <span class="math-container">$\hat{L}$</span> denotes the <span class="math-container">$\ell$</span>-completed Bousefield localization at topological K-theory, implies the natural isomorphism</p>
<p><span class="math-container">$$\pi_i\widehat{L}(KR) \cong \pi_i\text{Map}_{\Gamma_F'}(X_+^\theta, \hat{\mathcal{K}})$$</span></p>
<p>where here, <span class="math-container">$R$</span> is the ring of integers localized away from <span class="math-container">$\ell$</span> of some totally real field <span class="math-container">$F$</span>, <span class="math-container">$X$</span> is a space realizing the étale homotopy type of <span class="math-container">$R$</span>, <span class="math-container">$\theta$</span> is the character of <span class="math-container">$\pi_1(X)$</span> corresponding to the <span class="math-container">$\ell$</span>-adic cyclotomic character, <span class="math-container">$X^\theta$</span> is the cover of <span class="math-container">$X$</span> corresponding to the kernel of <span class="math-container">$\theta$</span>, with the corresponding action of <span class="math-container">$\theta$</span> of the Galois/fundamental group. The subscript plus, as usual, denotes the unreduced suspension spectrum. The Galois group of the <span class="math-container">$\ell$</span>-adic cyclotomic extension is in particular <span class="math-container">$\Gamma_F'$</span>, and lives inside <span class="math-container">$\Gamma'\cong \mathbb{Z}_l^\times$</span> (i.e. the corresponding Galois group over <span class="math-container">$\mathbb{Q}$</span>.) </p>
<p>The <span class="math-container">$\text{Map}$</span> is then an equivariant mapping spectrum, with the action of <span class="math-container">$\Gamma'$</span> via Adams operations on the target.</p>
<p>(It is assumed we have a rigid, i.e. not just up to homotopy, action of <span class="math-container">$\Gamma'$</span> on <span class="math-container">$\hat{\mathcal{K}}$</span>.)</p>
<p>So, <strong>how does this follow</strong>? There is no real explanation given, which makes me think it has to be simple. But I'm confused what the relationship of the localized algebraic K-theory groups to étale cohomology has to do with this twisted equivariant mapping spectrum. </p>
<p>Apologies if this is some very formal thing; I'm pretty un-versed in these matters. I'll delete this question if it turns out to just be very obvious somehow.</p>
http://www.4124039.com/q/3382292Reduced expression and Bruhat orderJames Cheunghttp://www.4124039.com/users/1102292019-08-13T03:03:19Z2019-08-17T18:49:48Z
<p>For <span class="math-container">$n\ge 3$</span>. Let <span class="math-container">$s_1\cdots s_n$</span> be a reduced expression of <span class="math-container">$x$</span>. Suppose <span class="math-container">$s_1\cdots s_{n-1}\le w$</span> and <span class="math-container">$s_2\cdots s_{n}\le w$</span>. </p>
<blockquote>
<p>Does this imply <span class="math-container">$x\le w$</span>?</p>
</blockquote>
http://www.4124039.com/q/3378332How to deduce the following map between Zariski tangent spaces is surjective?Takeshi Goudahttp://www.4124039.com/users/1373202019-08-07T14:18:47Z2019-08-17T16:54:17Z
<p>Let <span class="math-container">$f: X \rightarrow Y$</span> be a morphism of schemes: <span class="math-container">$X$</span>, <span class="math-container">$Y$</span> are regular schemes, let <span class="math-container">$Z_1, Z_2$</span> be two closed regular subschemes of <span class="math-container">$X$</span>, let <span class="math-container">$x \in X \times_Y Z_2$</span> such that <span class="math-container">$y = f(x) \in Z_1 \cap Z_2$</span>.
