Recent Questions - MathOverflowmost recent 30 from www.2874565.com2018-11-20T17:03:08Zhttp://www.2874565.com/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://www.2874565.com/q/3157940Relation of complex line bundles and circle bundleMustafahttp://www.2874565.com/users/1315932018-11-20T16:56:36Z2018-11-20T16:56:36Z
<p>What is the relation of complex line bundles to circle bundles?</p>
http://www.2874565.com/q/3157930Generalization of Komlós–Major–Tusnády Approximationuser35154http://www.2874565.com/users/1260792018-11-20T16:23:54Z2018-11-20T17:01:38Z
<p>The Komlós–Major–Tusnády Approximation (see <a href="https://en.wikipedia.org/wiki/Koml%C3%B3s%E2%80%93Major%E2%80%93Tusn%C3%A1dy_approximation" rel="nofollow noreferrer">Wikipedia</a>) considers the sum of uniform variables in <span class="math-container">$(0,1)$</span>. There are also version where instead the sum of equiprobable <span class="math-container">$0/1$</span> variables is used (<span class="math-container">$p=1/2$</span>).</p>
<p>Is there are generalization for arbitrary <span class="math-container">$p$</span>? That is, can it be generalized to the sum of i.i.d. <span class="math-container">$0/1$</span> variables with probability <span class="math-container">$p$</span>?</p>
http://www.2874565.com/q/3157920How can I find the value of x? [on hold]Giusyhttp://www.2874565.com/users/1315922018-11-20T16:22:23Z2018-11-20T16:22:23Z
<p>The four angles of a quadrilateral are 90°.
(x+15)°, (x+25)° and (x I 35)°.
Find the value of x.
To be honest, I don't know what it means.</p>
http://www.2874565.com/q/3157911An upper bound for the number of singularities of a transversal vector field isometric to the zero fieldAli Taghavihttp://www.2874565.com/users/366882018-11-20T16:19:16Z2018-11-20T16:51:32Z
<p>Let <span class="math-container">$(M,g)$</span> be a Riemannian manifold. We equip the tangent bundle <span class="math-container">$TM$</span> with the Sasaki metric <span class="math-container">$g_s$</span>.</p>
<p>A smooth vector field <span class="math-container">$X:M \to TM$</span> is called a transversal vector field if <span class="math-container">$X(M)$</span> is transverse to the zero section. We say that a vector field <span class="math-container">$X$</span> is "Isometric to the zero field" if there is an isometry <span class="math-container">$\phi$</span> of <span class="math-container">$(TM,g_s)$</span> which maps <span class="math-container">$X(M)$</span> to the zero section.</p>
<blockquote>
<p>Question:Let <span class="math-container">$(M,g)$</span> be a compact Riemannian manifold. Is there a uniform upper bound <span class="math-container">$N$</span>, depending only on <span class="math-container">$(M,g)$</span>, such that every transversal vector field <span class="math-container">$X$</span> isometric to the zero field has at most <span class="math-container">$N$</span> singularities?</p>
</blockquote>
http://www.2874565.com/q/3157891Laurent series expansion of Theta function expressionAranhttp://www.2874565.com/users/1169362018-11-20T15:28:05Z2018-11-20T15:28:05Z
<p>Using the product definition of the theta function</p>
<p><span class="math-container">$$ \theta(z;p) = \prod_{k=0}^{\infty}(1-p^k x)(1-p^{k+1}/x) $$</span></p>
<p>I would like to find the Laurent series expansion of the following:</p>
<p><span class="math-container">$$ \frac{\sqrt{\theta(qz^2)\theta(qw^2)}\theta(z)\theta(w)}{\theta(qz)\theta(qw)\theta(qzw)\theta(qz/w)}$$</span></p>
<p>where <span class="math-container">$q=\sqrt{p}$</span>.</p>
