Recent Questions - MathOverflowmost recent 30 from www.2874565.com2019-05-30T23:58:51Zhttp://www.2874565.com/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://www.2874565.com/q/3329100About the Zassenhaus's filtration of a group GMateus Figueiredohttp://www.2874565.com/users/1412902019-05-30T23:45:14Z2019-05-30T23:45:14Z
<p>The <span class="math-container">$n$</span>-th term of the filtration of Zassenhaus of a group <span class="math-container">$G$</span>, denoted by <span class="math-container">$D_n(G)$</span>, is the subgroup generated by all <span class="math-container">$p^k$</span>-th power of an element <span class="math-container">$x\in \gamma_i(G)$</span> such that <span class="math-container">$ip^k$</span> is greater or equal <span class="math-container">$n$</span>. In a text by Zelmanov, i saw that we can get the subgroup <span class="math-container">$D_n(G)$</span> using the same powers but only of the simple commutators in <span class="math-container">$G$</span>. </p>
<p>Can anyone explain me how to get this or tell me a reference (book or article) that contains the proof of this statement?</p>
http://www.2874565.com/q/3329071Non-standard tensor products of inner product spacesPierre Duboishttp://www.2874565.com/users/1266062019-05-30T22:24:48Z2019-05-30T22:24:48Z
<p>For two inner product spaces <span class="math-container">$(\mathcal{V}, (\cdot,\cdot)_V)$</span> and <span class="math-container">$(\mathcal{W}, (\cdot,\cdot)_W)$</span>, we can put an inner product on their tensor product in the obvious way:
<span class="math-container">$$
(1) ~~~~ \langle v \otimes w, v' \otimes w'\rangle := \langle v,v'\rangle_V \langle w,w'\rangle_W.
$$</span>
This then implies that
<span class="math-container">$$
(2) ~~~~ \|v \otimes w\| = \|v\|_V \|w\|_W.
$$</span>
Is there an example of an inner product on <span class="math-container">$\mathcal{V} \otimes \mathcal{W}$</span> such that (2) holds, but the inner product is not of the form (1)? Are such things of interest? Do they have a name?</p>
http://www.2874565.com/q/3329064Tricks for getting a creative ideauser7280899http://www.2874565.com/users/1035982019-05-30T22:05:44Z2019-05-30T23:15:40Z
<p>Caveat: I fear that people will criticize me for asking this potentially inappropriate question here, but I guess that the community here is quite unique in the ability of potentially answering my question (if you don't have an answer, then probably there is no), and that there is some (little) chance of getting some good answers to the question - and not asking this question here or banning it right away would reduce the chance of getting a good answer to 0.</p>
<p>The question is: If one tries to prove something, are there some tricks for getting a creative idea? Ok, I now what you think, yes, there is no algorithm to finding a creative idea, otherwise the idea wouldn't be <em>creative</em>. However, there are some general "tricks": if for minutes one stares at ones sheet of papers with no new ideas, just moving in the same thought cycles, it certainly helps to go and <em>talk</em> to a colleague, because somehow talking awakes the creative ability of the brain (and, additionally, together with a colleague one can mutually pick up an idea of the other and think it a bit further). Also, forgetting the problem for a moment and go and attend talks (even if they are about another topic) or even just rest helps. Do you have any other general "tricks" for getting creative ideas for solving mathematical problems?</p>
<p>To make the question a bit more concrete, do you know of any tricks for finding or looking for a good lemma (or several lemmas)? I have the feeling that often the most creativity in proving a theorem lies in finding the right lemma (not even the proof of it, but just the statement). I noticed that whenever I see a proof about which I afterwards say "wow, that's genius, I don't even rudimentally see how one could have come up with it", the crucial point was a lemma (or several lemmas). This also seems to me to be one difference between doing research and doing like homework problems: in homework assignments the proofs usually require only one or two main ideas, and if it requires a lemma, this lemma often is stated in the task as a subtask - while in research, one doesn't even know how much one has to "go down", how many levels of lemmas one has to show.</p>
http://www.2874565.com/q/3329052When is Fun(X,C) comonadic over C with respect to the colimit functor?Jonathan Beardsleyhttp://www.2874565.com/users/115462019-05-30T22:00:48Z2019-05-30T22:00:48Z
<p>Because I'm primarily interested in this question from the point of view of <span class="math-container">$\infty$</span>-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular, I'll just say "category" instead of <span class="math-container">$\infty$</span>-category or quasicategory, to keep things legible. I suspect that, up to adding in the word "homotopy," a lot of what I'm going to say holds very generally.</p>
<p>It's not hard to work out that if <span class="math-container">$Top$</span> is the category of <span class="math-container">$\infty$</span>-groupoids (or anima, according to Peter Scholze), then there is an equivalence between comodules over any <span class="math-container">$X\in Top$</span>, with respect to the Cartesian monoidal structure, and the slice category <span class="math-container">$Top/X$</span>. I'll write <span class="math-container">$Comod_X(Top)\simeq Top/X$</span>. Furthermore, by Lurie's straightening/unstraightening (a.k.a. the <span class="math-container">$\infty$</span>-categorical Grothendieck construction), we have an equivalence <span class="math-container">$Top/X\simeq Fun(X,Top)$</span>. The first category, <span class="math-container">$Comod_X(Top)$</span>, is clearly comonadic over <span class="math-container">$Top$</span> via the forgetful functor <span class="math-container">$U\colon Comod_X(Top)→Top$</span>. Moreover, the colimit functor <span class="math-container">$Fun(X,Top)→Top$</span> factors through <span class="math-container">$Comod_X(Top)$</span> and is the left adjoint of an adjoint equivalence realizing the composite equivalence <span class="math-container">$Comod_X(Top)?Fun(X,Top)$</span>. Thus, <span class="math-container">$Fun(X,Top)$</span> is comonadic over <span class="math-container">$Top$</span> via the colimit functor <span class="math-container">$colim\colon Fun(X,Top)→Top$</span>. </p>
<p>My question is whether or not this is more generally true, perhaps for some generic abstract reasons that I'm not aware or, or not seeing. Note that it is <em>not</em> in general true (I don't think) that <span class="math-container">$colim:Fun(D,C)→C$</span> is comonadic for arbitrary categories <span class="math-container">$D$</span> and <span class="math-container">$C$</span>. However, I'm happy with assuming that <span class="math-container">$D$</span> is an <span class="math-container">$∞$</span>-groupoid and that <span class="math-container">$C$</span> is "nice," i.e. at least (locally) presentable. </p>
<p>So really the question is, are there check-able conditions under which the colimit functor <span class="math-container">$colim:Fun(X,C)→C$</span> is comonadic, under the assumptions that <span class="math-container">$X$</span> is an <span class="math-container">$∞$</span>-groupoid and <span class="math-container">$C$</span> is presentable? </p>
http://www.2874565.com/q/3329040Sub-Gaussian decay of the measure of Euclidean ballsnevereverneverhttp://www.2874565.com/users/1230752019-05-30T21:22:40Z2019-05-30T21:56:05Z
<p>Let <span class="math-container">$X$</span> be a random vector in <span class="math-container">$\mathbb{R}^d$</span> satisfying the following property: there exists <span class="math-container">$C_1,C_2>0$</span> such that
<span class="math-container">$$\int_0^{+\infty}\mathbb{P}(\|X-\mu_0\|\leq\sqrt{t})\exp(-t)dt\leq C_1\exp(-C_2\|\mu_0\|^2)$$</span>
for any <span class="math-container">$\mu_0\in\mathbb{R}^d$</span>. Here <span class="math-container">$\|\|$</span> is the Euclidean norm in <span class="math-container">$\mathbb{R}^d$</span>.
