Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2015-03-04T19:20:06Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/1990740Establishing Duality in Tannakian CategoriesWill Sawinhttp://mathoverflow.net/users/180602015-03-04T19:14:44Z2015-03-04T19:14:44Z
<p>I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid. </p>
<p>However, I find the <a href="http://en.wikipedia.org/wiki/Rigid_category" rel="nofollow">definition</a> of rigid categories somewhat difficult. I don't know how to show that these morphisms are identities.</p>
<p>Is there a way of looking at this that makes it more clear when these identities hold?</p>
<p>In my cases it is obvious that there exists a functor $D$ and isomorphisms:</p>
<p>$$Hom(X,Y) = Hom(D(Y),D(X))$$</p>
<p>$$Hom(X,Y)=Hom(1, Y \otimes D(X) ) $$</p>
<p>$$ X = D(D(X)) $$</p>
<p>$$ D(X \otimes Y) = D(X) \otimes D(Y)$$</p>
<p>So in particular I get maps $1 \to X \otimes D(X)$ and $X \otimes D(X) \to 1$. Is there a simple way of expressing duality in terms of a property of this functor?</p>
http://mathoverflow.net/q/1990710Replacing functors by topologically or simplicially enriched functorsTom Goodwilliehttp://mathoverflow.net/users/66662015-03-04T19:01:08Z2015-03-04T19:01:08Z
<p>I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, perhaps continuous on morphisms in the sense that when a set map $K\to Top(X,Y)$ corresponds to a continuous map $K\times X\to Y$ then the resulting set map $K\to Map(F(X),F(Y))$ corresponds to a continuous map $K\times F(X)\to F(Y)$ when $K$ is a cell.</p>
<p>Question 1: Is this true? If so, what is a reference?</p>
<p>There are also simplicial analogues of this: instead of seeking to replace $F$ by a continuous functor, one could seek to replace it by a simplicial functor, i.e. a map of simplicially enriched categories from $Top$ to $Top$ (so that in addition to assigning an object $F(X)$ to each object $X$ it also assigns a map of ``simplicial hom-sets'' to each pair of objects). I believe this can be done by realizing the simplicial object
$$n\mapsto F(X^{\Delta^n}).$$
Perhaps another way is to use the cosimplicial object
$$n\mapsto F(X\times \Delta^n).$$
Here $F$ might be a functor from simplicial sets to simplicial sets rather than from spaces to spaces.</p>
<p>Question 2: Again, references please.</p>
<p>Question 3: Can a simplicially enriched replacement for $F$ be used to make a continuous replacement? I get a bit muddled here by the thought that while continuity is a property simplicial enrichment is a structure. </p>
http://mathoverflow.net/q/199069-1von Neumann algebras and bidualVeronica Neveshttp://mathoverflow.net/users/688132015-03-04T18:32:15Z2015-03-04T18:32:15Z
<p>Let $A$ be a C$^*$-algebra, $x \in A$ a non-zero positive element of $A$, $r(x)$ the range projection of $x$ in $A^{**}$ and $e = 1 - r(x)$.</p>
<p>How do I show that $I := A \cap e A^{**} e$ coincides with the set $\{ x \in A : exe = e \}$?</p>
<p>I am not sure what is $A \cap e A^{**} e$. It is a C$^*$-subalgebra of $A$ or $A^{**}$?</p>
<p>I need to show that exist a projection $f$ in $A^{**}$ such that $I^{**} = f A^{**} f$ or to show that $I^{**}$ is a closed hereditary C$^*$-subalgebra of $A^{**}$.</p>
<p>In many moments we take elements of the bidual and multiply by elements of the space. I don't know what it really means.</p>
http://mathoverflow.net/q/199068-1Motivation of $a_p$ for non-CM elliptic curvesSlimeonlinehttp://mathoverflow.net/users/664432015-03-04T18:15:45Z2015-03-04T18:15:45Z
<p>For an elliptic curve $E$ without CM let $\overline{E}$ be the good reduction of $E$ modulo $p$ prime. The value $a_p = p+ 1 - \mathbb{F}_p$ is referenced by DDT <a href="http://modular.math.washington.edu/edu/2011/581g/misc/Darmon-Diamond-Taylor-Fermats_Last_Theorem.pdf" rel="nofollow" > on p.19 </a> and Ribet <a href="https://math.berkeley.edu/~ribet/Articles/toulousela.pdf" rel="nofollow"> on p.5 </a>. However there is very little motivation (if any) provided about the reasoning for the study of $a_p$ and the equation which generates it. </p>
<p>I have not seen reference to this value elsewhere and am looking for the motivation and/or references to further my study.</p>
