Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2022-05-16T17:31:22Zhttps://mathoverflow.net/feedshttps://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/4226860Continuous modelingPicaPythonhttps://mathoverflow.net/users/4823882022-05-16T17:13:53Z2022-05-16T17:13:53Z
<p>I am reviewing for an exam and was reviewing last year's exam. Since our professor doesn't want to solve it in class, I come here to see if someone is so kind to solve it. The problem has to be modeled with differential equations. Attached is the problem:</p>
<p>at 23:08 hours alerting that a lifeless body has been found inside a cold room of the President Hotel. Two patrol cars are immediately dispatched to the hotel. Upon arrival of the first car at 23:11, one of the agents inspects the scene, confirms the death of the victim and takes his temperature, which is 12.40ºC. After 30 seconds, the second patrol car arrives and another officer takes the body temperature again, which is 11.85ºC. The officer in charge confirms that the victim is L. Palmer, one of his employees. The detective in charge of the case observes that access to the cold room is controlled by a magnetic lock and asks for the access log to the cold room, which each employee can only enter by swiping his or her identification card. The manager provides the police with the access list shown in the attached table.
At what time do you think L. Palmer could have died, taking into account that the cold room is at approximately 0°C?
Who would be your prime suspect?
Justify your answers.</p>
<p><a href="https://i.stack.imgur.com/sEnES.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sEnES.png" alt="enter image description here" /></a></p>
https://mathoverflow.net/q/4226823G-sheaves on spaces with a free G-actionMisha Verbitskyhttps://mathoverflow.net/users/33772022-05-16T16:18:06Z2022-05-16T16:18:06Z
<p>Let <span class="math-container">$X$</span> be a topological space
equipped with an action of a group <span class="math-container">$G$</span>. in the Tohoku paper, Grothendieck defined
"<span class="math-container">$G$</span>-sheaves" on <span class="math-container">$X$</span> as sheaves equipped with <span class="math-container">$G$</span>-action on the etale space,
compatible with the <span class="math-container">$G$</span>-action on <span class="math-container">$X$</span>. It is more or less the same
as <span class="math-container">$G$</span>-equivariant objects in the category of sheaves on <span class="math-container">$X$</span> with respect
to <span class="math-container">$G$</span> acting on this category by endofunctors, but the notion of "<span class="math-container">$G$</span>-equivariant sheaves" is taken over by sheaves equivariant with respect to a group scheme
or to a continuous action of a topological group.</p>
<p>Suppose now
that the action of <span class="math-container">$G$</span> on <span class="math-container">$X$</span> is free. Then the
category of <span class="math-container">$G$</span>-sheaves on <span class="math-container">$X$</span> is equivalent to the category
of sheaves on <span class="math-container">$X/G$</span>. I need this statement when <span class="math-container">$G$</span> acts on a manifold <span class="math-container">$X$</span> properly
discontinuously, but I suppose this is true in all generality.
I have a proof which works (at least for properly
discontinuous actions), but it's a bit too long for such a
classical-looking statement, so I am asking for a reference
I can use for this. I spent quite a few hours googling and browsing the Stacks Project, without avail.</p>
https://mathoverflow.net/q/4226780essential numerical range of an idempotentmathbeginnerhttps://mathoverflow.net/users/1531962022-05-16T15:45:20Z2022-05-16T15:45:20Z
<p>Notation: <span class="math-container">$W_e()$</span> denotes the essential numerical range of an operator in <span class="math-container">$L(H)$</span> and <span class="math-container">$\Bbb D$</span> is the unit disk of <span class="math-container">$\Bbb C$</span>.</p>
<p>I tried to prove the conclusion : <span class="math-container">$\Bbb D\subset W_e(Q)$</span> iff <span class="math-container">$\Bbb D\subset W_e(Q_{H_{ns}})$</span>.</p>
<p>It is easy to check that the "if part".</p>
<p>But for the converse direction, I met with some problems. For any <span class="math-container">$\lambda\in \Bbb D$</span>
,there exists a basis <span class="math-container">$\{e_n\}_{n=1}^{\infty}\cup\{f_n\}_{n=1}^{\infty}$</span> such that</p>
<p><span class="math-container">$$\lambda=\lim_{n}\langle\begin{pmatrix} e_n \\ f_n \end{pmatrix}, (P\oplus \begin{pmatrix}I & 0\\R& 0\end{pmatrix})
\begin{pmatrix} e_n \\ f_n \end{pmatrix}\rangle$$</span>, how to show that for any <span class="math-container">$\lambda\in \Bbb D$</span>, there is a basis <span class="math-container">$\{g_n\}_{n=1}^{\infty}$</span>such that</p>
<p><span class="math-container">$\lambda=\lim_{n}\langle g_n, \begin{pmatrix}I & 0\\R& 0\end{pmatrix} g_n\rangle$</span>,</p>
<p><a href="https://i.stack.imgur.com/BLeCr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BLeCr.jpg" alt="enter image description here" /></a></p>
https://mathoverflow.net/q/4226751Convergent algorithm for minimizing nonconvex smooth functiondohmatobhttps://mathoverflow.net/users/785392022-05-16T15:19:01Z2022-05-16T15:19:01Z
<p>Let <span class="math-container">$\Phi$</span> be the Gaussian CDF and for <span class="math-container">$\gamma\ge 0$</span> and <span class="math-container">$h>0$</span>, define a loss function <span class="math-container">$\ell_h:\{\pm 1\} \times \mathbb R$</span> by
<span class="math-container">$$
\ell_{\gamma,h}(y,y') := \phi_{\gamma,h}(yy') := \Phi((yy'-\gamma)/h).
$$</span></p>
<p>Note that</p>
<ul>
<li><span class="math-container">$\phi_{\gamma,h}$</span> is nonconvex.</li>
<li><span class="math-container">$\phi_{\gamma,h}$</span> is nondecreasing</li>
<li><span class="math-container">$\phi_{\gamma,h}$</span> is <span class="math-container">$O(1/h)$</span>-Lipschitz continuous and its derivative is <span class="math-container">$O(1/h^2)$</span>-Lipschitz.</li>
</ul>
<p>Let <span class="math-container">$(x_1,y_1),\ldots,(x_n,y_n)$</span> be <span class="math-container">$n$</span> points in <span class="math-container">$\mathbb R^{d} \times \{\pm 1\}$</span>, and for any <span class="math-container">$(w,b) \in \mathbb R^{d+1}$</span>, define</p>
<p><span class="math-container">$$
F(w,b) := \frac{1}{n}\sum_{i=1}^n\phi_{\gamma,h}(y_i(x_i^\top w - b)).
$$</span></p>
<p>Fix <span class="math-container">$p \in [1,\infty]$</span> and let <span class="math-container">$B_p$</span> be the unit-ball in <span class="math-container">$\mathbb R^d$</span> w.r.t the <span class="math-container">$\ell_p$</span>-norm. I'm particularly interested in the case <span class="math-container">$p \in \{1,2\}$</span>.</p>
<blockquote>
<p><strong>Question.</strong> <em>Is there a convergent gradient-based algorithm which minimizes <span class="math-container">$F$</span> over <span class="math-container">$B_p \times \mathbb R$</span>, or even <span class="math-container">$B_p \times \{0\}$</span> ?</em></p>
</blockquote>
<p>Ideally, an explicit rate of convergence would be provided too.</p>
https://mathoverflow.net/q/422672-2Cumulants of a sequence of variables with zero mean and variancegashttps://mathoverflow.net/users/4823832022-05-16T15:03:58Z2022-05-16T16:10:25Z
<p>Can one prove for a sequence of positive random variable <span class="math-container">$$X_{n}$$</span> such that <span class="math-container">$\lim_{n\to \infty}E[x_{n} = 0]$</span> and <span class="math-container">$\lim_{n\to \infty}E[x_{n}x_{n}]= 0$</span> all the cumulants go to zero once <span class="math-container">$n\to \infty$</span> ?</p>
https://mathoverflow.net/q/4226710Inf as a condition in the Kantorovich-Rubinstein dualityFranzFerdXhttps://mathoverflow.net/users/4823802022-05-16T14:35:41Z2022-05-16T14:38:25Z
<p>I am currently working on understanding the Kantorovich-Rubinstein duality and Wassertein loss.
