Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2016-05-27T20:25:14Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2399270On a count of certain number of primes in an intervalTurbohttp://mathoverflow.net/users/100352016-05-27T19:53:51Z2016-05-27T19:59:32Z
<p>Fix a prime $p$, $\alpha\in(0,1)$ and $\beta\in(1,2)$ and let $\mathcal U$ be primes in $[p^\alpha,\beta p^\alpha]$ such that if $b\in\mathcal U$ and if $d$ is multiplicative inverse of $b$ in $\Bbb Z_{2p-1}$ then $gcd(m,p-1)=1$ where $m$ satisfies $bd=1+m(2p-1)$.</p>
<p>What is the cardinality of $\mathcal U$ in worst case and in average case as we change prime $p$ at least under any conjectures?</p>
<p>Without $m$ criterion we have from pnt $\frac{(\beta-1)p^\alpha}{\alpha\log p}$ primes. Best case is if $p-1=2q$ for some prime $q$ then essentially this estimate $\frac{(\beta-1)p^\alpha}{\alpha\log p}$ does not change since $\alpha$ is fixed.</p>
<p>However what if $p-1$ has $(\log p)^{1-\gamma}$ factors for some fixed $\gamma\in(0,1)$ or if $p-1$ has $\frac{c\log p}{\log\log p}$ factors for some fixed $c\in(0,1)$? It is not clear how many such $m$ will avoid these factors.</p>
<p><strong>Conjecture</strong>: In average case there is a $\zeta\in(1,2)$ such that for any fixed $\alpha\in(0,1)$ and $\beta\in(1,2)$ number of such primes in $\mathcal U=[p^\alpha,\beta p^\alpha]$ is $\Theta\Big(\frac{(\beta-1)p^\alpha}{(\alpha\log p)^{\zeta}}\Big)$.</p>
http://mathoverflow.net/q/2399232Hadwiger number and minimal degreeDominic van der Zypenhttp://mathoverflow.net/users/86282016-05-27T19:01:42Z2016-05-27T19:52:58Z
<p>Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximal $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree, do we have $\delta(G)\leq\eta(G)$?</p>
http://mathoverflow.net/q/2399218Concept associated to the Eudoxus realsPhilhttp://mathoverflow.net/users/922262016-05-27T18:00:49Z2016-05-27T19:25:46Z
<p>I am aware of three different constructions of the field of real numbers : </p>
<ol>
<li><p>The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the completion of $\mathbb{Q}$ for the standard metric on $\mathbb{Q}$. </p></li>
<li><p>The Dedekind cut construction : in this case, the field $\mathbb{Q}$ is seen as a partially ordered set (with the standard order), and $\mathbb{R}$ is the completion of $\mathbb{Q}$ in the sense of the smallest complete lattice containing $\mathbb{Q}$.</p></li>
<li><p>The "Eudoxus" reals : in this case, we don't start from $\mathbb{Q}$ but directly from $\mathbb{Z}$. Reals numbers are identified as equivalent classes of "almost-homomorphisms" from $\mathbb{Z}$ into $\mathbb{Z}$ (functions $f$ from $\mathbb{Z}$ into $\mathbb{Z}$ such that $\{f(m+n)-f(m)-f(n): m, n\in\mathbb{Z}\}$ is finite and two almost-homomorphisms $f$, $g$ are equivalent iff the set $\{f(m)-g(m): m\in\mathbb{Z}\}$ is finite. As far as I understand, only the additive group structure on $\mathbb{Z}$ is used. </p></li>
</ol>
<p>My question is this : We see that the first two constructions are actually just specific examples of application of a general process (metric completion, order completion). What is the "completion" process associated to the Eudoxus real construction (if there is one)? Could it be applied to other abelian groups, was is already explored ?</p>
http://mathoverflow.net/q/2399150Becoming a Mature Mathematician [on hold]Rebecca Hardenbrookhttp://mathoverflow.net/users/922202016-05-27T15:40:07Z2016-05-27T15:40:07Z
