Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2015-12-01T09:31:38Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2249930Is this a sufficient condition for distributivity of a lattice?drhabhttp://mathoverflow.net/users/402632015-12-01T09:17:06Z2015-12-01T09:24:34Z
<p>I asked this question <a href="http://math.stackexchange.com/q/1548838/75923">here</a> on Math.SE but uptil now it was not answered. So I decided to give it a try here. Thank you in advance.</p>
<p>If a lattice $L$ is distributive then it can be shown that for $a,b,c\in L$: $$[a\wedge b=a\wedge c\text{ and }a\vee b=a\vee c]\implies b=c$$</p>
<p>So for fixed $a,u,v\in L$ there is at most one $b$ such that $u=a\wedge b$ and $v=a\vee b$.</p>
<p>Is the converse of this true? More precisely, is the following statement true?</p>
<blockquote>
<p>If for each triple $a,u,v$ in a lattice there is at most one $b\in L$ such that $u=a\wedge b$ and $v=a\vee b$ then the lattice is distributive.</p>
</blockquote>
http://mathoverflow.net/q/224992-1Some questions regarding the concatenation theory $TC$Thomas Benjaminhttp://mathoverflow.net/users/205972015-12-01T08:03:55Z2015-12-01T08:03:55Z
<p>Consider Grzegorczyk's concatenation theory $TC$, a "weak theory of words over the two letter alphabet $\Sigma$={a,b} [this from Grzegorczyk's and Zdanowski's paper "Undecidability and Concatenation"--my comment]:</p>
<p>$TC1$: x$\frown$(y$\frown$z)=(x$\frown$y)$\frown$z</p>
<p>$TC2$: x$\frown$y=z$\frown$w$\Rightarrow$((x=z$\land$y=w)$\lor$$\exists$u((x$\frown$u=z $\land$y=u$\frown$w)$\lor$(x=z$\frown$u$\land$u$\frown$y=w)))</p>
<p>$TC3$: $\lnot$($\alpha$=x$\frown$y)</p>
<p>$TC4$: $\lnot$($\beta$=x$\frown$y)</p>
<p>$TC5$: $\lnot$($\alpha$=$\beta$), where $\alpha$ and $\beta$ denote one letter words $a$ and $b$ respectively.</p>
<p>It should be noted that in the aforementioned paper Grzegorczyk and Zdanowski prove $TC$ essentially undecidable, however, they also note that $TC$ is also minimally essentially undecidable, e.g.:</p>
<p>"...Indeed, if we drop $TC5$ then we can interpret all axioms in the model for arithmetic without zero $($$\omega$$\setminus${0}, $+$, $1$,$1$$)$. By Presburger['s] theorem this model has a decidable theory."</p>
<p>Suppose now that one drops $TC5$ and adds the following axiom introducing the notion of subtext x$\lt$y, i.e. 'x is a subtext of y':</p>
<p>$TC5a$: x$\lt$y $\Longleftrightarrow$ y=x $\lor$ ($\exists$z,w )(x=y$\frown$z $\lor$ x=z$\frown$y $\lor$x=z$\frown$y$\frown$w)</p>
<p>Question 1: Is this new theory also decidable?</p>
<p>Question 2: Is this theory also consistent?</p>
<p>Question 3: If this theory is consistent, can the primitive recursive functions (appropriately recast in the language of concatenation) be consistently added? Can the first-order predicate calculus with only bounded numerical quantification be consistently added as well?</p>
<p>Question 4. How much of 'contentual number theory' does the resulting theory capture?</p>
<p>(It should be noted that semantical (i.e. model theoretic methods) can be used here, much as Hilbert and Bernays did in the <em>Grundlagen</em>, vol I,ch. 2, "Elementary Number Theory--Finitistic Inference and its Limits.) </p>
http://mathoverflow.net/q/2249913$SL_2(\mathbb{F})$, decomposing $\mathbb{C}\{X\}$ into irreducible $G$-representations and dimensionsuser75004http://mathoverflow.net/users/822142015-12-01T07:59:15Z2015-12-01T07:59:15Z
<p>Let $\mathbb{F}$ be a finite field with $q$ elements and $H = \mathbb{F}^\times$, the multiplicative group of $\mathbb{F}$. It is known that $H$ is a cyclic group of order $q - 1$, so $\widehat{H} = \text{Hom}(H, \mathbb{C}^*)$ is also a cyclic group of order $q - 1$.</p>