The following part <a href="https://courses.cs.washington.edu/courses/cse599i/20au/resources/L12_duality.pdf" rel="nofollow noreferrer">of these class notes</a>:</p>
<blockquote>
<p>Collecting the terms algebraically we can rewrite the Lagrangian as : <br />
<span class="math-container">$L(\pi, f, g) = \underset{x\sim p}{\mathbb{E}}[f(x)] + \underset{y\sim p}{\mathbb{E}}[g(y)] + {\displaystyle\int_{X \times X}\Big(||x-y|| - f(x) - g(y)\Big)\pi(x,y)dydx}$</span>
<br />
And we appeal to strong duality to write <br />
<span class="math-container">$W(p, p_g) = \underset{\pi}{\inf}\underset{f, g}{\sup}L(\pi, f, g) = \underset{f, g}{\sup}\underset{\pi}{\inf}L(\pi, f, g)$</span> <br />
<strong>Note that if <span class="math-container">$||x-y|| \leq f(x) + g(y)$</span> for some <span class="math-container">$x, y \in X$</span> then we can concentrate the mass of <span class="math-container">$\pi$</span> at <span class="math-container">$(x,y)$</span> and send <span class="math-container">$L(\pi, f, g)$</span> to <span class="math-container">$-\infty$</span></strong></p>
</blockquote>
<p><a href="https://courses.cs.washington.edu/courses/cse599i/20au/resources/L12_duality.pdf" rel="nofollow noreferrer">-Kantorovich-Rubinstein Duality,John Thickstun, p.1</a></p>
<p>I understand why the author tries to show that it goes to <span class="math-container">$-\infty$</span>, because then it becomes a constraint. I also understand how it concentrate mass at <span class="math-container">$(x, y)$</span> I think. <br />
<br />
The part I don't get is the part in bold : <strong>Why would the Lagrangian go to <span class="math-container">$-\infty$</span> when we concentrate the mass on a point <span class="math-container">$(x, y)$</span> such that <span class="math-container">$||x-y|| \leq f(x) + g(y)$</span> ?</strong><br />
I really can't see it. I could not find any explanation anywhere.</p>
https://mathoverflow.net/q/4226702Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?Elías Guisadohttps://mathoverflow.net/users/1018482022-05-16T14:13:55Z2022-05-16T15:29:36Z
<p>My question is about one of those several concepts in algebraic geometry who everybody uses but nobody defines or introduces properly.</p>
<p>Given a ringed space <span class="math-container">$(X,\mathcal{O}_X)$</span> and ideal sheaves <span class="math-container">$\mathcal{I},\mathcal{J}\subset\mathcal{O}_X$</span>, we define the ideal product presheaf <span class="math-container">$\mathcal{I}\cdot_p\mathcal{J}$</span> as the ideal presheaf
<span class="math-container">$$
U\mapsto(\mathcal{I}\cdot_p\mathcal{J})(U)=\mathcal{I}(U)\mathcal{J}(U)\subset\mathcal{O}_X(U).
$$</span>
My question is: is this presheaf a sheaf? Or is it necessary to sheafify to obtain the correct definition of the ideal product sheaf?</p>
<p>I was quite a while trying to look for a counterexample myself and I couldn't find any. After asking as well to the lecturer in the graduate algebraic geometry course I am following, he told me he could not find a counterexample. I've already asked this question <a href="https://math.stackexchange.com/q/4443091/394668">here on MSE</a>, but nobody has answered yet. There I explain the approaches I've tried. The problem is that I don't even have any probability argument to believe the answer to be positive or negative. I don't see why the presheaf shouldn't be a sheaf, but a positive proof seems to be unlikely (as I explain it in the MSE post). So if at least I receive an answer of the like "I don't think it's a sheaf" from an expert on the field I think I would be somewhat content.</p>
<p>From my experience, I think there is not that many people in MSE interested on scheme-theoretic algebraic geometry, so I am reasking the question here on mathoverflow hoping there's more people here that could give any comments.</p>
https://mathoverflow.net/q/4226691Is a fixed subgroup of a compact Lie group cotorally included in finitely many conjugacy classes?N.B.https://mathoverflow.net/users/1314532022-05-16T14:04:35Z2022-05-16T14:04:35Z
<p>Let <span class="math-container">$G$</span> be a compact Lie group, in the next discussion we consider only its closed subgroups without specifying it. We say that a subgroup <span class="math-container">$L$</span> is a cotoral subgroup of <span class="math-container">$K\leq G$</span> if <span class="math-container">$L \trianglelefteq K$</span> and the quotient <span class="math-container">$K/L$</span> is a torus (i.e. a connected abelian compact Lie group). We write <span class="math-container">$L\leq_{ct} K$</span> for this relation.</p>
<p>My question is the following: suppose we fix <span class="math-container">$H\leq G$</span> a subgroup, then are the conjugacy classes of the subgroups <span class="math-container">$K\leq G$</span> such that <span class="math-container">$H\leq_{ct}K$</span> finitely many?</p>
<p>If we look at the examples of rank <span class="math-container">$1$</span> then this is true. For <span class="math-container">$G=S^1$</span> we have <span class="math-container">$C_n \leq_{ct} S^1$</span> for any <span class="math-container">$n$</span> but no two <span class="math-container">$C_n$</span> and <span class="math-container">$C_m$</span> are cotoral. In <span class="math-container">$G=O(2)$</span> the dihedral component does not provide any new problem: we can fix dihedral subgroups <span class="math-container">$D_{2n}$</span> such that <span class="math-container">$N_G(D_{2n})=D_{4n}$</span>, hence up to conjugacy we only have the non-trivial cotoral inclusions <span class="math-container">$D_{2n}\leq_{ct}O(2)$</span>. The subgroup structure of <span class="math-container">$SO(3)$</span> is more complicated, but even in this case I could not find a closed subgroup cotorally included in infinitely many non-conjugated subgroups.</p>
<p>I am not so much familiar with compact Lie groups that I can show this should be true in general. For a fixed <span class="math-container">$H\leq G$</span> trivially the length of the chain of cotoral inclusions <span class="math-container">$H <_{ct} K_1 <_{ct} K_2 <_{ct}\dots <_{ct} K_n$</span> must be finite because at each step we are increasing the rank of the subgroup and this is bounded by the rank of the ambient group <span class="math-container">$G$</span>. But I cannot see why the subgroups <span class="math-container">$K$</span> with <span class="math-container">$H<_{ct} K$</span> and <span class="math-container">$K/H$</span> of fixed rank should be finitely many up to conjugacy.</p>
<p>I would think that either the claim is false or there is some theorem in the literature which leads to such result.</p>
https://mathoverflow.net/q/422668-4Minimal polynomial of square root 2 in mod 2 [closed]Mathfanhttps://mathoverflow.net/users/4823792022-05-16T14:03:44Z2022-05-16T14:03:44Z
<p>It is x^2-2 which is equal to x^2 in mod 2.</p>
<p>But it is not irreducible.</p>
<p>But minimal polynomial should be irreducible,what is wrong here?</p>
https://mathoverflow.net/q/4226660Functional relationship between two quantitiesdohmatobhttps://mathoverflow.net/users/785392022-05-16T13:09:26Z2022-05-16T16:05:58Z
<p>Let <span class="math-container">$\mu \in \mathbb R^n$</span> and let <span class="math-container">$\Sigma$</span> be a positive-definite matrix of order <span class="math-container">$n \ge 2$</span>. Fix <span class="math-container">$t \ge 0$</span> and define <span class="math-container">$\alpha(\mu,\Sigma,t) > 0$</span> by</p>
<p><span class="math-container">$$
\alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{1}{\|w\|_\Sigma}\varphi\left(\frac{w^\top \mu - t}{\|w\|_\Sigma}\right),
$$</span></p>
<p>where <span class="math-container">$\|w\|_\Sigma := \sqrt{w^\top \Sigma w}$</span> and <span class="math-container">$\varphi$</span> is the Gaussian pdf. Also define <span class="math-container">$\beta(\mu,\Sigma,t) \ge 0$</span> by
<span class="math-container">$$
\beta(\mu,\Sigma,t) := \inf_{\|z\| \le t}\|z-\mu\|_{\Sigma^{-1}}.
$$</span></p>
<blockquote>
<p><strong>Question.</strong> <em>Is there any functional relationship between <span class="math-container">$\alpha(\mu,\Sigma,t)$</span> and <span class="math-container">$\beta(\mu,\Sigma,t)$</span> ?</em></p>
</blockquote>
<h2>Example: isotropic case</h2>
<p>Suppose <span class="math-container">$\Sigma = I_n$</span>, the identity matrix. Then</p>
<p><span class="math-container">$\alpha(\mu,\Sigma,t) = \varphi(r_\star)$</span>, where
<span class="math-container">$$
\begin{split}
-r_\star := \inf_{\|w\| = 1}|w^\top \mu - t| &= \inf_{\|w\| = 1}\sup_{s \in \{\pm 1\}}s(w^\top \mu-t) = \sup_{s \in \{\pm 1\}}-st+\inf_{\|w\| = 1}sw^\top \mu\\
& = \sup_{s \in \{\pm 1\}}-st-\|\mu\|=t-\|\mu\|,
\end{split}
$$</span>
if <span class="math-container">$\|\mu\| \ge t$</span>, and <span class="math-container">$r_\star = 0$</span> otherwise. That is, <span class="math-container">$r_\star = (\|\mu\|-t)_+$</span>.</p>
<p>On the other hand, one computes
<span class="math-container">$$
\begin{split}
\inf_{\|z\| \le 1}\|z-\mu\|
&= \begin{cases}
0,&\mbox{ if }\|\mu\| \le t,\\
\|t\mu/\|\mu\|-\mu\| = |t-\mu| = \|\mu\|-t,&\mbox{ else}
\end{cases}\\
&=(\|\mu\|-t)_+ = r_\star.