http://mathoverflow.net/q/1990661Simple example game with 3 playersmathiasjhttp://mathoverflow.net/users/688112015-03-04T18:05:30Z2015-03-04T18:05:30Z
<p>I am currently writing an algorithm to compute different things as nash equilibria, dominated strategies etc for normal-form games. Since I am now trying to extend it to an infinite amount of players, I need a simple game with 3 players, with the nash equilibria in mixed strategies, pure strategies and dominated strategies. As we only handled 2 player games in our lessons I can't really come up with an own game as I don't really have my mind wrapped around the whole thing, while I can extend the code for more players without really knowing if this is the correct approach. I know that this is probably confusing, but I can extend the code to support more than 2 players while I cannot calculate it myself ;)
A really simple game would already be quite a progress in order to test my code. So 3 players with two strategies for each one would probably be enough. I didn't find anything in the internet as this probably is quite an advanced topic...</p>
<p>Thanks!</p>
http://mathoverflow.net/q/1990651Limits at infinity of fellow-travelling sequences in Teichmuller space,leone slavichhttp://mathoverflow.net/users/247682015-03-04T17:56:45Z2015-03-04T18:13:52Z
<p>I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.</p>
<p>Suppose that we have closed surface of genus $g\geq 2$, and a sequence $\{X_n\}_{n\in \mathbb{N}}$ of points in its Teichmuller space $T(S)$.</p>
<p>Suppose furthermore that, for $n\rightarrow \infty$ this sequence converges to a point $\lambda\in \mathbb{P}ML(S)$ in Thurston's compactification of Teichmuller space. For example, we can take a diverging sequence of points on a Teichmuller geodesic associated to a pseudo-Anosov automorphism of $S$.</p>
<p>Now suppose that we have another sequence $\{Y_n\}_{n\in \mathbb{N}}$ of points in $T(S)$ which fellow travels the first, meaning that $$d(X_n,Y_n)\leq r$$
for all $n$ and some $r>0$. Here, $d$ denotes the Teichmuller distance.</p>
<p>Here are the questions:</p>
<ol>
<li><p>Does the sequence $\{Y_n\}$ converge to some point in $\mathbb{P}ML(S)$?</p></li>
<li><p>If (1) is true, is the limit point of $\{Y_n\}$ the same as the limit point of $\{X_n\}$?</p></li>
</ol>
<p>It seems to me that the answer to both questions should be positive, but I am having trouble in proving the above statements, as convergence to a point in the boundary is expressed in terms of ratios between hyperbolic lengths and transverse measures of curves, while Teichmuller distance is related to the stretch factor of Teichmuller maps. These notions are (at least for me) easily related. </p>
<p>Any help is kindly appreciated!</p>
http://mathoverflow.net/q/1990571local rings with finite type maximal idealprochethttp://mathoverflow.net/users/273982015-03-04T16:11:46Z2015-03-04T16:54:37Z
<p>Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal).
Is there a sufficient condition for $A$ to be noetherian?</p>
<p>For example, we know that the completion $\hat{A}$ will be noetherian(<a href="http://stacks.math.columbia.edu/tag/05GH" rel="nofollow">http://stacks.math.columbia.edu/tag/05GH</a>). </p>
<p>How can we "descend"?</p>
http://mathoverflow.net/q/1990561Non-closability of an operatorGeno Whirlhttp://mathoverflow.net/users/338042015-03-04T15:59:42Z2015-03-04T16:29:11Z
<p>Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by
$$T(u_1,u_2) := (u_1' + au_2',0)$$
on $\textrm{Dom} \,T := \{u=(u_1,u_2) \in H \ \vert\ u_j \in C^1[0,1],j=1,2\}$ is supposed to be non-closable, since the domain of the adjoint operator is $\{0\} \oplus L^2(0,1)$, not dense in $H$.</p>
<p>I am having trouble proving that the domain of the adjoint is the one mentioned above, could you please help me?</p>
http://mathoverflow.net/q/1990511Root of positive function in Fourier algebraHannes Thielhttp://mathoverflow.net/users/249162015-03-04T14:26:50Z2015-03-04T19:18:27Z
<p>Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$.</p>
<blockquote>
<p>Question 1: Let $f\in A(G)$ be a function that is pointwise positive. Does the function $\sqrt{f}$ belong to $A(G)$?</p>
</blockquote>