<p>For context, this expression arises in a determinant calculation of the form</p>
<p><span class="math-container">$$ \prod_{i=1}^N \frac{\theta_4(x_i)\theta_1^2(x_i)}{\theta_4^2(x_i)} \det{\bigg(\frac{1}{\theta_4((x_i+x_j)/2)\theta_4((x_i-x_j)/2)}\bigg)}$$</span></p>
<p>Where the above uses an alternate definition for the theta functions that can be easily found.</p>
<p>Naturally there are multiple ways of bringing in the product factor in front of the determinant within the determinant, and I have chosen the expression that is symmetric under exchange of <span class="math-container">$z,w$</span>. Others may be easier to work with.</p>
<p>I found one potentially relevant expansion of the form </p>
<p><span class="math-container">$$ \frac{b_1\theta(b_0 b_1)\theta(b_0/b_1)z^{-1}\theta(z^2)}{\theta(b_0z)\theta(b_0/z)\theta(b_1z)\theta(b_1/z)}$$</span></p>
<p>which <em>almost</em> reduces to a result that could be useful, in the limit <span class="math-container">$z\rightarrow qz, b_1 \rightarrow 1$</span>, but what I really need has a function <span class="math-container">$\theta(qz^2)$</span> in the numerator, and not <span class="math-container">$\theta(z^2)$</span>.</p>
<p>Any help would be greatly, greatly appreciated.</p>
<p>Thank you.</p>
http://www.2874565.com/q/3157840Is $\ell_2(A)$ a Hilbert C$^*$-module with Opial property?Dadrahmhttp://www.2874565.com/users/1305892018-11-20T14:39:41Z2018-11-20T14:39:41Z
<p>If <span class="math-container">$A=Mat_{n\times n}(\mathbb{C}) $</span>, Is <span class="math-container">$\ell_2(A)$</span> a Hilbert <span class="math-container">$A$</span>-module with Opial property?</p>
<p>I tried to solve it for many hours, But I didn't solve it.</p>
http://www.2874565.com/q/3157834Effective BertiniGiuliohttp://www.2874565.com/users/488662018-11-20T14:29:00Z2018-11-20T14:47:53Z
<p>Let <span class="math-container">$X$</span> be a smooth complex projective manifold, and <span class="math-container">$L$</span> an ample line bundle. By Bertini's Theorem, for every integer <span class="math-container">$q$</span> big enough there exists an open dense subset <span class="math-container">$U_q\subset |qL|$</span> such that every divisor <span class="math-container">$D$</span> in <span class="math-container">$U_q$</span> is smooth.</p>
<p>Warm up question: is the complement of <span class="math-container">$U_q$</span> always a divisor?</p>
<p>We can define a bigger open subset <span class="math-container">$V_q\subset |qL|$</span> as
<span class="math-container">$$
V_q:=\{D\in |qL| \; \textrm{s. t.} \; (X,\frac{1}{q}D) \; \textrm{ is klt} \}
$$</span></p>
<p>My question is: what is the dimension of the complement of <span class="math-container">$V_q$</span> ? (or at least can we bound its asymptotic in <span class="math-container">$q$</span> ? e.g. is it upper-bounded by <span class="math-container">$aq^{\dim X}$</span> with <span class="math-container">$a$</span> a constant which is strictly smaller than the volume of <span class="math-container">$L$</span>? )</p>
http://www.2874565.com/q/3157810Calculate a realization of two stochastic variables as a function of the same realization of a stochastic process?Dr_Zaszu?http://www.2874565.com/users/1315892018-11-20T13:33:16Z2018-11-20T16:22:22Z
<p>I have the following parametric stochastic integral:
<span class="math-container">$$
I(\lambda) = \int_{t_i}^{t_{i+1}} \exp(\lambda (t_{i+1} - s)) dW_s,
$$</span>
where <span class="math-container">$W_t$</span> is a zero-average totally uncorrelated stationary white-noise process.
