Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2015-09-02T06:53:50Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2162080Current status of the linearity of mapping class groupuser6419http://mathoverflow.net/users/626472015-09-02T06:32:50Z2015-09-02T06:43:37Z
<p>In the paper <a href="http://arxiv.org/abs/1012.1032v2" rel="nofollow">A faithful linear-categorical action of the mapping class group of a surface with boundary</a> it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the authors provide a faithful linear-categorical action of the mapping class group of a surface with necessarily non-empty boundary using knot Floer homology that does not seem decategorifiable in a non-trivial way. </p>
<p>On the other hand, there exists the following paper, <a href="http://arxiv.org/abs/math/0312449v8" rel="nofollow">Mapping class groups are linear</a>, in which the author uses the non-commutative machinery to prove that they are. As I have asked in <a href="http://mathoverflow.net/questions/216204/are-all-transversely-oriented-foliations-given-by-closed-forms">this</a> question, there might perhaps be a mistake in one of the proof ingredients.</p>
<p>Hence the question: is it known that a mapping class group of some surface (possibly with non-empty boundary or non-zero number of punctures) is linear?</p>
http://mathoverflow.net/q/2162071unordered configuration space over spheres and Euclidean spaces好孩子http://mathoverflow.net/users/658002015-09-02T06:31:04Z2015-09-02T06:31:04Z
<p>For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then </p>
<p>$$
B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1},
$$
$$
B(S^n,2)\simeq \mathbb{R}P^n.
$$
Hence </p>
<p>$
(*)
$
$$
B(\mathbb{R}^{n+1},2)\simeq B(S^n,2).$$</p>
<p>Inspired by $(*)$, are there any relations between $B(\mathbb{R}^k,2)$ and $B(S^n,k)$ for general $k\geq 3$ or for some particular $k=2^i$?</p>
http://mathoverflow.net/q/2162040Are all transversely oriented foliations given by closed forms?user6419http://mathoverflow.net/users/626472015-09-02T06:18:08Z2015-09-02T06:18:08Z
<p>This <a href="http://arxiv.org/pdf/math/0110227v7.pdf" rel="nofollow">paper</a>, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the book <a href="http://rads.stackoverflow.com/amzn/click/3642690173" rel="nofollow">"Differential Geometry of Foliations: The Fundamental Integrability Problem"</a> by Reinhart places a restriction on the transversely oriented foliation to have trivial holonomy. On the other hand, the latter result pertains to $C^k$-foliations, while the first one refers to the smooth case.</p>
<p>Is the claim in the paper false?</p>
http://mathoverflow.net/q/2162030Fundamental Group of SL_2Pierre MATSUMIhttp://mathoverflow.net/users/357312015-09-02T06:11:59Z2015-09-02T06:11:59Z
<p>I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. </p>
<p>Because SL_2(R) is already a 3-dimensional manifold, so it is difficult to visualize the manifold like Riemann surface. </p>
<p>Question：Is there any simple method to know thefundamental groups of GL_n(F) for n≧2 with a field F?</p>
<p>I imagined Perelman's result could be used by generalizing base field from R to F. </p>
<p>Please teach me about the above question. Pierre Matsumi</p>
http://mathoverflow.net/q/2162020Connectedness of moduli of vector bundlesHongluhttp://mathoverflow.net/users/103332015-09-02T05:01:02Z2015-09-02T06:53:50Z
<p>Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? Or are there any counterexamples?</p>
<p>For example, if we have a short exact sequence of vector bundles</p>
<p>$$0\rightarrow E_0\rightarrow E_1\rightarrow E_2\rightarrow 0 $$,</p>
<p>then $E_1$ can be deformed to $E_0\oplus E_2$ by finding a curve between corresponding elements in $Ext^1(E_2,E_0)$. Since the relations in $K^0$ are generated by short exact sequences of vector bundles, I was attempting to generalize this kind of argument but failed. </p>
<hr>
<p>Another motivation for this question is that I want to know the possibility of identifying the Gromov-Witten theories of $\mathbb P(V_1)$ and $\mathbb P(V_2)$ for such $V_1$ and $V_2$ by deforming one to another. Therefore I think I would allow families of vector bundles over a reducible variety.</p>
http://mathoverflow.net/q/2161981Probability that an integer contains no $1\bmod 4$ prime factorArulhttp://mathoverflow.net/users/764792015-09-02T03:14:31Z2015-09-02T06:27:08Z
<p>$n$ represents integer variable.</p>
<p>What is the probability that and integer contains at most $r(n)$ prime factors of form $1\bmod 4$ where $r(n)$ is a function of $\omega(n)$ (number of distinct prime factors of $n$)?</p>
<p>What is the probability that and integer contains at most $r_t(n)$ prime factors of form $(2t+1)\bmod 2^k$ prime factor where $t\in\{0,\dots,2^{k-1}-1\}$ where $r_t(n)$ is a function of $\omega(n)$ (number of distinct prime factors of $n$)?</p>
http://mathoverflow.net/q/2161920floating point representation via the perspective of TTE/computable analysisSorcererofDMhttp://mathoverflow.net/users/274672015-09-02T01:10:47Z2015-09-02T01:10:47Z
<p>Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not another. However, is there a way to generalize TTE as to yield floating point numbers as a special case? For example via some form of multi-representation?</p>
<p>One obstruction would be that in any floating number implementation, there are only finitely many numbers, so the "natural" topology on the "resultant reals" would be discrete. But the current reality from the programmer's point of view is that the reals can be manipulated as easily as the integers, and this perspective seems unlikely to yield to the treatment of reals as infinite objects in the near future. So perhaps a more robust theory of real computation should take this finiteness into account (perhaps sacrificing some elegance of combining topology with computation in TTE).</p>
<p>Otherwise, is there a current theory that accounts for the behavior of floating point reals better than the real-RAM model, for example by integrating numerical stability as well?</p>
http://mathoverflow.net/q/216191-1Are polls good approximationsuser47958http://mathoverflow.net/users/479582015-09-02T01:00:33Z2015-09-02T01:00:33Z
<p>Let $X$ be a finite set and $A\subseteq X$ and $m$ be a natural number satisfying $m\le |X|$ and $\epsilon$ be a small positive number.</p>
<p>I'm interested to know if one selects a random $Y\subseteq X$ with $|Y|=m$, will $\frac{|Y\cap A|}{|Y|}$ be a good approximation for $\frac{|A|}{|X|}$. It is a common approximation used in many practical situations. </p>
<p>Is $\frac{|Y\cap A|}{|Y|}$ really a good approximation for $\frac{|A|}{|X|}$? </p>
<p>(I think the probability that a random $Y$ (with $m$ element), satisfies
$\left| \frac{|A|}{|X|}- \frac{|Y\cap A|}{|Y|} \right|\le \epsilon$ will reveal the answer. But its probability formula seems to be cumbersome.)</p>
http://mathoverflow.net/q/2161880Obtaining z-transform of a multivariate nonlinear difference equationuser2971034http://mathoverflow.net/users/787962015-09-02T00:27:18Z2015-09-02T00:27:18Z
<p>My research area is not Mathematics, but I am facing a conceptual mathematical issue, the answer to which I could not find in regular textbooks and other material that the internet fetched me and hence I seek expert advice. For my research, I want to obtain the z-transform of a difference equation. Now typically it is pretty straightforward and I know the procedure to obtain z-transform of difference equations that say look like:</p>
<p>x[k+2]=2x[k+1]-x[k]+8</p>
<p>My problem however is multivariate and looks like this:
x[k+1]=ay[k]+ ((x[k])^2)(y[k])
y[k+1]=b+(1-a)y[k] - ((x[k])^2)(y[k]); where a,b are positive parameters.</p>
<p>Can anybody provide any guidance on how to obtain the z-transform of the above difference equations? The ((x[k])^2)(y[k]) term is what has stumped me. If you could guide me to any relevant literature that addresses this issue as well I would be grateful.</p>
<p>Regards,
Shruti</p>
http://mathoverflow.net/q/2161761approximate diameter of polytopes in high dimensionsrjmhttp://mathoverflow.net/users/787882015-09-01T21:13:06Z2015-09-02T00:44:40Z
<p>I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}.