<p>The motivation for this Question is the following:</p>
<blockquote>
<p>Question 2: Given $f\in A(G)$, does the function of absolute values, $|f|$, belong to $A(G)$?</p>
</blockquote>
<p>Since $A(G)$ is closed under passing to complex conjugation, a positive answer to Question 1 would imply a positve answer to Question 2.</p>
<p>Additionally, if $f,|f|\in A(G)$, is there a relation between the norms of $f$ and $|f|$ in $A(G)$?</p>
http://mathoverflow.net/q/1990503Is every connected reductive group over a local field already defined over a global field?Timo Richarzhttp://mathoverflow.net/users/688052015-03-04T14:20:38Z2015-03-04T16:21:09Z
<p>Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does there exist a global field $F$, a place $v$ in $F$ with $F_v\simeq K$ and a connected reductive group $\tilde{G}$ over $F$ such that as groups
$$\tilde{G}\otimes_F F_v\simeq G?$$
I am particularly interested in the case where $K$ is of equal characteristic. Thanks in advance!</p>
http://mathoverflow.net/q/1990491Sections of inverse image sheaf of sheaf of sections of vector bundlerafaelmhttp://mathoverflow.net/users/152922015-03-04T13:27:04Z2015-03-04T13:35:47Z
<p>Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.</p>
<p>In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:</p>
<blockquote>
<p>Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$. </p>
</blockquote>
<p>There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.</p>
<p>On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.</p>
<p>My questions are:</p>
<blockquote>
<p>Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?</p>
<p>Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?</p>
</blockquote>
<p>Any help of reference is appreciated.</p>
<p>P.S. This question was also posted on <a href="http://math.stackexchange.com/questions/1168421/sections-of-inverse-image-sheaf-of-sheaf-of-sections-of-vector-bundle">math.stackexchange</a> few days ago, with no answer.</p>
http://mathoverflow.net/q/1990460Reducedness of scheme theoretic fibers of toric morphismsPedro Monterohttp://mathoverflow.net/users/317242015-03-04T13:10:50Z2015-03-04T15:11:48Z
<p>Let's consider $X=X(\Delta_X)$ and $Y=Y(\Delta_Y)$ two complete $\mathbb{Q}$-factorial toric varieties over an algebraically closed field of characteristic zero, and let $f:X\to Y$ be a flat toric morphism of pure relative dimension $n$.</p>
<blockquote>
<p>Are there combinatorial conditions under which <strong>all</strong> the scheme
theoretic fibers are reduced?</p>
</blockquote>
<p>If we denote by $X_y$ the fiber at the (closed) point $y\in Y$, and we assume that $\operatorname{red} X_y$ is normal (or at least $S_2$) then by the Theorem I.7.3.1 in Kollar's book "Rational Curves on Algebraic Varieties", sufficient condition will be that $X_y$ must be reduced at its generic points. </p>
<p>Maybe it is too naive, but if we can restrict the torus to the fiber then will have a dense group scheme (that must to be reduced in characteristic zero by a <a href="http://mathoverflow.net/questions/22553/are-group-schemes-in-char-0-reduced-yes">theorem of Cartier</a>) on the fiber. Can we do that? Are there examples of such morphisms admitting non reduced fibers?</p>
<p>Another remark is that if we restrict ourselves to the case $n=1$ then by the Theorem II.2.8 in the same book, if we assume that for every $y\in Y$ the 1-cycle theoretic fiber $f^{[-1]}(y)$ is an irreducible and reduced rational curve, then $f$ is a $\mathbb{P}^1$-bundle. So it might be interesting to ask,</p>
<blockquote>
<p>If $n=1$, when can we assure that the 1-cycle theoretic fibers are
reduced?</p>
</blockquote>
<p>Thank you in advance for your comments.</p>
http://mathoverflow.net/q/1990422Largeness, generic, random pointsPeva Blanchardhttp://mathoverflow.net/users/592392015-03-04T12:36:01Z2015-03-04T14:48:32Z
<p>As presented in Oxtoby's book ( <a href="http://link.springer.com/book/10.1007%2F978-1-4615-9964-7" rel="nofollow">http://link.springer.com/book/10.1007%2F978-1-4615-9964-7</a> ), there are two notions of largeness for subspace $Y$ of a given space $X$:</p>
<ol>
<li>Topology: $X$ is a topological space, and $Y$ is large if its complement is a countable union of nowhere dense subsets.</li>