\end{split}
$$</span></p>
<p>We conclude</p>
<blockquote>
<p><span class="math-container">$\alpha(\mu,I_n,t) = \varphi(\beta(\mu,I_n,t))$</span>.</p>
</blockquote>
https://mathoverflow.net/q/4226650Projectively normal and normal when the ring satisfies Serre's criterionFreePawnhttps://mathoverflow.net/users/3384562022-05-16T12:40:04Z2022-05-16T15:22:02Z
<p>In Hartshorne, Ex II 5.14, says that</p>
<blockquote>
<p>A closed subscheme <span class="math-container">$X \subseteq \mathbb{P}_{A}^r$</span> is projectively normal for the given embedding, if its homogeneous coordinate ring <span class="math-container">$S(X)$</span> is an integrally closed domain.</p>
</blockquote>
<p>and that</p>
<blockquote>
<p>a scheme <span class="math-container">$X$</span> is normal if its local rings are integrally closed domains.</p>
</blockquote>
<p>Now, assume <span class="math-container">$\mathbb{K}$</span> is an algebraically closed field of characteristic <span class="math-container">$0$</span>. Let <span class="math-container">$R=\mathbb{K}[x_0,x_1,x_2,\dotsc,x_n]/I$</span>, where <span class="math-container">$I\subseteq \mathbb{K}[x_0,x_1,\dotsc,x_n]$</span> is a homogeneous ideal. Also, assume that <span class="math-container">$R$</span> is an integral domain which satisfies Serre's criterion for normality (i.e., <span class="math-container">$R_1$</span> and <span class="math-container">$S_2$</span>).</p>
<p>Now, consider the projective variety (scheme) <span class="math-container">$X$</span>, defined by <span class="math-container">$R$</span> in <span class="math-container">$\mathbb{P}^{n}$</span>, i.e., <span class="math-container">$X=\operatorname{Proj}(R)$</span>.</p>
<p>Wouldn't that mean <span class="math-container">$X$</span> is normal and projectively normal?</p>
https://mathoverflow.net/q/4226466What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$BPKhttps://mathoverflow.net/users/1405742022-05-16T06:29:04Z2022-05-16T17:09:55Z
<p>Let <span class="math-container">$F$</span> be the non-archimedean local field <span class="math-container">$\mathbb{Q}_p$</span> for some prime <span class="math-container">$p$</span> and <span class="math-container">$D$</span> be a quaternion division algebra over <span class="math-container">$F$</span>. Let <span class="math-container">$\mathcal{O}_D$</span> and <span class="math-container">$\mathcal{P}_D$</span> denote the ring of integers of <span class="math-container">$D$</span> and its unique maximal ideal (resp.). Then, what is the finite group
<span class="math-container">$$ \frac{D^*}{F^*(1+ \mathcal{P}_D)} = ?$$</span> where <span class="math-container">$D^*=D-\{0\}$</span> and <span class="math-container">$F^*=F-\{0\}$</span> are multiplicative groups.</p>
<p>Consider the reduced norm map <span class="math-container">$N_{rd}:D \rightarrow F$</span>, then <span class="math-container">$N_{rd}(D^*)=F^*$</span> and if <span class="math-container">$D^1$</span> denotes the reduced norm one elements of <span class="math-container">$D$</span>, then we have an exact sequence
<span class="math-container">$$1 \rightarrow D^1 \rightarrow D^* \rightarrow F^* \rightarrow 1$$</span> but we have <span class="math-container">$D^1 \cap F^*=\{\pm 1\}$</span>. What we know from Carl Riehm's article that <span class="math-container">$$ \frac{D^1}{(1+ \mathcal{P}_D)} \cong {_N}(\mathbb{F}_{p^2})= \text{Finite cyclic group of order } (p+1).$$</span>
Here <span class="math-container">${_N}(\mathbb{F}_{p^2})$</span> is the subgroup of <span class="math-container">$\mathbb{F}_{p^2}$</span> consisting of norm 1 elements.</p>
<p>Question: Similarly, can we write <span class="math-container">$ \frac{D^*}{F^*(1+ \mathcal{P}_D)}$</span> in terms of finite fields?</p>
<p>Any comments or suggestions will be extremely helpful. Thanks in advance.</p>
https://mathoverflow.net/q/4226404Which varieties are sums of tensor powers of the Lefschetz motive?John Baezhttps://mathoverflow.net/users/28932022-05-16T03:53:59Z2022-05-16T16:47:34Z
<p>Any smooth projective variety <span class="math-container">$X$</span> gives an object <span class="math-container">$h(X)$</span> in the category of <a href="https://ncatlab.org/nlab/show/pure+motive" rel="nofollow noreferrer">pure Chow motives</a>. If <span class="math-container">$X$</span> is a <a href="https://en.wikipedia.org/wiki/Generalized_flag_variety" rel="nofollow noreferrer">generalized flag variety</a>, i.e. a quotient <span class="math-container">$G/P$</span> where <span class="math-container">$G$</span> is semisimple linear algebraic group and <span class="math-container">$P$</span> is a <a href="https://en.wikipedia.org/wiki/Borel_subgroup#Parabolic_subgroups" rel="nofollow noreferrer">parabolic subgroup</a>, I believe <span class="math-container">$h(X)$</span> is a direct sum of tensor powers of the <a href="https://mathoverflow.net/questions/14587/understanding-the-definition-of-the-lefschetz-pure-effective-motive">Lefschetz motive</a>, because <span class="math-container">$X$</span> can be decomposed into <a href="https://en.wikipedia.org/wiki/Schubert_variety" rel="nofollow noreferrer">Schubert varieties</a> which are copies of <span class="math-container">$\mathbb{A}^n$</span> for various <span class="math-container">$n$</span>.</p>
<p>If this is correct, I'd like to know: <strong>which other smooth projective varieties give pure Chow motives that are direct sums of tensor powers of the Lefschetz motive?</strong></p>
<p>My intuition is that any variety with something like a "Schubert decomposition" — roughly, a well-behaved way of expressing it as a disjoint union of copies of <span class="math-container">$\mathbb{A}^n$</span>'s — will have this property. I feel there should be plenty. But I don't actually know any varieties with this property, apart from flag varieties!</p>
<p>Any variety <span class="math-container">$X$</span> of dimension <span class="math-container">$d$</span> over <span class="math-container">$\mathbb{F}_p$</span> having this property will have an associated polynomial <span class="math-container">$N_X$</span> of degree <span class="math-container">$d$</span> with natural number coefficients:</p>
<p><span class="math-container">$$ N_X(q) = \sum_{n = 0}^d a_n q^n $$</span></p>
<p>such that <span class="math-container">$X$</span> has <span class="math-container">$N_X(q)$</span> points over <span class="math-container">$\mathbb{F}_q$</span> when <span class="math-container">$q$</span> is any power of <span class="math-container">$p$</span>.</p>
<p><strong>Is the converse true? Is a smooth projective variety <span class="math-container">$X$</span> over <span class="math-container">$\mathbb{F}_p$</span> with a polynomial <span class="math-container">$N_X$</span> having this property always a direct sum of tensor powers of the Lefschetz motive?</strong></p>