$$
Let $P$ be a polytope in $\Delta_c^n$ generated as the intersection of $m$ half-spaces.
Let us equip $\Delta^n$ with the distance defined with a positive-definite matrix $Q$.
I am looking to compute the diameter of the set $P$.</p>
<p>From a quick exploration, the diameter is the maximal distance between two extremal points of $P$.</p>
<p>The problem is that the number of these extremal points is typically exponential in $n$.</p>
<p>But if $Q$ has only a few eigenvalues that are not small, is there a smart way to approximate the diameter of $P$ ? </p>
<p>I am interested in any setting, even randomized ones, where the diameter can be approximated correctly numerically. The only hypothesis I do like to keep is the fact that $n$ being large. </p>
http://mathoverflow.net/q/2161742Conjectured new primality test for Mersenne numbersTony Reixhttp://mathoverflow.net/users/787892015-09-01T20:51:38Z2015-09-02T04:36:10Z
<p>How to prove that this <strong>conjecture</strong> about a new primality test for Mersenne numbers is true ?</p>
<p><strong>Definition</strong>: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$</p>
<p><strong>Conjecture:</strong> $$M_{q}\text{ is a prime iff: } \ S_{q-1} \equiv S_{0} \pmod{M_{q}}$$
$$\text{ and iff: } \prod_{0}^{q-2} S_i \equiv 1 \pmod{M_{q}}$$
$$\text{ and iff: } S_i \not\equiv S_0 \pmod{M_{q}} , 0 < i < q-1$$</p>
<p>The easy part part (if $M_{q}$ is prime then $S_{q-1}\equiv S_{0} \pmod{M_{q}}$), has already been proven. See: <a href="http://tony.reix.free.fr/Mersenne/ConjectureLLTCyclesMersenne.pdf" rel="nofollow">http://tony.reix.free.fr/Mersenne/ConjectureLLTCyclesMersenne.pdf</a> .</p>
<p>Now, a proof of the converse is needed.</p>
<p>The "classic" LLT by Lucas and Lehmer for Mersenne numbers is based on the binary tree of the digraph (under $x^2-2 \pmod{M_q}$).
This new conjectured test makes use of a cycle of the same digraph.</p>
<p>The goal of this question is to validate (or not) the method (use a cycle of the digraph) for Wagstaff numbers ($\dfrac{2^q+1}{3}$ , q prime) for which we also lack a proof for the converse. See: <a href="https://trex58.files.wordpress.com/2009/01/wagstaffandfermat.pdf" rel="nofollow">https://trex58.files.wordpress.com/2009/01/wagstaffandfermat.pdf</a> for a proof of the easy part, by Robert Gerbicz .</p>
<p>I and a team (Vincent and Paul) had found a very large Wagstaff PRP with this algorithm. See: <a href="http://www.primenumbers.net/prptop/prptop.php" rel="nofollow">http://www.primenumbers.net/prptop/prptop.php</a> (2^4031399+1)/3 , rank 9 for now.</p>
http://mathoverflow.net/q/2161581The Laplacian of an expression involving the Ricci tensorAlex M.http://mathoverflow.net/users/547802015-09-01T17:48:36Z2015-09-02T02:44:07Z
<p>While doing some computations on a <em>compact</em> Riemannian manifold I have reached the following expression:</p>
<p>$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$</p>
<p>where $\Delta_y$ is the Laplacian with respect to $y$, $Ric_y$ is the Ricci tensor in $y$ and $\exp_y : U_y \subset TyM \to M$ is the exponential map at $y$. Note that, if $r = d(x,y)$, then each argument of $Ric_y$ is at least $O(r)$ for $r \to 0$, if not even better. Thus, the whole expression inside the Laplacian is at least $O(r^2)$.</p>
<p>Can you tell whether the expression above is $0$ or not? Explicitly doing the calculations does not seem a feasible choice. (In particular, if the whole expression inside the Laplacian were $O(r ^{2 + \varepsilon})$ for some $\varepsilon > 0$, the answer would be affirmative.)</p>
http://mathoverflow.net/q/2161412Second order estimates of Monge-Ampere equationssvahttp://mathoverflow.net/users/161832015-09-01T14:26:55Z2015-09-02T00:19:34Z
<p>In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the steps includes the $C^0$-estimate of the Laplacian of the unknown function. The only known to me methods to get it are modifications of the Pogorelov's method introduced originally for the real MA-equation (its version was also used in the proof of the Calabi-Yau theorem, please correct me if I am wrong).</p>
<p><strong>My question is whether there are other methods to get the $C^0$-estimate of the Laplacian?</strong></p>
http://mathoverflow.net/q/2161270Are constructive characterisations of k-regular (simple) graphs known?user62562http://mathoverflow.net/users/625622015-09-01T11:28:28Z2015-09-02T00:09:45Z
<p>By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some sequence of these operations and every intermediate graph is also in the class.</p>
<p>For 3-regular graphs I found the paper "Inductive classes of cubic graphs" by V Batagelj in 1981. <a href="http://vlado.fmf.uni-lj.si/vlado/papers/cubicEger.pdf" rel="nofollow">http://vlado.fmf.uni-lj.si/vlado/papers/cubicEger.pdf</a></p>
<p>For 4-regular graphs there is a paper in French: "Construction of 4-regular graphs" by Bories and Jolivet in 1983.</p>