<li>Measure: $X$ is a probabilistic space, and $Y$ is large if $Y$ has measure $1$.</li>
</ol>
<p>These two notions do not coincide in general. They are often used to state how a property $P$ is "true", i.e., true on a large set (either topolgy or measure-wise).</p>
<p>It seems, in some cases, that a property $P$ being true on a large set is equivalent to $P$ being true of a single point. So far, I have seen the notions of</p>
<ol>
<li>Generic point in the context of algebraic geometry. For instance, a polynomial vanishes on a generic point iff it vanishes everywhere (see e.g. <a href="http://mathoverflow.net/a/92031/59239">http://mathoverflow.net/a/92031/59239</a>)</li>
<li>Martin-Löf random point: this has a more measure-theoretic flavour, as it can be defined as a point not satisfying any Martin-Löf test on a given (computable) probability space. Yet, here, I'm not sure if a property being true on a martin-löf random point is necessarily true almost everywhere, but my guess is this is true (modulo some constraints on the type of property).</li>
</ol>
<p>Are these similarities superficial ? Are there other examples of notions of largeness which also admit their own definitions of "generic point" ? Is there a systematic (or natural) association of "def of genericity" with "def of largeness" ?</p>
<p>Thanks</p>
http://mathoverflow.net/q/1990190A modified notion of ranksTurbohttp://mathoverflow.net/users/100352015-03-04T08:19:45Z2015-03-04T18:24:28Z
<p>Given $M\in\{0,1\}^{n\times n}$ of rank $r$.</p>
<p>Denote $p^{\Bbb Z_{\geq0}}$ to be collection of finite non-negative powers of prime $p$.</p>
<p>Denote $\mathscr{P}_r[M,p]=\{P\in\{0,p^{\Bbb Z_{\geq0}}\}^{n\times n}:P[i,j]\neq0\iff M[i,j]=1\text{, }\mathsf{rank}(P)=r\}$.</p>
<p>Denote $\mathscr{P}_r[M,a,p]=\{P\in\mathscr{P}_r[M,p]:1\leq\mathsf{rank}_{\Bbb F_p}(P^{[i]})\leq a\text{ in }P=\sum_{i=0}^{r[P]}P^{[i]}p^i\text{, }P^{[i]}\in\{0,1\}^{m\times n}\}$.</p>
<p>Is there sharp upper bound in terms of $r$ to
$$r(s,M,p)=\min_{P\in\mathscr{P}_r[M,(\log r)^s,p]}r[P]\text{ with }{0\leq s<1}?$$</p>
<p>More precisely, with every $\epsilon>0$, is $$r(s,M,p)=2^{o(r^\epsilon)}?$$</p>
http://mathoverflow.net/q/1989992Central limit theorem with degenerate covariance matrixAustenhttp://mathoverflow.net/users/251452015-03-04T00:55:03Z2015-03-04T15:04:40Z
<p>Are there known generalisations of the central limit theorem for several random variables when the covariance matrix is degenerate? </p>
<p>The usual proof of CLT based on characteristic functions (see e.g. <a href="http://en.wikipedia.org/wiki/Central_limit_theorem#Proof_of_classical_CLT" rel="nofollow">Wikipedia</a>) yields a degenerate multivariate normal distribution. I'm after any kind of result that resolves that degeneracy. Before taking the limit one evidently has a well behaved probability distribution for the sum of $N$ instances of each of the random variables, and one may anticipate that the width of the distribution of the sum in the degenerate directions is less than $\sqrt{N}$. But can more be said in general?</p>
http://mathoverflow.net/q/19899110What other books are like these?Clark Kimberlinghttp://mathoverflow.net/users/614262015-03-03T22:52:11Z2015-03-04T13:49:12Z
<p>A certain class of books is defined as follows: (1) the book was kept for years in a cafe or mathematics library; (2) the primary contents are research problems and comments, handwritten by resident and visiting mathematicians; (3) the book still exists. Examples include the <a href="http://kielich.amu.edu.pl/Stefan_Banach/e-duda.html" rel="nofollow">Lwów and Wrocław Scottish Books</a> in Poland and the Boneyard Book in the University of Illinois Archives. What are some others?</p>
http://mathoverflow.net/q/1989890Combinatorial support set in CRTTurbohttp://mathoverflow.net/users/100352015-03-03T22:03:44Z2015-03-04T18:29:54Z
<p>Is there a function $g(s)$ such that if there is a set of numbers $\{r_i\}_{i=1}^m$ such that $r_i\bmod p_j\in\{0,1\}$ at every prime in $\{p_j\}_{j=1}^n$ such that $2^t\bmod p_j\neq1$ at every $t\in\Bbb N_{\leq m}$ with condition that at every subset $\mathscr{X}\subseteq \{p_j\}_{j=1}^n$, there exists no subset of numbers $\mathscr{Y}\subseteq \{r_i\}_{i=1}^m$ such that