https://mathoverflow.net/q/4226377Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?Josh Lackmanhttps://mathoverflow.net/users/403232022-05-15T23:49:08Z2022-05-16T15:37:21Z
<p>I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that <span class="math-container">$\infty$</span>-groupoids, with <span class="math-container">$\infty$</span>-categorical equivalences as weak equivalences, are equivalent to topological spaces, with weak equivalences being weak homotopy equivalences.</p>
<p>Now for simplicity, let me focus on the version of the homotopy hypothesis that says that <span class="math-container">$1$</span>-groupoids, with categorial equivalences as weak equivalences, are equivalent to topological spaces, where weak equivalences are weak <span class="math-container">$1$</span>-equivalences, ie. maps which induce isomorphisms on <span class="math-container">$\pi_0\,,\pi_1\,.$</span></p>
<p>Now on the other hand, Lie's second and third theorems imply that there is an equivalence of categories between simply connected Lie groups and Lie algebras. These theorems generalize to Lie groupoids and Lie algebroids, where simply connected becomes source simply connected (well, in order for Lie's third theorem to hold completely one needs to use smooth spaces which are generalizations of manifolds, but we can always replace "Lie algebroids" with "integrable Lie algebroids", in any case).</p>
<p>Right now these two results may not seem related, for two reasons:</p>
<ol>
<li><p>There is no Lie algebroid present on the topological spaces. However, if we work with manifolds instead (or some appropriate class of infinite dimensional manifolds), then <span class="math-container">$M$</span> comes with a natural Lie algebroid, namely <span class="math-container">$TM\,.$</span> Therefore, there is a canonical Lie algebroid present.</p>
</li>
<li><p>The groupoids in the homotopy hypothesis are discrete, so there is no source simply connected condition. However, the source simply connected integration of <span class="math-container">$TM$</span> is the fundamental groupoid <span class="math-container">$\Pi_1(M)\,,$</span> and this is Morita equivalent to <span class="math-container">$\pi_1(M)\,.$</span> Therefore <span class="math-container">$\Pi_1(M)\,,$</span> with the topology associated with the smooth structure, is equivalent to <span class="math-container">$\Pi_1(M)$</span> with the discrete topology, and I believe this is the correct groupoid to compare <span class="math-container">$M$</span> with in the homotopy hypothesis for 1-types. So in a sense, the source simply connected condition is naturally present.</p>
</li>
</ol>
<p>Now, if we assume that we that we have a notion of weak equivalence of Lie algebroids which implies that a morphism <span class="math-container">$TM\to TN$</span> is a weak equivalence if the induced map <span class="math-container">$M\to N$</span> is a weak <span class="math-container">$1$</span>-equivalence, then we seem to get a connection between Lie's theorems and the homotopy hypothesis, ie. the homotopy hypothesis for (smooth) 1-types seems to be implied by a version of Lie's theorems (this wouldn't exactly be Lie's theorems since Lie's theorems use isomorphisms as weak equivalences, but we can use other weak equivalences instead).</p>
<p>I can go into more detail, but has this connection been written about elsewhere in the literature, or is there any reason to doubt this connection?</p>
https://mathoverflow.net/q/4226222Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditionsPaolo Bernuzzihttps://mathoverflow.net/users/4823372022-05-15T19:23:23Z2022-05-16T14:51:59Z
<p>I would like to ask a question with possibly a reference. If we have a Schrödinger operator <span class="math-container">$-\Delta+V$</span> on an interval <span class="math-container">$[0,L]$</span> with <span class="math-container">$V$</span> continous and Dirichlet conditions, can we state that the eigenfunctions of such operator are uniformly bounded, i.e. there exists <span class="math-container">$M>0$</span> such that the eigenfunctions <span class="math-container">$\{\phi_n\}_n$</span> satisfy</p>
<p><span class="math-container">\begin{equation*} \sup_n \lvert\lvert \phi_n \rvert\rvert_\infty\leq M \end{equation*}</span>
?</p>
https://mathoverflow.net/q/4226174Problems arising from the Trudinger's paper in 1968 "Remarks concerning the conformal deformation of riemannian structures on compact manifolds"TeenFromAlishanhttps://mathoverflow.net/users/4691292022-05-15T17:56:30Z2022-05-16T15:42:01Z
<p>I'm reading the paper <a href="http://www.numdam.org/item/ASNSP_1968_3_22_2_265_0" rel="nofollow noreferrer"><strong>Remarks concerning the conformal deformation of riemannian
structures on compact manifolds</strong></a> by NEIL S. TRUDINGER.</p>
<p>I'm stuck with the Theorem 3, which says that let <span class="math-container">$u$</span> be a <span class="math-container">$W_{2}^{1}(M)$</span> solution of an equation of the form <span class="math-container">$$\frac{4(n-1)}{n-2} \Delta u-R u=-\bar{R} u^{\frac{n+2}{n-2}}.$$</span> Then <span class="math-container">$u \in C^{\infty}(M)$</span>.</p>
<p>The proof is as follows:</p>
<p>The function <span class="math-container">$u$</span> satisfies
<span class="math-container">\begin{equation}
\int_{M}\left(\frac{4(n-1)}{n-2} g^{i j} u_{i} \xi_{j}+R u \xi\right) d V=\bar{R} \int_{M}|u|^{N-1} \xi d V \label{1}\tag{1}
\end{equation}</span>
for all <span class="math-container">$\xi \in W_{2}^{1}(M)$</span>, then construct a test function.</p>
<p>Define <span class="math-container">$\bar{u}=\sup (u, 0)$</span> and for a fixed <span class="math-container">$\beta>1$</span>define the functions
<span class="math-container">$$
\begin{gathered}
G(\bar{u})= \begin{cases}\bar{u}^{\beta} & \text { if } \bar{u} \leq l \\
l^{q-1}\left(q l^{q-1} \bar{u}-(q-1) l^{q}\right) & \text { if } \bar{u}>l\end{cases} \\
F(\bar{u})= \begin{cases}\bar{u}^{q} & \text { if } \bar{u} \leq l \\
q l^{q-1} \bar{u}-(q-1) l^{q} & \text { if } \bar{u}>l \end{cases}
\end{gathered}
$$</span>
where <span class="math-container">$2 q=\beta+1$</span>.</p>
<p>The function <span class="math-container">$G(\bar{u})$</span> is a uniformly Lipshitz continuous function of <span class="math-container">$u$</span> and hence belongs to <span class="math-container">$W_{2}^{1}(M)$</span>. Likeuise <span class="math-container">$F(\bar{u})$</span>. Observe also that <span class="math-container">$G$</span> and <span class="math-container">$F$</span> vanish for negative <span class="math-container">$u$</span> and that
<span class="math-container">$$
\left(F^{\prime}(\bar{u})\right)^{2} \leq q G^{\prime}(\bar{u}), \quad(F(\bar{u}))^{2} \geq \bar{u} G(\bar{u}).