<p>I am currently a sophomore in my undergraduate mathematics program. It has taken me a while to take school seriously; I was one of "those" students who just skated by without studying until Linear Algebra. However, I did not start to take my education seriously until about two months ago. I took the semester off to re-assess what I truly wanted to study and decided to follow my (difficult) passion of mathematics. I am now enrolled in summer semester at my university and am taking ODEs and Real Analysis 1. Like many students, I find analysis to be challenging yet very exciting. Sometimes, however, I catch myself getting frustrated with my own self because I feel I am not making enough progress. Have you ever felt self doubt in your career as a mathematician? How did you overcome those worries? Also, what are some good techniques or resources to advance one's skills as a undergraduate level mathematician? Thank you for taking the time to read this.</p>
<p>Sincerely,
Rebecca</p>
http://mathoverflow.net/q/2399131Are these moduli problems of curves "well-behaved"?Adnonhttp://mathoverflow.net/users/922182016-05-27T15:02:05Z2016-05-27T15:02:05Z
<p>Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.</p>
<p>Let $H_{X,d}$ be the Hilbert scheme of smooth hypersurfaces of degree $d$ in $X$.</p>
<p>Note that $H_{X,d}$ is a quasi-projective scheme over $\mathbb C$ equipped with an action of the automorphism group $G:=\mathrm{Aut}(X)$ of $X$.</p>
<p>Is the action of $G$ on $H_{X,d}$ proper? That is, is the DM-stack $[G\backslash H_{X,d}]$ separated?</p>
<p>There is a natural representable morphism of stacks</p>
<p>$$[G\backslash H_{X,d}] \to \mathcal M_g$$</p>
<p>What are some of its abstract properties? Is it quasi-finite?
Does it induce an open immersion on coarse spaces?</p>
http://mathoverflow.net/q/239912-6applications that involves the Legendre Polynomials [on hold]weam nourhttp://mathoverflow.net/users/905832016-05-27T14:58:48Z2016-05-27T14:58:48Z
<p>i am requested to make a basic research about the applications that involve the Legendre Polynomials.. thanks in advance </p>
http://mathoverflow.net/q/2399112harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?john mangualhttp://mathoverflow.net/users/13582016-05-27T14:46:17Z2016-05-27T15:41:31Z
<p>Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where </p>
<p>What happens in the $p$-adic case? Is there sphere still a compact manifold? By sphere I mean:</p>
<p>$$ S^2 = \{ (x,y,z) \in \mathbb{Q}_p^3: x^2 + y^2 + z^2 = 1 \}$$</p>
<p>In order to have harmonic analysis, what does it mean to integrate over the 2-sphere in this case? What is an example of an element of $L^2$ ?</p>
http://mathoverflow.net/q/2399100History of the Vertex Disjoint Cycle Cover with Minimal Edgeweight SumManfred Weishttp://mathoverflow.net/users/313102016-05-27T14:28:59Z2016-05-27T14:28:59Z
<p>Questions:</p>
<ul>
<li><p>who first posed the problem of determining a collection of (directed) cycles, whose edgeweight sum is minimal and, for which each vertex belongs to exactly one of the cycles?</p></li>
<li><p>who came up with the solution of duplicating the vertex set and thus reducing the problem to a minimum-weight bipartite matching? </p></li>
</ul>
http://mathoverflow.net/q/2399092If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?antoniov.joelhttp://mathoverflow.net/users/605042016-05-27T14:11:41Z2016-05-27T14:11:41Z
<p>Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles</p>
<p>$$
TM = E^{s} \oplus E^{c} \oplus E^{u}.
