Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2018-12-18T16:09:01Zhttps://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttps://mathoverflow.net/q/3189871The singularties of the dicriminant loci of the Lagrangian fibrationEdward Teachhttps://mathoverflow.net/users/982562018-12-18T15:31:24Z2018-12-18T15:31:24Z
<p>Let <span class="math-container">$X$</span> be a holomorphic symplectic variety of dimension <span class="math-container">$2n$</span> and <span class="math-container">$\pi: X \to \mathbb{P}^n$</span> be a Lagrangian fibration. It is known that <span class="math-container">$\pi$</span> is smooth outside of the discrimiant divisor <span class="math-container">$\Delta$</span>. The divisor <span class="math-container">$\Delta$</span> is not necessarily irreducible and hence it can be singular. Does somebody know some resctrictions on the singularities of <span class="math-container">$\Delta$</span>. For example, is <span class="math-container">$\Delta$</span> a simple normal crossing divisor?</p>
https://mathoverflow.net/q/3189861On a combinatorial set covering propertyDominic van der Zypenhttps://mathoverflow.net/users/86282018-12-18T15:17:34Z2018-12-18T15:17:34Z
<p>Let <span class="math-container">$\kappa < \lambda < \mu$</span> be infinite cardinals. Is there a collection <span class="math-container">${\cal U}\subseteq {\cal P}(\mu)$</span> of subsets of <span class="math-container">$\mu$</span> with the following properties?</p>
<ol>
<li>for all <span class="math-container">$U\in {\cal U}$</span> we have <span class="math-container">$|U| = \lambda$</span>;</li>
<li>every <span class="math-container">$S\subseteq \mu$</span> with <span class="math-container">$|S| = \kappa$</span> is contained in exactly one member of <span class="math-container">${\cal U}$</span>; and</li>
<li>for all <span class="math-container">$\alpha,\beta \in \mu$</span> we have <span class="math-container">$|{\cal U}_\alpha| = |{\cal U}_\beta|$</span>, where <span class="math-container">${\cal U}_\alpha = \{U\in {\cal U}: \alpha\in U\}$</span>, and <span class="math-container">${\cal U}_\beta$</span> is defined similarly.</li>
</ol>
https://mathoverflow.net/q/3189851Expander mixing lemma in combinatoric expandersArtur Riazanovhttps://mathoverflow.net/users/940562018-12-18T15:17:12Z2018-12-18T15:17:12Z
<p>There is a well-known relation between combinatoric expansion and the gap between the first and the second largest eigenvalues (Dodziuk 1984)
:
<span class="math-container">$$ h(G) \le d \sqrt{2 (1 - \alpha)} $$</span>
where
<span class="math-container">$$h(G) = \min\limits_{S\subseteq V;\; |S|\le|V|/2} \frac{|E(S,V\setminus S)|}{|S|},$$</span>
<span class="math-container">$\lambda_2 = \alpha \lambda_1 = \alpha d$</span> and <span class="math-container">$E(X,Y) = E \cap (X \times Y)$</span>. </p>
<p>The problem is that this connection holds only for <span class="math-container">$d$</span>-regular graphs, for non-regular graphs the largest eigenvalue is harder to compute.</p>
<p>However the expansion <span class="math-container">$h(G)$</span> is defined for non-regular graphs. <span class="math-container">$G$</span> is <span class="math-container">$(n/2, d, c)$</span>-expander if <span class="math-container">$h(G) \ge c$</span> and all degrees of the vertices of <span class="math-container">$G$</span> is at most <span class="math-container">$d$</span>. I would like to somehow convert such graph into an algebraic expander. What I really need from an algebraic expander that combinatoric one lacks is the mixing lemma:
<span class="math-container">$$ \left|{|E(S,T)| \over |V|} - {d |S| |T| \over |V|}\right| \le \alpha d \sqrt{|S||T|}. $$</span></p>
<p>What I can do is add loops to <span class="math-container">$G$</span> such that it is <span class="math-container">$d$</span>-regular. This modification does not affect <span class="math-container">$h(G)$</span> so the resulting graph <span class="math-container">$G'$</span> is <span class="math-container">$(n,d,\alpha)$</span> algebraic expander for some <span class="math-container">$\alpha$</span>. Therefore the mixing lemma works for <span class="math-container">$G'$</span>. But for disjoint <span class="math-container">$S$</span> and <span class="math-container">$T$</span> <span class="math-container">$E_G(S,T) = E_{G'}(S,T)$</span> so the mixing lemma works for disjoint <span class="math-container">$S$</span> and <span class="math-container">$T$</span> in <span class="math-container">$G$</span> as well.</p>
<p>Is there a flaw in this argument? And is there a better way to extract an algebraic properties from a combinatoric expander?</p>
<p><em>Dodziuk, Jozef</em>, <a href="http://dx.doi.org/10.2307/1999107" rel="nofollow noreferrer"><strong>Difference equations, isoperimetric inequality and transience of certain random walks</strong></a>, Trans. Am. Math. Soc. 284, 787-794 (1984). <a href="https://zbmath.org/?q=an:0512.39001" rel="nofollow noreferrer">ZBL0512.39001</a>.</p>
https://mathoverflow.net/q/3189840PCA with zero and high correlation in datathileepanhttps://mathoverflow.net/users/1336372018-12-18T15:13:41Z2018-12-18T15:13:41Z