<p>I have been unable to find corresponding results for 5-regular graphs or 6-regular graphs or higher. It seems like a quite natural question, so I'd be surprised if nothing was known but I haven't been able to find any references myself.</p>
<p>Are there known constructive characterisations of k-regular (simple) graphs for any integer $k\geq 5$?</p>
<p>If not, can anyone give some intuition for why such results would be hard? Any interesting applications?</p>
http://mathoverflow.net/q/21611512Are there open problems for primes which are known for probable primes?jorohttp://mathoverflow.net/users/124812015-09-01T08:14:09Z2015-09-02T02:28:50Z
<p>Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.</p>
<p>Probable primes are the union of the primes and base two pseudoprimes.</p>
<p>This definition is much simpler than the definition for primes
and the primes are sufficiently large subset of probable primes.</p>
<blockquote>
<p>Are there open problems for primes which are known for probable
primes?</p>
</blockquote>
<p>Positive answer doesn't necessarily mean the problem is
solved for the primes (e.g. infinitely many twin PP hypothetically might
mean finitely many twin primes and infinitely many twin base 2 pseudoprimes).</p>
http://mathoverflow.net/q/2160607About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$user40276http://mathoverflow.net/users/408832015-08-31T14:14:44Z2015-09-02T01:36:45Z
<p>I've asked this question <a href="http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text">http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text</a> on math.stackexchange , however I don't think I will receive any response there (because of the current activity in my post). Therefore I'm asking it here. If the question seems inconvenient because of the excess of questions, I can split this question into other ones (let me know).</p>
<p>I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ respectively). The motivation is that I'm trying to compute some (co)limits explicitly in the category of schemes.</p>
<p>There's this excellent answer here <a href="http://math.stackexchange.com/questions/102973/on-limits-schemes-and-spec-functor">http://math.stackexchange.com/questions/102973/on-limits-schemes-and-spec-functor</a> , but I still have some doubts.</p>
<p>More precisely, I want to know about the following assertions:</p>
<p>1) Is the category of locally ringed spaces (co)complete? <strong>The answer is yes by prop 1.6 in Demazure and Gabriel's "Groupes Algébriques" (I didn't notice that they proved the general case and not just the case of filtered colimits when I posted this question, sorry)</strong></p>
<p>In the answer cited above, the references implies the existence of cofiltered limits and filtered colimits, however as I understand the notion of filtered in these cases is restricted to the case where the index category is a poset.</p>
<p>2)Is the category of ringed spaces (co)complete?</p>
<p>3)What can be said about the underlying topological space of the (co)limit of locally ringed spaces? (Is it the (co)limit of the topological spaces?)</p>
<p>4)What can be said about the underlying topological space of the (co)limit of ringed spaces? (Is it the (co)limit of the topological spaces)</p>
<p>5)What can be said about the underlying topological space of the colimit of schemes? (Is it the (co)limit of the topological spaces)</p>
<p>Obviously, the underlying topological space of the pullback of schemes is not the pullback of the topological spaces (for instance, $\text{Spec} (\mathbb{C}) \times_{\text{Spec} (\mathbb{R})}\text{Spec} (\mathbb{C}) \cong \text{Spec} (\mathbb{C}\times\mathbb{C})$ by $a \otimes z \mapsto (az, a\overline{z})$), but the case of push outs seems to be true.</p>
<p>6) Are (co)limits preserved under the inclusions $\text{Sch} \hookrightarrow\text{LRS} \hookrightarrow \text{RS}$?</p>
<p><strong>The inclusion $\text{LRS} \hookrightarrow \text{RS}$ preserves colimits since it's a left adjoint (see below)</strong></p>
<p>7) For each forgetful functor $U : \mathcal{C} \rightarrow \mathcal{D}$, where $\mathcal{C}$ and $\mathcal{D}$ are equal to $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ (for all possible coherent substitutions), are there adjoint functors?</p>
<p>8) For each inclusion $\mathcal{C} \hookrightarrow \mathcal{D}$, where $\mathcal{C}$ and $\mathcal{D}$ are equal to $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ (for all possible coherent substitutions), are there adjoint functors?</p>
<p>According to <a href="http://arxiv.org/abs/1103.2139" rel="nofollow">http://arxiv.org/abs/1103.2139</a> [Cor. 6], the inclusion $\text{LRS} \hookrightarrow \text{RS}$ have a right adjoint given by localization of the terminal prime system.</p>
<p>Thanks in advance.</p>
http://mathoverflow.net/q/215947-1Lebesgue-integrability of piecewise function with random variable [on hold]Geremiahttp://mathoverflow.net/users/693182015-08-30T06:23:48Z2015-09-02T05:26:02Z
<p>This function is Lebesgue-integrable:$$\chi(x)= \left\{
\begin{array}{ll}
1 & \text{if}~x~\text{is rational}\\
0 & \text{if}~x~\text{is irrational}.