<p>Let $G = SL_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0 \in \mathbb{F}^2$. Hence, $G$ acts on $X := \mathbb{F}^2 \setminus \{0\}$. This is a finite set, so we have the $G$-representation in the $\mathbb{C}$-vector space $\mathbb{C}\{X\}$. The space $\text{End}_G\,\mathbb{C}\{X\} := \text{Hom}_G(\mathbb{C}\{X\}, \mathbb{C}\{X\})$, of $G$-intertwiners, is a $\mathbb{C}$-algebra with multiplication defined as a composition of intertwiners.</p>
<p>We also let $H$ act on $X$ by scalar multiplication $H \ni z: x \mapsto z \cdot x$. For each $\chi \in \widehat{H}$, we define a subspace of $\mathbb{C}\{X\}$ as follows:$$\mathbb{C}\{X\}^\chi := \{f \in \mathbb{F}\{X\} : f(z \cdot x) = \chi(z) \cdot f(x) \text{ for all }z \in \mathbb{F}^\times\}.$$What is the decomposition of $\mathbb{C}\{X\}$ into irreducible $G$-representations, and what are the dimensions of these simple representations?</p>
http://mathoverflow.net/q/224990-2Unable to track the 8 letter word [on hold]Xian Xuhttp://mathoverflow.net/users/834932015-12-01T06:54:20Z2015-12-01T06:54:20Z
<p><a href="http://i.stack.imgur.com/7514r.jpg" rel="nofollow">enter image description here</a></p>
<p>The answer would be like world XXXXXXXX</p>
<p>Please help me identify 8 letter word</p>
http://mathoverflow.net/q/2249890completion and convergence of spectral sequenceuser83492http://mathoverflow.net/users/834922015-12-01T06:41:33Z2015-12-01T06:57:41Z
<p>I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly convergent spectral sequence (of abelian groups) that comes from some Postnikov tower, say. What is the obstruction to having a strongly convergent spectral sequence after $p$-adic completion of both sides? If the groups are finitely generated, then this amounts to tensoring by $\mathbb{Z}_p$ which is flat...so everything is fine. What about other cases?</p>
http://mathoverflow.net/q/2249873Goodwillie's notes from MSRI Lecture SeriesJuan Villeta-Garciahttp://mathoverflow.net/users/705012015-12-01T06:27:47Z2015-12-01T06:27:47Z
<p>Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, McCarthy, Hesselholt-Madsen, etc. They should have stuff on FSP's, Algebraic K-Theory, computation of Tate Maps, Calculus, (TC maybe?). Seriously, they're cited in so much stuff, I just need to get my hands on them. </p>
http://mathoverflow.net/q/224986-3Decryption of the image attached [on hold]Xian Xuhttp://mathoverflow.net/users/834912015-12-01T06:23:05Z2015-12-01T06:23:05Z
<p><a href="http://i.stack.imgur.com/YTTA1.jpg" rel="nofollow">enter image description here</a></p>
<p>May you please help getting the message</p>
http://mathoverflow.net/q/2249851Generate harmonic polynomials for a finite groupThomashttp://mathoverflow.net/users/760182015-12-01T05:07:06Z2015-12-01T05:07:06Z
<p>Let $G$ be a finite group and $H_G$ be the set of harmonic polynomials. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. However, is there a similar concept for other finite groups?</p>
http://mathoverflow.net/q/2249821Is Carlos Simpson's Descent available online?Zhaoting Weihttp://mathoverflow.net/users/249652015-12-01T03:53:32Z2015-12-01T03:53:32Z
<p>I am not sure whether this question is suitable for MO. Is the paper "Descent" by Carlos Simpson in the book "Alexandre Grothendieck: A Mathematical Portrait" page 83-142 (or a similar version of that paper) available anywhere online?</p>
http://mathoverflow.net/q/2249811The separated uniform space associated with $(X,\mathfrak{U})$M. A.http://mathoverflow.net/users/830602015-12-01T03:52:06Z2015-12-01T03:52:06Z
<p>If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \mathfrak{U}$. Let $\pi: X\rightarrow X/R$ denote the quotient map and put $(Y,\mathfrak{W})= (X/R,\mathfrak{U}/R)$.</p>