$$</span>
Let us now substitute in \eqref{1} test functions
<span class="math-container">$$
\xi=\eta^{2} G(\bar{u})
$$</span>
we have (this is my calculation, I don't know how to get the result in paper)
<span class="math-container">$$\int_{M}\left(\frac{4(n-1)}{n-2} g^{i j} u_{i} (2\eta \eta_{j}G(\bar{u})+\eta^{2}G^{\prime}(\bar{u})u_{j})+R u \eta^{2} G(\bar{u})\right) d V=\bar{R} \int_{M}|u|^{N-1} \eta^{2} G(\bar{u}) d V.$$</span></p>
<p>Then
<span class="math-container">$$\frac{4(n-1) \mu}{n-2} \int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V = \int_{M}\bar{R} |u|^{N-1} \eta^{2} G(\bar{u})-\frac{4(n-1)}{n-2} g^{i j} u_{i}\left(2 \eta \eta_{j} G(\bar{u})\right)-Ru\eta ^{2} G(\bar{u}) dV.$$</span></p>
<p>Then he gets
<span class="math-container">$$\begin{aligned} \frac{4(n-1) \mu}{n-2} \int_{\dot{M}} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq \int_{M}\left(\frac{4(n-1)}{n-2} \sup \left|g^{i j}\right| \eta\left|\eta_{i}\right|+\sup |R| u \eta^{2}+\bar{R} u^{N-2} \eta^{2}\right) G d V. \end{aligned}$$</span></p>
<p>Did this term miss something? Should it be:
<span class="math-container">$$\begin{aligned} \frac{4(n-1) \mu}{n-2} \int_{\dot{M}} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq \int_{M}\left(\frac{4(n-1)}{n-2} \sup \left|g^{i j}\right| \eta\left|\eta_{j}\right|u_{i} +\sup |R| u \eta^{2}+\bar{R} u^{N-1} \eta^{2}\right) G d V? \end{aligned}$$</span>
Then he gets
<span class="math-container">$$\int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq C \int_{M}\left\{\left(\left|\eta_{i}\right|^{2}+\eta^{2}\right)(\bar{u}) G(\bar{u})+\eta^{2} \bar{u}^{ N-2} \bar{u} G(\bar{u})\right\} d V.$$</span>
I'm stuck with these steps.</p>
<p><strong>I tried like this:</strong></p>
<p>Since we have
<span class="math-container">$$\frac{4(n-1) \mu}{n-2} \int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V = \int_{M}\bar{R} |u|^{N-1} \eta^{2} G(\bar{u})-\frac{4(n-1)}{n-2} g^{i j} u_{i}\left(2 \eta \eta_{j} G(\bar{u})\right)-Ru\eta ^{2} G(\bar{u}) dV.$$</span>
And since M is compact we have</p>
<p><span class="math-container">$$-\int_{M} \eta u_{i} G=\int_{M} u\left(\eta_{i} G+\eta G_{i}\right)=\int_{M} u \eta_{i} G+\int_{M} u \eta G_{i}$$</span></p>
<p>then I compute the last term</p>
<p><span class="math-container">$$\int_{M} u \eta G_{i}=\int_{M} u \eta G^{\prime}(\bar{u}) u_{i}\leqslant \int_{M}\left(\frac{u^{2}+u_{i}^{2}}{2}\right) \eta G^{\prime}(\bar{u})$$</span></p>
<p>then put the <span class="math-container">$\int_{M} \frac{u_{i}^{2}}{2} \eta G^{\prime}(\bar{u})$</span> to the LHS.</p>
<p>And notice that <span class="math-container">$$G^{\prime}(\bar{u})= \begin{cases}\beta\bar{u}^{\beta-1} & \text { if } \bar{u} \leq l \\ l^{q-1}\left(q l^{q-1}\right) & \text { if } \bar{u}>l\end{cases}$$</span></p>
<p>So we have</p>
<p><span class="math-container">$$\left\{\begin{array}{l}\frac{\bar{u} G^{\prime}(\bar{u})}{\beta}=G(\bar{u}) \quad \bar{u} \leqslant l \\ \bar{u} G^{\prime}(\bar{u})-C_{0}=G(\bar{u}) \quad \bar{u}>l\end{array}\right.$$</span></p>
<p>so we can turn <span class="math-container">$u^{2}$$\eta G^{\prime}(\bar{u})$</span> to be less than something about <span class="math-container">$\eta (F(\bar{u}))^{2}$</span>.</p>
https://mathoverflow.net/q/4225954Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$jorohttps://mathoverflow.net/users/124812022-05-15T14:02:55Z2022-05-16T15:40:53Z
<p>Working with precision 500 decimal digits, mpmath in sage computes:</p>
<p><span class="math-container">$$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$</span></p>
<p>We believe the LHS of \eqref{1} diverges, so this isn't true.</p>
<blockquote>
<p>Q1. Are there theoretical reasons for mpmath to compute \eqref{1}?</p>
</blockquote>
<p><a href="https://sagecell.sagemath.org/?z=eJxVjkEOwiAQRfck3IElkLYWEjc23MKdcYG2WpICIwwLby-JmNTdvPf_T8Z5iAmZB29xpQTSYo7jOH158DDMkE21O1MJ8W3OqSyU5KJMi0Iunm_W32bLwokx3ishZZAt3uKTB9FddNeEC4-roAS12TXUQf8G-ZWQa_GH7QYn6rLBfY3A6x89avEB80BGGA==&lang=sage&interacts=eJyLjgUAARUAuQ==" rel="nofollow noreferrer">online code</a></p>
<p><strong>Added</strong> Despite the interesting answers, I am ready bet mpmath
doesn't do any analytic stuff not related to summation, it
works purely numerically and the function is treated as black box,
returning real number.</p>
https://mathoverflow.net/q/42258220Is spherical trigonometry a dead research area?Dave Shulmanhttps://mathoverflow.net/users/1288762022-05-15T12:15:06Z2022-05-16T15:47:15Z
<p>When I was an undergrad, the field of <a href="https://en.wikipedia.org/wiki/Spherical_trigonometry" rel="nofollow noreferrer">spherical trigonometry</a> was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for contemporary research?</p>
https://mathoverflow.net/q/4224501Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomialuser142929https://mathoverflow.net/users/1429292022-05-13T14:20:53Z2022-05-16T14:41:32Z
<p>In the post (cross-posted in Mathematics Stack Exchange with identificator MSE <a href="https://math.stackexchange.com/questions/4244256/conjectures-inspired-in-the-context-of-casas-alvero-conjecture-via-the-logarith"><strong>4244256</strong></a> and same title) we assume that <span class="math-container">$P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$</span> is a polynomial of degree <span class="math-container">$1<\deg(P)=d=n$</span> defined over a field <span class="math-container">$K$</span> of characteristic zero. We denote its derivatives as <span class="math-container">$P^{(i)}(x)$</span> (writting <span class="math-container">$P^{(0)}(x)=P(x)$</span>), and <span class="math-container">$a_n$</span> denotes the leading coefficient of <span class="math-container">$P(x)$</span>.</p>
<p>I've stated two conjectures (as speculations from the fact) inspired in that I've proven inductively that each polynomial <span class="math-container">$p(x)$</span> of degree <span class="math-container">$1<\deg(p)$</span> (and with corresponding <span class="math-container">$a_n\neq 0$</span>) satisfies <span class="math-container">$$p(x)=a_n\cdot\left(\frac{n-l}{\frac{d}{dx}\log p^{(l)}(x)} \right)^n\tag{1}$$</span>
for each integer <span class="math-container">$l$</span> with <span class="math-container">$0\leq l<n-1$</span>.</p>
<p><strong>Conjecture 1.</strong> <em>Let</em> <span class="math-container">$P(x)$</span> <em>a polynomial of degree</em> <span class="math-container">$1<n$</span>, <em>thus we assume</em> <span class="math-container">$P^{(n)}(0)\neq 0$</span>. <em>If the equation</em> <span class="math-container">$$P(x)=\frac{P^{(n)}(0)}{n!}\cdot\left(\frac{n-l}{\frac{d}{dx}\log P^{(l)}(x)} \right)^n$$</span> <em>holds for each integer</em> <span class="math-container">$0\leq l<n-1$</span>, <em>then</em> <span class="math-container">$P(x)$</span> <em>has the form</em> <span class="math-container">$$P(x)=\frac{P^{(n)}(0)}{n!}\cdot(x-\alpha)^n$$</span> <em>for some element</em> <span class="math-container">$\alpha\in K$</span>.</p>
<p>I know the statement of the called Casas-Alvero conjecture from Wikipedia <a href="https://en.wikipedia.org/wiki/Casas-Alvero_conjecture" rel="nofollow noreferrer"><em>Casas-Alvero conjecture</em></a>. I've speculated (while I don't know if this has a good mathematical content, or if these ideas are in the literature in some way more or less explicit) if from previous simple idea one can to state an equivalent form of Casas-Alvero conjecture.</p>
<p><strong>Conjecture 2.</strong> <em>Let</em> <span class="math-container">$P(x)$</span> <em>a polynomial of degree</em> <span class="math-container">$1<n$</span> <em>and leading coefficient</em> <span class="math-container">$a_n\neq 0$</span>. <em>The Casas-Alvero conjecture is equivalent to that the equation</em> <span class="math-container">$$P(x)\cdot\left(P^{(l+1)}(x)\right)^n=a_n\left((n-1)P^{(l)}(x)\right)^n,\tag{2}$$</span>
<em>holds for each integer</em> <span class="math-container">$0\leq l<n-1$</span>.</p>
<blockquote>
<p><strong>Question.</strong> Can you prove or refute (providing a counterexample, or reasoning if my conjectures have not a good mathematical content) these conjectures Conjcture 1 and Conjecture 2? <strong>Many thanks.</strong></p>
</blockquote>
<p>I don't know if these conjectures Conjecture 1 and Conjecture 2 are in the literature, my idea was very simple: that is to study the logarithmic derivative of derivatives of a given polynomial of the cited form, if it is in the literature please refer it in a comment or in your answer and I try to search and study this from the corresponding articles.</p>
<p>Remark: recently a professor solve a related question [1].</p>
<h2>References:</h2>
<p>[1] <em>Iterated derivatives and polynomials that are the power of a linear polynomial</em>, Mathematics Stack Exchange (Sep 12, 2021), post with identificator MSE <a href="https://math.stackexchange.com/questions/4248459/iterated-derivatives-and-polynomials-that-are-the-power-of-a-linear-polynomial"><strong>4248459</strong></a></p>
https://mathoverflow.net/q/4224290Decompose a Hilbert space into two invariant subspacesmathbeginnerhttps://mathoverflow.net/users/1531962022-05-13T07:34:47Z2022-05-16T17:16:27Z
<p><a href="https://i.stack.imgur.com/l8pi2.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/l8pi2.jpg" alt="enter image description here" /></a>The following conclusion is from Bourin, Lee, Pinchings and positive linear maps <a href="https://arxiv.org/abs/1505.02341" rel="nofollow noreferrer">arXiv:1505.02341 [math.FA]</a> <a href="https://www.zbmath.org/?q=an%3A1345.46050" rel="nofollow noreferrer">zbMath</a></p>
<p>Let <span class="math-container">$Q$</span> be an idempotent in <span class="math-container">$L(H)$</span>.Then we have a decomposition <span class="math-container">$H=H_s\oplus H_{ns}$</span> in two invatiant subspaces of <span class="math-container">$Q$</span> such that <span class="math-container">$Q$</span> acts on <span class="math-container">$H_s$</span> as a self-adjoint projection <span class="math-container">$P$</span>, and <span class="math-container">$Q$</span> acts on <span class="math-container">$H_{ns}$</span> as a purely non-self-adjoint idempotent , that is <span class="math-container">$Q_{H_{ns}}$</span> is unitarily equivalent to an operator on <span class="math-container">$F\oplus F$</span> of the form <span class="math-container">$
Q_{H_{ns}}\cong\begin{pmatrix}I & 0\\R& 0\end{pmatrix}$</span>.</p>
<p>My question : how to construct <span class="math-container">$H_s$</span> and <span class="math-container">$H_{ns}$</span>?. It is natrual to think of the following decomposition :<span class="math-container">$H=QH\oplus (I-Q)H$</span>, but <span class="math-container">$Q|_{QH}$</span> is not a projection.</p>
https://mathoverflow.net/q/4224026A curious $q$-series identity on a truncated Euler functionHenryhttps://mathoverflow.net/users/455532022-05-12T19:47:09Z2022-05-16T16:23:05Z
<p>Recall that a <a href="https://en.wikipedia.org/wiki/Q-Pochhammer_symbol" rel="noreferrer"><span class="math-container">$q$</span>-Pochhammer symbol</a> is defined as
<span class="math-container">$$
(x)_n = (x;q)_n := \prod_{l=0}^{n-1}(1-q^l x).