\end{array}
\right.$$</p>
<p>¿But is this function:$$\chi_2(x)= \left\{
\begin{array}{ll}
x & \text{if}~x~\text{is rational}\\
-x & \text{if}~x~\text{is irrational}
\end{array}
\right.$$?</p>
http://mathoverflow.net/q/2159323largest subgroup of $Out(\hat{F_2})$ which preserves the Nielsen invariantoxeimonhttp://mathoverflow.net/users/152422015-08-29T20:36:46Z2015-09-02T02:47:10Z
<p>Let $x,y$ be generators for the free group $F_2$. It's known that $Aut(F_2)$, and hence $Out(F_2)$ preserves the conjugacy class of the subgroup $\langle[x,y]\rangle$ generated by $[x,y]$ (This conjugacy class is in some contexts called the nielsen invariant)</p>
<p>On the other hand, if we view $x,y$ as topological generators of $\hat{F_2}$ (hence fixing an embedding $F_2\hookrightarrow\hat{F_2}$), then $Out(\hat{F_2})$ does not have the same property of preserving the conjugacy class of $\langle [x,y]\rangle$ (this follows from the fact that $Aut(\hat{F_2})$ has the <em>strong lifting property</em>).</p>
<p>My question is - What is the stabilizer of the conjugacy class of $\langle[x,y]\rangle$ in $Out(\hat{F_2})$?</p>
<p>The stabilizer should be a closed subgroup, lets call it $S$ - could it be $\overline{GL_2(\mathbb{Z})}\cong\overline{Out(F_2)}\cong \widehat{Out(F_2)}\subset Out(\hat{F_2})$?</p>
<p>If not, can we describe the difference $S/\overline{GL_2(\mathbb{Z})}$ somehow?</p>
http://mathoverflow.net/q/21592820Advanced Differential Geometry TextbookDiffGeomInteresthttp://mathoverflow.net/users/786732015-08-29T19:43:08Z2015-09-02T03:06:39Z
<p>I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.</p>
<p>In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses.</p>
<p>They are Switzer <em>Algebraic Topology: Homology and Homotopy</em> and Whitehead <em>Elements of Homotopy Theory</em>. These are both excellent books that (theoretically) give you overviews and introduction to most of the main topics that you need for becoming a modern researcher in algebraic topology.</p>
<p>Differential Geometry seems replete with excellent introductory textbooks. From Lee to do Carmo to so many others.</p>
<p>Now you might be thinking that Kobayashi/Nomizu seems natural. But the age of those books is showing in terms of what people are really doing today compared to what you learn from using those books. They just aren't the most efficient way to learn modern differential geometry (or so I've heard).</p>
<p>I am looking for a book that covers topics like Characteristic Classes, Index Theory, the analytic side of manifold theory, Lie groups, Hodge theory, Kahler manifolds and complex geometry, symplectic and Poisson geometry, Riemmanian Geometry and geometric analysis, and perhaps some relations to algebraic geometry and mathematical physics. But none of these topics completely, just as Switzer does with a unifying perspective and proofs of legitimate results done at an advanced level, but really as an introduction to each of the topics (Switzer does this with K-theory, spectral sequences, cohomology operations, Spectra...).</p>
<p>The only book I have found that is sort of along these lines is Nicolaescu's <em>Lectures on the Geometry of Manifolds</em>, but this book misses many topics.</p>
<p>This was inspired by page viii of Lee's excellent book: <a href="https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PR8&dq=lee+smooth+manifolds&hl=en&sa=X&ved=0CB4Q6AEwAGoVChMIoeufxJnAxwIVhjw-Ch2w8gaP#v=onepage&q=lee%20smooth%20manifolds&f=false" rel="nofollow">link</a> where he lists some of these other topics and almost implies that they would take another volume. I'm wondering whether that advanced volume exists.</p>
<p>Any recommendations for great textbooks/monographs would be much appreciated!</p>
<p><strong>Edit</strong>: there are many excellent recommendations (I particularly like the Index theory text mentioned by Gordon Craig in the comments as it doesn't shy away from analysis, and does so many things in geometry plus has extensive references) below.
One other reference that I found which people may find interesting is the following: <a href="http://www.ams.org/bookstore-getitem/item=PSPUM-54" rel="nofollow">link</a> and <a href="http://www.ams.org/books/pspum/054.3/pspum054.3-endmatter.pdf" rel="nofollow">link2</a> where Prof. Greene and Yau say: "It is our hope that the three volumes of these proceedings, taken as a whole, will provide a broad overview of geometry and its relationship to mathematics
in toto, with one obvious exception; the geometry of complex manifolds...Thus the reader seeking a complete view of geometry would do well to add
the second volume on complex geometry from the 1989 Proceedings to the
present three volumes". However most of the articles are research level articles and lack the coherence and unified vision of a textbook/monograph. </p>
http://mathoverflow.net/q/2154440Reference request for the focussing exampleGeorge Simpsonhttp://mathoverflow.net/users/710462015-08-23T13:30:02Z2015-09-02T02:26:48Z
<p>I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. 48, No 3. </p>
<p>There is a mention on energy estimates being fixed time estimates at a specified time, but requiring a lot of regularity in $L^p.$ Then there is a mention of the focussing example.</p>
<p>The article talks of how it is an example with initial data dispersed near the unit sphere and the solution $u$ focuses at the origin at time $t=1$ with high $L^{\infty}$ norm. </p>
<p>I was wondering if anyone can provide any references for the Focussing example, or if they know anywhere I can find out more about it?</p>
<p>Thank you in advance for you comments. </p>
http://mathoverflow.net/q/21492752Important formulas in CombinatoricsGil Kalaihttp://mathoverflow.net/users/15322015-08-17T08:59:32Z2015-09-02T05:32:13Z
<h2>Motivation:</h2>
<p>The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See <a href="https://gilkalai.wordpress.com/2015/08/10/nogafest-nogaformulas-and-amazing-cash-prizes/">this post</a>, for the poster, and cash prizes offered for identifying the formulas.) This demonstrates that sometimes (but certainly not always) a major research progress, even areas, can be represented by a single formula. Naturally, following Alon's poster, I thought about representing other people's works through formulas. (My own work, Doron Zeilberger's, etc. Maybe I will pursue this in some future posts.) But I think it will be very useful to collect major formulas representing major research in combinatorics. </p>
<h2>The Question</h2>
<h3>The question collects important formulas representing major progress in combinatorics.</h3>
<p>The rules are:</p>
<h2>Rules</h2>
<p>1) one formula per answer</p>
<p>2) Present the formula <em>explicitly</em> (not just by name or by a link or reference), and briefly explain the formula and its importance, again not just link or reference. (But then you may add links and references.) </p>
<p>3) Formulas should represent important research level mathematics. (So, say $\sum {{n} \choose {k}}^2 = {{2n} \choose {n}}$ is too elementary.) </p>
<p>4) The formula should be explicit as possible, moving from the formula to the theory it represent should also be explicit, and explaining the formula and its importance at least in rough terms should be feasible.</p>
<p>5) I am a little hesitant if classic formulas like $V-E+F=2$ are qualified. </p>
http://mathoverflow.net/q/19838710Most dense subset of numbers that avoids arbitrarily long arithmetic progressionsJoseph O'Rourkehttp://mathoverflow.net/users/60942015-02-25T01:25:54Z2015-09-02T02:43:25Z
<p>The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
arbitrarily long sequences of elements of $S$ in arithmetic progression?