<p>$(Y,\mathfrak{W})$ is called he separated uniform space associated with $(X,\mathfrak{U})$.</p>
<p>I am trying to show that $(Y,\mathfrak{W})$ is a hausdorff uniform space.</p>
<p>$\mathfrak{L}=\{ (\pi\times\pi)(M):M\in \mathfrak{U}\}$ is a basis for a uniformity ($\mathfrak{W}$) on $X/R$.</p>
<p>how can we show that: for every $(\pi\times\pi)(M)\in \mathfrak{L}$; $(\pi\times\pi)(N)\circ(\pi\times\pi)(N) \subset (\pi\times\pi)(M)$ for some $(\pi\times\pi)(N) \in \mathfrak{L}$. </p>
<p>and why $(\pi\times\pi)^{-1}((\pi\times\pi)(M))= R\circ M\circ R$; for every $M\in \mathfrak{U}$ ?</p>
<p>$(Y,\mathfrak{W})$ Used for construct the hausdorff completion of a hausdorff uniform space $(X,\mathfrak{U})$.</p>
http://mathoverflow.net/q/2249780A variance-preserving Boolean functionmath-Studenthttp://mathoverflow.net/users/416662015-12-01T02:51:10Z2015-12-01T02:51:10Z
<p>Let a random variable $X$ be given with $P_X$ supported over $\mathcal{X}$. What are the necessary conditions for the existence of a boolean function $f:\mathcal{X}\to \{0,1\}$ such that $\mathsf{var}(f(X))=\mathsf{var}(X)$? </p>
http://mathoverflow.net/q/2249771Construction of $n$ makes $s_2(nk)<s_2(n)$Peter. Yanghttp://mathoverflow.net/users/799422015-12-01T01:56:31Z2015-12-01T05:02:03Z
<p>$s_2(n)$ denotes the sum of the standard base-2 digits of $n$.</p>
<p>For a fixed odd number $k>1$, can we construct $n\in \mathbb{Z}^+$, to make $s_2(nk)<s_2(n)$?</p>
<p>To clarify, that's not $s_2(nk) \lt s_2(k)$. For example, if $k=7$, we can take $n=23=10111_2$ and $nk=161=10100001_2$.</p>
http://mathoverflow.net/q/2249760References about Hasse diagrams of root systemsuserhttp://mathoverflow.net/users/504372015-12-01T01:48:01Z2015-12-01T03:30:59Z
<p>This is to ask about references of <em>Hasse diagrams</em> of irreducible root systems. I found <a href="http://www.rug.nl/research/portal/files/14628596/03_c3.pdf" rel="nofollow">here</a> and <a href="https://en.wikipedia.org/wiki/E6_%28mathematics%29#/media/File:E6HassePoset.svg" rel="nofollow">there</a> nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical root systems ($A_n, B_n, C_n, D_n$).</p>
<p>Thanks in advance.</p>
http://mathoverflow.net/q/2249682Are there other integer solutions to the equation $9x^3 -1 = y^3$ besides $(x,y) =(1,2)$ and $(0, -1)$?Tatendahttp://mathoverflow.net/users/832362015-12-01T00:30:27Z2015-12-01T07:30:00Z
<p>Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?</p>
http://mathoverflow.net/q/2249530Problem related to Frobenius coin problemTurbohttp://mathoverflow.net/users/100352015-11-30T19:30:57Z2015-12-01T08:28:45Z
<p>Call a pair of integers $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if we have the property that
$$\mbox{ if }ax+by=rs\mbox{ and }ax'+by'=tu\mbox{ for some }r,s,t,u>1,x,y,x',y'>0\mbox{ with }rs,tu,ru,st,rt,su<g(a,b)$$
$$\mbox{then we should have } av+bw=ru\mbox{ and }av'+bw'=st\mbox{ satisfies }vw<0\mbox{ and }v'w'<0$$
$$\mbox{ and }av+bw=rt\mbox{ and }av'+bw'=su\mbox{ satisfies }vw<0\mbox{ and }v'w'<0$$</p>
<p>$g(a,b)$ is Frobenius number.</p>
<p>Is there an $n_0\in\Bbb N$ such that for every $n\in\Bbb N_{>n_0}$ we have a good coprime pair in $[n,2n]$?</p>
<hr>
<p>A bad pair example:</p>
<p>$a=22,b=21,s = 16, t = 17,r = 19,u = 15$</p>
<p>$$10a+4b=rs$$
$$8a+7b=rt$$
$$9a+2b=su$$
$$3a+9b=tu$$</p>
http://mathoverflow.net/q/2249452Positivity of semiclassical pseudodifferential operatorsLiren Linhttp://mathoverflow.net/users/41192015-11-30T18:32:13Z2015-12-01T02:33:01Z
<p>Let me first give some background. (My reference is Martinez's book <em>An introduction to semiclassical and microlocal analysis</em>)</p>