$$g(s)>\log mn - \log |\mathscr{X}||\mathscr{Y}|\text{ and }$$
$$\big(\prod_{r_i\in\mathscr{Y}}r_i\big)\bmod p_j=1\text{ or }\big(\prod_{r_i\in\mathscr{Y}}(r_i+1)\big)\bmod p_j=1$$</p>
<p>then real matrix given by $M[i,j]=r_i\bmod p_j$ cannot be rank $s$? </p>
<p>What if we relax condition $2^t\bmod p_j\neq1$ at every $t\in\Bbb N_{\leq m}$?</p>
http://mathoverflow.net/q/1989803Balancing real numbers in one dimensionfundahttp://mathoverflow.net/users/393592015-03-03T20:18:37Z2015-03-04T16:26:40Z
<p>Given numbers $0 \leq d_i \leq 1$ for $i=1,\ldots,m$, it is easy to see that you can always find signs $\varepsilon_i \in \{-1,1\}$ such that the partial sums $\sum_{i=1}^k \varepsilon_i d_i/2$, for $k=0,\ldots,m$(where $\sum_{i=1}^0 = 0$) lie in an interval of size at most 1. Is this best possible? i.e. is there a choice of $m$ and $d_i$ such that for any choice of signs the strip must be of size at least 1-$\delta$?</p>
<p>Such an example for $\delta = 1/4$ is to take $m = 3$, $d_1 = 1 = d_3$ and $d_2 = 1/2$. What about for smaller $\delta$?</p>
http://mathoverflow.net/q/1989663Are isometric homorphisms of C* algebras *-homorphismsSimon Henryhttp://mathoverflow.net/users/221312015-03-03T17:32:17Z2015-03-04T19:07:33Z
<p>Here is my precise question:</p>
<p>Let $A$ and $B$ be two $C^*$ algebras. Let $f: A \rightarrow B$ a morphism of algebras which is isometric (on its image). Is $f$ necessary a $*$-homomorphism ?</p>
<p>It sounds like a basic question, but I haven't found any counterexample nor any basic reference mentioning this kind of result, so I hope it is non trivial and suitable for MO.</p>
<p>Another equivalent question is the following:</p>
<p>Let $A$ be a $C^*$-algebra and $x$ an element of $A$ such that:</p>
<p>1) the spectrum of $x$ is included in $\mathbb{R}$</p>
<p>2) for any polynomial $P$ (with coefcients in $\mathbb{C}$) the norm of $P(x)$ is the supremum of $|P(t)|$ for $t \in \text{Spec}(x)$.</p>
<p>Is $x$ necessarily self-adjoint ?</p>
<p>Indeed if the answer to this second question is yes, then any isometric algebra homorphism send self adjoint element to self adjoint element hence is a $*$-homorphism, and conversely if the answer to the first question is yes then for such an element $x$ one can construct an isometric morphism from $\mathcal{C}(\text{Spec}(X))$ to $A$ which is hence a $*$-homomorphism and hence $x$ is self adjoint as the image of a self adjoint element.</p>
<p>Moreover, If I'm not mistaken the answer to these two questions is yes at least for finite dimensional algebras.</p>
<p>I don't really have a precise motivation for this question except curiosity, but it might be interesting to have such a "$*$-free" characterization of morphisms of $C^*$-algebras if one want to develop a satisfying analogue to $C^*$ algebras for other valued field than $\mathbb{R}$ and $\mathbb{C}$.</p>
http://mathoverflow.net/q/1989593Staircase Schur functions squaredZach Hamakerhttp://mathoverflow.net/users/77172015-03-03T16:09:30Z2015-03-04T15:20:55Z
<p>Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by the partition $\lambda$. I would also be interested in objects corresponding to $s^{\Delta_n} s^{\Delta_{n+1}}$.</p>
<p>By non-obvious, I mean to avoid simple simple characterizations that avoid of any particular structure of $\Delta_n$ such as:</p>
<ul>
<li><p>The monomial generating function over semi-standard Young tableaux of skew-shape $\Delta_{2n} \setminus (n)^n$ where $(n)^n$ is the square partition.</p></li>
<li><p>The generating function over Gessel's fundamental quasisymmetric functions indexed by descent sets of the standard Young tableau of the same skew shape.
Essentially these are pairs of standard Young tableaux of shape $\Delta_n$ sharing the index set $\{1,,2,\dots,2n\}$. There are many equivalent objects using standard bijections such as RSK and Edelman-Greene.</p></li>
</ul>
<p>Note the number of standard Young tableaux of skew-shape $\Delta_{2n} \setminus (n)^n$ is
$$ f^{\Delta_{2n} \setminus (n)^n} = {2 {n \choose 2} \choose {n\choose 2}} (f^{\Delta_n})^2 $$
where $f^{\lambda}$ is the number of standard Young tableau of shape $\lambda$.