To make this more precise (following a comment by Robert Israel), </p>
<blockquote>
<p><strong><em>Q</em></strong>. What is the cardinality of the largest subset $S_n$ of $[1,n]=\{1,2,3,\ldots,n\}$
that avoids $k$-term arithmetic progressions of elements in $S_n$,
as a function of $n$ and $k$?</p>
</blockquote>
<p>As $n \to \infty$, can the density be significantly more dense than the primes density, $n / \log_e n$? </p>
<p>I suspect this is a well-studied question, in which case quotes and/or pointers would suffice. Thanks! </p>
http://mathoverflow.net/q/1954423Convergence of random variables with hypergeometric distributionuser2173168http://mathoverflow.net/users/416382015-02-02T02:03:20Z2015-09-02T00:07:36Z
<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/223882014-12-02T02:11:31Z2015-09-02T04:17:43Z
<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/13590215Higher level analogs of Nicolas-Serre theorypaul Monskyhttp://mathoverflow.net/users/62142013-07-06T04:06:54Z2015-09-01T23:18:28Z
<p>NICOLAS-SERRE THEORY</p>
<p>Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke operators $T_n$, $n$ odd, on $Z/2[[x]]$ and show:</p>
<ul>
<li><p>1) $V$ is stable under the $T_n$. If $n$ is not a square, then $T_{n}(F^{k})$, $k$ odd, is a sum of various $F^j$ with $j$ odd and smaller than $k$. (This comes from modular form theory for the full modular group).</p></li>
<li><p>2) The only elements of $V$ annihilated by both $T_3$ and $T_5$ are $0$ and $F$.</p></li>
<li><p>3) There are $m_{a,b}$ forming a basis of $V$ such that $m_{0,0}$ is $F$, the $m_{a,b}$ with $a$ and $b$ not both $0$ are divisible by $x^2$, and $T_3$ and $T_5$ reduce $a$ by 1 and $b$ by 1 respectively.</p></li>
<li><p>4) In the action of the algebra spanned by the $T_n$, each $T_n$ "<em>acts as a power series in $T_3$ and $T_5$</em>". This tells us that a certain completion of this Hecke algebra is a $2$-variable power series ring over $Z/2$ in $T_3$ and $T_5$.</p></li>
<li><p>The proofs of (2)-(4) are rather technical and rely on a certain "code".</p></li>
</ul>
<h1>CONJECTURAL ANALOGS</h1>
<p>I've experimentally found some analogs to the above related to modular forms of level $N$ when $N$ is 3, 5, 7, or 11. The situation when $N=3$ is particularly nice. Explicitly let $D \in Z/2[[x]]$ be $x+x^{25}+x^{49}+...$, the exponents being the squares prime to $6$, and let $V$(plus) be spanned by the $D^k$ with $k \equiv 1 (\text{mod}\ 6)$. Then:</p>
<ul>
<li><p>1*) V(plus) is stable under the $T_n$ with $n \equiv 1(\text{mod}\ 6)$. If $n$ is a non-square and $\equiv 1 (\text{mod}\ 6)$ then $T_n(D^k)$ is a sum of $D^j$ for various $j$ with $j\equiv 1 (\text{mod}\ 6)$ and $j$ smaller than $k$. (This is a theorem, whose proof uses modular forms of level 3).</p></li>
<li><p>2*) I believe that the only elements of V(plus) annihilated by $T_7$ and $T_{13}$ are $0$ and $D$.</p></li>
<li><p>3*)I believe there are $m_{a,b}$ precisely as in 3) with $F, T_3$ and $T_5$ replaced by $D, T_7$, and $T_{13}$. (I've calculated these when $a+b$ is at most 6. For example, $m_{4,2}$ is $(D^{73})+(D^{121})+(D^{145})$).</p></li>
<li><p>4*)I believe that in the action of the algebra spanned by the $T_n$ with $n\equiv 1 (\text{mod}\ 6)$ on V(plus) each such $T_n$ acts as a formal power series in $T_7$ and $T_{13}$. Also if $n$ is 1 mod 24, then $T_n$ acts as a power series in $(T_7)^2$ and $(T_{13})^2$, and in particular $(T_1)+(T_{25})=(T_5)^2$ acts by $(T_7)^2+(T_{13})^2$+(higher order terms). (This has implications for the structure of a completion of the algebra spanned by the $T_n$ with $(n,6)=1$ on the space spanned by the D^k with $(k,6)=1)$.</p></li>
</ul>
<p><strong>QUESTION:</strong> posed to those who understand the Nicolas-Serre code. Can one use some modification of the code to establish the truth of my beliefs stated above? Any thoughts about proving these conjectures would be welcome.</p>
<p>Remark 1: When $N=5$ there is a rather more complicated analog of V(plus) on which the T_n with n=1 or 3 mod 8 act. Results like (1) can again be proved. Results like (2)-(4) seem to hold with $T_3$ and $T_5$ replaced by $T_3$ and $T_{11}$. And if $n$ is 1 or 9 mod 40 then $T_n$ seems to be a power series in $(T_3)^2$ and $(T_{11})^2$. In particular $(T_7)^2$ acts by $(T_3)^2+(T_{11})^2$+(higher order terms). Again there would be implications for the structure of a completed Hecke algebra.</p>
<p>Remark 2: When $N=3$ there are similar apparent results for V(minus), the space spanned by the $D^k$ with k=5 mod 6. And when $N=5$, there is an analog of this V(minus) with similar apparent results. </p>
<p>EDIT___A GENERAL FRAMEWORK</p>
<p>I'll place the calculations I've made into a general setting, and then explain how the case N=3 described under CONJECTURAL ANALOGS fits into this expanded framework.</p>
<p>Let N be an odd prime and HE the "level N shallow Hecke algebra" spanned by the T_k: Z/2[[x]]-->Z/2[[x]] with(k,2N)=1. Let F be as above and G be F(x^N). I'll adopt the notation of my question 138495--"Are these two subspaces of Z/2[[x]] the same?" In particular, M is the integral closure of Z/2[G] in Z/2(F,G) viewed as a subspace of Z/2[[x]]. M has a modular forms interpretation which shows that it (and its subspaces M(odd) and C) are stable under HE. Indeed each of these is an increasing union of finite dimensional HE stable subspaces. Each of these finite dimensional subspaces is evidently annihilated by a product of powers of maximal ideals of HE. It's known, I believe, that only finitely many of these maximal ideals appear altogether. So in particular, C is a direct sum of finitely many HE stable subspaces, C(J), with the maximal ideal J acting locally nilpotently on C(J).(One case of interest is when J=Ann(F), the annihilator of F in HE. This ideal contains T_p for all primes p other than 2 or N. When N=3,5, or 7, I'm informed that Ann(F) is the only maximal ideal that appears, so that C=C(J).)</p>
<p>Now let pr:Z/2[[x]]-->Z/2[[x]] be the map which removes from each h in Z/2[[x]] all terms in which N divides the exponent. Then pr(C(J)) is HE stable with J acting locally nilpotently on it. As in Nicolas-Serre one can then construct a "J-completion" of HE acting faithfully on pr(C(J)). A general question is:</p>
<p>WHAT IS THE STRUCTURE OF THIS J-COMPLETION?</p>
<p>The hope that, as in Nicolas-Serre, the J-completion is a 2 variable power series ring proves mistaken. But in some cases the following seems to hold:</p>