<p>Let $m\in\mathbb{R}$, and $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\to\mathbb{C}$ be a symbol in $S_{2n}(\langle\xi\rangle^{m})$. That is, regarded as a function on $\mathbb{R}^{2n}$, $p$ is smooth, and for every multiindex $\alpha=(\alpha_1,\ldots,\alpha_{2n})$, there is a constant $C_\alpha$ such that $|\partial^\alpha p|\le C_\alpha\langle\xi\rangle^m$. Here $\langle\xi\rangle:=(1+|\xi|^2)^{1/2}$. Associate to such $p$ and small parameters $h>0$ we define the (Weyl) semiclassical pseudodifferential operator ($h$-PDO) $$\mathrm{Op}_h^W(p)u\,(x)=\frac{1}{(2\pi h)^n}\int e^{i(x-y)\cdot\xi/h}p(\frac{x+y}{2},\xi)u(y)\,dy\,d\xi\quad(u\in C^\infty_c(\mathbb{R}^n)).$$ I'm concerned with the validity of $\mathrm{Op}_h^W(p)\ge 0$, that is $\langle\mathrm{Op}_h^W(p) u,u\rangle_{L^2}\ge 0$ for all $u\in C^\infty_c(\mathbb{R}^n)$. The following result is well-known.</p>
<blockquote>
<p><strong>Sharp Gårding inequality</strong> If $p\ge 0$, then there exists a constant $C>0$ such that $$\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge -Ch\|u\|_{H^{m/2}}^2\tag{1}$$ for all small enough $h>0$ (i.e. for all $h$ smaller than some positive constant depending on $C$) and for all $u\in C^\infty_c(\mathbb{R}^n)$.</p>
</blockquote>
<ul>
<li>When $m=0$ (in fact, I don't know if this is needed), the <strong>Fefferman-Phong Inequality</strong> improves the result to $\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge -Ch^2\|u\|_{L^2}^2$. However, $\mathrm{Op}_h^W(p)\ge 0$ is still not guaranteed. </li>
<li>On the other hand, the <strong>Easy Gårding inequality</strong> replaces the assumption $p\ge 0$ by $p\ge p_0\langle\xi\rangle^m$ for some constant $p_0>0$, and the conclusion is $\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge C\|u\|_{H^{m/2}}^2$ for every $0<C<p_0$ (the larger $C$ is chosen, the smaller $h$ is required). Thus, $\mathrm{Op}_h^W(p)\ge 0$.</li>
</ul>
<blockquote>
<p><strong>Question.</strong> If $p\ge 0$, and there exist constants $p_0>0$ and $R>0$ such that $p\ge p_0\langle\xi\rangle^m$ for $|\xi|\ge R$, is it possible to add some "smallness" assumption on the size of the zero set of $p$ to guarantee $\mathrm{Op}_h^W(p)\ge 0$? </p>
</blockquote>
<p>The model example in my mind is $p\in |\xi|^2\in S_{2n}(\langle\xi\rangle^2)$. In this case $p=0$ only at $\xi =0$ (precisely, on $\mathbb{R}^n_x\times\{\xi=0\}$). We cannot apply the Easy Gårding inequality, while $\mathrm{Op}_h^W(|\xi|^2) = -h^2\Delta\ge 0$, since $$\langle-h^2\Delta u,u\rangle_{L^2} = h^2\int |\nabla u|^2.$$ </p>
<p>Here are some more concrete questions: </p>
<ul>
<li>Is it true that $\mathrm{Op}_h^W(p)\ge 0$ if, besides the assumptions given in <strong>Question</strong>, $p=0$ only at a single $\xi$? </li>
<li>How about $p=0$ on a small set in $\mathbb{R}^n_x\times\{|\xi|<R\}$, say with small Hausdorff dimension? </li>
<li>If yes, are there generalizations to systems (matrix-valued $p$)? </li>
<li>Or there are counterexamples for these questions? </li>
</ul>
<p>Indeed, the case of matrix-valued $p$ is crucial for my current research. But any partial answer or relevant references will be appreciated.</p>
http://mathoverflow.net/q/224930-1On the Theory of Infinite Step Processes of Sequential Decision Making [on hold]Boris Modelhttp://mathoverflow.net/users/834692015-11-30T15:44:29Z2015-12-01T06:32:50Z
<p>On the border of Set Theory and Game Theory there is a broad class of Infinite Step Processes of Sequential Decision Making that can be characterized by the main following property: a Future development of the process depends on the process Present state and does not depend directly on the process Past [1] (these processes have the same nature as for example chess and checkers have: a game Future depends on the game Present state and does not depend directly on the game Past).