$$</p>
<p>We say that $f$ is dynamically coherent if there exist $f$-invariant center stable and center unstable foliations $\mathcal{W}^{s}$ and $\mathcal{W}^{u}$, tangent to the bundles $E^{cs}$ (i.e. $E^{c} \oplus E^{s}$) and $E^{cu}$ (i.e. $E^{c} \oplus E^{u}$), respectively; intersecting their leaves one obtains an invariant center foliation $W^{c}$ as well.</p>
<blockquote>
<p>Question: If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?</p>
</blockquote>
<p>At local level there is an anwser in the paper <a href="http://arxiv.org/pdf/1409.0738v1.pdf" rel="nofollow">A non-dynamically coherent example on $\mathbb{T}^{3}$</a>, that is, if $E$ is a distribution, $\mathcal{W}$ a foliation tangent to $E$ and $W(x)$ is a leaf of $\mathcal{W}$ through the point $x$, $E$ is locally uniquelly integrable at $x$ if any embedded arc through $x$ and tangent to $E$ is contained in $W(x)$.</p>
<p>But, what about the non-local level?</p>
http://mathoverflow.net/q/2398891(quasi)metric on Riemannian manifolds via Brownian Motion?guslhttp://mathoverflow.net/users/922032016-05-27T09:03:04Z2016-05-27T18:24:55Z
<p>Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to the average commute time from $a$ to $b$ under Brownian Motion (or rather, to $\epsilon$-ball $B = \{ x : |x - b| < \epsilon\}$), via a function $h_M$ such that $d(a,b) = g(\epsilon) h_M(T_{aB})$, where $T_{aB}$ is the average commute time from $a$ to $B$, and $g(\epsilon)$ is the normalization function.</p>
<p>Is $d$ a metric? Is it very different from geodesic distance?</p>
http://mathoverflow.net/q/239883-1Hadwiger number of total graphDominic van der Zypenhttp://mathoverflow.net/users/86282016-05-27T06:28:57Z2016-05-27T15:45:48Z
<p>Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its <a href="http://mathoverflow.net/questions/239806/total-chromatic-number-and-total-clique-number">total graph</a>. The <em>Hadwiger number</em> $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$.</p>
<p>Is there an example of a graph $G$ such that $\eta(T(G)) > \Delta(G) + 2$ (where $\Delta(G)$ is the maximum degree)?</p>
http://mathoverflow.net/q/2398772s(n) = kn or s(n) = n/k? [on hold]goodboyhttp://mathoverflow.net/users/920592016-05-27T03:19:19Z2016-05-27T13:52:07Z
<p><strong>This is not an important question, just for fun.</strong></p>
<p><strong>Definition:</strong></p>
<p>$\sigma (n)$ = sum of the positive divisors of $n$.<br>
$s(n)$ = sum of the proper positive divisors of $n$. </p>
<p><strong>For $s(n) = kn$ , where $k$ is a natural number:</strong></p>
<p>When $k = 1,$ then $n$ is a perfect number which has been discussed a lot.</p>
<p>How about $k = 2,3,4,5,\ldots$?</p>
<p>Based on some computations ( $n < 1.5\cdot 10^9$), I <strong>haven't found</strong></p>
<ol>
<li>Any odd number satisfying $s(n) = kn$.</li>
<li>$s(n) \geq 5n$</li>
<li>$s(n) = 4n$</li>
</ol>
<p><strong>for $s(n) = n/k$ with natural $ k >1$:</strong></p>
<p>$n$ must be prime (and thus $k=n$.)</p>
<p><strong>Conjecture:</strong></p>
<ol>
<li>If $s(n) = kn$, then $n$ must be even.</li>
<li>$s(n) < 5n$</li>
<li>$s(n) = n/k$ for a natural $ k >1$ $\iff n$ is prime . </li>
</ol>
<p><strong>Question:</strong>
<strong>Could you provide a counterexample or prove it?</strong></p>
http://mathoverflow.net/q/2398661Divisibility of Dirichlet L-functionsuser92196http://mathoverflow.net/users/921962016-05-26T22:24:41Z2016-05-27T17:26:48Z
<p>Let $k$ be an even integer and $p$ a prime number such that $p-1|k$.</p>
<p>Suppose that $p$ does not divide $L(1-k,\chi)$ and $L(1-k,\psi)$, where $\chi$ and $\psi$ are quadratic characters.</p>
<p>Can we deduce that $p$ does not divide $L(1-k,\chi.\psi)$?</p>
http://mathoverflow.net/q/2398655Normal subgroups of Aut(M)Ioannis Souldatoshttp://mathoverflow.net/users/136942016-05-26T22:23:37Z2016-05-27T14:24:37Z
<p>Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations. </p>
<p><strong>Theorem</strong> The normal subgroups of $S_\infty$ are exactly the following four (in increasing order): $\{id\}$, $A$, $S$, $S_\infty$.</p>
<p>Now, consider a model $M$ with domain $\mathbb{N}$ and let $Aut(M)$ be the group of automorphisms of $M$. $Aut(M)$ is a closed subgroup of $S_\infty$ and every closed subgroup of $S_\infty$ equals $Aut(M)$, for some model $M$.</p>