I want to avoid numerical integration, so I calculated the following distribution <span class="math-container">$I(\lambda)$</span>:
<span class="math-container">$$
I(\lambda) \propto N(0, \sigma^2),
$$</span>
with
<span class="math-container">$$
\sigma^2 = \frac{\exp(2\lambda \Delta t) - 1}{2 \lambda},
$$</span>
from which I can draw the realizations of the integral.</p>
<p>However, I need to calculate not just one <span class="math-container">$I(\lambda)$</span>, but two integrals <span class="math-container">$I(\lambda_0), I(\lambda_1)$</span> for the <strong>same realization</strong> of the stochastic process. These integrals are correlated stochastic variables, so I cannot just draw their values independently. </p>
<p>It is easy to calculate the distribution of <span class="math-container">$I(\lambda_0) - I(\lambda_1)$</span>, but it doesn't give me access to <span class="math-container">$P(I(\lambda_1) \mid I(\lambda_0))$</span> or <span class="math-container">$P(I(\lambda_0), I(\lambda_1))$</span> that I would need to draw from the joint distribution. I have also thought about a Fourier approach, but the white noise process has a flat spectral density, which creates its own problems.</p>
<p><strong>Question:</strong></p>
<p>Does anyone have ideas on how I could sample my two integrals for the same realization of the stochastic process?</p>
<p><strong>Edit:</strong></p>
<p>This questions seems to have been asked <a href="http://www.2874565.com/questions/287107/conditional-stochastic-integration">before</a>, but the accepted answer is wrong.</p>
http://www.2874565.com/q/3157805Divisors of the regular character of a finite groupJohn Murrayhttp://www.2874565.com/users/207642018-11-20T13:33:09Z2018-11-20T15:02:08Z
<p>Recall that the <em>regular character</em> <span class="math-container">$\rho=\hspace{-.2cm}\sum\limits_{\chi\in\operatorname{Irr}(G)}\hspace{-.2cm}\chi(1)\chi$</span> of a finite group <span class="math-container">$G$</span> takes values
<span class="math-container">$$
\rho(g)=
\left\{\begin{array}{cl}
|G|,&\quad\mbox{if $g=1$.}\\
0,&\quad\mbox{if $g\in G\backslash\{1\}$.}
\end{array}
\right.
$$</span>
Given <span class="math-container">$\alpha\in\operatorname{Irr}(G)$</span>, we shall say that <strong><span class="math-container">$\alpha$</span> divides <span class="math-container">$\rho$</span></strong> if <span class="math-container">$\alpha\beta=\rho$</span>, for some generalized character <span class="math-container">$\beta$</span>.</p>
<p>Now <span class="math-container">$\alpha\rho=\alpha(1)\rho$</span>. So linear characters of <span class="math-container">$G$</span> divide <span class="math-container">$\rho$</span>. Notice that if <span class="math-container">$\alpha$</span> is faithful, then there is an integer polynomial <span class="math-container">$p$</span> such that <span class="math-container">$\alpha p(\alpha)=n_\alpha\rho$</span>, for some positive integer <span class="math-container">$n_\alpha$</span>. But I don't know anything about <span class="math-container">$n_\alpha$</span>. I have found only a small number of cases of <span class="math-container">$G$</span> and <span class="math-container">$\alpha$</span> where <span class="math-container">$\alpha$</span> does not divide <span class="math-container">$\rho$</span> (see below). My question is a small variant on <a href="https://math.stackexchange.com/questions/2933550/tensor-complement-of-representations-of-finite-groups">https://math.stackexchange.com/questions/2933550/tensor-complement-of-representations-of-finite-groups</a>:</p>
<p><strong>Question:</strong> Does there exist a <span class="math-container">$p$</span>-group <span class="math-container">$G$</span> and <span class="math-container">$\alpha\in\operatorname{Irr}(G)$</span>, such that <span class="math-container">$\alpha$</span> does not divide <span class="math-container">$\rho$</span>?</p>
<p>Note that in my original question solvable group was in place of <span class="math-container">$p$</span>-group. As Jeremy Rickard has pointed out, there are 3 groups of order <span class="math-container">$72$</span> with irreducible characters which do not divide <span class="math-container">$\rho$</span> (SmallGroups(72,n), for <span class="math-container">$n=22,23,24$</span>, in GAP notation).</p>