$$</span></p>
<p>I found the following curious <span class="math-container">$q$</span>-series identity that seems to hold for any <span class="math-container">$n\geq 0$</span>:
<span class="math-container">$$
(-1)^{n+1}q^{\frac{(n+1)(3n+2)}{2}}\sum_{j\geq 0}q^{j}(q^{j+1})_{n}(q^{j+2n+2})_{\infty} \overset{?}{=} \sum_{\substack{k\in \mathbb{Z}\\ |k| > n}}(-1)^{k}q^{\frac{k(3k-1)}{2}}.
$$</span>
Note, the right-hand side is a truncated version of the <a href="https://en.wikipedia.org/wiki/Euler_function" rel="noreferrer">Euler function</a>
<span class="math-container">$$
\phi(q) := (q)_{\infty} = \sum_{k\in \mathbb{Z}}(-1)^{k}q^{\frac{k(3k-1)}{2}}.
$$</span></p>
<p>How can we prove the above identity? Any suggestions/ideas would be greatly appreciated!</p>
https://mathoverflow.net/q/4223030Centralizer of an element in a matrix Lie group whose Jordan form is givenuser837898https://mathoverflow.net/users/3502972022-05-11T15:34:12Z2022-05-16T16:27:04Z
<p><span class="math-container">$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}$</span>Let <span class="math-container">$G\subset \GL_n(\mathbb{C})$</span> be a complex matrix Lie group, (e.g., <span class="math-container">$\SO_n$</span>, <span class="math-container">$\Sp_{n}$</span>), and let <span class="math-container">$x\in G$</span> with Jordan form <span class="math-container">$u$</span> (<span class="math-container">$u$</span> may not belong to <span class="math-container">$G$</span>) given, I wonder how to compute the centralizer <span class="math-container">$Z_{G}(x)$</span>.</p>
<p>I know the Lie algebra of <span class="math-container">$Z_{G}(x)$</span> is <span class="math-container">$\{Y\in \mathfrak{g}\mid \Ad(x)Y=Y\}$</span>. But it's hard for me to find a matrix <span class="math-container">$t\in \GL_n(\mathbb{C})$</span> such that <span class="math-container">$tut^{-1}\in G$</span>. I tried to find <span class="math-container">$U\in \mathfrak{gl}_n(\mathbb{C})$</span> such that <span class="math-container">$\exp(U)=u$</span> and then find a <span class="math-container">$T\in \mathfrak{gl}_n(\mathbb{C})$</span> such that <span class="math-container">$\ad(T)(U)\in \mathfrak{g}$</span>, but I don't think that means <span class="math-container">$\exp(T)u(\exp(T))^{-1}\in G$</span>.</p>
<p>Any help will be appreciated.</p>
https://mathoverflow.net/q/4222782Counting Euler circuits through labelled trees where $v_1$ and $v_2$ have distance twoTardishttps://mathoverflow.net/users/4094122022-05-11T11:01:42Z2022-05-16T14:20:02Z
<p>Let <span class="math-container">$T_n$</span> be the set of all labelled trees with <span class="math-container">$n$</span> vertices. For any <span class="math-container">$T \in T_n$</span> let <span class="math-container">$D(T)$</span> be the 'doubled tree', where each edge of <span class="math-container">$T$</span> is replaced by one directed edge in each direction. <span class="math-container">$D(T)$</span> is now an Eulerian directed graph with <span class="math-container">$2(n-1)$</span> edges and by the B.E.S.T theorem it has <span class="math-container">$\prod\limits_{i=1}^n (\deg_{D(T)}(v_i)-1)!$</span> many Euler circuits.</p>
<p>Let <span class="math-container">$A_n \subset T_n$</span> be the set of all labelled trees with <span class="math-container">$n$</span> vertices, with the property that vertices <span class="math-container">$v_1$</span> and <span class="math-container">$v_2$</span> have a distance of <span class="math-container">$2$</span>, i.e. there is no connection between <span class="math-container">$v_1$</span> and <span class="math-container">$v_2$</span>, but there is a <span class="math-container">$t \in \{3,...,n\}$</span> such that there are edges <span class="math-container">$(v_1,v_t)$</span> and <span class="math-container">$(v_2,v_t)$</span>.</p>
<p><strong>Question:</strong> Is there a simpler formula for the expression
<span class="math-container">$$
\sum\limits_{T \in A_n} \prod\limits_{i=1}^n (\deg_{D(T)}(v_i)-1)! \ ,
$$</span>
which counts the number of Euler circuits through trees from <span class="math-container">$A_n$</span>.</p>
<p>If the distance is prescribed as <span class="math-container">$1$</span> instead of <span class="math-container">$2$</span>, then by using the fact that there are <span class="math-container">${n-2 \choose d_1-1,...,d_{n}-1}$</span> many <span class="math-container">$T \in T_n$</span> with <span class="math-container">$d_i = \deg_{T}(v_i) = \deg_{D(T)}(v_i)$</span> the formula is found to be <span class="math-container">$2 \frac{(2n-3)!}{n!}$</span>.</p>
<p>Another way of formulating the question would be: For given <span class="math-container">$d_1,...,d_{n} \in \{1,...,n-1\}^n$</span> with <span class="math-container">$d_1+...+d_n = 2n-2$</span>, how many of the <span class="math-container">${n-2 \choose d_1-1,...,d_{n}-1}$</span> many <span class="math-container">$T \in T_n$</span> with <span class="math-container">$d_i = \deg_{T}(v_i) = \deg_{D(T)}(v_i)$</span> satisfy the condition that <span class="math-container">$v_1$</span> and <span class="math-container">$v_2$</span> have distance two?</p>
<p>Any help is much apprechiated.</p>
https://mathoverflow.net/q/4204740VC dimension of a certain derived class of binary functionsdohmatobhttps://mathoverflow.net/users/785392022-04-16T08:32:05Z2022-05-16T15:02:10Z
<p>Let <span class="math-container">$X$</span> be a measurable space and let <span class="math-container">$P$</span> be a probability distribution on <span class="math-container">$X \times \{\pm 1\}$</span>. Let <span class="math-container">$F$</span> be a function class on <span class="math-container">$X$</span>, i.e., a collection of (measurable) functions from <span class="math-container">$X$</span> to <span class="math-container">$\mathbb R$</span>. Fix <span class="math-container">$\alpha \in \mathbb R$</span> and <span class="math-container">$\beta > 0$</span>, and cobsider a derived function class on <span class="math-container">$H := \{\ell_f \mid f \in F\}$</span> on <span class="math-container">$X \times \{\pm 1\}$</span>, where for each <span class="math-container">$f \in F$</span>, the new function <span class="math-container">$\ell_f:X \times \{\pm 1\} \to \{0,1\}$</span> is defined by
<span class="math-container">$$
\ell_f(x,y) = \begin{cases}
1,&\mbox{ if }|yf(x)-\alpha| \ge \beta,\\
0,&\mbox{ otherwise.}
\end{cases}
$$</span></p>
<blockquote>
<p><strong>Question.</strong> <em>What is the a good upper-bound on VC-dimension of <span class="math-container">$H$</span> in terms of some complexity measure associated with <span class="math-container">$F$</span> (e.g., Rademacher complexity of <span class="math-container">$F$</span>, VC-dimension of <span class="math-container">$\mathrm{subgraph}(F) := \{A_f \mid f \in F\}$</span>, where <span class="math-container">$A_f := \{x \in X \mid f(x) \le 0\}$</span>, etc.) ?</em></p>