<p>How would the eigen values look like when we apply PCA to a dataset with zero correlation between variables and when there is very high correlation between variables</p>
https://mathoverflow.net/q/3189831When Stone–Čech compactification is totally disconnectedArenahttps://mathoverflow.net/users/1336362018-12-18T15:09:08Z2018-12-18T15:09:08Z
<p>A topological space <span class="math-container">$X$</span> is totally disconnected if the connected components in <span class="math-container">$X$</span> are the one-point sets, and a topological space, <span class="math-container">$X$</span> is called completely regular exactly in case points can be separated from closed sets via continuous real-valued functions. Let <span class="math-container">$X$</span> be a totally disconnected and completely regular topological space. Can we deducd that <span class="math-container">$\beta X$</span> is also totally disconnected, where <span class="math-container">$\beta X$</span> is the Stone–Čech compactification of <span class="math-container">$X$</span>?</p>
https://mathoverflow.net/q/3189820An integral involving the argument of the Gamma functionOneTwoOnehttps://mathoverflow.net/users/1336342018-12-18T14:51:27Z2018-12-18T14:51:27Z
<blockquote>
<p>Evaluate <span class="math-container">$$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$</span>
where <span class="math-container">$\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$</span></p>
</blockquote>
<p>Note that <span class="math-container">$I$</span> converges since <span class="math-container">$\Gamma(s)\sim s\log s$</span>. I tried Wolfram Alpha, but it hasn't given me an answer after almost 90 minutes of computation, hence perhaps will never do.</p>
<p>PS: Migrated from <a href="https://math.stackexchange.com/q/3045147">https://math.stackexchange.com/q/3045147</a></p>
https://mathoverflow.net/q/3189800Simple way to generate (or characterize) real $N \times N$ matrices $K$ satisfying $(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \subseteq [\![N]\!]$dohmatobhttps://mathoverflow.net/users/785392018-12-18T14:39:38Z2018-12-18T14:39:38Z
<p>Let <span class="math-container">$N$</span> be a large positive integer.</p>
<h1>Question</h1>
<p>What is a simple way to generate (or characterize) real <span class="math-container">$N \times N$</span> matrices <span class="math-container">$K$</span> such that
<span class="math-container">$$
(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \subseteq [\![N]\!],
$$</span>
where <span class="math-container">$A$</span> is masked version of the <span class="math-container">$N$</span>-by-<span class="math-container">$N$</span> identity matrix defined by <span class="math-container">$(I_A)_{i,j}=1$</span> if <span class="math-container">$i=j \in A$</span> and <span class="math-container">$0$</span> otherwise.</p>
https://mathoverflow.net/q/318978-1Order Statistics and High Dimension GeometryAbhishekhttps://mathoverflow.net/users/111832018-12-18T14:13:59Z2018-12-18T14:13:59Z
<p>Suppose that I have iid random variables <span class="math-container">$\mathrm U_n \sim \mathrm U(0,1)$</span>. Then, for <span class="math-container">$\mathrm Y_m$</span> defined as,</p>
<p><span class="math-container">$$\mathrm Y_m = \min_{n \in [1,m]} \mathrm U_n$$</span></p>
<p>it is easy to compute <span class="math-container">$\mathbb{E}[\mathrm Y_m] = \frac1{m+1}$</span>.</p>
<p>Now consider a compact convex region <span class="math-container">$\mathcal{C}$</span> in higher dimensions <span class="math-container">$\mathbb{R}^k$</span> and let <span class="math-container">$\mathbf p_n$</span> be fixed points in <span class="math-container">$\mathcal C$</span> where <span class="math-container">$\mathbf p_n$</span> may lie on the boundary of <span class="math-container">$\mathcal C$</span>. Let <span class="math-container">$\mathrm D_n$</span> denote iid random variables in <span class="math-container">$\mathcal C$</span> such that <span class="math-container">$\mathbb P_{\mathrm D_n}(S) \geq \kappa \frac{\mathrm{vol}(S)}{\mathrm{vol}(\mathcal C)}~\forall S\subseteq \mathcal C$</span> for some<br>
fixed <span class="math-container">$\kappa < 1$</span>. Thus <span class="math-container">$\mathrm D_n$</span> denotes a "skewed" uniform distribution of points in <span class="math-container">$\mathcal C$</span>.</p>
<p>Suppose that <span class="math-container">$\mathrm U_n$</span> now corresponds to the distance of <span class="math-container">$\mathrm D_n$</span> from <span class="math-container">$\mathbf p_n$</span> ie. <span class="math-container">$\mathrm U_n \triangleq ||\mathrm D_n - \mathbf p_n||$</span>. Can someone point me in the right direction to find upper bounds on <span class="math-container">$\mathbb E[\mathrm Y_n]$</span> in such cases ( or direct me to references ) ?</p>
https://mathoverflow.net/q/3189771Representation as $n=p^2+q^2-r^2$toshihttps://mathoverflow.net/users/958382018-12-18T14:13:42Z2018-12-18T14:28:51Z
<p>What is known about the number of representations of a positive integer <span class="math-container">$n$</span> as
<span class="math-container">$$
\rho(n) = \# \{ (p,q,r): n=p^2+q^2-r^2\},
$$</span>
where all the variables are primes?