Some examples of such Infinite Step Processes give us Differential Games in certain posing [2] and Infinite Stage Games of Search and Completion [3,4].</p>
<p>For these Processes with the use of Axiom of Choice some generalizations of basic results, which are well known facts in case of Finite Step Decision Making Processes (for example, existence of a uniformly optimal strategy), can be proved [1,2].</p>
<p>But the question when information about process Past additional to knowledge of process Present state is important for these Infinite Step Processes from the point of view of process optimal result and when this additional information is not important has remained an open question.</p>
<p>For example, for infinite chess and checkers (without rules of stopping) information of game Past history additional to information of game Present state does not improve best guaranteed result but for Infinite Stage Games of Search and Completion, having just the same structure as infinite chess and checkers [1], information of game Past history additional to information of game Present state does improve best guaranteed result [1,3].</p>
<p>So what criteria are dividing having the same structure games, on games, for which information of game Past history additional to information of game Present state does not improve best guaranteed result and on games, for which information of game Past history additional to information of game Present state does improve best guaranteed result?</p>
<p>References</p>
<ol>
<li><p>B. I. Model’, The existences of an overall έ-optimal strategy and validity of Bellman’s
functional equation in an extended class of dynamic processes. I; II, Engineering Cybernetics, No.5, 1975, pp. 13 – 19; No. 6, 1975, pp. 12 – 19.</p></li>
<li><p>B. I. Model’, A certain class of differential games, Engineering Cybernetics, No.2, 1978, pp. 32 – 38.</p></li>
<li><p>B. I. Model’, Games of search and completion, Journal of Mathematical Sciences, Vol. 80, No 2, 1996, pp. 1699 - 1744, Plenum Publishing Corporation, New York.</p></li>
<li><p>U. Abraham, R. Schipperus, Infinite games on finite sets, Israel Journal of Mathematics 159, 2007, pp. 205-219.</p></li>
</ol>
http://mathoverflow.net/q/2249244Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$Feldmann Denishttp://mathoverflow.net/users/171642015-11-30T14:59:30Z2015-12-01T07:48:43Z
<p>Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result :
$$\sum_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{1+ \pi\coth\pi}{2}$$
is known by WolframAlpha. But my question is : does there exists a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?</p>
http://mathoverflow.net/q/22491313"Fractally self-similar" numbersმამუკა ჯიბლაძეhttp://mathoverflow.net/users/412912015-11-30T13:12:17Z2015-12-01T08:36:00Z
<p>This is another question about visualization of <a href="https://en.wikipedia.org/wiki/Ford_circle" rel="nofollow">Ford circles</a>, the previous one being <a href="http://mathoverflow.net/q/224816/41291">Confusion with practically implementing rational approximations</a>. Here is an output of zooming into Ford circles at $\frac1{\sqrt2}$</p>
<p><a href="http://i.stack.imgur.com/0bpLv.gif" rel="nofollow"><img src="http://i.stack.imgur.com/0bpLv.gif" alt="enter image description here"></a></p>
<p>Empirically I found that the pattern repeats periodically and arranged the number of frames so that one gets impression of a continuous infinite zoom (however if one looks very attentively there is a slight jump where the animation restarts).</p>
<p>What I want to know is whether the pattern indeed repeats rigorously for quadratic irrationalities, and whether for other irrational numbers the pattern occasionally becomes "almost" the same (clearly if the number is rational then the picture eventually starts degenerating into the horizontal line).</p>
<p>Since I do not actually know how exactly to formulate it, let me ask the question in form that practically occurred to me: for a real $x$ denote by $P_r(c)$ the picture of Ford circles in the rectangle $(x-c,x+c)\times(0,c)$ with resolution $r$. That is, features of size less than $rc$ cannot be observed; in particular, only the Ford circles of radius $>rc$ are visible, and moreover a circle cannot be distinguished from another one if they are both contained in an annulus of width $<rc$.</p>