<p><strong>Non-solvable example:</strong> Let <span class="math-container">$\alpha$</span> be one of the two degree 4 irreducible characters of <span class="math-container">$\operatorname{SL}(2,5)$</span>. Then <span class="math-container">$\alpha$</span> has a single conjugacy class of zeros <span class="math-container">$4a$</span>. Let <span class="math-container">$\beta$</span> be a class function such that <span class="math-container">$\alpha\beta=\rho$</span>. Then <span class="math-container">$\beta(1a)=30$</span>, <span class="math-container">$\beta(4a)$</span> is odd and <span class="math-container">$\beta$</span> vanishes on all other classes. Let <span class="math-container">$\psi\in\operatorname{Irr}(\operatorname{SL}(2,5))$</span>, with <span class="math-container">$\psi(1)=2$</span>. Then <span class="math-container">$\psi(4a)=0$</span>. So <span class="math-container">$\langle\beta,\psi\rangle=\frac{1}{2}$</span>, showing that <span class="math-container">$\beta$</span> is not a generalized character.</p>
<p>A similar example is provided by a degree <span class="math-container">$9$</span> irreducible character of the group <span class="math-container">$3.A_6.2_2$</span> (the second degree 2 extension of the triple cover of the alternating group <span class="math-container">$A_6$</span>, in Atlas notation). The groups <span class="math-container">$\operatorname{SL}(2,5)$</span> and <span class="math-container">$3.A_6.2_2$</span> are in the small list of non-solvable groups with an irreducible character which vanishes on only one conjugacy class. See S.~Madanha, <em>On a question of Dixon and Rahnamai Barghi</em>, arXiv:1811.03972 [math.GR].</p>
<p>On a positive note, <span class="math-container">$\alpha$</span> divides <span class="math-container">$\rho$</span> in the following cases:</p>
<ol>
<li><p><span class="math-container">$\alpha(1)=|G|_p$</span>, for some prime <span class="math-container">$p$</span> (take <span class="math-container">$\beta=1_S^G$</span>, for <span class="math-container">$S\in\operatorname{Syl}_p(G)$</span>).</p></li>
<li><p><span class="math-container">$\alpha$</span> is totally ramified with respect to an irreducible character of <span class="math-container">$Z(G)$</span>.</p></li>
<li><p><span class="math-container">$\alpha$</span> is induced from a linear character of a normal subgroup of <span class="math-container">$G$</span> with cyclic quotient (R. Gow).</p></li>
<li><p><span class="math-container">$G$</span> is a <span class="math-container">$2$</span>-group with <span class="math-container">$|G|\leq256$</span> or a <span class="math-container">$3$</span>-group with <span class="math-container">$|G|\leq729$</span> (GAP). In fact for all such <span class="math-container">$G$</span> and <span class="math-container">$\alpha$</span>, it seems that there is a character <span class="math-container">$\beta$</span> with <span class="math-container">$\alpha\beta=\rho$</span>.</p></li>
<li><p><span class="math-container">$G=A_n$</span> or <span class="math-container">$S_n$</span>, for <span class="math-container">$n\leq15$</span> (GAP).</p></li>
<li><p><span class="math-container">$G=\operatorname{SL}(2,3),\operatorname{GL}(2,3)$</span> or the binary octahedral group (`fake <span class="math-container">$\operatorname{GL}(2,3)$</span>').</p></li>
<li><p><span class="math-container">$G$</span> is a small finite simple group (GAP).</p></li>
</ol>
http://www.2874565.com/q/3157791On the maximal value of the valuation at infinity of elements in the ring of integers of a global function fieldAndryhttp://www.2874565.com/users/232072018-11-20T13:25:54Z2018-11-20T13:30:59Z