<p>There are one or more index 2 subgroups of Z/(8N)* such that when one replaces HE by the subalgebra HE# spanned by the T_k with k in one of these subgroups, and C(J) by the subspace C(J)# consisting of those h in C(J) for which all the exponents that appear are either divisible by N or lie in the subgroup, then the resulting J-completion of HE# is a 2 variable power series ring. Furthermore, in a number of these cases, the J-completion of HE appears to be isomorphic to the non-reduced ring Z/2[[x,y,z]]/(z^2).</p>
<p>In the examples I've looked at, N=3,5,7 and 11. For each of these N, C is the set of elements of M(odd) whose trace from Z/2(F,G) to Z/2(G) is 0, and there is a Z/2[G^2] basis Ck of C with k odd, k between 0 and 2N, given in question 138495 with nice properties. Set Dk equal to pr(Ck), and extend the definition of Dm to all odd m so that whenever m=k+2N, then Dm=(G^2)Dk. Then Dm=0 when N divides m, while the remaining Dm are a Z/2 basis of pr(C). Calculations with this basis are very convenient. I won't go into the details of the calculations now, but I'll explain how the case N=3 under CONJECTURAL ANALOGS fits into this new setting. </p>
<p>Suppose then that N=3 and J=Ann(F). Look at the N=3 paragraph under SOME REMARKABLE FACTS in question 138495. Since C1=F, D1=pr(F) which is the D described above. Then if r=6k+1, Dr=(G^2k)*D. But classical results show that G=D^3, so that Dr=D^r for all r that are 1 mod 6. Similarly, C5=(F^2)G, so that D5=pr(F^2)G=(D^2)G=D^5, and it follows that Dr=D^r for all r that are 5 mod 6. Thus pr(C) is just the space V spanned by the D^r with (r,6)=1.</p>
<p>Now as I've noted, C=C(J) in this setting. Now {1,7,13,19} is an index 2 subgroup of (Z/24)*, and a Z/2 basis of the corresponding subgroup pr(C#) of pr(C) is given by the D^k with k=1 mod 6. It follows that pr(C(J)#) is just the space V(plus) spanned by these D^k, described under CONJECTURAL ANALOGS, and we are in precisely the situation given there.</p>
<p>EDIT(Sept. 1)__WHAT THE COMPUTER SUGGESTS IN LEVEL 11</p>
<p>The 1-dimensional subspace {0,F} of C is always HE stable. When N=11 there is another HE stable 1-dimensional subspace {0,t} of C, with t as in the N=11 paragraph of my question 138495. (In fact, t=C1+C3+C5+C9+C15). One way to see HE stability is the following--t^12=FG, the reduction of the expansion of the modular form delta(z)delta(11z). So t is the reduction of the expansion of (eta(z)eta(11z))^2, and this last is the weight 2 newform for Gamma_0 (11). Now Ann(t) and Ann(F) are maximal ideals in HE. Write C(t) and C(F) for C(Ann(t)) and C(Ann(F)). I've been informed that C is the direct sum of C(t) and C(F). Here's what the computer suggests for the Ann(t)-completion of HE acting on pr(C(t)).</p>
<p>(*) The above completion is a 2-variable power series ring over Z/2, with an element of square 0 adjoined. More precisely the map from Z/2[[X,Y,Z]] to the Ann(t)-completion that sends X,Y and Z to T_3 +I, T_5 +I and T_7 is onto and the kernel is generated by f^2 where
f=Z+X+X^2+XY+Y^2+(X^2)Y+X(Y^2)+Y^3+higher degree stuff in X and Y.</p>
<p>Remarks__See my question 137260--Questions(related to deformation theory?)... for what the above tells us about a very natural attempted generalization of Nicolas-Serre to level 11. I have conjectures similar to (*), supported by the computer, for the Ann(F)-completion of HE acting on pr(C(F)) when N=3,5 or 11. But N=7 is more complicated; I don't understand it at this time.</p>
<p>Assume now that N=11. Since T_3(t)=t, T_3+ I acts nilpotently on C(t), while T_3 acts nilpotently on C(F). So we can use the calculations made of the various T_3(Dm) as sums of various Dk's to decompose each Dm into its pr(C(t)) and pr(C(F))-components for a wide range of m, and then calculate the effect of T_3, T_5 and T_7 on each of the pr(C(t))-components.</p>
<p>I now make use of two index 2 subgroups of (Z/88)*, G1 and G2. The first consists of the k that are squares mod 11, and the second of the 4n+1 that are squares mod 11 and the 4n+3 that are non-squares. Use G1 and G2 to define subalgebras HE(1#) and HE(2#) of HE, as well as subspaces C(t,1#) and C(t,2#) of C(t). Using the calculations outlined in the last paragraph, I empirically find:</p>
<p>(A)--If we replace F by pr(t)=[1,3,5,9,15] (this is shorthand for D1+D3+D5+D9+D15), T3 and T5 by T3 +I and T_5 +I, the algebra spanned by the T_n by HE(1#), and V by pr(C(t,1#)), then 2),3) and 4) under NICOLAS-SERRE THEORY still hold.</p>
<p>(B)--If we replace F by [1,5,9], T3 and T5 by T5 +I and T7, the algebra spanned by the T_n by HE(2#), and V by pr(C(t,2#)), then 2),3) and 4) under NICOLAS-SERRE THEORY still hold.
(I've calculated many m_(a,b) both for A and for B. For example in A, m_(1,7)=[23,31,47,71,191,223]).</p>
<p>Now let x,y and z be the images of T3 +I, T5 +I and T7 in the Ann(t)-completion of HE, acting on pr(C(t)). From A and B one should be able to show:</p>
<p>1) z^2 is a power series in x^2 and y^2. (First one shows that this is true with HE replaced by HE(1#) and C(t) by C(t,1#)).</p>
<p>2) More precisely f^2=0, where f=z+x+x^2+xy+y^2+(x^2)y+x(y^2)+y^3+higher degree stuff in x and y. (For this one needs to write z^2 as a power series in x^2 and y^2 to the needed accuracy. This is accomplished by seeing what (T_7)^2 does to m_(7,1), m_(5,3), m(3,5) and m(1,7)).</p>
<p>3) Every element of the J-completion of HE acting on pr(C(t) is a power series in x,y and z.(For the elements of the J-completion of HE(1#)(resp. HE(2#)) acting on C(t) are power series in x and y(resp. y and z). And the two subalgebras generate HE.</p>
<p>Putting 2) and 3) together should give the desired (conjectural) structure theorem for our Ann(t)-completion of HE.</p>
<p>EDIT(Sept. 10) WHAT THE COMPUTER SUGGESTS IN LEVELS 3 and 5</p>
<p>When N=3 or 5, J can only be Ann(F), and so C=C(J). The computer indicates that in each case there is a triple of integers {n_1,n_2,n_3} with the following properties:</p>
<p>I.--Let a and b be two elements of the triple and G be the index 2 subgroup of (Z/8N)* generated by a,b and the squares. Use G to define the subalgebra HE# of HE and the subspace
C# of C, as under WHAT IS THE STRUCTURE OF THIS J-COMPLETION? Then 2),3) and 4) under the heading NICOLAS-SERRE THEORY hold provided we replace:</p>
<p>The space spanned by the T_n by HE#, V by C#, T_3 by T_a and T_5 by T_b.</p>
<p>In particular, the Ann(F)-completion of HE# acting on C# is a power series ring in T_a and T_b over Z/2.</p>
<p>II.--The Ann(F)-completion of HE acting on C is a power series ring in x=T_(n_1), y=T_(n_2) and z =T_(n_3) with a single relation f^2=0, where f=x+y+z+(higher degree stuff).</p>
<p>IIa.--When N=3, n_1=5, n_2=7, n_3=13 and f=x+g where g=y+z+y^3+y*z^2+z^3+(higher degree stuff in y and z.</p>
<p>IIb.--When N=5, n_1=3, n_2=7, n_3=11 and f=y+g where g=x+z+x^3+x*z^2+z^3+(higher degree stuff in x and z.</p>
<p>In support of (I), I calculated the m_(i,j) for each HE# and C# whenever i+j is 6 or less.