Then I would like to know how I can deduce that the map
<span class="math-container">$$
T_xX \rightarrow (T_yY/T_yZ_2) \otimes_{k(y)} k(x)
$$</span>
is surjective knowing that </p>
<p>1) the image of <span class="math-container">$T_{x} (X \times_Y Z_1)$</span> generates <span class="math-container">$T_yY/T_yZ_2$</span> </p>
<p>2) <span class="math-container">$T_x(X \times_Y Z_1) \rightarrow T_y Z_1 \otimes_{k(y)} k(x)$</span> is surjective. </p>
<p>It is supposed to follow from these things but not seeing how... I would appreciate any explanation on this. Thank you!</p>
http://www.4124039.com/q/3363911Relation between the decomposition invariants of a projective reduced curve and its normalizationwindsheafhttp://www.4124039.com/users/981292019-07-18T09:33:41Z2019-08-17T17:30:39Z
<p>Let <span class="math-container">$X$</span> be a reduced projective scheme over <span class="math-container">$k$</span> which is of pure
dimension 1. Let <span class="math-container">$\pi: X \to \mathbb{P}_k^1$</span> be a finite (hence
affine, surjective and flat) morphism of schemes having degree
<span class="math-container">$n$</span>. Since <span class="math-container">$X$</span> is Cohen-Macaulay, <span class="math-container">$\pi_*\mathcal{O}_X$</span> is a free
<span class="math-container">$\mathcal{O}_{\mathbb{P}_k^1}$</span>-module of finite rank <span class="math-container">$n$</span> and hence
decomposes into a direct sum of Serre's twisted sheaves:
<span class="math-container">$$
\pi_*\mathcal{O}_X \cong \bigoplus_{i=1}^n \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_X|_i)
$$</span>
The integers <span class="math-container">$|\mathcal{O}_X|_1 \geq \ldots \geq |\mathcal{O}_X|_n$</span>
are uniquely determined by <span class="math-container">$\mathcal{O}_X$</span>. Now we have a similar
situation for the structure sheaf of any irreducible component <span class="math-container">$X_i$</span> of
<span class="math-container">$X$</span>: The closed immersion <span class="math-container">$j_i: X_i \to X$</span> is a finite morphism and
hence <span class="math-container">$\pi_i = \pi \circ j_i: X_i \to X \to \mathbb{P}_k^1$</span> is also
finite of degree <span class="math-container">$n_i < n$</span> if <span class="math-container">$X_i \neq X$</span>. Hence
<span class="math-container">$$
(\pi_i)_*\mathcal{O}_{X_i} \cong \pi_* ((j_i)_* \mathcal{O}_{X_i}) \cong
\pi_* (\mathcal{O}_X / \mathcal{I}_i) \cong \bigoplus_{j=1}^{n_i}
\mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_{X_i}|_j)
$$</span>
where <span class="math-container">$\mathcal{I}_{X_i}$</span> denotes the ideal sheaf cutting out <span class="math-container">$X_i$</span> in
<span class="math-container">$X$</span>. </p>
<p>Since <span class="math-container">$X$</span> is reduced, we have a finite morphism
<span class="math-container">$X' = \bigoplus_{i=1}^m X'_i \to X$</span> where <span class="math-container">$X'$</span> is the normalization of
<span class="math-container">$X$</span> (and <span class="math-container">$X_i'$</span> is the normalization of the component <span class="math-container">$X_i$</span>) and a
corresponding injective morphism
<span class="math-container">$\mathcal{O}_X \hookrightarrow \bigoplus_{i=1}^n
\mathcal{O}_{X_i}/\mathcal{I}_i$</span>
with finite index <span class="math-container">$\chi(\mathcal{S})$</span> where <span class="math-container">$\mathcal{S}$</span> makes the following sequence exact
<span class="math-container">$$
0 \to \mathcal{O}_X \hookrightarrow \bigoplus_{i=1}^n
\mathcal{O}_{X_i}/\mathcal{I}_i \to \mathcal{S} \to 0.
$$</span>
It is not hard to see that
<span class="math-container">$$
\pi_* \mathcal{O}_X \hookrightarrow \pi_*\left( \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i\right) \cong \bigoplus_{i=1}^{m} \bigoplus_{j=1}^{n_i}
\mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_{X_i}|_j).