If the above property holds, is the following statement true: there exists a sequence of vectors <span class="math-container">$\mu_n$</span> in <span class="math-container">$\mathbb{R}^d$</span> and a sequence of real numbers <span class="math-container">$t_n\to+\infty$</span> (<span class="math-container">$t_n$</span> may depend on <span class="math-container">$\mu_n$</span> for example <span class="math-container">$t_n=\|\mu_n\|^2/4$</span>) such that:
<span class="math-container">$$\lim_{n\to+\infty}\frac{\mathbb{P}(\|X-\mu_n\|\leq1)}{\mathbb{P}(\|X-\mu_n\|\leq \sqrt{t_n})\exp(-t_n)}=0$$</span></p>
<p>If this is not true, is there a counter example?</p>
http://www.2874565.com/q/332903-1How can I solve this issue?Dev.Hranthttp://www.2874565.com/users/1412842019-05-30T21:22:15Z2019-05-30T21:22:15Z
<p>I have a problem related to this theme system analysis and action research. If you can please solve this, in another case please take me more resource for understanding this.</p>
<p>The system consists of N-quantity of sequential-connected elements. To each of those elements, there are also simultaneously connected similar-typed, mutually replaceable elements. There is an assumption, that we know the <span class="math-container">$P_j$</span> probability of uninterrupted work for each of the each j-numbered elements. Thus, if we connect mj similar-typed mutual-replaceable elements to the <span class="math-container">$j$</span>-numbered element, the probability of uninterrupted work of <span class="math-container">$j$</span>-numbered subsystem will be</p>
<p><span class="math-container">$r_j(m_j) = 1 - (1-P_j)$</span>^<span class="math-container">$(1+m_j)$</span>, </p>
<p><span class="math-container">$j = \vec{1,N}$</span>, <span class="math-container">$m_j$</span> = 0,1,2…
Assuming that the weight of the <span class="math-container">$j$</span>-numbered elements is <span class="math-container">$W_j$</span>, and the value <span class="math-container">$C_j$</span>, it is required to count the amount of <span class="math-container">$j = \vec{1,N}$</span> mutual-replaceable elements in a way so that the weight of system does not exceed <span class="math-container">$W$</span>, and the value - does not exceed the point of <span class="math-container">$C$</span>.</p>
<p><strong>See</strong></p>
<p><a href="https://i.stack.imgur.com/36DiQ.jpg" rel="nofollow noreferrer">DATA</a></p>
http://www.2874565.com/q/3329013Greatest prime factor of n and n+1Dmitry Krachunhttp://www.2874565.com/users/1287412019-05-30T21:09:02Z2019-05-30T21:09:02Z
<p>For a positive integer <span class="math-container">$n$</span> we denote its largest prime factor by <span class="math-container">$\operatorname{gpf}(n)$</span>. Let's call a pair of distinct primes <span class="math-container">$(p,q)$</span> <span class="math-container">$\textbf{nice}$</span> if there are no natural numbers <span class="math-container">$n$</span> such that <span class="math-container">$\operatorname{gpf}(n)=p, \operatorname{gpf}(n+1)=q$</span> or <span class="math-container">$\operatorname{gpf}(n)=q, \operatorname{gpf}(n+1)=p$</span>. For example, <span class="math-container">$(2,19)$</span> is nice. </p>
<p>Are there nice pairs <span class="math-container">$(p,q)$</span> with <span class="math-container">$p,q>100$</span>?</p>
http://www.2874565.com/q/332897-6HTTP GET request in math [on hold]someonehttp://www.2874565.com/users/1412862019-05-30T20:32:20Z2019-05-30T20:43:59Z
<p>How would I go about making an HTTP GET request? I've tried looking at Curl for math, but it just doesn't provide enough information. Can I do something like this?</p>
<p><span class="math-container">$d = g([S,O,M,E,U,R,L])$</span></p>
<p>Thanks.</p>
http://www.2874565.com/q/3328960The cobordism hypothesis, and the bordism n-category as a free constructionJamie Vicaryhttp://www.2874565.com/users/7992019-05-30T20:30:47Z2019-05-30T20:43:26Z
<p>In this question, I write <span class="math-container">$\text{Bord}_n$</span> for the symmetric monoidal <span class="math-container">$n$</span>-category of framed n-bordisms, and <span class="math-container">$\text{D}_n$</span> for the free symmetric monoidal <span class="math-container">$n$</span>-category with all duals on the terminal category.</p>
<p>The Cobordism Hypothesis is often stated informally as follows:</p>
<blockquote>
<p>STATEMENT 1: <span class="math-container">$\text{Bord}_n$</span> is equivalent to <span class="math-container">$\text{D}_n$</span>.</p>
</blockquote>
<p>However, this seems to be a poor match for the formal statement, which is more like this:</p>
<blockquote>
<p>STATEMENT 2: for all symmetric monoidal <span class="math-container">$n$</span>-categories <span class="math-container">$\mathcal{C}$</span>, the <span class="math-container">$n$</span>-groupoid <span class="math-container">$\text{Bord}_n \to \mathcal{C}$</span> is equivalent to the <span class="math-container">$n$</span>-groupoid <span class="math-container">$\text{D}_n \to \mathcal{C}$</span>.</p>
</blockquote>
<p>This apparent inconsistency can be seen, for example, in <a href="https://ncatlab.org/nlab/show/cobordism+hypothesis" rel="nofollow noreferrer">the <span class="math-container">$n$</span>Lab page on the cobordism hypothesis</a>. The first statement is the one that seems intended from early work, such as <a href="https://arxiv.org/abs/q-alg/9503002" rel="nofollow noreferrer">Baez and Dolan's original paper</a>. However, it is only the second statement that I am reasonably comfortable with, in the sense of understanding how it works in some simple cases (such as the 2d case analyzed in detail in <a href="https://arxiv.org/abs/1411.6691" rel="nofollow noreferrer">the thesis of Piotr Pstragowski</a>; see in particular Theorem 3.17.)</p>
<p>My question, then, is the following. <strong>Is there some sense in which Statement 1 is true?</strong> For example, this could involve an appropriate technical definition of "free".</p>
<p>As an aside, I find it uncomfortable that there is even a gap between these notions. It feels intuitively like some sort of Yoneda argument should be sufficient to show that Statement 2 implies Statement 1. If anybody has some insight on that, it could give some useful perspective.</p>