<p>For which $x$ do there for any resolution $r$ exist two different $c$ and $c'$ such that the pictures $P_r(c)$ and $P_r(c')$ cannot be distinguished? (Well, they must be "sufficiently different" - say, there is still another $c<c''<c'$ such that $P_r(c'')$ can be distinguished from both.)</p>
<p>And the question about this question - what is (if any) a rigorous mathematical statement behind it?</p>
<p>(Later - decided to add the zoom for the golden ratio, here the jump is almost impossible to notice</p>
<p><a href="http://i.stack.imgur.com/w7Me8.gif" rel="nofollow"><img src="http://i.stack.imgur.com/w7Me8.gif" alt="enter image description here"></a></p>
<p>Circles closest to the center alternate between left and rigth, ratios of their sizes must be something like consecutive Fibonacci numbers...</p>
http://mathoverflow.net/q/224899-1Maximal induced cycles on $n$-clique graphsDominic van der Zypenhttp://mathoverflow.net/users/86282015-11-30T09:07:03Z2015-12-01T09:00:06Z
<p>For any set $X$ we set $[X]^2 = \big\{\{a,b\}: a, b\in X\text{ and } a\neq b\big\}$.</p>
<p>We say a simple undirected graph $G=(V,E)$ is an $n$-<em>clique graph</em> if there are $S_1,\ldots,S_n\subseteq V$ such that</p>
<ol>
<li>$|S_k| = n$ for all $k=1,\ldots, n$;</li>
<li>$V = \bigcup_{k=1}^n S_k$;</li>
<li>$i\neq k \in \{1,\ldots,n\}$ implies $|S_i \cap S_k| = 1$.</li>
<li>$E = \bigcup_{k=1}^n [S_k]^2$ (that is, all the $S_k$ are complete, and there are no edges between different $S_k$.)</li>
</ol>
<p>Let $c(n)$ be the maximum length of an <a href="https://en.wikipedia.org/wiki/Induced_path" rel="nofollow">induced cycle</a> that any $n$-clique graph $G$ can have. Is there an explicit formula for $c(n)$, and if not, what is $\lim_{n\to\infty}\frac{c(n)}{n}$?</p>
http://mathoverflow.net/q/2248071Complexity theory and closed form formulas in analysisplmhttp://mathoverflow.net/users/65752015-11-29T06:54:44Z2015-12-01T05:59:18Z
<p>My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ given as integrals, for given initial conditions $(q(0),p(0))$, of algebraic functions in the hamiltonian $H(p,q)$. (In a broad sense that is, for instance derivatives of polynomials should be considered algebraic operators.) In particular computing approximate solutions of completely integrable hamiltonian systems can be carried out in polynomial time in the time variable, with an oracle for the computation of the hamiltonian to any given finite accuracy (maybe some more conditions on the hamiltonian should be imposed).</p>
<p>More generally the notion of closed form solution should correspond to "relatively easily computable". Or at least computable using well-known functions (e.g. elementary, elliptic, etc.).</p>
<p>Are there articles giving and studying such general definitions of closed forms? Are there articles studying the link between complexity, closed forms, and "chaos"?</p>
<p>Thank you.</p>
http://mathoverflow.net/q/2245810Existence of a fixed point [on hold]Nilanhttp://mathoverflow.net/users/545072015-11-26T07:43:11Z2015-12-01T09:28:58Z
<p>Let $[a,b]$, $[c,d]$ be two intervals of $\mathbb{R}$ with $a\lt c\lt b\lt d$ and $f:[a,b]\to[c,d]$ be a bijective, smooth function. </p>
<p>My question is:</p>
<blockquote>
<p>Which condition(s) would guarantee existence of a fixed point of $f$ ?</p>
</blockquote>
http://mathoverflow.net/q/2241353Assuming $depth M\ge depth N$, what can one say about $depth M_p$ and $depth N_p$?user 1http://mathoverflow.net/users/477632015-11-20T19:46:54Z2015-12-01T07:49:36Z
<p><strong>Definition</strong>. Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite R-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$. Then the common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$ denoted by $grade(I,M)$.<br>
$grade(m,M)$ is denoted by $depth\ M$. So by $depth\ M_p$, we mean $grade(pR_p,M_p)$.<br>
Assuming $depth\ M\ge depth\ N$, what can one say about $depth\ M_p$ and $depth\ N_p$? Is there any inequality between them?<br>