What about the average number of representations
<span class="math-container">$$\sum_{n \le x}\rho(n) \; ?$$</span></p>
https://mathoverflow.net/q/3189744Normalizers of subsystem subgroups of Lie groupsColumbus1https://mathoverflow.net/users/1336282018-12-18T13:35:28Z2018-12-18T15:29:14Z
<p>Let <span class="math-container">$G$</span> be a semisimple complex Lie group, and let <span class="math-container">$H$</span> be a subgroup corresponding to a subset of the extended Dynkin diagram of <span class="math-container">$G$</span> (à la Borel - de Siebenthal). I would like to know if there is a recipe for computing the normalizer of <span class="math-container">$H$</span>. My feeling is that this must be known, but I could not find anything.</p>
<p>For concreteness, consider this example. Let <span class="math-container">$G=E_7$</span> (simply connected, say). There is a subgroup of type <span class="math-container">$A_7$</span>, which has index <span class="math-container">$2$</span> inside its normalizer. This corresponds to the involution of the Dynkin diagram of type <span class="math-container">$A_7$</span>, that one can check respects the highest root of <span class="math-container">$E_7$</span> and so respects the extended Dynkin diagram of type <span class="math-container">$E_7$</span>. On the other hand, consider the subgroup of type <span class="math-container">$D_6\times A_1$</span>. The index of the subgroup in its normalizer is at most <span class="math-container">$2$</span>, because the Dynkin diagram has automorphism group <span class="math-container">$2$</span>. It is clear that the obvious involution of the Dynkin diagram does not extend to an involution of the extended Dynkin diagram of <span class="math-container">$E_7$</span>, because the highest root is sent to some other root. Does this imply that the subgroup is self-normalizing? I would expect that this subgroup is not self-normalizing.</p>
https://mathoverflow.net/q/3189725Percentage of Ramanujan's conjectures that were proven correctAidan Rockehttps://mathoverflow.net/users/563282018-12-18T13:06:07Z2018-12-18T14:43:28Z
<p>Today I read the following brief but insightful account of Ramanujan's approach to mathematics: <a href="https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf" rel="noreferrer">https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf</a> and while reading this I wondered whether we have a lower-bound on the percentage of Ramanujan's conjectures which are correct. </p>
<p>I'm planning to get a copy of Ramanujan's notebooks. Meanwhile, the above question intrigues me. </p>
https://mathoverflow.net/q/3189712Invariant submanifolds tangent to isotypic subrepresentationsAnna Abashevahttps://mathoverflow.net/users/883852018-12-18T13:01:36Z2018-12-18T14:42:39Z
<p>Let <span class="math-container">$G$</span> be a Lie group acting on a complex manifold <span class="math-container">$M$</span>. Let <span class="math-container">$p$</span> be an isolated fixed point. Let us look at the representation of <span class="math-container">$G$</span> on <span class="math-container">$T_pM$</span>. Suppose <span class="math-container">$T_pM = \bigoplus V_i^{\oplus n_i}$</span> where <span class="math-container">$V_i$</span> is an irreducible complex representation and <span class="math-container">$V_i\ne V_j$</span>. Let us denote <span class="math-container">$V_i^{\oplus n_i}$</span> by <span class="math-container">$W_i$</span> and call <span class="math-container">$W_i$</span> an isotypic component of the representation.</p>
<p>Question 1: Under what conditions on <span class="math-container">$G$</span> and/or on the isotypic component <span class="math-container">$W_i$</span> there locally exists a <span class="math-container">$G$</span>-invariant complex submanifold <span class="math-container">$M_i$</span> such that <span class="math-container">$T_pM_i = W_i$</span>? If it exists, how one can construct it? It is clear that we should look exactly on the isotypic components and not on arbitrary subrepresentations as one can easily imagine an example when the representation is trivial but for every complex subspace of <span class="math-container">$T_pM$</span> there doesn't exist an invariant submanifold tangent to it. In fact, when <span class="math-container">$W_i$</span> is trivial, I suppose, many bad things can happen.</p>
<p>Question 2: If such a submanifold exists, is it unique?</p>
<p>I'd be grateful if you can answer these questions just for <span class="math-container">$\mathbb G_m^n$</span> but of course it is much more interesting for arbitrary Lie groups. In the case of <span class="math-container">$\mathbb G_m^n$</span> one can look on the attractor and the repellent of the fixed point but that gives as only invariant submanifolds tangent the direct sum of subrepresentations with negative and positive weights accordingly.</p>
https://mathoverflow.net/q/3189682Dimension of hermitian rank at most $k$ matrices over quaternionsJosiah Parkhttps://mathoverflow.net/users/1187312018-12-18T11:55:34Z2018-12-18T12:24:16Z
<p>In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the <a href="https://core.ac.uk/download/pdf/82151498.pdf" rel="nofollow noreferrer">spectral theorem</a> (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in <span class="math-container">$\mathbb{H}^{m\times m}$</span>, allowing for a convenient notion of rank of a normal matrix over quaternions (the number of positive singular values). Vectors have images under the <span class="math-container">$*$</span>-homomorphism, <span class="math-container">$\Psi:\mathbb{H}^m\rightarrow \mathbb{C}^{2m}$</span> given by <span class="math-container">$$\xi=\xi_{1}+\xi_{2}j \mapsto \begin{pmatrix} \ \ \xi_{1} \\ -\overline{\xi_{2}}\end{pmatrix},$$</span> where <span class="math-container">$\xi_{1},\xi_{2}$</span> are complex quaternions (that is, with zero <span class="math-container">$j$</span> and <span class="math-container">$k$</span> parts), and the image of (right) linearly independent vectors over <span class="math-container">$\mathbb{H}$</span> remain linearly independent over <span class="math-container">$\mathbb{C}$</span> under this mapping. Further for matrices over quaternions the mapping <span class="math-container">$\Phi:\mathbb{H}^{m\times m}\rightarrow\mathbb{C}^{2m\times 2m}$</span> <span class="math-container">$$\Gamma_{1}+\Gamma_{2}j\mapsto \begin{pmatrix} \Gamma_{1} & \Gamma_{2} \\ -\overline{\Gamma_{2}} & \overline{\Gamma_{1}} \end{pmatrix}$$</span> the rank of the image under <span class="math-container">$\Phi$</span> of a matrix is doubled.</p>
<p>Without taking consideration the form of matrices in the image of <span class="math-container">$\Phi$</span>, the set of <span class="math-container">$2m\times 2m$</span> matrices of rank at most <span class="math-container">$2k$</span> over <span class="math-container">$\mathbb{C}$</span> is an irreducible projective algebraic variety of co-dimension <span class="math-container">$4(m−k)^2$</span>. Is there a way to meaningfully say something about the dimension of the space of hermitian <span class="math-container">$m\times m$</span> matrices over quaternions with rank less than or equal to <span class="math-container">$k$</span>? </p>
