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curious little things - aggregated feedsenMath Overflow Recent Questions: Why every affine variety admit $C^\infty$-exhaustion which admit the complex homogeneous Monge-Amphre equation and a non-degenerate condition
https://mathoverflow.net/questions/273065/why-every-affine-variety-admit-c-infty-exhaustion-which-admit-the-complex-hom
<p>Why we have always the following fact for affine variety </p>
<p>If $M$ is an affine variety of dimension $n$ then there exists an $C^\infty$ exhaustion $\tau: M\to [0,r)$, $0<r\leq \infty$ such that </p>
<p>on $M^*=M\setminus \tau^{-1}\{0\}$ the function $u=\log \tau$ satisfies</p>
<p>1) $(\partial\bar\partial u)^n=0$</p>
<p>2)$\sqrt{-1}\partial\bar\partial u\geq 0$ and $(\sqrt{-1}\partial\bar\partial u)^{n-1}\neq 0$</p>
<p>outside of the ramification divisor</p>Mon, 26 Jun 2017 10:24:54 -0600Math Overflow Recent Questions: Partitioning a rectangle into different isosceles triangles
https://mathoverflow.net/questions/273064/partitioning-a-rectangle-into-different-isosceles-triangles
<p>After all the discussion raised by <a href="https://mathoverflow.net/questions/45008/partitioning-a-rectangle-into-congruent-isosceles-triangles">this old question</a>, I am wondering about a somewhat complementary one: </p>
<blockquote>
<p>For any given rectangle, does there exist a finite set of pairwise <em>different</em> isosceles triangles which tile it? </p>
</blockquote>
<p>It is easy to tile e.g. a $1\times a$ rectangle for $1<a<2$ by four isosceles triangles, but with two of them being equal. In the case that $a=\sqrt{\frac{5-\sqrt{5}}2}$, we are lucky and can split one of those into two smaller ones, obtaining a tiling into 5 different isosceles triangles (with all occurring angles being multiples of $\frac\pi{10}$). BTW, we can iterate that by splitting the blue triangle again etc., getting tilings of the same rectangle into $k$ different isosceles triangles for all $k\ge5$.<br>
<a href="https://i.stack.imgur.com/imXES.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/imXES.jpg" alt="enter image description here"></a></p>
<p>I am quite sure the answer to the initial question is no, and it may even be interesting to restrict it to the following: </p>
<blockquote>
<p>For which other rectangles is such a tiling known to exist?</p>
</blockquote>
<p>And possibly, it doesn't even make a difference if we allow an <em>infinite</em> set of pairwise different isosceles triangles! </p>Mon, 26 Jun 2017 09:48:13 -0600Math Overflow Recent Questions: Sobolev trace theorem on Lipschitz domains
https://mathoverflow.net/questions/273062/sobolev-trace-theorem-on-lipschitz-domains
<p>Supposing that D is a bounded Lipschitz domain (and not smooth) in $\mathbb{R}^d$. From what I know, it is known that the trace operator is well-defined and continuous from $H^s(D)$ to $H^l(\partial D)$ when $l=s-1/2$ and $ 1/2<s<3/2$. My question is what happens when $s>3/2$, is the above result true? Also I am interested in versions concerning more general spaces like Besov or Tribel-Lizorkin.
Any answer or reference will be appreciated. </p>Mon, 26 Jun 2017 08:31:53 -0600Math Overflow Recent Questions: Categories whose auto-equivalences are naturally isomorphic to the identity
https://mathoverflow.net/questions/273059/categories-whose-auto-equivalences-are-naturally-isomorphic-to-the-identity
<p>Are there any useful characterisation of categories whose auto-equivalences are all naturally isomorphic to the identity? For example, I read in this thread, <a href="https://mathoverflow.net/questions/7793/what-are-the-auto-equivalences-of-the-category-of-groups">What are the auto-equivalences of the category of groups?</a> , that the category of groups is one such category. Is there some nice general property that I can use to check whether or not a category admits of auto-equivalences that are not naturally isomorphic to the identity? If not, it'd be useful just to find some more interesting examples of categories with this property. </p>
<p>More specifically, what examples (if any) are there of auto-equivalences that send every object to an isomorphic object but are not naturally isomorphic to the identity? For instance, I know that any auto-equivalence of SETS must send every object to an isomorphic object, but is it true that any such equivalence is naturally isomorphic to the identity? If so, is this a general property of toposes or just a special case? </p>Mon, 26 Jun 2017 07:56:42 -0600Math Overflow Recent Questions: Understanding poincare conjecture for high dimensions