If (I) holds, an argument similar to the one I made in the last edit, should give the remaining results.</p>
<p>Remark__I think there are similar results when N=11, J=Ann(F), though I haven't carried out the calculations of the m_i,j very far. But N=7 is, as I've noted, very much different. </p>
http://mathoverflow.net/q/1023316Sparse ramsey theoryShahabhttp://mathoverflow.net/users/148752012-07-16T05:38:09Z2015-09-02T03:36:20Z
<p>It is known that for any graph H and all $k∈N$, there exists a graph $G$ such that any $k$-coloring of the edges of $G$ yields a monochromatic copy of H and ω(G)=ω(H) (the two graphs have the same clique numbers). </p>
<p>My question is: Given any graph $H$ with finite girth, is there a $G$ with the same girth as $H$ such that any $2$-coloring of the edges of $G$ yields a monochromatic copy of $H$? </p>
<p>I think this is an open problem but if someone can confirm that and give some references concerning this I would be most obliged. </p>
http://mathoverflow.net/q/8407417undecidable sentences of first-order arithmetic whose truth values are unknownsymplectomorphichttp://mathoverflow.net/users/30922011-12-22T06:34:48Z2015-09-02T06:25:27Z
<p>Godel's undecidable sentences in first-order arithmetic were guaranteed to be true, by construction. But are there examples of specific sentences known to be undecidable in first-order arithmetic whose truth values <em>aren't</em> known? I'm thinking, by contrast, of the situation in set theory: CH is undecidable in ZFC, but its truth value is, in some sense, unknown.</p>
<p>Paris and Harrington <a href="http://en.wikipedia.org/wiki/Paris-Harrington_theorem" rel="nofollow">showed</a> the strengthened finite Ramsey theorem is true (in the sense of provable in second-order arithmetic) but undecidable in first-order arithmetic. I'm asking for "natural" examples in this general vein -- but whose truth values haven't yet been settled.</p>
<p>EDIT. Let me clarify my interest in the question, which is more philosophical than mathematical. I asked it on the basis of the following passage in Peter Koellner's <a href="http://logic.harvard.edu/koellner/QAU_reprint.pdf" rel="nofollow">paper</a> "On the Question of Absolute Undecidability":</p>
<blockquote>
<p>The above statements of analysis [i.e. all projective sets of reals are Lebesgue measurable] and set theory [i.e. CH] differ from the early arithmetical instances of incompleteness in that their independence does <em>not</em> imply their truth. Moreover, it is not immediately clear whether they are settled at any level of the hierarchy. They are much more serious cases of independence.</p>
</blockquote>
<p>What I'm asking is whether there <em>are</em> "much more serious cases" of independence even in first-order arithmetic -- and not in the trivial case of full-on ZFC, like V=L, etc. By a sentence with "unknown truth value," I just mean a sentence that hasn't been proved in a theory stronger than first-order arithmetic. (For example, Paris and Harrington proved the strengthened finite Ramsey theorem in second-order arithmetic.)</p>
http://mathoverflow.net/q/7522020Physicist's request for intuition on covariant derivatives and Lie derivativesIgor Rivinhttp://mathoverflow.net/users/111422011-09-12T13:34:26Z2015-09-02T02:38:14Z
<p>A friend of mine is studying physics, and asks the following question which, I am sure, others could respond to better:</p>
<p>What is the difference between the covariant derivative of $X$ along the curve $(t)$ and a Lie derivative of $X$ along $y(t)?$ I know the technical stuff about not needing to define a connection with a Lie derivative, needing to define the fields $X$ and $Y$ over a greater neighborhood, etc.</p>
<p>I am looking for a more physical sense. If a Lie derivative gives the sense of the change of a vector field along the direction of another field, how does the covariant derivative differ?</p>
http://mathoverflow.net/q/2964411Enumerating ways to decompose an integer into the sum of two squaresMathMonkeyhttp://mathoverflow.net/users/71072010-06-26T22:05:28Z2015-09-02T03:32:24Z
<p>The well known <a href="http://mathworld.wolfram.com/SumofSquaresFunction.html">"Sum of Squares Function"</a> tells you <strong>the number</strong> of ways you can represent an integer as the sum of two squares. See the link for details, but it is based on counting the factors of the number N into powers of 2, powers of primes = 1 mod 4 and powers of primes = 3 mod 4.</p>
<p>Given such a factorization, it's easy to find the <strong>number</strong> of ways to decompose N into two squares. But how do you efficiently <strong>enumerate</strong> the decompositions?</p>
<p>So for example, given N=2*5*5*13*13=8450 , I'd like to generate the four pairs:</p>
<p>13*13+91*91=8450</p>
<p>23*23+89*89=8450</p>
<p>35*35+85*85=8450</p>
<p>47*47+79*79=8450</p>
<p>The obvious algorithm (I used for the above example) is to simply take i=1,2,3,...,$\sqrt{N/2}$ and test if (N-i*i) is a square. But that can be expensive for large N. Is there a way to generate the pairs more efficiently? I already have the factorization of N, which may be useful.</p>
<p>(You can instead iterate between $i=\sqrt{N/2}$ and $\sqrt{N}$ but that's just a constant savings, it's still $O(\sqrt N)$.</p>
http://mathoverflow.net/q/922028What does the generating function $x/(1 - e^{-x})$ count?Theo Johnson-Freydhttp://mathoverflow.net/users/782009-12-18T01:44:01Z2015-09-02T04:13:56Z
<p>Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty \frac{(-x)^n}{(n+1)!} \right)^m = 1 + \frac12 x + \frac1{12}x^2 + 0x^3 - \frac1{720}x^4 + \dots $$
where I might have made an arithmetic error in expanding it out.</p>
<ol>
<li><p>Are all the coefficients <em>egyptian</em>, in the sense that they are given by $A^{(n)}(0)/n! = 1/N$ for $N$ an integer? The answer is no, unless I made an error, e.g. the third coefficient. But maybe every non-zero coefficient is egyptian?</p></li>
<li><p>If all the coefficients were positive eqyptian, then the sequence of denominators might count something — one hopes that the $n$th element of any sequence of nonnegative integers counts the number of ways of putting some type of structure on an $n$-element set.</p></li>