What if we impose additional assumptions on $M$ and $N$? for example if $M=R/I$ and $N=R/J$?</p>
<p>Thank you.</p>
http://mathoverflow.net/q/21435411Nice sign-expansions of special surreal numbersJames Propphttp://mathoverflow.net/users/36212015-08-09T02:28:10Z2015-12-01T04:03:34Z
<p>What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?</p>
<p>I can think of more than one natural way to generalize the notion of a rational number, but "ratio of omnific integers" is not one of them, since every real number is a ratio of omnific integers. Maybe ratios of ordinals are the thing to look at (as a generalization of the non-negative rationals). Or maybe we should look at the Field-closure of the ordinals. Or perhaps we should consider the Class containing every surreal number whose normal form involves only rational numbers at all levels. </p>
<p>I can also think of more than one way to generalize the notion of eventual periodicity to sign-sequences indexed by a general ordinal alpha. One of them is a variant of "Kaufman decimals" (see <a href="https://mchouza.wordpress.com/2013/08/25/kaufman-decimals/">https://mchouza.wordpress.com/2013/08/25/kaufman-decimals/</a> and <a href="http://www.jefftk.com/p/decimal-inconsistency">http://www.jefftk.com/p/decimal-inconsistency</a>) in which the digit-set {0,...,9} is replaced by {$+$,$-$} and every over-bar is assigned an ordinal.</p>
<p>A seemingly different but possibly equivalent notion generalizing eventual periodicity involves a kind of symbolic dynamics I haven't seen before, where the Monoid of ordinals acts on ordinal-indexed sequences: if $s$ is a sequence indexed by some initial segment of the ordinals, and $\iota$ is some ordinal, define $T^{\iota}(s)$ to be the sequence obtained by omitting the first $\iota$ terms of $s$ (with $T^{\iota}(s)$ defined to be the empty sequence if $\iota$ is greater than or equal to the length of $s$). Then eventual periodicity (in the case where $s$ is indexed by the natural numbers) is seen to be a special case of the condition that the orbit of $s$ under the action of the Monoid of all ordinals is finite. (See Joel Hamkins' recent post, showing that constant sequences satisfy this finiteness condition: <a href="http://jdh.hamkins.org/every-ordinal-has-only-finitely-many-order-types-for-its-final-segments/">http://jdh.hamkins.org/every-ordinal-has-only-finitely-many-order-types-for-its-final-segments/</a>.)</p>
<p>If my original question seems too vague (what does "right generalization" mean?), here are two very concrete ones that are relevant: does the surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) lie in the field generated by $\omega$? and, is it expressible as a ratio of ordinals? (Note that this sign-sequence does not satisfy the aforementioned finiteness property, although perhaps it satisfies some weaker regularity condition.) Here $-^{\omega}$ denotes a string of $\omega$-many $-$'s and $\cdots$ denotes that the pattern continues $\omega$ times.</p>
<p>I'd be interested in any implications that might hold between these various properties. </p>
http://mathoverflow.net/q/1954423Convergence of random variables with hypergeometric distributionuser2173168http://mathoverflow.net/users/416382015-02-02T02:03:20Z2015-12-01T02:47:00Z
<p>This is a very interesting conjecture of large scale property of hypergeometric distribution. </p>
<p>Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in a bag, in which $ax$ of them are colored white and the other $N-x$ are colored black. Now randomly select $N$ balls without replacement from the bag. Let random variable $X$ denote the number of white balls selected. For another $y=x+1$, we also select $N$ balls from a bag with $N+(a-1)y$ balls, in which $ay$ are colored white, then we get a similar random variable $Y$.</p>
<p><strong>In short,</strong> $X \sim H(N+(a-1)x,ax,N)$ and $Y \sim H(N+(a-1)y,ay,N)$, with $y=x+1$. </p>
<p>We want to prove that $\mathbb{E}(\frac{xy}{X}-\frac{xy}{Y})\rightarrow \frac{1}{a}$, when $N\rightarrow\infty$. The convergence is uniform w.r.t to $x$, namely, the convergence rate is independent of $x$. (For large enough $N$, the expectation can be arbitrarily close to $1/a$, regardless of which $x<N-1$ we choose. See the comment of the first answer.)</p>