<p><strong>My question</strong>: Are there any (partial?) results generalizing the above theorem to $Aut(M)$? I.e. what are the normal subgroups of $Aut(M)$, for various $M$? Equivalently, if $A$ is a closed subgroup of $S_\infty$, what are the normal subgroups of $A$?</p>
http://mathoverflow.net/q/2398352How are $\alpha$ and $f(\alpha)$ obtained from the calculated values of $h(q)$ in the Multifractal analysis? [on hold]ADGhttp://mathoverflow.net/users/513862016-05-26T13:08:14Z2016-05-27T15:31:56Z
<p>In most papers they mention some kind of Legendre transform but I'm not entirely clear about the process, sine they don't mention enough details, at least for me, I tried reading the cited references but I was unsuccessful until now, I think. You could look at <a href="https://arxiv.org/pdf/physics/0202070v1.pdf" rel="nofollow">Kantelhardt's paper</a>. </p>
<p>Another thing, some papers refer to $f(\alpha)$ vs. $\alpha$ as the multi-fractal spectrum whereas some refer to $D(q)$ vs $h(q)$, does it make any difference, how, why not?</p>
<hr>
<h1>Addendum (27 May 16):</h1>
<p>$f(\alpha),\alpha,D(q),h(q)$ are mentioned in <a href="https://arxiv.org/pdf/physics/0202070v1.pdf" rel="nofollow">Kantelhardt's paper</a> and <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3366552/pdf/fphys-03-00141.pdf" rel="nofollow">Ihlen's paper</a>. You can see the multifractal spectrum as $f(\alpha)$ vs. $\alpha$ at page 11 of Kantelhardt's paper whereas and the multifractal spectrum as $D(q)$ vs. $h(q)$ at page 11 of Ihlen's paper.</p>
http://mathoverflow.net/q/2397960Convergence of unitary products on a Hilbert spaceALBhttp://mathoverflow.net/users/914832016-05-26T02:48:07Z2016-05-27T15:47:36Z
<p>First: I'm sorry for the basic question--I can move it to Math SE if necessary...</p>
<p>Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $||\cdot||$ be the operator norm on $X$. Let $S_n = \{U_k\}_k^n$ and $p(S_n) = U_nU_{n-1}\cdots U_2U_1$.</p>
<p><strong>Conjecture:</strong> $\{p(S_n)\}_n$ converges uniformly if $\{U_n\}_n\rightarrow I$ ($I$ is the identity), also uniformly. </p>
<p><strong>My current progress:</strong> If necessary, the assumption can be relaxed to $\{p(S_{nm})\}_m\rightarrow I$ for all $n > m$ as $m\rightarrow\infty$ with $S_{nm} := \{U_k\}_{k=m}^n$. Clearly $p(S_{nm}) = p(S_n)(p(S_m))^*$ so $||p(S_n)-p(S_m)||$ converges. Hence $p(S_n)$ is Cauchy and thus converges uniformly since $X$ is a Hilbert space, thus complete. I'm not 100% my reasoning about this however--viz. does the completeness of $X$ imply convergence for sequences of operators as well as points in $X$? Furthermore, I can't think of a good place to start on the conjecture as stated. </p>
<p>Advice is appreciated!</p>
<p><strong>Update:</strong></p>
<p>Prof. Isreal and Sebastian Goette have provided the counterexample of $U_n = e^{\alpha i/n}$ ($\alpha\in \mathbb{R}$) for the conjecture. Thus the premise should be relaxed. Thank you all for the help!</p>
http://mathoverflow.net/q/2397820Mean curvature and submanifoldGio712http://mathoverflow.net/users/739922016-05-25T22:17:27Z2016-05-27T16:22:58Z
<p>Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap
$$
G=S^{N-1}\cap\{x_N>0\}
$$
with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a submanifold of $S^{N-1}$. Consider $\mbox{div}_{\partial G}$ the tangential divergence on the boundary $\partial G$ as
$$
\mbox{div}_{\partial G} Y = \mbox{div}_{S^{N-1}}Y - \nu . DY. \nu
$$
where
$$
\nu= (0,-1) \in R^N\times R
$$
be the outward normal field to $\partial G$ in $S^{N-1}$. How can i compute $\mbox{div}_{\partial G} \nu$? Is this related to some mean curvature value? The idea is to use this value in order to apply a divergence type formula for $\partial G$ as a subscape of $S^{N-1}$.</p>
http://mathoverflow.net/q/2397620Finding the right σ-algebra. Question on uncertainty related to the secretary problemThomas Ehttp://mathoverflow.net/users/921272016-05-25T18:38:45Z2016-05-27T13:34:59Z
<p>Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.</p>
<p>In this setting it is relevant what is the distribution of the values of the presented items $(X_i)_{i \leq n}$ the problem is to what $\sigma$-algebra filtration $(\natural_i)_{i \leq n}$ the best stopping time $T$ has to be adapted to.