http://www.2874565.com/q/3328950Holomorphic maps from upper half plane to itself (or equivalently Poincare disc to itself)nGlacTOwnShttp://www.2874565.com/users/1111382019-05-30T20:19:53Z2019-05-30T20:19:53Z
<p>Suppose I parametrize complex plane by coordinates,<span class="math-container">$$z = x+i y,\ \bar z=x-i y$$</span>
then the upper half plane, <span class="math-container">$\mathbb H_+$</span> is given by <span class="math-container">$y>0$</span>. I am looking for chiral coordinate transformations, <span class="math-container">$f(z)$</span>, such that </p>
<ol>
<li>they map the boundary (<span class="math-container">$y=0$</span>) to itself,<span class="math-container">$$f(z)=\bar f(\bar z) \ \text{whenever } z=\bar z~.$$</span></li>
<li><span class="math-container">$z+\bar z >0 \Leftrightarrow f(z)+\bar f(\bar z)>0~.$</span></li>
</ol>
<p>Since there is a conformal map between <span class="math-container">$\mathbb H_+$</span> and unit disc (Poincare disc), <span class="math-container">$\mathbb D = \{w:|w|<1\}$</span>, where the above conditions become:</p>
<ol>
<li>the coordinate transformations map the boundary of <span class="math-container">$\mathbb D$</span> (<span class="math-container">$|w|=1$</span>) to itself,<span class="math-container">$$g(w) \bar g(\bar w)=1 \ \text{whenever } w\bar w=1~.$$</span></li>
<li><span class="math-container">$w \bar w<1 \Leftrightarrow g(w)\bar g(\bar w)<1~.$</span></li>
</ol>
<p>The <a href="https://en.wikipedia.org/wiki/Schwarz_lemma" rel="nofollow noreferrer">Schwarz-Pick Lemma</a> seems to suggest that a general holomorphic transformation brings the boundary of the disc closer than <span class="math-container">$1$</span> in the Poincaré metric (I am interested in AdS<span class="math-container">$_2$</span> so I can equivalently say that the <em>new</em> boundary after the coordinate transformation is at a finite distance from any interior point),<span class="math-container">$$\frac{g'(w)\bar g'(\bar w)}{\left(1-g(w)\bar g(\bar w)\right)^2}dw d\bar w \le \frac1{(1-w\bar w)^2}dw d\bar w$$</span> and the equality folds only for Mobius transformations (which can be seen as isometries of AdS<span class="math-container">$_2$</span>).</p>
<ul>
<li>Is it correct to deduce that there are no (non-trivial, of course not the Mobius transformation) holomorphic transformations that satisfy the conditions 1 and 2 above, or am I interpreting the Schwarz-Pick lemma incorrectly?</li>
<li>If my interpretation of Schwarz-Pick lemma is correct and there are no <em>holomorphic</em> maps with the given constraints, then what are the less restrictive class of functions that obey the above constraints?</li>
</ul>
<p><strong>PS</strong>: I have also asked the same question in Physics.SE <a href="https://physics.stackexchange.com/q/483215/12225">here</a>. I hope this doesn't violate any rules.</p>
http://www.2874565.com/q/3328932Projection of an invariant almost complex structure to a non-integrable oneAli Taghavihttp://www.2874565.com/users/366882019-05-30T19:55:03Z2019-05-30T21:24:30Z
<p>My apologies in advance if my question is obvious or elementary.</p>
<p>We identify elements of <span class="math-container">$S^3$</span> with their quaternion representation <span class="math-container">$x_1 + x_2i + x_3j + x_4k$</span>. We consider two independent vector fields <span class="math-container">$S_1(a) = ja$</span> and <span class="math-container">$S_2(a) = ka$</span> on <span class="math-container">$S^3$</span>. On the other hand <span class="math-container">$P: S^3\to S^2$</span> is a <span class="math-container">$S^1$</span>-principal bundle with the obvious action of <span class="math-container">$S^1$</span> on <span class="math-container">$S^3$</span>. Then the span of <span class="math-container">$S_1, S_2$</span> is the standard horizontal space associated to the standard connection of the principal bundle <span class="math-container">$S^3 \to S^2$</span>. Then each horizontal space has an almost complex structure <span class="math-container">$J$</span>. This is the standard structure associated to <span class="math-container">$S_1, S_2$</span> coordinates which is defined by <span class="math-container">$J(S_1) = S_2, J(S_2) = -S_1$</span>.</p>
<blockquote>
<p>Is this structure invariant under the action of <span class="math-container">$S^1$</span>? If yes, we can define a unique almost complex structure on <span class="math-container">$S^2$</span> which is <span class="math-container">$P$</span> related to the structure on the total space. Now is this structure on <span class="math-container">$S^2$</span> integrable?</p>
<p>As a similar question, is there an example of a principal bundle <span class="math-container">$P\to X$</span>, with <span class="math-container">$P$</span> a real manifold, <span class="math-container">$X$</span> a complex manifold, and a connection which admits an invariant almost complex structure projecting to a non-integrable structure?</p>
</blockquote>
http://www.2874565.com/q/3328921Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain?wellfedgremlinhttp://www.2874565.com/users/949592019-05-30T19:46:54Z2019-05-30T19:46:54Z
<p>A Fatou-Bieberbach domain <span class="math-container">$\Omega \subseteq \mathbb{C}^n$</span> is a domain that is a proper subset of <span class="math-container">$\mathbb{C}^n$</span> and is biholomorphic to <span class="math-container">$\mathbb{C}^n$</span>. A domain is said to be Runge if for each holomorphic function defined on it and each compact set in it, there exists a sequence of polynomials that converges uniformly to the holomorphic function on the compact set. Here is my question:</p>
<blockquote>
<p>Given a Fatou-Bieberbach domain <span class="math-container">$\Omega$</span>, does there exists a Runge Fatou-Bieberbach domain <span class="math-container">$\Omega^\prime$</span> such that <span class="math-container">$\Omega^\prime \subseteq \Omega$</span>?</p>
</blockquote>
<p>It was shown by Wold in 2007 that non-Runge Fatou-Bieberbach domains exists.</p>