<p>It is likely that the convergence must use a suitable coupling of $X$ and $Y$. </p>
http://mathoverflow.net/q/1885700Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.Youngsuhttp://mathoverflow.net/users/223882014-12-02T02:11:31Z2015-12-01T04:48:56Z
<p>I would like to know a reference of the following statement (or counter example). </p>
<p>Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is Noetherian local and $S_1$ is finitely generated over $S_0$. If $S$ is an integral domain, then $X = \operatorname{Proj} S$ is a irreducible and reduced. Hence $\mathcal{O}_{X,x}$ is an integral domain for any $x \in X$. But what about equidimensionality? That is</p>
<blockquote>
<p>Let $S$ be a Noetherian standard graded equidimensional ring over a Noetherian local ring. Then is the local ring $\mathcal{O}_{X,x}$ equidimensional for all $x \in X$?</p>
</blockquote>
<p>A ring $R$ is called $\textit{equidimensional}$ if $\dim R = \dim R/p$ for any minimal prime $p$ of $R$.</p>
http://mathoverflow.net/q/1425700Chain rule for fractional derivative defined via Fourier transformMiloshttp://mathoverflow.net/users/401992013-09-19T07:58:48Z2015-12-01T04:59:04Z
<p>It is well known that in the case of integer order differentiation the formula $\partial_{x}f(x,u(x))=\partial_{u}f\cdot \partial_{x}u+\partial_{x}f\cdot u$ holds. If we define fractional derivative via Fourier transform <strong><em>F</em></strong> as $D_{x}^{\alpha}u(x)=F^{-1}[{(i\xi)^{\alpha}\widehat{u}(\xi)}]$, where ^ denotes the Fourier transform, is there a similarly formula for $D_{x}^{\alpha}f(x,u(x))$? </p>
http://mathoverflow.net/q/811944Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?John Maidenshttp://mathoverflow.net/users/193412011-11-17T19:23:04Z2015-12-01T05:48:56Z
<p>I am attempting to solve the argument maximization problem</p>
<p>$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$</p>
<p>where the functions $f_1$ and $f_2$ are concave but difficult to evaluate but their convex conjugates $f^∗_1$ and $f^∗_2$ are easy to evaluate. We can further assume that $f^∗_1$ and $f^∗_2$ are differentiable and that we can evaluate their gradients. Since the sum operation is dual to the infimal convolution (or epi-sum) operation</p>
<p>$$(g\#h)(x) = \inf_w \{ g(x−w) + h(w) \} $$</p>
<p>the standard maximization problem is easy to compute by duality using the identity</p>
<p>$$\sup_x\{⟨x,l⟩−f_1(x)−f_2(x)\}=\inf_w \ \{ f^*_1(l−w) + f^∗_2(w) \}.$$</p>
<p>Is it possible to compute the solution to problem $(1)$ is an analogous manner, making only calls to the conjugate functions $f^∗_1$ and $f^∗_2$ or their gradients?</p>
http://mathoverflow.net/q/5552616Example of a variety with $K_X$ $\mathbb Q$-Cartier but not CartierAnonymoushttp://mathoverflow.net/users/129922011-02-15T16:14:38Z2015-12-01T07:05:36Z
<p>I know the definition of $K_X$ on a normal, singular variety, but I don't have a good set of examples in my mind. What's an example of a variety where $K_X$ is $\mathbb Q$-Cartier but not Cartier? Are there any conditions under which an adjunction formula lets me compute the canonical class of a singular divisor?</p>
http://mathoverflow.net/q/2878849nontrivial theorems with trivial proofsMichael Hardyhttp://mathoverflow.net/users/63162010-06-20T01:04:53Z2015-12-01T09:22:21Z
<p>A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because its proof is trivial.</p>
<p>I don't remember who said that, and the person whose door it was posted on didn't remember either.</p>
<p>This leads to two questions:</p>
<p>(1) Who was it? And where do I find it in print---something citable? (Let's call that one question.)</p>
<p>(2) What are examples of nontrivial theorems whose proofs are trivial? Here's a wild guess: let's say for example a theorem of Euclidean geometry has a trivial proof but doesn't hold in non-Euclidean spaces and its holding or not in a particular space has far-reaching consequences not all of which will be understood within the next 200 years. Could that be an example of what this was about? Or am I just missing the point?</p>