<p>Quaternions being non-commutative means that the condition of all size <span class="math-container">$k+1$</span> minors vanishing which can define a rank at most <span class="math-container">$k$</span> matrix over <span class="math-container">$\mathbb{C}$</span> is not handled similarly for <span class="math-container">$\mathbb{H}$</span>, with the closest thing resembling being 'quasideterminants' (see the <a href="https://arxiv.org/abs/math/0208146" rel="nofollow noreferrer">paper</a> by Gelfand et al from 2002 ). Does this question make sense in non-commutative algebraic geometry, or is it more natural to come from another perspective? </p>
https://mathoverflow.net/q/3189662IC sheaves and formal neighbourhoodsAlexander Bravermanhttps://mathoverflow.net/users/38912018-12-18T11:12:33Z2018-12-18T11:12:33Z
<p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be two schemes of finite type over a finite field <span class="math-container">$\mathbb F_q$</span>. Let <span class="math-container">$x$</span> (resp. <span class="math-container">$y$</span>) be an <span class="math-container">$\mathbb F_q$</span>-point of <span class="math-container">$X$</span> (resp. of <span class="math-container">$Y$</span>). </p>
<p>Let now <span class="math-container">$l$</span> be a prime which is prime to <span class="math-container">$q$</span>. Let <span class="math-container">$IC(X), IC(Y)$</span> denote the corresponding <span class="math-container">$l$</span>-adic intersection cohomology sheaves.
Assume now that we are given an isomorphism between the formal neighbourhoods of <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. </p>
<p><span class="math-container">$\mathbf{Question:}$</span> Does it give rise to an isomorphism between the stalk of <span class="math-container">$IC(X)$</span> at <span class="math-container">$x$</span> and the stalk of <span class="math-container">$IC(Y)$</span> at <span class="math-container">$y$</span> (compatible with the action of Frobenius on both stalks)? In fact, for my purposes it is enough to know that they are abstractly isomorphic (in a way which is compatible with the action of Frobenius). </p>
https://mathoverflow.net/q/3189592Bounded holomorphic functions on a Riemann surface separating pointsJaikrishnanhttps://mathoverflow.net/users/360382018-12-18T10:16:38Z2018-12-18T15:18:28Z
<p>Let <span class="math-container">$R$</span> be a Riemann surface that admits a non-constant bounded holomorphic function. Then is it true that any two points of <span class="math-container">$R$</span> can be separated by a bounded holomorphic function? This is easy to see when <span class="math-container">$R$</span> is a planar domain. </p>
https://mathoverflow.net/q/3189525What do we know about the ramification of the modularity map $X_0(N)\to E$?Bonbonhttps://mathoverflow.net/users/1084862018-12-18T07:05:02Z2018-12-18T14:26:21Z
<p>Let <span class="math-container">$E$</span> be an elliptic curve over <span class="math-container">$\mathbb{Q}$</span>, and let <span class="math-container">$N$</span> be its conductor. By the modularity of elliptic curves over <span class="math-container">$\mathbb{Q}$</span>, there exists a surjective map <span class="math-container">$f:X_0(N)\to E$</span>, where <span class="math-container">$X_0(N)$</span> is the modular curve. </p>
<p>I want to know about the ramification of this map, is there any results or examples or references?</p>
<p>New Edit: By the way, what do we know about the degree of <span class="math-container">$f$</span> ?</p>
https://mathoverflow.net/q/3188980Can reflection in $V$ and reflection out of $V$ principles be used in the same theory?Zuhair Al-Joharhttps://mathoverflow.net/users/953472018-12-17T19:26:54Z2018-12-18T11:39:01Z
<p>I've noticed that most of the work in Ackermann set theory is primarily concerned with constructing sets in <span class="math-container">$V$</span>, the rest of the classes are just excess material, carrying no comprehension over them. There is a try of Muller in which he strengthen the class existence principle of Ackermann into Separation over classes, the resultant theory is <span class="math-container">$A$</span>, and adding Regularity <span class="math-container">$R$</span>, and Choice <span class="math-container">$C$</span>, he gets into <span class="math-container">$ARC$</span>, a theory claimed [see <a href="http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF" rel="nofollow noreferrer">here] </a>to serve as a foundation of both category and set theory, and thus for most of mathematics.</p>
<p>This gave me the idea of <em>reflecting-out of <span class="math-container">$V$</span></em> principle, since Ackermann's set theory can be interpeted in systems using reflection [see <a href="https://mathoverflow.net/questions/317658/what-is-the-strength-of-adding-limitation-of-size-and-a-simple-version-of-reflec">here] </a>, so if to any of the two systems appearing in that posting (with reflection in them re-named as <em>reflection in <span class="math-container">$V$</span></em>), we add the following principle:</p>
<p><strong>Reflection out of <span class="math-container">$V$</span> schema:</strong> if <span class="math-container">$\varphi$</span> is a sentence in <span class="math-container">$FOL(=,\in)$</span>, i.e. doesn't use the symbol <span class="math-container">$V$</span>, and <span class="math-container">$\varphi^V$</span> is the bounded by <span class="math-container">$V$</span> sentence of <span class="math-container">$\varphi$</span>, i.e. the sentence obtained by merely bounding every quantifier in <span class="math-container">$\varphi$</span> by <span class="math-container">$V$</span>, then: <span class="math-container">$ \varphi^V \to \varphi $</span>, is an axiom.</p>
<p>In other words we are reversing the reflection process, so we are concluding things about classes in general by reflecting from the inside of <span class="math-container">$V$</span> to outside it. By that, all set axioms (i.e. sentences in the language of set theory that are satisfied in <span class="math-container">$V$</span>), would generalize over all classes. This way we easily get to interpret Muller's theory.</p>
<blockquote>
<p>Question: is there an obvious inconsistency with a theory that both uses reflection in <span class="math-container">$V$</span> and reflection out of <span class="math-container">$V$</span> principles?</p>
</blockquote>
https://mathoverflow.net/q/3188692If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non-positive sectional curvature?Jialong Denghttps://mathoverflow.net/users/905122018-12-17T13:23:51Z2018-12-18T14:00:15Z
<p>If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature?