http://www.2874565.com/q/3328872Snaith splitting for operads in spectra?Tim Campionhttp://www.2874565.com/users/23622019-05-30T19:01:42Z2019-05-30T23:46:29Z
<p>Let <span class="math-container">$O$</span> be an augmented operad. Then there is a functor <span class="math-container">$J^O: Top_\ast \to Top_\ast$</span> which takes <span class="math-container">$X$</span> to the free <span class="math-container">$O$</span>-algebra on <span class="math-container">$X$</span> subject to the condition that the nullary operation coincides with the basepoint. The Snaith splitting tells us that <span class="math-container">$\Sigma^\infty J^O(X) = \Sigma^\infty \vee_{n \geq 1} O(n)_+ \wedge_{\Sigma_n} X^{\wedge n}$</span>. Is there a version of this where <span class="math-container">$X$</span> is a pointed spectrum rather than a pointed space?</p>
<p>That is, let <span class="math-container">$\mathbb S \to X$</span> be a spectrum equipped with a map from the sphere spectrum. If <span class="math-container">$O$</span> is an augmented operad, there should be an <span class="math-container">$O$</span>-algebra <span class="math-container">$J^O(X)$</span> such that the data of a map of <span class="math-container">$O$</span>-algebra spectra <span class="math-container">$J^O(X) \to A$</span> is equivalent to a map of spectra <span class="math-container">$X \to A$</span> commuting with the maps from <span class="math-container">$\mathbb S$</span>.</p>
<p><strong>Question:</strong> Is there an easy formula for <span class="math-container">$J^O(X)$</span> when <span class="math-container">$X$</span> is a spectrum equipped with a map <span class="math-container">$\mathbb S \to X$</span>? Can one at least say when <span class="math-container">$J^O(X)$</span> is nonzero?</p>
<p>Apparently such a formula will have to be a little more complicated than when <span class="math-container">$X$</span> is a space -- in particular, the dependence on the choice of map <span class="math-container">$\mathbb S \to X$</span> will be more subtle. For instance, if <span class="math-container">$O$</span> is an <span class="math-container">$E_n$</span> operad and <span class="math-container">$\mathbb S \to X$</span> is the zero map (or more generally smash-nilpotent), then <span class="math-container">$J^O(X) = 0$</span>, rather than some large sum of wedges of <span class="math-container">$X$</span>.</p>
<p>I think there's at least a colimit formula analogous to the James construction (which is, after all, the case <span class="math-container">$O = E_1$</span>). For instance, when <span class="math-container">$O = E_\infty$</span>, it should be the case that <span class="math-container">$J^O(X)$</span> is a certain colimit indexed over the category of finite sets and injections sending <span class="math-container">$n \mapsto X^{\wedge n}$</span>. However, what I'd really like is a formula which (like the formula of the Snaith splitting) allows one to easily see that <span class="math-container">$J^O(X) \neq 0$</span> in reasonable cases, and so far this colimit formula has been a little too complicated for me to say this.</p>
http://www.2874565.com/q/3328855Number rings with elements of norm $-1$Tinahttp://www.2874565.com/users/1412822019-05-30T18:33:33Z2019-05-30T18:33:33Z
<p>Is there a nice characterization of number rings <span class="math-container">$\mathcal{O}$</span> such that there exists an element <span class="math-container">$x \in \mathcal{O}$</span> whose norm is <span class="math-container">$-1$</span>?</p>
<p>One obvious necessary condition is that <span class="math-container">$\mathcal{O}$</span> must have a real embedding. For real quadratic number rings, norm <span class="math-container">$-1$</span> elements sometimes exist and sometimes don't. For instance:</p>
<p><strong>Example</strong>: In <span class="math-container">$\mathbb{Z}[\sqrt{2}]$</span>, the norm of <span class="math-container">$1+\sqrt{2}$</span> is <span class="math-container">$-1$</span>.</p>
<p><strong>Example</strong>: In <span class="math-container">$\mathbb{Z}[\sqrt{3}]$</span>, there is no element of norm <span class="math-container">$-1$</span>. Indeed, the norm of <span class="math-container">$a+b\sqrt{3}$</span> is <span class="math-container">$a^2-3b^2$</span>. If this were <span class="math-container">$-1$</span>, then we would have that <span class="math-container">$a^2 \equiv -1$</span> mod <span class="math-container">$3$</span>, which is impossible.</p>
http://www.2874565.com/q/3328811Composition of monads induces tensor product in the category of modulesBen Sprotthttp://www.2874565.com/users/100072019-05-30T18:14:26Z2019-05-30T18:40:51Z
<p>I have recently <a href="http://www.2874565.com/questions/332874/quantum-scattering-experiments-c-modules-n-modules-and-their-monads">asked a question</a> about the composition of two monads, namely <span class="math-container">$\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$</span>. I am conjecturing that the cateogory of <span class="math-container">$\mathbb{C}$</span>-Modules and the category of <span class="math-container">$\mathbb{N}$</span>-Modules also live in a category of categories, namely a category of categories of modules, <span class="math-container">$\mathcal{C}_{Mod}$</span>. Since the monads compose, I am guessing that the modules themselves compose in the category <span class="math-container">$\mathcal{C}_{Mod}$</span>. We should, then, expect some kind of composition in this category. A first guess is that of a tensor product, so my first question is just whether this is true: monad composition induces tensor product. What is the standard name for the inducement where a composition of monads induces a tensor product of modules?</p>
<p>Edit:
I think this is a monoidal functor between the category of modules and the category Mon_set, of monads on SET.</p>
http://www.2874565.com/q/3328792Homotopy type of transversal families of submanifolds through deformationBrianThttp://www.2874565.com/users/1245922019-05-30T17:39:05Z2019-05-30T21:36:14Z