</ol>
<p>Of course, generating functions really come in two types: ordinary and exponential. The difference is whether you think of the coefficients as $\sum a_n x^n$ or as $\sum A^{(n)} x^n/n!$. If it makes more sense as an exponential generating function, that's cool too.</p>
<p>So my question really is: is there a way of computing the $n$th coefficient of $A(x)$, or equivalently of computing $A^{(n)}(0)/n!$, without expanding products of power series the long way?</p>
<h3>Where you might have seen this series</h3>
<p>Let $\xi,\psi$ be non-commuting variables over a field of characteristic $0$, and let $B(\xi,\psi) = \log(\exp \xi \exp \psi)$ be the Baker-Campbell-Hausdorff series. Fixing $\xi$ and thinking of this as a power series in $\psi$, it is given by
$$B(\xi,\psi) = \xi + A(\text{ad }\xi)(\psi) + O(\psi^2)$$
where $A$ is the series above, and $\text{ad }\xi$ is the linear operator given by the commutator: $(\text{ad }\xi)(\psi) = [\xi,\psi] = \xi\psi - \psi\xi$.</p>
<p>More generally, $B$ can be written entirely in terms of the commutator, and so makes sense as a $\mathfrak g$-valued power series on $\mathfrak g$ for any Lie algebra $\mathfrak g$. It converges in a neighborhood of $0$ when $\mathfrak g$ is finite-dimensional over $\mathbb R$, in which case $\mathfrak g$ is a (generally noncommutative) "partial group".</p>
<p>(More generally, you can consider the "formal group" of $\mathfrak g$. Namely, take the commutative ring $\mathcal P(\mathfrak g)$ of formal power series on $\mathfrak g$; then $B$ defines a non-cocommutative comultiplication, making $\mathcal P = \mathcal P(\mathfrak g)$ into a Hopf algebra. Or rather, $B(\mathcal P)$ does not land in the algebraic tensor product $\mathcal P \otimes \mathcal P$. Instead, $\mathcal P$ is <em>cofiltered</em>, in the sense that it is a limit $\dots \to \mathcal P_2 \to \mathcal P_1 \to \mathcal P_0 = 0$, where (over characteristic 0, anyway) $\mathcal P_n = \text{Poly}(\mathfrak g)/(\mathfrak g \text{Poly}(\mathfrak g))^n$, where $\text{Poly}(\mathfrak g)$ is the ring of polynomial functions on $\mathfrak g$, and $\mathfrak g \text{Poly}(\mathfrak g)$ is the ideal of functions vanishing at $0$. Then $B$ lands in the <em>cofiltered tensor product</em>, which is just what it sounds like. (In arbitrary characteristic, $\mathcal P$ is the cofiltered dual of the filtered Hopf algebra $\mathcal S \mathfrak g$, the symmetric algebra of $\mathfrak g$, filtered by degree.))</p>
<h3>Why I care</h3>
<p>When $\mathfrak g$ is finite-dimensional over $\mathbb R$, and $U$ is the open neighborhood of $0$ in which $B$ converges, then $\mathfrak g$ acts as left-invariant derivations on $U$, where by <em>left-invariant</em> I mean under the multiplication $B$. Hence there is a canonical identification of the universal enveloping algebra $\mathcal U\mathfrak g$ with the algebra of left-invariant differential operators on $U$. Since $\mathfrak g$ is in particular a vector space, the "symbol" map gives a canonical identification between the algebra of differential operators on $U$ and the algebra of functions on the cotangent bundle $T^{\ast} U$ that are polynomial (of uniformly bounded degree) in the cotangent directions. Left-invariance then means that the operators are uniquely determined by their restrictions to the fiber $T^{\ast}_0\mathfrak g = \mathfrak g^{\ast}$, and the space of polynomials on $\mathfrak g^{\ast}$ is canonically the symmetric algebra $\mathcal S \mathfrak g$. This gives a canonical PBW map $\mathcal U \mathfrak g \to \mathcal S \mathfrak g$, a fact I learned from J. Baez and J. Dolan.</p>
<p>(In the formal group language, the noncocommutative cofiltered Hopf algebra $\mathcal P(\mathfrak g)$ is precisely the cofiltered dual to the filtered algebra $\mathcal U\mathfrak g$, whereas with its cocommutative Hopf structure $\mathcal P(\mathfrak g)$ is dual to $\mathcal S \mathfrak g$. But as algebras these are <em>the same</em>, and unpacking the dualizations gives the PBW map $\mathcal U\mathfrak g \cong \mathcal S \mathfrak g$, and explains why it is actually an isomorphism of coalgebras.)</p>
<p>Anyway, in one direction, the isomorphism $\mathcal U\mathfrak g \cong \mathcal S \mathfrak g$ is easy. Namely, the map $\mathcal S \mathfrak g \to \mathcal U \mathfrak g$ is given on monomials by the "symmetrization map" $\xi_1\cdots \xi_n \mapsto \frac1{n!} \sum_{\sigma \in S_n} \prod_{k=1}^n \xi_{\sigma(k)}$, where $S_n$ is the symmetric group on $n$ letters, and the product is ordered. (In this direction, the isomorphism of coalgebras is obvious. In fact, the corresponding symmetrization map into the full tensor algebra is a coalgebra homomorphism.)</p>
<p>In the reverse direction, I can explain the map $\mathcal U \mathfrak g \to \mathcal S \mathfrak g$ as follows. On a monomial $\xi_1\cdots \xi_n$, it acts as follows. Draw $n$ dots on a line, and label them $\xi_1,\dots,\xi_n$. Draw arrows between the dots so that each arrow goes to the right (from a lower index to a higher index), and each dot has either 0 or 1 arrow out of it. At each dot, totally order the incoming arrows. Then for each such <em>diagram</em>, evaluate it as follows. What you want to do is collapse each arrow $\psi\to \phi$ into a dot labeled by $[\psi,\phi]$ at the spot that was $\phi$, but never collapse $\psi\to \phi$ unless $\psi$ has no incoming arrows, and if $\phi$ has multiple incoming arrows, collapse them following your chosen total ordering. So at the end of the day, you'll have some dots with no arrows left, each labeled by an element of $\mathfrak g$; multiply these elements together in $\mathcal S\mathfrak g$. Also, multiply each such element by a numerical coefficient as follows: for each dot in your original diagram, let $m$ be the number of incoming arrows, and multiply the final product by the $m$th coefficient of the power series $A(x)$. Sum over all diagrams.</p>
<p>Anyway, the previous paragraph is all well and cool, but it would be better if the numerical coefficient could be read more directly off the diagram somehow, without having to really think about the function $A(x)$.</p>