If one has full information on the distribution it would be $(\sigma(X_1,...,X_i))_{i \leq n}$. The winning chance then is asymptotically $0.58$ if $X_1$ has a continuous c.d.f.</p>
<p>For no information on the possible distributions the right filtration is:
$(\sigma(R_1,...,R_i))_{i \leq n}$ with $R_j = \sum^{j}_{k=1}1_{X_k \leq X_j}$, that is the rank of $X_j$.</p>
<p>What is the right $\sigma$-algebra to model the case where for an example one knows $X_1$~$N(\mu,\sigma^2)$ but nothing on $\mu$ and $\sigma^2$?</p>
<p>This relates here, and I have made proposals and submitted an example to deal with. </p>
<p><a href="http://math.stackexchange.com/questions/1798208/finding-the-right-sigma-algebra-question-on-uncertainty-related-to-the-secre">http://math.stackexchange.com/questions/1798208/finding-the-right-sigma-algebra-question-on-uncertainty-related-to-the-secre</a></p>
http://mathoverflow.net/q/23972810List of Applications of the $\partial\overline{\partial}$-lemmaHan Jin Mahttp://mathoverflow.net/users/761042016-05-25T12:13:18Z2016-05-27T15:05:32Z
<p>Quoting from Huybrecht's book <strong>Complex Geometry</strong> on the $\partial\overline{\partial}$-lemma for Kaehler manifolds:</p>
<blockquote>
<p>Although it looks like a rather innocent technical statement, it is
crucial for many results.</p>
</blockquote>
<p>Later in the book the result is used in the study of formality of Kaehler manifolds, but what other important applications does it have? </p>
http://mathoverflow.net/q/2396831Conditioned sum of n Poissons versus unconditioned PoissonsMatthew Jungehttp://mathoverflow.net/users/528962016-05-24T22:52:20Z2016-05-27T16:39:08Z
<p>Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$:
$$\mathbf P[Z_k^* = j] = \mathbf P\left [Z_k = j \; \Big| \; \sum_1^n k Z_k =n\right ].$$</p>
<p>This is known as Ewen's measure. Notice that $\mathbf E \sum_1^n k Z_k = \theta n$, hence conditioning this to be $n$ should "squish" the $Z^*_k$, and make them "smaller" than their independent counterparts $Z_k$. </p>
<p>Stochastic dominance $Z^*_k \preceq Z$ does not appear to hold, but it would be sufficient for our purposes to prove a statement like
$$\liminf_{n >0} \; \textbf P\left[ \bigcap_{k= 1}^{n} \{Z_k^* \leq Z_k\}\right] > 0.$$</p>
<p>Is there a standard approach for such a bound?</p>
http://mathoverflow.net/q/23849212Duality between topology and bornologyBipolar Mindshttp://mathoverflow.net/users/582112016-05-10T17:33:04Z2016-05-27T14:56:43Z
<p>I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:</p>
<p>Let $X$ be a set and let $\mathcal{P}(X)$ be the bounded lattice induced by the power set $P(X)$ together with unions, intersections, $X$ and $\emptyset$ as joints, meets, $1$ and $0$. An $\textbf{ideal}$ $\beta \subseteq \mathcal{P}(X)$ is called a $\textbf{bornology}$ if $\bigvee_{B \in \beta} \, B=1.$</p>
<p>This should be equivalent to the usual definition. An obvious dualization of this is a $\textbf{filter}$ $\nu \subseteq \mathcal{P}(X)$, s.t. $\bigwedge_{N \in \nu}=0$. </p>