<p>Let <span class="math-container">$A,B \subset M$</span> be two transversal submanifolds of a compact manifold <span class="math-container">$M$</span>. It seems rather intuitive that if <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are deformed (say smoothly) in a way that they remain transversal to each other, the homotopy type of the intersection remains unchanged during the deformation. </p>
<p>I would like to know if there is a way to write such a homotopy in the following situations:</p>
<ol>
<li><span class="math-container">$A = f^{-1}(0)$</span> is the zero level set of some smooth function <span class="math-container">$f : M \to \mathbb{R}$</span>, and <span class="math-container">$B = B_s$</span> is a smooth family of submanifolds such that <span class="math-container">$A$</span> and <span class="math-container">$B_s$</span> are always transverse. Is it true that the homotopy type of <span class="math-container">$A \cap B_s$</span> is independent of <span class="math-container">$s$</span> ?</li>
<li><span class="math-container">$A_s := f_s^{-1}(0)$</span> is the zero set of a smooth function <span class="math-container">$f_s : M \to \mathbb{R}$</span>, and the family <span class="math-container">$A_s$</span> is always transversal to a fixed submanifold <span class="math-container">$B$</span>. Is it true that the homotopy type of <span class="math-container">$A_s \cap B$</span> is independent of <span class="math-container">$s$</span> ?</li>
</ol>
<p>Any help will be appreciated. Thanks in advance.</p>
http://www.2874565.com/q/3328743Quantum Scattering Experiments: C-Modules, N-Modules and their monadsBen Sprotthttp://www.2874565.com/users/100072019-05-30T16:26:12Z2019-05-30T23:05:42Z
<p>I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check.</p>
<ol>
<li>The cateogory of <span class="math-container">$\mathbb{C}$</span>-Modules is monadic over set</li>
<li>The category of <span class="math-container">$\mathbb{N}$</span>-Modules is monadic over Set</li>
<li>Since the monad, <span class="math-container">$\mathcal{M}_C$</span>, which factors through the category of <span class="math-container">$\mathbb{C}$</span>-Modules is seated on Set and likewise for the monad, <span class="math-container">$\mathcal{M}_N$</span>, that factors through the category of <span class="math-container">$\mathbb{N}$</span>-Modules, these monads compose according to functor composition. Their composition, <span class="math-container">$\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$</span>, is also a monad on SET.</li>
</ol>
<p>The physical interpretation:
The monad, <span class="math-container">$\mathcal{M}_P$</span>, can be interpreted as complex combinations of collections of particle types. Thus, this monad can be used to represent quantum states for the input/output of a particle scattering experiment.</p>
<p>Edit:</p>
<p>Someone has pointed out a very important fact, which is that I have to specify a distributive law for the functor composition to become a monad. Unfortunately, I don't know how to define this. Perhaps the question can be phrased better as follows. Can we define a distributive law to make the composition into a monad? What is the distributive law? If there is a large freedom, perhaps we can select a law that captures some aspect of the physics we are trying to model.</p>
http://www.2874565.com/q/3328610How to compute means μ1 and μ2 knowing sum of Skellam distributions f(k;μ1,μ2) and sum μ1+μ2, where k is from 2 to n?Sergei Dvindenkohttp://www.2874565.com/users/1412752019-05-30T13:56:56Z2019-05-30T19:28:47Z
<p>The probability mass function for the <a href="https://en.wikipedia.org/wiki/Skellam_distribution" rel="nofollow noreferrer">Skellam distribution</a> for a count difference <span class="math-container">$ k=n_1-n_2 $</span> from two <a href="https://en.wikipedia.org/wiki/Poisson_distribution" rel="nofollow noreferrer">Poisson-distributed</a> variables with means <span class="math-container">$\mu_1$</span> and <span class="math-container">$\mu_1$</span> is given by:</p>
<p><span class="math-container">$$ f(k;\mu_1,\mu_2)= e^{-(\mu_1+\mu_2)}
\left({\mu_1\over\mu_2}\right)^{k/2}I_{|k|}(2\sqrt{\mu_1\mu_2}) $$</span>
where <span class="math-container">$I_k(z)$</span> is the modified <a href="https://en.wikipedia.org/wiki/Bessel_function" rel="nofollow noreferrer">Bessel function</a> of the first kind.</p>
<p>I know the sum of these functions for k from 2 to n
<span class="math-container">$$ \sum_{k=2}^n f(k;\mu_1,\mu_2) = a$$</span></p>
<p>I also know the sum <span class="math-container">$\mu_1 + \mu_2 = \mu$</span>. So i can substitute <span class="math-container">$\mu_1$</span> for <span class="math-container">$\mu - \mu_2. $</span>
So i have <span class="math-container">$$ a = \sum_{k=2}^n e^{-(\mu_1+\mu_2)}
\left({\mu_1\over\mu_2}\right)^{k/2}I_{|k|}(2\sqrt{\mu_1\mu_2}) = \sum_{k=2}^n e^{-(\mu)}
\left({\mu - \mu_2\over\mu_2}\right)^{k/2}I_{|k|}(2\sqrt{(\mu - \mu_2)\mu_2})$$</span> </p>
<p>My question is how can i compute <span class="math-container">$\mu_2$</span> from this equation if it's possible.</p>
http://www.2874565.com/q/33285212Is differential topology a dying field? [on hold]James Baxterhttp://www.2874565.com/users/1324462019-05-30T11:42:52Z2019-05-30T23:04:18Z
<p>I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field.</p>
<p>I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!</p>
http://www.2874565.com/q/3327964Hermitian sectional curvatureseubhttp://www.2874565.com/users/255902019-05-29T17:22:24Z2019-05-30T23:14:27Z
<p>Let <span class="math-container">$N$</span> be a Riemannian manifold, denote <span class="math-container">$R$</span> its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is <span class="math-container">$K(X,Y) = R(X,Y,X,Y)$</span> for an orthonormal pair.</p>
<p>Consider the complexified tangent space <span class="math-container">$TM \otimes \mathbb{C}$</span> and the complex-linear extension of <span class="math-container">$R$</span>, which we still denote <span class="math-container">$R$</span>. By definition, <span class="math-container">$N$</span> has nonpositive <em>Hermitian sectional curvature</em> if <span class="math-container">$R(X, Y, \bar{X}, \bar{Y}) \leqslant 0$</span> for all <span class="math-container">$X, Y \in TM \otimes \mathbb{C}$</span>.</p>