I wish to use the result about the question and find Leeb's work <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1358977" rel="nofollow noreferrer"><em>3-manifolds with(out) metrics of nonpositive curvature</em></a> relate it. But I can not understand the paper since I am a novice in 3-dimension manifold, so I ask for help .</p>
<p>Leeb asked in 1995: <em>which compact aspherical
3-manifolds admit Riemannian metrics of nonpositive sectional curvature?</em> Do we know more results about the question though the progress of those years in 3-manifold?</p>
<p>Add: Thanks to Thiku's answer, it is clearly for aspherical 3-manifold. It may too ambitious to ask <em>which compact aspherical smooth
n-manifolds (n>3) admit Riemannian metrics of nonpositive sectional curvature?</em>, since Davis and Januszkiewicz proved some exotic aspherical closed manifolds in the paper <em><a href="https://mathscinet.ams.org/mathscinet/search/publdoc.html?extend=1&l=100&pg1=INDI&s1=93735&sort=Newest&vfpref=html&r=34&mx-pid=1131435" rel="nofollow noreferrer">Hyperbolization of polyhedra</em>.</a> For example, <em>for each n≥4 there exists an aspherical closed n-dimensional manifold such that its universal covering is not homeomorphic to n-dimension Euclid space.</em> So I reduce it to the following questions:</p>
<p>1) If the total space of circle bundle over closed manifold with Riemannian metric of non-positive sectional curvature also admits Riemannian metric of non-positive sectional curvature?</p>
<p>2) If the total space of fiber bundle over circle, which the fiber are closed manifold with Riemannian metric of non-positive sectional curvature, also admits Riemannian metric of non-positive sectional curvature?</p>
<p>3) In general, given the fiber bundle with fiber and base space are closed manifolds with Riemannian metric of non-positive sectional curvature, if its total space also admits Riemannian metric of non-positive sectional curvature?</p>
https://mathoverflow.net/q/3188435Unconditionaly convergent series in some functional spacesDuchamp Gérard H. E.https://mathoverflow.net/users/252562018-12-17T09:00:09Z2018-12-18T11:37:49Z
<p>Linked with [this question and discussion](
<a href="https://mathoverflow.net/questions/289760/bilinear-product-of-two-summable-families">Bilinear product of two summable families</a>), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall that a
family <span class="math-container">$(a_i)_{i\in I}$</span> in a topological abelian group <span class="math-container">$(G,+)$</span> is called <em>summable</em> with sum <span class="math-container">$S$</span> iff
for all neighbourhood of zero <span class="math-container">$W$</span> it exists <span class="math-container">$J_W\subset_{finite} I$</span> such that for all <span class="math-container">$J$</span> with
<span class="math-container">$J_W\subset J\subset_{finite} I$</span>, <span class="math-container">$(S-\sum_{i\in J}a_i)\in W$</span>. It amounts to the same to say that the
net <span class="math-container">$J\mapsto \sum_{i\in J}a_i$</span> (from <span class="math-container">$2^{(I)}$</span>, the set of finite subsets of <span class="math-container">$I$</span>, ordered by
inclusion, to <span class="math-container">$G$</span>) converges to <span class="math-container">$S$</span>. </p>
<p>It is known that, when <span class="math-container">$I=\mathbb{N}$</span> (series <span class="math-container">$\sum_{n\geq 0}\,a_n$</span>) the series
<span class="math-container">$\sum_{n\geq 0}\,a_n$</span> is summable iff it is unconditionaly convergent, i.e. the sequence of partial
sums
<span class="math-container">$$
N\to \sum_{n=0}^N\,a_{\sigma(n)}
$$</span><br>
converges for all permutation <span class="math-container">$\sigma$</span> of <span class="math-container">$\mathbb{N}$</span>. </p>
<p><strong>Question(s)</strong> I am particularly interested in counterexamples/results about series
<span class="math-container">$\sum_{n\geq 0}\,a_n$</span> which are unconditionaly convergent but not absolutely convergent
in the following frameworks </p>
<ol>
<li> <span class="math-container">$K=[0,1]\subset \mathbb{R}$</span> and a series of continuous real functions <span class="math-container">$\sum_{n\geq 0}\,f_n$</span> unconditionaly convergent but not absolutely convergent i.e.