<p>This should be seen as the set of all neighbourhoods with respect to some topology on $X$ and the question is now how to extract a topology from it. I know that there is a unique topology for every <a href="https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)#Topology_from_neighbourhoods" rel="nofollow">neighbourhood system</a> $\{N(x)\}_{x \in X}$ and of course I can define a trivial neighbourhood system from a given filter $\nu \subseteq \mathcal{P}(X)$ by setting $N(x)$ to be the subfilter of sets in $\nu$ containing $x$. This leads to the finest possible topology, but what I actually want is a unique coarsest neighbourhood system (s.t. $\nu = \cup_x \, N(x) $)! Unfortunately, I don't really have an idea how to start showing the existence of such a thing.. do you? Of course, any other idea how to define open sets is welcome! </p>
<p><strong>EDIT 1:</strong> it seems that <em>coarsest</em> is not the right property here, since for a given topology, the neighbourhoodfilter is in general not the coarsest neighbourhood system</p>
<p><strong>EDIT 2:</strong> I guess I made a really stupid mistake here: the filter $\nu$ cannot distinguish between topologies $\tau,\tilde{\tau}$ with neighbourhood filters $N(x),\tilde{N}(x)$, s.t. $\bigcup_x \, N(x) =\bigcup_x \, \tilde{N}(x)= \nu$..</p>
<p><strong>EDIT 3:</strong> For topological vector spaces $(X,\tau),(X,\tilde{\tau})$ it should be true that from $\bigcup_x \, N(x) =\bigcup_x \, \tilde{N}(x)$ follows $N(0)=\tilde{N}(0)$, thus $\tau = \tilde{\tau}$. I used both continuity of multiplication and addition.
Moreover, if $\tau$ is nontrivial, we have $\bigcap_{N \in \bigcup_x N(x)} \, N = \emptyset$. I didn't use that $\tau$ is Hausdorff.</p>
http://mathoverflow.net/q/2045002Rate of convergence of Riemann sum of quasi-regular functionsdavidgontierhttp://mathoverflow.net/users/711902015-05-02T09:44:04Z2016-05-27T15:52:41Z
<p>The following result is well-known (I consider the 3-dimensional case only):</p>
<p><strong>Theorem:</strong> if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then
$$
\left| \int_{\mathbb{R}^3} f - \frac{1}{N^3} \sum_{k \in \mathbb{Z}^3} f \left (\frac{k}{N}\right)\right| \le C_s \dfrac{\| f \|_{H^s}}{N^s}.
$$
Recall that $H^{3/2}(\mathbb{R}^3) \hookrightarrow C^0(\mathbb{R}^3)$ so that the point-wise estimates make sense.</p>
<p>Now, I consider the following function. Let $\Psi$ be a $C^\infty(\mathbb{R}^3)$ radial cut-off function with $\Psi(0) = 1$ and $\Psi(x) = 0$ if $| x | > 1$, and let
$$
f(x) = \frac{(x_1)^4}{| x |^4} \Psi(x).
$$
I also set $f(0) = 0$ to simplify the notation. Note that $f$ is not continuous at the origin (hence $f \notin H^{3/2}(\mathbb{R}^3)$).</p>
<p><strong>The question is:</strong> what is the best constant $s$ so that
$$
\left| \int_{\mathbb{R}^3} f - \frac{1}{N^3} \sum_{k \in \mathbb{Z}^3} f \left (\frac{k}{N}\right)\right| \le C_s N^{-s} \quad ?