<p>Obviously, nonpositive Hermitian sectional curvature is stronger than nonpositive sectional curvature. </p>
<p><strong>QUESTION.</strong> Is nonpositive Hermitian curvature <em>strictly</em> stronger than nonpositive curvature? </p>
<p>In other words, are there examples of Riemannian manifolds with nonpositive sectional curvature, but not nonpositive Hermitian sectional curvature?</p>
<p>I expect the answer easily yes, in fact it is claimed in e.g. [1] or [8], but I couldn't find an example in the relevant literature, e.g. [1][2][3][4][5][6][7][8][9].</p>
<p>NB: Yau-Zheng [8] showed that the answer is no for manifolds with negative <span class="math-container">$\delta$</span>-pinched sectional curvature with <span class="math-container">$\delta \geqslant 1/4$</span>. According to [9, Theorem 9.26], the answer is no for Kähler surfaces. </p>
<p><span class="math-container">$$$$</span></p>
<hr>
<p><strong>FOLLOW UP QUESTIONS</strong></p>
<p>Following (almost) the terminology of Siu [6], a Riemannian manifold with nonpositive Hermitian sectional curvature has "strongly nonpositive curvature". He also introduces other notions of curvature such as "very strongly nonpositive" as follows. Consider the curvature operator
<span class="math-container">$$
\begin{aligned}
Q \colon \Lambda^2 TM \times \Lambda^2 TM \to \mathbb{R}
\end{aligned}
$$</span>
such that <span class="math-container">$Q$</span> is defined for decomposable tensors by <span class="math-container">$Q(X\wedge Y, Z \wedge W) = R(X , Y, Z , W)$</span>. We still denote <span class="math-container">$Q$</span> its complex-linear extension to complexified vectors. By definition, <em><span class="math-container">$N$</span> has very strongly nonpositive curvature</em> if <span class="math-container">$Q(\sigma, \bar{\sigma}) \leqslant 0$</span> for all tensors <span class="math-container">$\sigma \in \Lambda^2 TM \otimes \mathbb{C}$</span> (not just decomposable ones).</p>
<p><strong>Question 2.</strong> Is there an example showing that very strongly nonpositive curvature is strictly stronger than strongly nonpositive curvature? </p>
<p><strong>Question 3.</strong> What about the condition that <span class="math-container">$Q(\sigma, \sigma) \leqslant 0$</span> for all <span class="math-container">$\sigma \in \Lambda^2 TM$</span> (no complexification)? Is it stronger than nonpositive curvature?</p>
<p>Finally, just to be thorough, there is a notion of (very) strongly negative curvature, but it's not simply something like <span class="math-container">$Q(\sigma, \bar{\sigma}) < 0$</span> for all nonzero <span class="math-container">$\sigma$</span>, that is too much to ask. Assume now that <span class="math-container">$N$</span> is a Kähler manifold. Then <span class="math-container">$Q(\sigma, \bar{\sigma}) = 0$</span> for any <span class="math-container">$\sigma$</span> of type <span class="math-container">$(2,0)$</span> or <span class="math-container">$(0,2)$</span>, e.g. <span class="math-container">$X \wedge Y$</span> with <span class="math-container">$X, Y \in T^{1,0} M$</span>. By definition, <span class="math-container">$N$</span> has very strongly negative curvature if <span class="math-container">$Q(\sigma, \bar{\sigma}) < 0$</span> for all nonzero tensors <span class="math-container">$\sigma$</span> of type <span class="math-container">$(1,1)$</span>, and <span class="math-container">$N$</span> has strongly negative curvature if <span class="math-container">$Q(\sigma, \bar{\sigma}) < 0$</span> for all nonzero tensors <span class="math-container">$\sigma$</span> of type <span class="math-container">$(1,1)$</span> and of length <span class="math-container">$\leqslant 2$</span>, e.g. <span class="math-container">$\sigma = X \wedge \bar{Y} + Z \wedge{\bar{W}}$</span>.</p>
<p>It is clear that
<span class="math-container">$$\text{very strongly negative} ~\Rightarrow~ \text{strongly negative} ~\Rightarrow~ \text{negative sectional curvature}$$</span></p>
<p><strong>Question 4.</strong> Are there examples proving that the converse implications are false?</p>
<p>Again, according to [9, Theorem 9.26], the answer is no for Kähler surfaces.</p>
<p>Remark: Of course, there are similar notions of (very) strong nonnegative / positive curvature and one could ask the same questions.</p>
<p><span class="math-container">$$$$</span></p>
<hr>
<p>[1] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fundamental groups of compact Kähler manifolds. 1996.</p>
<p>[2] Eells and Lemaire. Two reports on harmonic maps. 1995</p>
<p>[3] Jost and Yau. Harmonic mappings and Kähler manifolds. 1983.</p>
<p>[4] Mostow and Siu. A compact Kähler surface of negative curvature not coveredby the ball. 1980.</p>
<p>[5] Ohnita and Udagawa. Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds. 1990.</p>
<p>[6] Siu. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. 1980.</p>
<p>[7] Xin. Geometry of harmonic maps. 1996</p>
<p>[8] Yau and Zheng. Negatively <span class="math-container">$\frac14$</span>-pinched Riemannian metric on a compact Kähler manifold.</p>
<p>[9] F. Zheng, Complex differential geometry, 2000.</p>
http://www.2874565.com/q/3327645Italian-style algebraic geometry in homotopy theory?user141225http://www.2874565.com/users/02019-05-29T12:24:53Z2019-05-30T20:08:56Z
<p>In homotopy theory, stacks <a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture19.pdf" rel="noreferrer">can be</a> occasionally useful (i.e. in the chromatic story). I come from a differential geometry background, so when people say that algebraic geometry is useful in homotopy theory, I have mixed feelings (like the stack classifying formal groups of height <span class="math-container">$n$</span> is basically a quotient of a single point, not a particularly geometric object from some perspectives).</p>
<p>Has Italian-style algebraic geometry ever shed light on homotopy theory? A caricaturistic (and probably wrong) example of how an an answer should look like: "27 lines on a cubic surface are actually in bijection with this stable homotopy group of spheres!"</p>
<p>P.S. "Italian-style" can be understood in many ways; one possible interpretation is "the study of non-trivial facts about separated schemes of finite type over <span class="math-container">$\mathbb{C}$</span>."</p>