</blockquote>
<p>I'm particularly interested in the case where <span class="math-container">$X=$</span> euclidean <span class="math-container">$\mathbb R^d$</span> (or unit-sphere in <span class="math-container">$\mathbb R^d$</span>) and <span class="math-container">$F$</span> is the collection of functions <span class="math-container">$f:\mathbb R^d \to \mathbb R$</span> of the form <span class="math-container">$f(x) \equiv x^\top w + c$</span>, for some <span class="math-container">$b \in \mathbb R$</span> and unit vector <span class="math-container">$w \in \mathbb R^d$</span>.</p>
<p><strong>Related: <a href="https://mathoverflow.net/q/420435/78539">Rademacher complexity of function class $\{(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$</a></strong></p>
https://mathoverflow.net/q/4204680What is the best way to choose initial basis when applying simplex method to an equality form of LP?sansaquahttps://mathoverflow.net/users/4806642022-04-16T06:12:43Z2022-05-16T16:04:00Z
<p>Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not familiar with how to choose the initial basis for simplex method, and would like some tips.</p>
<p>Think about a standard form of LP: maximize <span class="math-container">$cx$</span> s.t. <span class="math-container">$Ax \le b, x \ge 0$</span>. In this case we can choose a obvious basis, i.e., columns of slack variables. For example, if <span class="math-container">$A = \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix}$</span>, then we extend <span class="math-container">$A$</span> with slack variables (<span class="math-container">$A' = \begin{bmatrix} 1 & 2 & 1 & 0\\ 3 & 4 & 0 & 1\end{bmatrix}$</span>) and can choose the columns 3 and 4 as an initial basis. (It doesn't necessarily provide a basic <em>feasible</em> solution but it doesn't matter here.)</p>
<p>A rough version of my question is, what is the best way to choose the initial basis for <em>equality</em> constraints? I know a transform from an equality constraint to inequality ones at least works. For example, we can replace an equality constraint <span class="math-container">$x_1 + 2x_2 + 3x_3 = 4$</span> with two inequality contraints <span class="math-container">$x_1 + 2x_2 + 3x_3 \le 4, x_1 + 2x_2 + 3x_3 \ge 4$</span> and add two columns for slack variables. Naively thinking, however, we can directly apply simplex method against an equality form of LP (i.e. maximize <span class="math-container">$cx$</span> s.t. <span class="math-container">$Ax = b, x \ge 0$</span>) without any transform, if we can choose a set of linearly independent columns of matrix <span class="math-container">$A$</span>. Of course <span class="math-container">$A$</span> can be rank-deficient, but we can at least reduce the number of extended columns if we know the maximal linearly independent columns of <span class="math-container">$A$</span>. Is this `basis' problem as hard as e.g. LP?</p>
<p>The reasons I'd like to avoid a transform to two constraints are:</p>
<ul>
<li>It increases the size of instance,</li>
<li>and it produces yet another problem related to numerical errors. (That is, we need to choose appropriate epsilon to avoid infeasibility caused by two inequality constraints with the same bound.)</li>
</ul>
<p>A bit detailed version of my questions are as follows:</p>
<ol>
<li>How do well-known LP solvers deal with equality constraints? Do they just replace the equality with inequality? Are my concerns above of little practical importance?</li>
<li>(If the answer to 1 requires solving <span class="math-container">`</span>basis' problem,) is there any practically efficient way to solve the basis problem above for a <em>sparse</em> instance? For a dense instance, I know e.g. Gaussian elimination works.</li>
</ol>
https://mathoverflow.net/q/4204350Rademacher complexity of function class $\{(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$dohmatobhttps://mathoverflow.net/users/785392022-04-15T16:25:28Z2022-05-16T14:06:05Z
<p>Let <span class="math-container">$X$</span> be a measurable space and let <span class="math-container">$P$</span> be a probability distribution on <span class="math-container">$X \times \{\pm 1\}$</span>. Let <span class="math-container">$F$</span> be a function class on <span class="math-container">$X$</span>, i.e., a collection of (measurable) functions from <span class="math-container">$X$</span> to <span class="math-container">$\mathbb R$</span>. Fix <span class="math-container">$\alpha \in \mathbb R$</span> and <span class="math-container">$\beta > 0$</span>, and cobsider a derived function class on <span class="math-container">$H := \{\ell_f \mid f \in F\}$</span> on <span class="math-container">$X \times \{\pm 1\}$</span>, where for each <span class="math-container">$f \in F$</span>, the new function <span class="math-container">$\ell_f:X \times \{\pm 1\} \to \{0,1\}$</span> is defined by
<span class="math-container">$$
\ell_f(x,y) = \begin{cases}
1,&\mbox{ if }|yf(x)-\alpha| \ge \beta,\\
0,&\mbox{ otherwise.}
\end{cases}
$$</span></p>
<p>Let <span class="math-container">$\sigma_1,\ldots,\sigma_n$</span> be an iid sequence of Rademacher <span class="math-container">$\pm 1$</span> random variables, independent of the <span class="math-container">$x_i$</span>'s and <span class="math-container">$y_i$</span>, and define</p>
<p><span class="math-container">$$
\begin{split}
R_n(F) &:= \mathbb E_{\sigma_1,\ldots,\sigma_n}\left[\sup_{f \in F}\sum_{i=1}^n \sigma_i f(x_i)\right]\\
R_n(H) &:= \mathbb E_{\sigma_1,\ldots,\sigma_n}\left[\sup_{h \in H}\sum_{i=1}^n \sigma_i h(x_i,y_i)\right].
\end{split}
$$</span>
Note that <span class="math-container">$R_n(F)$</span> (resp. <span class="math-container">$R_n(H)$</span>) is nothing but the <em>Rademacher complexity</em> of <span class="math-container">$F$</span> (resp. <span class="math-container">$H$</span>).</p>
<blockquote>
<p><strong>Question.</strong> <em>What is a good upper-bound for <span class="math-container">$\mathbb E\ R_n(H)$</span> in terms of <span class="math-container">$\mathbb E\,R_n(F)$</span>, <span class="math-container">$\alpha$</span>, and <span class="math-container">$\beta$</span> ?</em></p>
</blockquote>
<p>I'm particularly interested in the case where <span class="math-container">$F := F_{\mathrm{lin}}$</span>, a function class on <span class="math-container">$\mathbb R^d$</span> defined by
<span class="math-container">$$
F_{\mathrm{lin}} := \{x \mapsto x^\top w + b \mid b \in \mathbb R, \,w \in \mathbb R^d\}.