<span class="math-container">$$
\sum_{n\geq 0}\,||f_n||_K<+\infty
$$</span>
(where <span class="math-container">$\|f\|_K=\sup_{s\in K}|f_s|$</span>)
<li> <span class="math-container">$\mathcal{H}(\Omega)$</span> (space of holomorphic functions <span class="math-container">$\Omega\to \mathbb{C}$</span>, where
<span class="math-container">$\Omega\subset \mathbb{C}$</span> is not empty and open). In this context, absolutely convergent,
for a series <span class="math-container">$\sum_{n\geq 0}\,f_n$</span>, means that for all <span class="math-container">$K\subset_{compact} \Omega$</span>, one has
<span class="math-container">$$
\sum_{n\geq 0}\,||f_n||_K<+\infty
$$</span>
</ol>
<p>are there (counter-)examples or general results in these directions ?</p>
https://mathoverflow.net/q/31883950Old books you would like to have rewritten with high-quality typesettingC.F.Ghttps://mathoverflow.net/users/906552018-12-17T08:12:38Z2018-12-18T13:21:36Z
<p>There are some questions on <a href="http://mathoverflow.net">mathoverflow</a> such as</p>
<ul>
<li><a href="https://mathoverflow.net/q/18271/90655">What out-of-print books would you like to see re-printed?</a></li>
<li><a href="https://mathoverflow.net/q/117415/90655">Old books still used</a></li>
</ul>
<p>with answers that tell us things such as:</p>
<p><em>Mathematicians prefer to use older books because of some old books are full of amazing ideas and some of them are comprehensive (such as books of Spivak).</em></p>
<blockquote>
<p><strong>Question:</strong> What older books (with low quality typesetting) would you like to see reprinted with high quality typesetting?</p>
</blockquote>
<p>My question is not just a question. We are a group of math students (most of them are geometry students) that want to re-write popular old books using <span class="math-container">$\LaTeX$</span>.</p>
<p>One can search for most cited books such as: <em>Curvature and Betti numbers</em> (K Yano, S Bochner) or <em>Einstein manifolds</em> (AL Besse).</p>
https://mathoverflow.net/q/3188273Flat solvmanifolds?user60933https://mathoverflow.net/users/473362018-12-17T05:26:30Z2018-12-18T14:15:47Z
<p>I am trying to look for some reference for solvmanifolds and come up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds by Bieberbach's Theorem. I am almost sure I missed something trivial, but what's the special of studying flat solvmanifolds?</p>
https://mathoverflow.net/q/3155670Reference from the article "Random Ordinary Differential Equations", by J.L. Stranduser131499https://mathoverflow.net/users/1314992018-11-17T22:33:13Z2018-12-18T12:03:25Z
<p>In the article <a href="https://core.ac.uk/download/pdf/82447522.pdf" rel="nofollow noreferrer"><em>Random Ordinary Differential Equations</em></a>, Journal of differential equations 7, 538-553 (1970), by J.L. Strand, reference number 6 refers to his PhD thesis: <em>Stochastic Ordinary Differential Equations</em>, University of California (Berkeley), 1968. Reference number 5 is also a PhD thesis: <em>Random Ordinary Differential Equations</em>, University of California (Berkeley), 1968, by R. Edsinger.</p>
<p>Do you know how to obtain these two PhD theses?</p>
https://mathoverflow.net/q/2945271Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?Peterhttps://mathoverflow.net/users/493982018-03-06T11:48:02Z2018-12-18T11:25:27Z
<p>In <a href="https://math.stackexchange.com/questions/2635516">this question on MSE</a>, Enzo Creti asks for a prime number formed by concatenating the Mersenne numbers <span class="math-container">$2^n-1$</span> and <span class="math-container">$2^{n-1}-1$</span>, for example, 40952047. For all residues modulo 7, he found primes except for the residue 6. This is somewhat surprising because the residue 1 occurs only with half frequency.</p>
<blockquote>
<p>Is there any hidden structure forcing a non-trivial factor in the case of residue 6, or was it just "bad luck" that no prime was found despite an enormous search range?</p>
</blockquote>
<p>I invite everyone to join in the search for a prime. I posted the necessary details <a href="https://github.com/gnufinder/special-prime/issues" rel="nofollow noreferrer">on github</a>.</p>
<p>The following vector contains all numbers n<=366800 leading to a prime</p>
<p>[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]</p>
<p>Exponent <span class="math-container">$541456$</span> leads to another probable prime with residue 5 mod 7 and 325990 digits, but it need not be the next in increasing order.
More details can be found on the github-site.
Heuristically, for every k>=3 , the range [10^k..10^(k+1)] should contain 5.4 numbers leading to a prime, so the range [10^4..10^5] with 8 primes is "above average", whereas the range [10^3..10^4] is within the expectation.
The sequences of exponents and associated primes are here: <a href="https://oeis.org/A301806" rel="nofollow noreferrer">A301806</a> and here: <a href="https://oeis.org/A298613" rel="nofollow noreferrer">A298613</a>.