The first five terms of this sequence are 1, 2, 80, 236544, 108973522944, unless I screwed something up. Note this does not appear in OEIS.</p>
<p>While my interest is primarily in combinatorial objects, I would also be quite interested in cases where $s_{\Delta_n}^2$ arises from geometry or representation theory in a non-obvious way.</p>
http://mathoverflow.net/q/1989273regular polyhedra (and polytopes) in hyperbolic geometry, and generalisationsFeldmann Denishttp://mathoverflow.net/users/171642015-03-03T08:22:55Z2015-03-04T13:27:33Z
<p>While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably very well-known (even if I couldn't find any reference).</p>
<p>Actually, if we define a regular graph as a planar graph which (drawn on a sphere) has "faces" all surrounded by $n\ge 3$ edges and vertices all with $k\ge 3$ edges, there exist only the five regular graphs corresponding to the usual platonic solids, and now this result is a pure combinatorial result, using only $S+F=A+2$.</p>
<p>All this implies that the usual metric proofs of the impossibility of more than 5 platonic solids are quite misleading, but is there a general theory?</p>
<p>For example, with a similar definition of a regular graph on a surface of genus $g$, we get $S+F=A+2-2g$, so $nF/k+F=nF/2 +2-2g$, implying for $g=2$, $n=3$ and $k=7$ the uniqueness of a graph with 12 vertices, 42 edges and 28 triangles. It is not completely obvious that such a graph is impossible, but I am almost sure of it...</p>
http://mathoverflow.net/q/1988785A balls and urns model for a hashing problemMark Wildonhttp://mathoverflow.net/users/77092015-03-02T19:43:13Z2015-03-04T19:01:13Z
<p>Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c \in \{1,\ldots, N\}$ uniformly at random. Then choose further $b_1, \ldots, b_r \in \{1,\ldots, N\}$, so that $b_i$ is chosen uniformly at random from
$\{1,\ldots,N\} \backslash \{b_1,\ldots,b_{i-1}\}$, stopping as soon as ball $b_r$ and ball $c$ are in the same urn.</p>
<blockquote>
<p>What is the expected value of $r$?</p>
</blockquote>
<p>I can get some <a href="https://wildonblog.wordpress.com/2015/02/23/how-much-do-hash-collisions-help-an-attacker/">fairly crude upper and lower bounds</a>. I would like an asymptotically correct answer.</p>
<p>One possible approach to the problem is to approximate the number of balls in urn $j$ by a Poisson random variable with mean $1$. So I would also be interested in the answer to the following question.</p>
<blockquote>
<p>Let $B_1,\ldots, B_N$ be independent Poisson random variables with mean $1$. What is the expected value of $r$ if we start with $B_j$ balls in urn $j$, for each $j$? </p>
</blockquote>
<p><b>Motivation.</b> Suppose $\{1,\ldots, N\}$ are permitted passwords, and that passwords are hashed using an idealized hash function $h : \{1,\ldots, N\}\rightarrow \{1,\ldots, N\}$, constructed so that each $h(b)$ is chosen uniformly at random from $\{1,\ldots, N\}$. Then $r$ is the expected number of hashes we must compute to obtain a password $b \in \{1,\ldots,N\}$ with the same hash as a randomly chosen $c \in \{1,\ldots, N\}$. </p>
<p>Very possibly the answer to my question is out there in the cryptography literature, but if so, I'm finding it hard to find among all the papers dealing with the birthday paradox or other types of hash collision.</p>
http://mathoverflow.net/q/1988193Galois correspondence subgroups/subsystemsSébastien Palcouxhttp://mathoverflow.net/users/345382015-03-02T07:46:09Z2015-03-04T15:11:33Z
<p>In <a href="http://www.sciencedirect.com/science/article/pii/S0022123697932286#" rel="nofollow">this paper</a> (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: </p>
<p><img src="http://i.stack.imgur.com/S5YgO.png" alt="enter image description here"> </p>
<p>By applying this result to finite groups, we get a Galois correspondence between subgroups and subsystems (of unitary representations). </p>
<p>Such a result seems "fundamental" in the finite group theory but I don't know an "old" reference for it. </p>
<p><em>Question</em>: What's the first reference for this result in finite group theory? </p>
http://mathoverflow.net/q/1987701Tensor calculus on the frame bundleS.S.http://mathoverflow.net/users/666882015-03-01T12:45:18Z2015-03-04T14:53:37Z
<p>Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its derivative respect to a connection $\nabla$, evaluating it at a point, taking its Lie derivative, obtaining the curvature of the Levi-Civita connection etc.</p>
<p>However, there is a dual formulation on the frame bundle $F(M)$ of $M$, but I never knew how to do the same calculations on the frame bundle, and as I understand it is sometimes simpler to work on the frame bundle. I would like to know how a tensor on $M$ is represented from the point of view of the frame bundle, and how are the typical operations (curvature, Lie derivative etc) implemented. A tensor in $M$ is a section of the corresponding tensor vector bundle. How is this mapped to the frame bundle? </p>
<p>For example, given an open set $U$ of the atlas of $M$ I can write $g$ in coordinates as follows</p>