$$</span></p>
<blockquote>
<p><strong>My question is:</strong> What are the relations between
<span class="math-container">$L_{X'} := (|\mathcal{O}_{X_i}|_j)_{i,j}$</span> and <span class="math-container">$L_X := (|\mathcal{O}_{X}|_\ell)_\ell$</span> both
arranged in descending order? To be more specific: Does
<span class="math-container">$$
L_{X'}[i] - L_X[i] \in O\left( \frac{\chi(\mathcal{S})}{n}\right)
$$</span>
hold, i.e. are the differences balanced. Does someone know a good
read for this kind of situations or any references at all? Any good idea is also welcome.</p>
</blockquote>
<p><em>What I do know so far:</em></p>
<ol>
<li><span class="math-container">$L_{X} \leq L_{X'}$</span>, that is for all <span class="math-container">$i=1,\ldots,n : L_{X}[i] \leq L_{X'}[i]$</span>,</li>
<li><span class="math-container">$\sum_{\ell=1}^n |\mathcal{O}_{X}|_\ell = \chi(\mathcal{O}_X) -n$</span>,</li>
<li><span class="math-container">$\sum_{i,j} |\mathcal{O}_{X_i}|_j = \sum_{i=1}^m
(\chi(\mathcal{O}_{X_i}) -n_i) = \chi(\mathcal{O}_X) +
\chi(\mathcal{S}) -n$</span></li>
<li>Combining 2. and 3. we have: <span class="math-container">$\sum_{i=1}^n L_{X'}[i] - L_X[i] = \chi(\mathcal{S})$</span>.</li>
</ol>
http://www.4124039.com/q/3095836Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?Taras Banakhhttp://www.4124039.com/users/615362018-09-01T04:40:34Z2019-08-18T03:17:51Z
<p><strong>Definition 1.</strong> A compactification <span class="math-container">$c\mathbb N$</span> of the discrete space <span class="math-container">$\mathbb N$</span> is called <em>soft</em> if for any disjoint sets <span class="math-container">$A,B\subset\mathbb N$</span> with <span class="math-container">$\bar A\cap\bar B\ne\emptyset$</span> there exists a homeomorphism <span class="math-container">$h:c\mathbb N\to c\mathbb N$</span> such that <span class="math-container">$h(x)=x$</span> for all <span class="math-container">$x\in c\mathbb N\setminus\mathbb N$</span> and the set <span class="math-container">$\{x\in A:h(x)\in B\}$</span> is infinite.</p>
<p><strong>Definition 2.</strong> A compact Hausdorff space <span class="math-container">$X$</span> is called <em>Parovichenko</em> (resp. <em>soft Parovichenko</em>) if <span class="math-container">$X$</span> is homeomorphic to the remainder <span class="math-container">$c\mathbb N\setminus\mathbb N$</span> of some (soft) compactification <span class="math-container">$c\mathbb N$</span> of <span class="math-container">$\mathbb N$</span>?</p>
<p><strong>Remark 1.</strong> By a classical Parovichenko Theorem, each compact Hausdorff space of weight <span class="math-container">$\le\aleph_1$</span> is Parovichenko. Hence, under CH a compact Hausdorff space is Parovichenko if and only if it has weight <span class="math-container">$\le\mathfrak c$</span>. By a result of Przymusinski, each perfectly normal compact space is Parovichenko. On the other hand, Bell constructed an consistent example of a first-countable compact Hausdorff space, which is not Parovichenko. More information and references on Parovichenko spaces can be found in <a href="https://fa.its.tudelft.nl/~hart/37/publications/the_papers/open_problems_on_betaN.pdf" rel="nofollow noreferrer">this survey of Hart and van Mill</a> (see <span class="math-container">$\S$</span>3.10), </p>
<blockquote>
<p><strong>Problem 1.</strong> Is each Parovichenko compact space soft Parovichenko?</p>
</blockquote>
<p><strong>Remark 2.</strong> The Stone-Cech compactification <span class="math-container">$\beta\mathbb N$</span> of <span class="math-container">$\mathbb N$</span> is soft, but there are <a href="http://www.4124039.com/questions/309458/is-each-compactification-of-mathbb-n-soft">simple examples</a> of compactifications which are not soft. A compactification <span class="math-container">$c\mathbb N$</span> of <span class="math-container">$\mathbb N$</span> is soft if for any disjoint sets <span class="math-container">$A,B\subset\mathbb N$</span> with <span class="math-container">$\bar A\cap\bar B\ne\emptyset$</span> there are sequences <span class="math-container">$\{a_n\}_{n\in\omega}\subset A$</span> and <span class="math-container">$\{b_n\}_{n\in\omega}\subset B$</span> that converge to the same point <span class="math-container">$x\in\bar A\cap\bar B$</span>. This implies that a compactification <span class="math-container">$c\mathbb N$</span> is soft if the space <span class="math-container">$c\mathbb N$</span> is Frechet-Urysohn or has sequential square. This also implies that <em>each first-countable Parovichenko space is soft Parovichenko</em> (more generally, <em>a Parovichenko space <span class="math-container">$X$</span> is soft Parovichenko if each point <span class="math-container">$x\in X$</span> has a neighborhood base of cardinality <span class="math-container">$<\mathfrak p$</span></em>).</p>