<p>Let <span class="math-container">$F$</span> be a global function field with full constant field <span class="math-container">$\mathbb{F}_q$</span>. We fix a place <span class="math-container">$\infty$</span> and let <span class="math-container">$A$</span> be the ring of elements of <span class="math-container">$F$</span> regular away from <span class="math-container">$\infty$</span>. We denote by <span class="math-container">$v_\infty$</span> the normalized valuation associated to <span class="math-container">$\infty$</span>. If <span class="math-container">$a\in A$</span> and <span class="math-container">$v_\infty(a)\geq 0$</span> then <span class="math-container">$a$</span> has no poles. Therefore (see for example Corollary 1.1.20 in Stichtenoth's Algebraic Function Fields and Codes) <span class="math-container">$a$</span> is algebraic over <span class="math-container">$\mathbb{F}_q$</span> and then <span class="math-container">$a\in \mathbb{F}_q$</span> since it is the full constant field. We then have <span class="math-container">$v_\infty(a) < 0$</span> for all <span class="math-container">$a\in A\setminus \mathbb{F}_q$</span>. My questions are: </p>
<ol>
<li>Is <span class="math-container">$D_{F,\infty} := $</span>Max<span class="math-container">$\{v_\infty(a)/ a\in A, v_\infty(a)<0\}$</span> equal to -1? i.e. is there an element of <span class="math-container">$A$</span> of valuation -1 at infinity?</li>
<li>If <span class="math-container">$D_{F,\infty}$</span> is not -1, can it be computed explicitly, for instance in terms of the genus <span class="math-container">$g_F$</span> of <span class="math-container">$F$</span> (I am just really making a guess here, thinking about Riemann-Roch and its corollaries)?</li>
<li>Is there any algorithm to compute <span class="math-container">$D_{F,\infty}$</span>?</li>
</ol>
<p>Thank you.</p>
http://www.2874565.com/q/315777-1unbalanced Expander Graph Algorithmsujit dashttp://www.2874565.com/users/1315882018-11-20T13:02:08Z2018-11-20T13:02:08Z
<p>How can I make unbalanced expander graph explicitly, randomly or deterministically ? Is there any algorithm to do that?</p>
http://www.2874565.com/q/3157763A moment inequalityhopelesshttp://www.2874565.com/users/1315872018-11-20T12:59:37Z2018-11-20T15:14:03Z
<p>Let <span class="math-container">$\chi(s)=\int_{0}^{1}x(t)^{s}f(t)dt$</span>,
where <span class="math-container">$x(t)$</span> and <span class="math-container">$f(t)$</span> are real valued continuous functions for
<span class="math-container">$t\in[0,1]$</span>, and <span class="math-container">$f(t)\geq0$</span>. </p>
<p>Is it possible to show that</p>
<p><span class="math-container">$\left(\chi(0)\chi(2)-\chi(1)^{2}\right)\left(\chi(4)\chi(2)-\chi(3)^{2}\right)-\left(\chi(3)\chi(1)-\chi(2)^{2}\right)^{2}\geq0$</span></p>
<p>Note: I believe that</p>
<p><span class="math-container">$\chi(0)\chi(2)-\chi(1)^{2}\geq0$</span></p>
<p><span class="math-container">$\chi(4)\chi(2)-\chi(3)^{2}\geq0$</span></p>
<p>follows from the Cauchy-Schwarz inequality.
(correct me if I am wrong about this).</p>
http://www.2874565.com/q/3157750Distribution of dot product of two unit complex random vectorsQiangLihttp://www.2874565.com/users/1315862018-11-20T12:40:37Z2018-11-20T13:00:29Z
<p>Consider <span class="math-container">$u,v∈S^{M-1}\subset \mathbb{C}^M$</span> to be two independent unit norm random vectors on the <span class="math-container">$M?1$</span> dimensional complex sphere <span class="math-container">$S^{M?1}$</span>. In addition, <span class="math-container">$u$</span> follows an isotropic distribution, i.e., <span class="math-container">$u$</span> is uniformly distributed on the complex sphere <span class="math-container">$S^{M?1}$</span>. What is the distribution of <span class="math-container">$Z=|u?v|^2$</span>?</p>
http://www.2874565.com/q/3157741Does this element belong to all powers of the augmentation ideal of the group algebra.Meisam Soleimani Malekanhttp://www.2874565.com/users/847002018-11-20T12:35:00Z2018-11-20T12:35:00Z
<p>Let <span class="math-container">$G$</span> be a torsion free group, and let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> are elements in the augmentation ideal, <span class="math-container">$I$</span>, of <span class="math-container">$\mathbb CG$</span>, the group algebra of <span class="math-container">$G$</span>. Assume that there exists complex numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> such that <span class="math-container">$\alpha\beta=a\alpha+b\beta$</span>. Does it imply that <span class="math-container">$a\alpha+b\beta\in I^n$</span>, for all <span class="math-container">$n\in\mathbb N$</span>? </p>