<p>$g = g_{ab}\,dx^{a}\otimes dx^{b}$</p>
<p>What would be the analog local expression from the point of view of the frame bundle?</p>
<p>Finally, I would like to know a reference where these things are explained in detail.</p>
<p>Thanks.</p>
http://mathoverflow.net/q/1977655Why are pushouts the right tool in these setupsRoman Brucknerhttp://mathoverflow.net/users/203562015-02-17T13:52:34Z2015-03-04T13:45:01Z
<p>$\newcommand{\cat}[1]{\mathcal{#1}}$
$\newcommand{\cod}{\operatorname{cod}}$
$\DeclareMathOperator{\dom}{dom}$
$\DeclareMathOperator{\colim}{colim}$</p>
<p>The question is about two pushout constructions G.M. Kelly is using in his paper <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4759448&fulltextType=RA&fileId=S0004972700006353" rel="nofollow" title="A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on">A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on</a> (§14.1) to construct colimts and adjoints in a comma category setting. Sadly, he neither gives a motivation for these constructions, nor does he prove they do what they are supposed to do. And I can't figure out why pushouts are the right tool in this context.</p>
<p>I don't know weather there is a common answer for both constructions, but they appear side by side in a similar setting, so I figured I might as well put them into the same question.</p>
<p>The first one is</p>
<blockquote>
<p>Given a category $\cat{C}$ and an endofunctor $T\colon\cat{C}\to\cat{C}$, a diagram $D\colon I\to T\downarrow\cat{C}$ is given by two functors $X,Y\colon I\to\cat{C}$, and a natural transformation $\kappa\colon T\circ X\Rightarrow Y$. Then the colimit $\colim_I D$ is given by $(\colim X,f,x)$, where $f$ and $x$ are given by the pushout
$$
\require{AMScd}
\begin{CD}
\colim T\circ X@>{\colim\kappa}>> \colim Y\\
@V{\tilde T}VV @VhVV \\
T\colim X@>{f}>> x
\end{CD}
$$
where $\tilde T$ is the canonical comparison map.</p>
</blockquote>
<p>whereas the second is</p>
<blockquote>
<p>Given a category $\cat{C}$, two endofunctors $T,T'\colon\cat{C}\to\cat{C}$ and a natural transformation $\alpha\colon T'\to T$, we obtain a functor $\alpha^*\colon T\downarrow\cat{C}\to T'\downarrow\cat{C}$, which sends $(x,f,y)$ to $(x,f\circ\alpha_x,y)$. This functor has a left adjoint $\alpha_*\colon T'\downarrow \cat{C}\to T\downarrow \cat{C}$ which sends $(x',f',y')$ to $(x',\bar f' ,\bar y' )$, where $\bar f'$ and $\bar y'$ are given by the pushout
$$
\require{AMScd}
\begin{CD}
T'(x)@>{f'}>> y\\
@V{\alpha_x}VV @V\hat f' VV \\
T(x)@>{\bar f'}>> \bar y'
\end{CD}
$$</p>
</blockquote>
<p>I'm not asking for a calculation why these constructions do what they suppose to do, but rather some kind of motivation/explanation, why the author considers using pushouts in the first place and why they are supposed to work in this setting.</p>
<p><strong>[Edit]</strong>
Ok, so the second diagram seems to use what the nlab calls a <a href="http://ncatlab.org/nlab/show/base+change#pullback" rel="nofollow">cobase change</a> along every component of $\alpha$. The respective nlab article itself isn't very useful, though.</p>
http://mathoverflow.net/q/1962381Curve associated to bipartite graphTurbohttp://mathoverflow.net/users/100352015-02-11T06:49:08Z2015-03-04T19:03:00Z
<p>Given real biadjacency matrix $A\in\{0,1\}^{n\times n}$ of a bipartite graph with rank $r\in[2,n-1]$, denote $A(x)$ to be matrix where $0$ is replaced by $x$ and $1$ by $1-x$. Denote $$p_1(t,x)=Det(tI-A(x))$$ $$p_2(t,x)=Det(tI-A(x)A'(x))$$ where $A'(x)$ is transpose of $A(x)$.</p>
<p>One can read off rank of $A$ from highest $m$ such that $t^m|p_2(t,0)$. Roots of polynomial $p_1(t,0)$ have special meaning to graph properties.</p>
<p>Could considering curves $p_1(t,x)=0$ or $p_2(t,x)=0$ shed more light on bipartite graph properties?</p>
http://mathoverflow.net/q/1881744Teichmuller geodesics vs. geodesics in the hyperbolic planeyangleehttp://mathoverflow.net/users/412192014-11-26T21:24:13Z2015-03-04T17:54:16Z
<p>Geodesics in $\mathbb H^2$ have the following properties: </p>
<ol>
<li><p>For every two points in the plane there exists a unique geodesic joining them. </p></li>
<li><p>Every geodesic determines exactly two points on the boundary of $\mathbb H^2$. </p></li>
<li><p>Conversely, every pair of points on $\partial \mathbb H^2$ determine a unique geodesic </p></li>
<li><p>Any two different geodesics that travels at bounded distance are asymptotic. </p></li>
</ol>
<p>How many of the above properties are true for Teichmuller geodesics on the Teichmuller space
of a closed surface with Thurston's boundary? Could you also provide references, please? </p>
http://mathoverflow.net/q/1857277Triangle inequality for $L^1$-norm with respect to a stateMateusz Wasilewskihttp://mathoverflow.net/users/249532014-10-29T18:22:36Z2015-03-04T18:34:18Z
<p>It is well-known that the naive construction of non-commutative $L^p$-spaces is performed only in tracial case. I would like to know if it is really a necessity.</p>
<p>To wit, let $\varphi$ be a normal state on a von Neumann algebra $M$. Suppose that the triangle inequality for the $L^1$-norm induced by $\varphi$ holds, i.e.