<blockquote>
<p><strong>Problem 2.</strong> Is each (Frechet-Urysohn) sequential Parovichenko space soft Parovichenko? </p>
</blockquote>
<p>The following concrete version of Problem 1 describes an example of a Parovichenko space for which we do not know if it is soft Parovichenko.</p>
<blockquote>
<p><strong>Problem 3.</strong> Let <span class="math-container">$X$</span> be a compact space that can be written as the union <span class="math-container">$X=A\cup B$</span> where <span class="math-container">$A$</span> is homeomorphic to <span class="math-container">$\beta\mathbb N\setminus\mathbb N$</span>, <span class="math-container">$B$</span> is homeomorphic to the Cantor cube <span class="math-container">$\{0,1\}^\omega$</span> and <span class="math-container">$A\cap B\ne\emptyset$</span>. Is the space <span class="math-container">$X$</span> soft Parovichenko?</p>
</blockquote>
http://www.4124039.com/q/11791015Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?Adam Epsteinhttp://www.4124039.com/users/158192013-01-02T22:15:45Z2019-08-17T21:34:16Z
<p>The customary formulation of the Axiom of Infinity within Zermelo-Fraenkel set theory asserts the existence of an inductive set: a set $ I$ with $\varnothing\in I$ such that $x\in I$ implies $x\cup\{x\}\in I$. Since the intersection of any nonempty set of inductive sets is itself inductive, an instance of the Axiom Schema of Separation implies the existence of a smallest inductive set, namely the set of <em>von Neumann naturals</em> $$\mathbb{N}_{\bf vN} = \{\varnothing, \{\varnothing\},\{\varnothing,\{\varnothing\}\},\ldots\}.$$</p>
<p>Any inductive set is infinite (in fact, Dedekind infinite) but this formulation of the axiom asserts more, namely the existence of a specific countably infinite set. Given one such set, the existence of others, for example the set of <em>Zermelo naturals</em> $$\mathbb{N}_{\bf Zer}=\{\varnothing,\{\varnothing\},\{\{\varnothing\}\},\ldots\}$$ follows from appropriate instances of the Axiom Schema of Replacement. </p>
<p>Consider the subsystem of Zermelo-Fraenkel set theory with axioms Extensionality, Separation Schema, Union, Power Set, Pair. Augment this Basic System with an Axiom of Infinity which asserts the existence of an infinite set, but not any particular one. Such a formulation requires that the notion of 'finite' be defined prior to that of 'natural number', following Kuratowski for example. Any infinite set $ I$ determines a Dedekind-infinite set of <em>local naturals</em> $$\mathbb{N}_I=\{\mbox{equinumerosity classes of finite subsets of } I\}$$ which (duly equipped with initial element and successorship) yields a Lawvere natural number object, as in the Recursion Theorem. The existence of $\mathbb{N}_{\bf vN}$ and $\mathbb{N}_{\bf Zer}$ then follow from appropriate instances of Replacement.</p>
<p>One might wonder if there is some clever way to specify an infinite set without recourse to Replacement. That is, does there exist (in the language of set theory) a formula $\boldsymbol \phi$
with one free variable $x$ such that<br>
$$ \mbox{Basic+Infinity+Foundation } \vdash\; \exists y ( \forall x (x\in y \leftrightarrow \boldsymbol \phi)\,\wedge \, y \mbox{ is infinite})\>?$$</p>
<p>I'm inclined to guess no, on the following circumstantial grounds:</p>
<ul>
<li><p>For $\mathbb{N}_{\bf vN}$ and $\mathbb{N}_{\bf Zer}$ the use of Replacement is essential: Mathias has shown (Theorem 5.6 of <em>Slim Models of Zermelo Set Theory</em> that there exist transitive models ${\mathfrak M}_{\bf vN}$ and ${\mathfrak M}_{\bf Zer}$ of Basic+Infinity+Foundation with ${\mathbb N}_{\bf vN}\in {\mathfrak M}_{\bf vN}$ and ${\mathbb N}_{\bf Zer}\in{ \mathfrak M}_{\bf Zer}$, but such that every element of ${\mathfrak M}_{\bf vN}\cap {\mathfrak M}_{\bf Zer}$ is hereditarily finite.</p></li>
<li><p>The usual definitions of $\mathbb{N}_{\bf vN}$ and $\mathbb{N}_{\bf Zer}$ involve unstratified formulas. Coret has shown (Corollary 9 of <em>Sur les cas stratifiés du schéma du replacement</em>) that this is unavoidable:
$$ \mbox{Basic+Infinity } \vdash\; \forall y ( \forall x (x\in y \leftrightarrow \boldsymbol \phi)\,\rightarrow \, y \mbox{ is hereditarily finite})$$
for any stratified $\boldsymbol \phi$. Using the same technique he has shown (Corollary 10) that Basic+Infinity proves every stratified instance of Replacement.</p></li>
</ul>
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