$$
Numerically, I observe $s = 3$ (with no doubt possible). I even observe that there exists a constant $C$ such that
$$
\forall N, \quad \left| \int_{\mathbb{R}^3} f - \frac{1}{N^3} \sum_{k \in \mathbb{Z}^3} f \left (\frac{k}{N}\right) - \frac{C}{N^3}\right| \approx 0.
$$
Unfortunately, I have a hard time to prove this fact (I got a proof for all $s < 3$,...).</p>
<p><strong>Remark:</strong> if we change the power $4$ to the power $2$, then a simple symmetry argument shows that $s = 3$, and the constant $C$ corresponds to a missing part in the Riemann-sum (<strong>i.e.</strong> the $k = 0$ term).</p>
http://mathoverflow.net/q/1986212Octonions product: inversion in the right and identity in the leftJjmhttp://mathoverflow.net/users/623672015-02-27T09:30:46Z2016-05-27T15:16:11Z
<p>Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance <a href="http://www.math.duke.edu/~bryant/Spinors.pdf" rel="nofollow">Robert Bryant's notes</a>), one realises that the key fact bringing all the phenomena of triality etc. is the following: that there exist some octonions $u_1,\ldots,u_k$ such that</p>
<p>$$R_{u_k}\ldots R_{u_1}=-Id\qquad\qquad L_{u_k}\ldots L_{u_1}=Id,$$</p>
<p>where $L_u$ and $R_u$ denote left and right product with $u$ respectively.</p>
<p>This may be found by the Lie group properties of $Spin(8)$ and $SO(8)$. But</p>
<p><strong>May some set $u_1,\ldots,u_k$ be described explicitely?</strong></p>
<p>Any idea is welcome.</p>
<p><strong>EDIT:</strong></p>
<p>Although it is not directly related to the question, it is interesting to note the vital importance of the claim. Following Briant's notes, we consider the group of maps $\mathbb{O}\oplus\mathbb{O}\longrightarrow\mathbb{O}\oplus\mathbb{O}$ generated by the elements $L_u\oplus R_u:(a,b)\longmapsto(ua,bu)$ with $\|u\|=1$. Of course, we are dealing with some group $G\subset SO(8)\times SO(8)$. Then: if we read the previous paragraphs, it happens that $G$ is the group $Spin(8)$ defined in terms of Clifford algebras, and the key property of this $Spin(8)$ being really a double cover of $SO(8)$ is supported partially by the fact that <em>the inclussion $G\subset SO(8)\times SO(8)$ is not a vacuous one, as could be the diagonal $SO(8)\subset SO(8)\times SO(8)$, because $(Id,-Id)\in G$</em>. That is why the sign is crucial.</p>
http://mathoverflow.net/q/1942793Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$Jjmhttp://mathoverflow.net/users/623672015-01-19T16:22:56Z2016-05-27T19:50:27Z
<p>I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$). More concretely, it would be very useful to know:</p>
<ul>
<li><p>The image of the basis elements $e_k$</p></li>
<li><p>The image of the volume element $v=e_1e_2e_3e_4e_5e_6e_7e_8$</p></li>
<li><p>The equations of the $\pm$-eigespaces of $v$, $\Delta_{\pm}$</p></li>
</ul>
http://mathoverflow.net/q/1447240Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?Hamedhttp://mathoverflow.net/users/134312013-10-13T14:52:06Z2016-05-27T15:59:59Z
<p>Concerning my previous question <a href="http://mathoverflow.net/questions/138937/non-simply-connected-Hyper-Kahler-4-manifolds-without-ale-metrics">Non simply connected HyperKähler 4-manifolds without ALE metrics</a> the following question occurred to me:</p>
<p>Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?</p>
http://mathoverflow.net/q/1229832Calculate channel capacity of general channel under constraintuser31757http://mathoverflow.net/users/317572013-02-26T14:35:35Z2016-05-27T19:51:17Z
<p>Given a conditional distribution $P_{Y|X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y|X}(y|x)P_X(x)\text{dx}$ (this corresponds to finding the channel capacity $C(X;Y):=\max_{P_X}I(X;Y)$) subject to the constraint $E_{P_X}[-\log(X)]=a$.</p>