I noticed these things:
let's call ec(n)=<span class="math-container">$2^n-1$</span>||<span class="math-container">$2^{n-1}-1$</span> where || denotes the concatenation in base 10. </p>
<p>ec(43*5) is prime. 5 (odd) is congruent to -8 mod 13.
ec(43*1620) is prime. 1620 (even) is congruent to 8 mod 13.
ec(43*2140) is prime. 2140 (even) is congruent to 8 mod 13.
ec(43*12592) is prime. 12592 (even) is congruent to 8 mod 13.
ec(67*1) is prime. 1 is congruent to 1 mod 13.
ec(67*768) is prime. 768 is congruent to 1 mod 13.</p>
https://mathoverflow.net/q/2848994Intersection of iterated powerset in NFUMarcos Cramerhttps://mathoverflow.net/users/51992017-10-31T17:03:34Z2018-12-18T16:02:45Z
<p>I am interested in the existence of the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ for any given set $x$, in the context of NFU (New Foundations with Urelements). It seems to me that the comprehension axiom in NFU does not allow to prove that $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists for any set $x$. On the other hand, it seems to me that the model of NFU described in the SEP article <a href="https://plato.stanford.edu/entries/settheory-alternative/#MathNFUInfiChoi" rel="nofollow noreferrer">Alternative Axiomatic Set Theories</a> satisfies the property that for every $x$, $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists. </p>
<p>Let me informally describe my reasons for believing that in that model, $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists for every $x$. The model described in the SEP article is defined with the help of a non-standard model of ZFC with an automorphism $j$ that moves rank $\alpha$ to a lower rank. In that case, the $V_\alpha$ of that model of ZFC is a model of NFU, if $\in_{NFU}$ is suitably defined as in the linked article. Under those conditions, the interpretation of $\mathcal{P}(x)$ in the model will be an element of $V_{j(\alpha)+1}$. But then the interpretation of $\mathcal{P}^n(x)$ in the model will be an element of $V_{j(\alpha)+1}$ for any $n \in \mathbb{N}$, so the interpretation of $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ in the model will also be an element of $V_{j(\alpha)+1}$, so $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists according to the model.</p>
<p>I have four interdependent questions:</p>
<ul>
<li>Is it correct that the statement "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" is not provable in NFU? </li>
<li>Is my above sketch of the claim that the statement "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" could be added to NFU without losing consistency correct?</li>
<li>If the answer to the second question is "no", are there other known ways to prove that the statement "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" could be added to NFU without losing consistency?</li>
<li>If the answer to the fourth question is "no", is it known that adding "for every $x$, the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ exists" to NFU leads to an inconsistency?</li>
</ul>
https://mathoverflow.net/q/27899612Precise form of the mean motion theoremAlexandre Eremenkohttps://mathoverflow.net/users/255102017-08-17T23:42:17Z2018-12-18T14:52:39Z
<p>Consider an exponential polynomial
<span class="math-container">$$f(t)=\sum_{k=1}^na_k\exp(i\lambda_kt),$$</span>
where <span class="math-container">$a_k$</span> are complex and <span class="math-container">$\lambda_k, t$</span> real. The usual form of the Mean Motion Theorem says that the limit
<span class="math-container">$$\lim_{t\to+\infty}\frac{\arg f(t)}{t}$$</span>
exists. (If <span class="math-container">$f$</span> has real zeros one defines <span class="math-container">$\arg f$</span> by bypassing them along small
half-circles in the upper half-plane).</p>
<p>All books that I know mention that this was conjectured by Lagrange, and proved by
P. Bohl for <span class="math-container">$n=3$</span> and by B. Jessen and H. Tornehave (1945) in general, after earlier incomplete proofs by H. Weyl and P. Hartman. However in the paper of Bohl, a much more subtle question is actually studied, namely whether we have
<span class="math-container">$$\arg f(t)=ct+O(1).$$</span>
He shows that for <span class="math-container">$n=3$</span> this is sometimes the case, sometimes not, and gives an
exact condition in terms of <span class="math-container">$a_k,\lambda_k$</span>. My question is:</p>
<blockquote>
<p>Has anyone ever continued this line of inquiery? Can
<span class="math-container">$$\arg f(t)=ct+o(t)$$</span>
be improved: a) in general, b) under some additional conditions on <span class="math-container">$a_k,\lambda_k$</span>?</p>
</blockquote>
<p>Of course one such condition is known since Lagrange: if <span class="math-container">$|a_1|>\sum_{k=2}^n|a_k|$</span>, then the error term is <span class="math-container">$O(1)$</span>.</p>
<p>Ref. P. Bohl, "<a href="https://eudml.org/doc/149304" rel="nofollow noreferrer">Über ein in der Theorie der säkularen Störungen vorkommendes Problem</a>", J. reine angew Math. 135 (1909) 189-283. There is a Russian translation
in P. Bohl, Collected Works, Riga, Znanie, 1974. </p>
https://mathoverflow.net/q/2508453extending continuous functions from dense subsets to quasicompactsuser97621https://mathoverflow.net/users/976212016-09-27T14:22:46Z2018-12-18T14:01:10Z
<p>I am interested under what assumptions one can always extend continuously a function defined on a dense subset; the range of the function is compact but not necessarily Hausdorff. </p>
<p>That is, I am interested in generalisations of the following theorem [Engelking, General Topology] to non-Hausdorff compact spaces:</p>
<blockquote>