http://www.2874565.com/q/3327194Trace of inverse of random positive-definite matrix in high dimension?Goulifethttp://www.2874565.com/users/392612019-05-28T22:27:13Z2019-05-30T20:17:04Z
<p>Consider a random matrix <span class="math-container">$A \in \mathbb{R}^{n\times n}$</span> with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of <span class="math-container">$$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1})$$</span> when the size of the matrix <span class="math-container">$n$</span> goes to infinity. Here,
<span class="math-container">$\lambda > 0$</span> is fixed and ensures that <span class="math-container">$A^T A + \lambda \mathrm{Id}$</span> is invertible as a positive definite matrix.</p>
<p>Typically, I am wondering if this quantity behaves asymptotically like <span class="math-container">$n^{\gamma}$</span> for some <span class="math-container">$\gamma$</span>. </p>
http://www.2874565.com/q/3326510Stability analysis of a differential equationhwathttp://www.2874565.com/users/1265782019-05-28T02:54:19Z2019-05-30T19:56:42Z
<p>My question is about stability analysis and whether the equilibrium point is well-defined in example 2?. If they are not defined, how can one approach the stability analysis for those cases.</p>
<p>Example 1:</p>
<p><span class="math-container">$\frac{dx}{dt} = -x + c$</span></p>
<p>In this example, <span class="math-container">$c$</span> is a constant and <span class="math-container">$x=c$</span> is an asymptotically stable equilibrium point. If <span class="math-container">$x<c$</span>, <span class="math-container">$\frac{dx}{dt}$</span> will be positive, pushing <span class="math-container">$x$</span> toward <span class="math-container">$c$</span> and a similar statement holds for the negative case. In this case, the equilibrium point is well defined. </p>
<p>Example 2:
We have a slightly modified equation: </p>
<p><span class="math-container">$\frac{dx}{dt} = -x + c(1-\frac{1}{M}\exp(-2t))$</span></p>
<p>and M is a very large positive constant.
No matter where we start from, we'll always approach <span class="math-container">$c$</span>.
Can we say <span class="math-container">$x=c$</span> is also an equilibrium point in this case? The problem is when we plug this value in the equation <span class="math-container">$\frac{dx}{dt}$</span> will not be zero at any time. My understanding is for an equilibrium point, we must be able to plug in the state at the equilibrium point and observe that it doesn't change. Is my understanding correct and if yes, how can one approach the stability analysis in the second case.</p>
http://www.2874565.com/q/3326311Construction of elliptic equation with Neumann boundary condition from a minimization problemmnmn1993http://www.2874565.com/users/879222019-05-27T20:14:32Z2019-05-30T18:51:57Z
<p>My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.</p>
<p>Let <span class="math-container">$B=B_1 \subset \mathbb{R}^3$</span> and <span class="math-container">$E : H^1(B) \to \mathbb{R}$</span>
<span class="math-container">$$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$</span>
We assume that <span class="math-container">$u_0 \in W^{1,2}$</span> to be the minimizer of the functional <span class="math-container">$E$</span> in the configuration space
<span class="math-container">$$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$</span>
Since <span class="math-container">$u_0$</span> is the critical point of the functional, we let <span class="math-container">$\xi \in K$</span>, we obtain the equation
<span class="math-container">$$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$</span>
If we further require that <span class="math-container">$\xi$</span> vanishes on the boundary, we have the EL equation
<span class="math-container">$$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$</span>
Suppose we also have that <span class="math-container">$u \in H^2(B)$</span>, we have
<span class="math-container">$$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$</span>
We finally obtain the equation
<span class="math-container">$$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$</span></p>
<p>My main goal is to prove the minimizer <span class="math-container">$u_0$</span> solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the <span class="math-container">$H^2$</span> bound of <span class="math-container">$u$</span>? I think we can apply standard estimate to obtain <span class="math-container">$H^2_{loc}$</span>. If we do not have the fact that <span class="math-container">$u \in H^2(B)$</span>, we may hard to have the existence of <span class="math-container">$\dfrac{\partial u}{\partial n}$</span> on the boundary by trace theorem.</p>
http://www.2874565.com/q/3325405Symmetry of the distribution of prime gapsSylvain JULIENhttp://www.2874565.com/users/136252019-05-26T19:10:31Z2019-05-30T21:42:06Z
<p>Following <a href="http://www.2874565.com/questions/332500/positive-proportion-of-logarithmic-gaps-between-consecutive-primes">Positive proportion of logarithmic gaps between consecutive primes</a> let for given <span class="math-container">$\lambda$</span>, <span class="math-container">$\alpha$</span> and for any <span class="math-container">$x$</span> all positive the quantities <span class="math-container">$S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\leqslant x\colon p_{n+1}-p_{n}\leqslant\lambda\log^{\alpha}x\}$</span> and <span class="math-container">$S^{+}_{\lambda,\alpha}(x):=\#\{p_{n+1}\leqslant x\colon p_{n+1}-p_{n}\geqslant\lambda\log^{\alpha}x\}$</span>.</p>
<p>Is it presentely known, 6 years after Yitang Zhang's 2013 breakthrough, whether <span class="math-container">$S_{1,1-\alpha}^{-}(x)\sim S_{1,1+\alpha}^{+}(x)$</span>? If not, is it a consequence of some widely believed conjecture such as Hardy-Littlewood <span class="math-container">$k$</span>-tuple conjecture?</p>
http://www.2874565.com/q/3325004Positive proportion of logarithmic gaps between consecutive primesKellohttp://www.2874565.com/users/1410192019-05-26T09:31:31Z2019-05-30T22:24:20Z