$$</span></p>
https://mathoverflow.net/q/4191853A generalization of the law of tangentsEmmanuel José Garcíahttps://mathoverflow.net/users/947292022-03-29T11:59:39Z2022-05-16T15:30:34Z
<p>The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.</p>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> be the lengths of the three sides of a triangle, and <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span> be the angles opposite those three respective sides. The law of tangents states that</p>
<p><span class="math-container">$$\frac{\tan\frac12(\alpha-\beta)}{\tan\frac12(\alpha+\beta)}=\frac{a-b}{a+b}.\tag{1}\label{1}$$</span></p>
<p>The law of tangents can be used in any case where two sides and the included angle, or two angles and a side, are known.</p>
<p>Although Viète gave us the modern version of the law of tangents, it was Fincke who stated the law of tangents for the first time and also demonstrated its application by solving a triangle when two sides and the included angle are given (see <a href="http://www.geocities.ws/galois_e/pdf/mollweide%20article.pdf" rel="nofollow noreferrer">Wu - The Story of Mollweide and Some Trigonometric Identities</a>).</p>
<p>A proof of the law of tangents is provided by Wikipedia (see <a href="https://en.wikipedia.org/wiki/Law_of_tangents#Proof" rel="nofollow noreferrer">here</a>).</p>
<p><strong>Generalization</strong>. Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> and <span class="math-container">$d$</span> be the sides of a cyclic convex quadrilateral. Let <span class="math-container">$\angle{DAB}=\alpha$</span> and <span class="math-container">$\angle{ABC}=\beta$</span>, then the following identity holds</p>
<p><span class="math-container">$$\frac{\tan\frac12(\alpha-\beta)}{\tan\frac12(\alpha+\beta)}=\frac{(a-c)(b-d)}{(a+c)(b+d)}.\tag{2}\label{2}$$</span></p>
<p><strong>Proof</strong>. Using the sum-to-product formulas and then the double angle formula for sine we can rewrite the left-hand side of <span class="math-container">$(2)$</span> as follows</p>
<p><span class="math-container">$$\frac{\tan\frac12(\alpha-\beta)}{\tan\frac12(\alpha+\beta)}=\frac{\sin\frac12(\alpha-\beta)\cos\frac12(\alpha+\beta)}{\cos\frac12(\alpha-\beta)\sin\frac12(\alpha+\beta)}=\frac{\sin{\alpha}-\sin{\beta}}{\sin{\alpha}+\sin{\beta}}=\frac{\sin{\frac{\alpha}{2}}\cos{\frac{\alpha}{2}}-\sin{\frac{\beta}{2}}\cos{\frac{\beta}{2}}}{\sin{\frac{\alpha}{2}}\cos{\frac{\alpha}{2}}+\sin{\frac{\beta}{2}}\cos{\frac{\beta}{2}}}.$$</span></p>
<p>Keeping in mind the <a href="https://en.wikipedia.org/wiki/Brahmagupta%27s_formula" rel="nofollow noreferrer">Brahmagupta's formula</a>, substituting from the half-angle formulas (see (1) at <a href="https://geometriadominicana.blogspot.com/2020/07/killing-three-birds-with-one-stone.html" rel="nofollow noreferrer">Killing three birds with one stone</a>) and then factorizing and simplifying we have</p>
<p><span class="math-container">$$\begin{align*}\frac{\tan\frac12(\alpha-\beta)}{\tan\frac12(\alpha+\beta)}&=\frac{\frac{\Delta}{ad+bc}-\frac{\Delta}{ab+cd}}{\frac{\Delta}{ad+bc}+\frac{\Delta}{ab+cd}}=\frac{ab-ad+cd-bc}{ab+ad+cd+bc}=\frac{(a-c)(b-d)}{(a+c)(b+d)}\end{align*}.$$</span></p>
<p><span class="math-container">$\square$</span></p>
<p>The formula \eqref{2} reduces to the law of tangents for a triangle when <span class="math-container">$c=0$</span>.</p>
<p>While I am sure this approach is not the simplest, it has the merit of having suggested to me (when applied in other contexts) what is possibly the <em>first</em> generalization of the law of tangent in over 400 years. This approach also led me to generalize (perhaps also its <em>first</em> generalization in 315 years) Mollweide's (rather Newton's) formula (see <a href="https://geometriadominicana.blogspot.com/2022/01/generalization-of-mollweides-formulas.html" rel="nofollow noreferrer">A generalization of Mollweide's Formula (rather Newton's)</a>).</p>
<p>Crossposted at <a href="https://math.stackexchange.com/questions/4404991/generalization-of-the-law-of-tangents-for-a-cyclic-quadrilateral">MathSE</a>.</p>
<p><strong>Question: Is this generalization known?</strong></p>
https://mathoverflow.net/q/39337411A remarkable identity involving $\chi^2$ random variablesIon Nechitahttps://mathoverflow.net/users/301382021-05-21T08:12:05Z2022-05-16T16:12:38Z
<p>In the process of computing inclusion constants for the complex matrix cube (which is a free spectrahedron), the following identity was proven: for all <span class="math-container">$n \geq 1$</span>,
<span class="math-container">$$\mathbb E \Big| \sum_{i=1}^{2n} x_i^2 - \sum_{j=1}^{2n} y_j^2 \Big| = \mathbb E \Big| \sum_{i=1}^{2n} x_i^2 - \sum_{j=1}^{2n-2} y_j^2 \Big| = 4^{1-n}n \binom{2n}{n},$$</span>
where <span class="math-container">$x_i$</span>,<span class="math-container">$y_j$</span> are i.i.d. standard real Gaussian random variables. The proof we have at this moment is by using the explicit form of the density of the difference of two <span class="math-container">$\chi^2$</span> random variables, see <a href="https://math.stackexchange.com/questions/85249/distribution-of-difference-of-chi-squared-variables">here</a> and also <em>Klar, Bernhard</em>, <a href="http://dx.doi.org/10.1080/00949655.2014.996566" rel="nofollow noreferrer"><strong>A note on gamma difference distributions</strong></a>, <a href="https://zbmath.org/?q=an:07183251" rel="nofollow noreferrer">ZBL07183251</a>.</p>
<p>Since the result is so simple, there should be a more direct and more insightful proof of it.</p>
<p><strong>Question 1:</strong> give an easy, conceptual proof of the identity above.</p>
<p>Consider now the function
<span class="math-container">$$k \mapsto \mathbb E \Big| \sum_{i=1}^{2k} x_i^2 - \sum_{j=1}^{2(n-k)} y_j^2 \Big|$$</span>
from <span class="math-container">$\{0,1,\ldots, n\} \to \mathbb R_+$</span>; as above, the <span class="math-container">$x_i$</span> and <span class="math-container">$y_j$</span> are standard i.i.d. Gaussians.</p>
<p><strong>Question 2:</strong> Show that the function above is unimodular, and that its minimum is attained at <span class="math-container">$k = \lfloor n/2 \rfloor$</span>.</p>
<p>There is a pretty involved proof of the second fact above in <em>Helton, J. William; Klep, Igor; McCullough, Scott; Schweighofer, Markus</em>, <a href="http://dx.doi.org/10.1090/memo/1232" rel="nofollow noreferrer"><strong>Dilations, linear matrix inequalities, the matrix cube problem and beta distributions</strong></a>, Memoirs of the American Mathematical Society 1232. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3455-7/pbk; 978-1-4704-4947-6/ebook). vi, 106 p. (2019). <a href="https://zbmath.org/?q=an:1447.47009" rel="nofollow noreferrer">ZBL1447.47009</a>.</p>
<p><strong>Question 3:</strong> Does the claim in ``Question 2'' hold for arbitrary probability distributions?</p>
<p>Any help or insight about these questions would be appreciated!</p>
https://mathoverflow.net/q/3639433Bounds on number of "non-metric" entries in matricesManfred Weishttps://mathoverflow.net/users/313102020-06-23T17:40:12Z2022-05-16T17:04:25Z
<p><strong>Question:</strong><br />
what upper bounds are known on the number of <em>non-metric</em> entries of finite dimensional square matrices <span class="math-container">$\boldsymbol{A}\in\mathbb{R}^{n\times n}$</span> with strictly positive off-diagonal elements <span class="math-container">$a_{ij}$</span>?</p>
<p>In this context <span class="math-container">$a_{ij}$</span> is defined to <em>metric</em> iff <span class="math-container">$\quad a_{ij}\leqq a_{ik}+a_{kj}\,\forall k\notin\lbrace i, j\rbrace\quad $</span> and <em>non-metric</em> otherwise.</p>
https://mathoverflow.net/q/1028651Solving an arbitrary polynomial in $Z_m$LordLinghttps://mathoverflow.net/users/252682012-07-22T12:00:37Z2022-05-16T16:16:52Z
<p>Say I have a polynomial <span class="math-container">$F$</span> of degree <span class="math-container">$n$</span> with coefficients in <span class="math-container">$Z_m$</span> and I wish to find <span class="math-container">$x$</span> such that <span class="math-container">$F(x)=0$</span> (mod <span class="math-container">$m$</span>). For instance if <span class="math-container">$F(x)=x^{2}-a$</span> the solution would be the modulo <span class="math-container">$m$</span> squareroot of <span class="math-container">$a$</span> (if there is a solution).</p>
<p>I'm primarily interested in solving the general case. We can assume <span class="math-container">$F$</span> and <span class="math-container">$m$</span> may be factored "easily."</p>
<p>One method similar to Hensel lifting I've already roughly considered would involve factoring <span class="math-container">$m=p_0^{a_0}\cdots p_k^{a_k}$</span> and brute forcing <span class="math-container">$x$</span> (mod <span class="math-container">$p_i$</span>) for each <span class="math-container">$i$</span> and lifting them to solutions mod <span class="math-container">$p_i^{a_i}$</span>. This would be problematic if <span class="math-container">$m$</span> contains a large prime power however.</p>
<p>So any of the following would be very useful:</p>
<ul>
<li>Fast algorithms for when <span class="math-container">$m=p^k$</span> is large</li>
<li>Methods to prove there are no solutions</li>
<li>Any proofs or conjectures that would indicate efficient algorithms cannot exist</li>
<li>Any relevant research</li>
</ul>