<p>In my particular case, $P_{Y|X}$ is a Bernoulli distribution with $X$ as its parameter and I'm looking for a distribution over the parameter.</p>
<p>My intuition would tell me this should be some Beta-distribution (something like $\text{Beta}(1/a,1)$?!) in my particular case, but I don't know how to approach such a problem, much less in the general case.</p>
<p>Could anyone point me in the right direction?</p>
http://mathoverflow.net/q/11252416Smooth curves on smooth varietiesAntoine Ducroshttp://mathoverflow.net/users/281432012-11-15T21:44:16Z2016-05-27T14:14:45Z
<p>Let $X$ be a smooth, proper algebraic variety over a field $k$, of positive dimension. </p>
<p>Is it true that $X$ contains a smooth Zariski-closed curve? </p>
<p>If it is projective, this is true by Bertini. But is it true in general?</p>
http://mathoverflow.net/q/493730Existence of flat models of a smooth finite type algebra over $R((t))$Samuelhttp://mathoverflow.net/users/115032010-12-14T10:44:12Z2016-05-27T14:07:05Z
<p>Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary),
$B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth.</p>
<p>Up to this generality, can one construct a flat model of $B$ over ring of formal power series $R[[t]]$ (i.e. a flat algebra $\tilde{B}$ over $R[[t]]$ such that $\tilde{B}\otimes_{R[[t]]}R((t))=B$)?</p>
<p>If not, what could be the weakest assumption that will allow this?</p>
http://mathoverflow.net/q/4518564Pseudonyms of famous mathematiciansDenis Serrehttp://mathoverflow.net/users/87992010-11-07T17:56:35Z2016-05-27T20:01:06Z
<p>Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles Lutwidge) Dodgson. Needless to say, L. Carroll was his pseudonym, used in literature.</p>
<p>Another (alive) mathematician writes under his real name and under a pseudonym (John B. Goode). (That person, by the way, is Bruno Poizat: it's no secret, even MathSciNet knows it.)</p>
<blockquote>
<p>What other mathematicians (say dead ones) had a pseudonym, either within their mathematical activity, or in a parallel career ?</p>
</blockquote>
<p>Of course, don't count people who changed name at some moment of their life because of marriage, persecution, conversion, and so on.</p>
<hr>
<p><strong>Edit</strong>.
The answers and comments suggest that there are at least four categories of pseudonyms, which don't exhaust all situations.</p>
<ul>
<li>Professional mathematicians, who did something outside of mathematics under a pseudonym (F. Hausdorff - <em>Paul Mongré</em>, E. Temple Bell - <em>John Taine</em>),</li>
<li>People doing mathematics under a pseudonym, and something else under their real name (Sophie Germain - <em>M. Le Blanc</em>, W. S. Gosset - <em>Student</em>)),</li>
<li>Professional mathematicians writing mathematics under both their real name and a pseudonym (B. Poizat - <em>John B. Goode</em>),</li>
<li>Collaborative pseudonyms (<em>Bourbaki, Blanche Descartes</em>)</li>
</ul>
http://mathoverflow.net/q/3713615Classification of finite groups of isometriesMathieu Dutour Sikirichttp://mathoverflow.net/users/88832010-08-30T09:33:42Z2016-05-27T16:32:43Z
<p>Consider the problem of classifying the finite groups of isometries of $\mathbb{R}^n$.</p>
<ul>
<li>For $n=2$ it is cyclic and dihedral groups.</li>
<li>For $n=3$ they are well known, probably from Kepler and are related to ade-classification.</li>
<li>For $n=4$ we can get them by taking the universal cover of $\mathrm{SO}(4)$ which is isomorphic to $\mathrm{SU}(2) \times \mathrm{SU}(2)$, though I do not know where the classification is available.</li>
</ul>
<p>But my main question is for dimension $n\geq 5$. Does anybody knows the state of the art? A reference would be most helpful.</p>
<p>Note that the finite subgroups of $\mathrm{GL}_n(\mathbb{Z})$ are classified for $n\leq 10$.</p>