<p>For <span class="math-container">$x, \lambda > 0$</span>, define
<span class="math-container">$$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$</span>
where <span class="math-container">$p_n$</span> is the <span class="math-container">$n$</span>th prime number. It is known [1] that an uniform version of the Hardy-Littlewood prime k-tuples conjecture implies that for fixed <span class="math-container">$\lambda > 0$</span>,
<span class="math-container">$$S_\lambda(x) \sim e^{-\lambda} \frac{x}{\log x}$$</span>
as <span class="math-container">$x \to +\infty$</span>.</p>
<p>My question is: If we want only a lower bound of the form
<span class="math-container">$$S_\lambda(x) \gg_\lambda \frac{x}{\log x}, \quad x > 2,$$</span>
for every fixed <span class="math-container">$\lambda > 0$</span>, has this been proved unconditionally?</p>
<p>Thank you for any reference or suggestion.</p>
<p>[1] Funkhouser,Goldston, Ledoan, Distribution of Large Gaps Between
Primes, <a href="https://doi.org/10.1007/978-3-319-92777-0_3" rel="nofollow noreferrer">https://doi.org/10.1007/978-3-319-92777-0_3</a></p>
http://www.2874565.com/q/3324851Galois representations associated to the algebraic cycles and transcendental cycles of K3 surfacesWenzhehttp://www.2874565.com/users/879102019-05-26T03:15:07Z2019-05-30T19:35:51Z
<p>Given a K3 surface <span class="math-container">$X$</span>, the cup product defines a non-degenerate even unimodular structure on the lattice <span class="math-container">$H^2(X,\mathbb{Z})$</span>. Inside this lattice we have the Neron-Severi group <span class="math-container">$\text{NS}(X)$</span>, which is also a primitive lattice. The rank of <span class="math-container">$\text{NS}(X)$</span>, denoted by <span class="math-container">$\rho(X)$</span>, is called the Picard number of <span class="math-container">$X$</span>. The orthogonal complement of <span class="math-container">$\text{NS}(X)$</span> is by definition the transcendental lattice
<span class="math-container">\begin{equation}
T(X):=\text{NS}(X)^\perp \subset H^2(X,\mathbb{Z}).
\end{equation}</span></p>
<p>In the note "Arithmetic of K3 surfaces" by Matthias Schutt, the author says that</p>
<p>"If <span class="math-container">$X$</span> is defined over some number field, the lattices of algebraic and transcendental cycles give rise to Galois representations of dimension <span class="math-container">$\rho(X)$</span> resp. <span class="math-container">$22-\rho(X)$</span>."</p>
<p>Does he mean that the Galois representation arise from the etale cohomology <span class="math-container">$H^2_{et}(X,\mathbb{Q}_\ell)$</span> splits into the direct sum of two sub-representations with dimension <span class="math-container">$\rho(X)$</span> (associated to the algebraic cycles) and <span class="math-container">$22-\rho(X)$</span> (associated to the transcendental cycles)? </p>
<p>I guess this statement might be true generally for algebraic surfaces. Could anyone explain it more carefully, and give a reference if possible?</p>
http://www.2874565.com/q/3319802The minimal volume of the intersection of two $\mathscr{l}_1$-ball in high dimensionnowherehttp://www.2874565.com/users/1408552019-05-20T07:23:14Z2019-05-30T20:41:25Z
<p>We define
<span class="math-container">$$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$</span></p>
<p>Now for arbitrary constant <span class="math-container">$r \ge s > 0$</span>, given constant <span class="math-container">$\epsilon \in (r-s,r+s)$</span>, considering <span class="math-container">$v \in \mathbb{R}^d$</span> such that <span class="math-container">$\Vert{v}\Vert_1 = \epsilon$</span>,
what is the minimal volume of <span class="math-container">$B_1(r,0) \cap B_1(s,v)$</span> when <span class="math-container">$v$</span> varying?</p>
<p>Is there any idea or suggested reference?</p>
<p>I am sure that it goes to minimal when <span class="math-container">$v = (\epsilon, 0, ..., 0)$</span>
after considering situations where <span class="math-container">$d = 2, 3$</span>.
But have no idea for strict proof.
Any idea for proof?</p>
<p>Thank you very much.</p>
http://www.2874565.com/q/3302796complex manifold with boundaryAndré Henriqueshttp://www.2874565.com/users/56902019-04-29T13:45:49Z2019-05-30T22:01:50Z
<p>My question is of local nature.<br>
Let <span class="math-container">$$f:\mathbb C^n\to\mathbb R$$</span> be a <span class="math-container">$C^\infty$</span> function that vanishes at <span class="math-container">$0\in \mathbb C^n$</span>, with non-zero derivative.<br>
Then, around <span class="math-container">$0\in \mathbb C^n$</span>, <span class="math-container">$$M:=f^{-1}(0)$$</span> is a CR manifold. Let me assume that <span class="math-container">$M$</span> is the simplest possible kind of CR manifold, namely that it is foliated by real-codimension-one complex submanifolds.</p>
<p>[Equivalently, for those who don't know what CR manifolds are, consider the hyperplane distribution <span class="math-container">$L:=TM\cap i\cdot TM\subset TM$</span>. I require the distribution <span class="math-container">$L$</span> to be integrable, i.e., to come from a (real codimention <span class="math-container">$1$</span>) foliation of <span class="math-container">$M$</span>.]</p>
<blockquote>
<p>Under the above assumptions, is <span class="math-container">$f^{-1}\big([0,\infty)\big)$</span> locally isomorphic to <span class="math-container">$$\big\{(z_1,...z_n)\in\mathbb C^n\,:\,\mathrm{im}(z_1)\ge 0\big\}?$$</span></p>
</blockquote>
<p>I.e., does there exist a neighbourhood <span class="math-container">$U\subset f^{-1}([0,\infty))$</span> of zero and an isomorphism <span class="math-container">$\varphi:U\to \big\{z\in\mathbb C^n\,:\,\sum|z_i|^2<1,\,\mathrm{im}(z_1)\ge 0\big\}$</span> which is holomorphic in the interior and smooth all the way to the boundary.</p>
http://www.2874565.com/q/372127Programming Languages Based on Category TheoryPatrick Tamhttp://www.2874565.com/users/13692009-11-02T01:14:53Z2019-05-30T19:30:23Z
<p>Since some computer scientists use category theory, I was wondering if there are any programming languages that use it extensively.</p>
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