<p>3.2.1. THEOREM. Let $A$ be a dense subspace of a topological space $X$ and $f$ a continuous mapping of $A$ to a compact space $Y$. The mapping $f$ has a continuous extension over $X$ if and
only if for every pair $B_1,B_2$ of disjoint closed subsets of $Y$ the inverse images $f^{-l}(B_1)$ and
$f^{-l}(B_2)$ have disjoint closures in the space $X$.</p>
</blockquote>
<p>I am mostly interested in sufficient conditions. </p>
<p>For example, is the following sufficient?</p>
<blockquote>
<p>(i) For each $Z_1, Z_2\subset A$, it holds $cl_X(Z_1) \cap cl_X( Z_2) = cl_X( cl_A(Z_1)\cap cl_A(Z_2))$</p>
<p>(ii) For each $Z\subset X$ closed, each closed subsets $Z_1, Z_2\subset A$ such that $Z\cap A=Z_1\cup Z_2$,
there are $Z'_1, Z'_2 \subset X$ closed such that $Z=Z'_1\cup Z'_2$, and $Z_1=Z'_1\cap A$, and $Z_2=Z'_2\cap A$, and $Z'_1\cap Z'_2=cl_X(Z_1\cap Z_2)$.</p>
<p>(iii) $A$ is an open dense subset of $X$</p>
</blockquote>
https://mathoverflow.net/q/1499505Moreau-Yosida regularization in Banach spacesGenHhttps://mathoverflow.net/users/432702013-11-25T22:41:48Z2018-12-18T14:05:53Z
<p>For a seminar I am working on a Moreau-Yosida regularization in Banach spaces.</p>
<p>The regularization is defined by</p>
<p><span class="math-container">$$f_\lambda(x) := \inf \left \{ \frac{\|x-y\|^2}{2\lambda} +f(y) : y \in X \right \}, ~ x \in X$$</span></p>
<p>where <span class="math-container">$X$</span> is a reflexive and strictly convex Banach space, <span class="math-container">$f: X \rightarrow \mathbb{R} \cup \{+\infty\}$</span> lower-semicontinuous, proper and convex, <span class="math-container">$\lambda > 0$</span>.</p>
<p>In <strong>Convexity and Optimization in Banach Spaces</strong>, V. Barbu and T. Precupanu, Springer, 2012, the authors prove that the infimum defining <span class="math-container">$f_\lambda(x)$</span> is attained for all <span class="math-container">$x \in X$</span>. </p>
<p>From that they deduce that <span class="math-container">$f_\lambda$</span> is convex and lower-semicontinious, but, unfortunately, they don't elaborate the argument. However, I do not see how it follows from what is given.</p>
<p>As far as I am aware, most literature deals with the case where <span class="math-container">$X$</span> is Hilbert, I could not finde the result stated above for the general case in another publication.</p>
<blockquote>
<p><strong>Question:</strong> Maybe someone can recommed me a paper where this result is proven? Or is there an obvious argument which I am missing at the moment?</p>
</blockquote>
<p>Thank you in advance</p>
https://mathoverflow.net/q/3247947What are some mathematical sculptures?Gerald Edgarhttps://mathoverflow.net/users/4542010-07-19T12:37:37Z2018-12-18T13:35:40Z
<p>Either intentionally or unintentionally.
Include location and sculptor, if known.</p>
https://mathoverflow.net/q/23478738Examples of common false beliefs in mathematicsgowershttps://mathoverflow.net/users/14592010-05-04T21:02:58Z2018-12-18T12:56:42Z
<p>The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while they are learning mathematics, but quickly abandon when their mistake is pointed out -- and also in why they have these beliefs. So in a sense I am interested in commonplace mathematical mistakes.</p>
<p>Let me give a couple of examples to show the kind of thing I mean. When teaching complex analysis, I often come across people who do not realize that they have four incompatible beliefs in their heads simultaneously. These are</p>
<p>(i) a bounded entire function is constant;<br/>
(ii) $\sin z$ is a bounded function;<br/>
(iii) $\sin z$ is defined and analytic everywhere on $\mathbb{C}$;<br/>
(iv) $\sin z$ is not a constant function.<br/></p>
<p>Obviously, it is (ii) that is false. I think probably many people visualize the extension of $\sin z$ to the complex plane as a doubly periodic function, until someone points out that that is complete nonsense.</p>
<p>A second example is the statement that an open dense subset $U$ of $\mathbb{R}$ must be the whole of $\mathbb{R}$. The "proof" of this statement is that every point $x$ is arbitrarily close to a point $u$ in $U$, so when you put a small neighbourhood about $u$ it must contain $x$. </p>
<p>Since I'm asking for a good list of examples, and since it's more like a psychological question than a mathematical one, I think I'd better make it community wiki. The properties I'd most like from examples are that they are from reasonably advanced mathematics (so I'm less interested in very elementary false statements like $(x+y)^2=x^2+y^2$, even if they are widely believed) and that the reasons they are found plausible are quite varied.</p>
https://mathoverflow.net/q/826037Proof assistants for mathematicsLSpicehttps://mathoverflow.net/users/23832009-12-08T22:25:03Z2018-12-18T12:07:34Z
<p>This question is related to (maybe even the same in intent as) <a href="https://mathoverflow.net/questions/1017/intro-to-automatic-theorem-proving-logical-foundations">Question 1017</a>, but none of the answers seem to address what I'm looking for.</p>
<p>There are a lot of resources available for people who want to use proof assistants like Coq, Isabelle, …, to prove properties about programs—and that's no surprise, since a lot of the development of these programs is done by computer scientists. However, I am interested in resources, and <em>especially</em> in course materials (because I'm trying to put together an independent study for a CS student), involving the use of proof assistants to prove <em>mathematical</em> statements—see the work of <a href="http://sites.google.com/site/thalespitt" rel="noreferrer">Hales</a> and <a href="http://www.cs.ru.nl/~freek" rel="noreferrer">Weedijk</a> for examples. Does anyone know of any such?</p>