$$
\varphi(|x+y|)\leqslant \varphi(|x|) + \varphi(|y|),
$$
where $x,y \in M$ and $|x|:= \sqrt{x^{\ast}x}$. Is it true that $\varphi$ is a trace ($\varphi(x^{\ast}x)=\varphi(xx^{\ast})$)? What if $\varphi$ is only a normal weight?</p>
http://mathoverflow.net/q/1387570Congruences for generalized Franel numbersZurab Silagadzehttp://mathoverflow.net/users/323892013-08-07T07:42:04Z2015-03-04T17:41:32Z
<p>Let us define generalized Franel numbers $f^{(m)}_n$ through recurrence relations:
$f^{(1)}_n=1$ for all $n$, and $$f^{(m)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3f^{(m-1)}_k.$$ In fact $$f^{(m)}_n=\sum\limits_{k_1,\ldots,k_m}\left(\frac{n!}{k_1!\cdots k_m!}\right)^3, \quad \quad \quad \quad \quad \quad \quad \quad (1)$$ where the sum is over all nonnegative $k_1,\ldots,k_m$, such that $k_1+\cdots+k_m=n$. $f^{(2)}_n$ are ordinary Franel numbers: $$f^{(2)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3.$$ Numerical evidence suggests the following congruences (I have checked only for smal values of $n$, $m$ and $p$) $$f^{(m)}_n\equiv 0 \quad (\mathrm{mod}\;m)$$ and
$$ \quad \quad \quad \quad \quad \quad f^{(m)}_p\equiv m \quad (\mathrm{mod}\;p^3)\quad \quad \quad \quad (2)$$ if $p$ is prime. How these congruences can be proved?</p>
<p>Note that (2) is a generalization of the Gary Detlefs conjecture (see <a href="http://oeis.org/A000172" rel="nofollow">http://oeis.org/A000172</a>) $$ \quad \quad \quad \quad \quad \quad f^{(2)}_p\equiv 2 \quad (\mathrm{mod}\;p^3)\quad \quad \quad \quad (3)$$ if $p$ is prime.
Was (3) ever proved?</p>
http://mathoverflow.net/q/1050568$6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomialsSatoshi Nawatahttp://mathoverflow.net/users/176442012-08-19T21:17:52Z2015-03-04T17:34:19Z
<p>Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin. </p>
<p>My Question:</p>
<blockquote>
<p>How much are known about quantum $6j$-symbolos for $U_q({\mathfrak{sl}_n})$? Are there partial results, say when the highest weights are given by small rank symmetric representation (a few horizontal boxes of Young Tableaux)?</p>
</blockquote>
<p>I do not know how one can calculate colored HOMFLY-PT polynomials for non-torus knots without information about quantum $6j$-symbols for $U_q({\mathfrak{sl}_n})$. Physicists guessed the form of HOMFLY-PT polynomials colored by symmetric representation for the figure-eight (See the <a href="http://arxiv.org/abs/1203.5978" rel="nofollow">paper</a> by the ITEP group and the recent <a href="http://arxiv.org/abs/1205.1515" rel="nofollow">paper</a>). However, I want to know more examples for colored HOMFLY-PT polynomials.</p>
<blockquote>
<p>Are there explicit formulae of colored HOMFLY-PT polynomials of non-torus knots? </p>
</blockquote>
<p>In addition, </p>
<blockquote>
<p>is there any way to calculate colored HOMFLY-PT polynomials of non-torus knots without using quantum $6j$-symbolos for $U_q({\mathfrak{sl}_n})$?</p>
</blockquote>
<p>A similar question can be found <a href="http://mathoverflow.net/questions/15800/calculating-6j-symbols-aka-racah-wigner-coefficients-for-quantum-groups">here</a></p>