https://mathoverflow.net/questions/273055/understanding-poincare-conjecture-for-high-dimensions
<p>I am a masters student of mathematics. Me and my friends wish to organize a small seminar, with aim of understanding the poincare conjecture. </p>
<p>We do not wish to delve into the case of 3-manifolds, but only understand it for high dimension. </p>
<p>My question is:
What resources (articles, books) should we use in order to understand the proof and the specific tools that are used in the proof? </p>Mon, 26 Jun 2017 06:51:59 -0600Math Overflow Recent Questions: Share of fortunate people in some pie splitting setting
https://mathoverflow.net/questions/273054/share-of-fortunate-people-in-some-pie-splitting-setting
<p>(This question is a follow-up on an <a href="https://mathoverflow.net/questions/272679/limit-of-biggest-share-of-the-pie">older one</a>.)</p>
<p>A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest gets all of what is left.) </p>
<p>A guest is said to be <strong>fortunate</strong> if his share of pie is strictly greater than the average share (which is $1/N$ of the original pie). Let $f(N)$ denote the number of fortunate guests out of the total of $N$ guests. What is the value of $$\lim\sup_{N\to\infty}\frac{f(N)}{N}$$
?</p>Mon, 26 Jun 2017 06:45:28 -0600Math Overflow Recent Questions: Lower Bound for $\sum_{\{i,j\}\subseteq \partial S \\ (g_i-g_j)^2 \le1}{(g_i-g_j)^2}$
https://mathoverflow.net/questions/273053/lower-bound-for-sum-i-j-subseteq-partial-s-g-i-g-j2-le1g-i-g-j
<p>Let $\emptyset \subsetneq S \subsetneq \{1,\cdots,n\}$ be a set with cardinality $s$, and $g\in\mathbb{R}^n$ be a vector such that
$$\sum_{\{i,j\}\subseteq \{1,\cdots,n\}}{(g_i-g_j)^2} = s(n-s).$$ </p>
<blockquote>
<p><strong>Question.</strong> Is this true?
$$\sum_{\{i,j\}\subseteq \partial S \\(g_i-g_j)^2\le1}{(g_i-g_j)^2} \ge \frac{s(n-s)}{n}$$
where $\partial S$ is the set of all $2$-subsets $\{i,j\}\subseteq \{1,\cdots,n\}$ that exactly one of $i$ or $j$ is in $S$. </p>
</blockquote>Mon, 26 Jun 2017 05:53:55 -0600Math Overflow Recent Questions: On certain sums involving the reciprocals of primes
https://mathoverflow.net/questions/273051/on-certain-sums-involving-the-reciprocals-of-primes
<p>Let $p$ be a prime and $f(x,k)= \sum_{p^{}\leq x} \frac{1}{p^k\log p}$ where $k\geq 1$ is an integer. Is there a known asymptotic expression for $f(x,k)$, even for $k=1$ ?</p>
<p>Motivation: If no such results are known, i'm intending to take this as my Bachelor's thesis problem.</p>Mon, 26 Jun 2017 05:36:52 -0600Math Overflow Recent Questions: What is a fat point?
https://mathoverflow.net/questions/273049/what-is-a-fat-point
<p>In our scriptum we're talking about singularities. And there is the term "fat point" (for example of "tangent of fat point") . I cannot find any definition :-/ Has somebody an idea?</p>Mon, 26 Jun 2017 05:27:44 -0600Math Overflow Recent Questions: Does the Krylov subspace exponential preserve structure?
https://mathoverflow.net/questions/273045/does-the-krylov-subspace-exponential-preserve-structure
<p>It is possible to approximate the action of a matrix exponential $exp(A)$ on a vector $v$ in the corresponding Krylov subspace, i.e. calculate $exp_{Kr}(A)v$ [<a href="http://www-users.cs.umn.edu/~saad/PDF/RIACS-90-ExpTh.pdf" rel="nofollow noreferrer">Saad</a>].</p>
<p>Some sources claim that it is possible to compute the action of $exp_{Kr}(A)V$, where $V$ is a matrix, preserving symplecticity [<a href="https://link.springer.com/article/10.1007/s10543-006-0096-6" rel="nofollow noreferrer">Lopez</a>] (however, the algorithm they provide does not produce the results described).</p>
<p>Are there any works showing that the Krylov-type exponential $exp_{Kr}(A)v$ preserves structure (especially, symplecticity) as does $exp(A)v$, when $v$ is a vector? Or, maybe, there is a way to prove it which I do not see?</p>
<p>For example, if $A\in sp(2n)$, then $\Psi := exp(A) \in Sp(2n)$, and for any two vectors $v, w$ holds true $\omega(v,w) = \omega(\Psi v, \Psi w)$, where $\omega(v,w) = v^T J w$ is the symplectic form. Does it holds for Krylov subspace exp?</p>Mon, 26 Jun 2017 05:02:05 -0600