http://www.2874565.com/q/3157730Shooting Method implementation, clarification for Newton's methodFarid Hasanovhttp://www.2874565.com/users/1315852018-11-20T12:26:29Z2018-11-20T12:26:29Z
<p>reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.</p>
<p><span class="math-container">$\dot x_1(t)=x_2(t),$</span></p>
<p><span class="math-container">$\dot x_2(t)=p_2(t)?\sqrt 2 x_1(t)e^{-αt},$</span></p>
<p><span class="math-container">$\dot p_1(t)=\sqrt 2p_2(t)e^{-αt}+x_(t)$</span></p>
<p><span class="math-container">$\dot p_2(t)=?p_1(t)$</span></p>
<p>with initial and boundary values of: <span class="math-container">$x_1(0)=1, p_2(0)=0,p_1(1)=0,p_2(1)=0$</span>
Following classic shooting method strategy, I suggest some values for <span class="math-container">$x_2(0), p_1(0)$</span>, solve the system of diff equations using Dormand Prince method, and obtain some values at the boundary. Here I check, if the solution on my boundaries meets the criteria of <span class="math-container">$p_1(1)=0,p_2(1)=0$</span>. If not, I have to suggest new values for initial <span class="math-container">$x_2(0), p_1(0)$</span>, to be iterated again. Since this is two-parametric shooting, I can't use simple linear interpolation as in one-parametric shooting, so I have to use Newton's method for that. Here is my question: How to implement Newton's method to iterate towards the right solution? I tried myself, and doesn't work. Can you at least help me with defining a Jacobian, for the given problem of iterative solution to <span class="math-container">$x_2(0), p_1(0)$</span>?</p>
http://www.2874565.com/q/3157692why the division field of an abelian variety contains a cyclotomic field?A. GMhttp://www.2874565.com/users/1031212018-11-20T10:28:13Z2018-11-20T16:15:15Z
<p>Given an abelian variety <span class="math-container">$A$</span> defined over <span class="math-container">$\mathbb{Q}$</span>, for a positive integer (we can suppose prime) <span class="math-container">$\ell$</span>, let <span class="math-container">$A[\ell]$</span> denote the group
of points of <span class="math-container">$A$</span> that are annihilated by <span class="math-container">$\ell$</span>, the division field <span class="math-container">$\mathbb{Q}(A[\ell])$</span> is obtained by adjoining to <span class="math-container">$\mathbb{Q}$</span> the <strong>coordinates</strong> of the points of <span class="math-container">$A[k]$</span>.</p>
<p>Why the <span class="math-container">$\ell$</span>-th cyclotomic field is contained in <span class="math-container">$\mathbb{Q}(A[\ell])$</span>?</p>
<p>I saw somewhere that this is because of the existenece of the Weil pairing (for simplicity, we can suppose that <span class="math-container">$A$</span> has a principal polarization)</p>
<p><span class="math-container">$$
e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell
$$</span></p>
<p>From <a href="http://www.2874565.com/questions/208386/n-th-root-of-unity-in-n-th-division-field-of-abelian-variety/208405">here</a>
I know that there exist points <span class="math-container">$P,Q\in A[\ell]$</span> such that <span class="math-container">$e_\ell(P,Q)$</span> is a primitive <span class="math-container">$\ell$</span>-th root of unity, but I dont understand how this is related to the <strong>coordinates</strong> of the points in <span class="math-container">$A[\ell]$</span>.
Thanks in advance.</p>
http://www.2874565.com/q/3157346Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$annie hearthttp://www.2874565.com/users/1064972018-11-19T23:03:54Z2018-11-20T16:58:34Z
<p>We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of
<span class="math-container">$$
\Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}),
$$</span>
where the <span class="math-container">$\mathbb{Z}/(8\mathbb{Z})$</span> is generated by a 2-manifold <span class="math-container">$M^2$</span> generator, such as
<span class="math-container">$$
\mathbb{RP}^2,
$$</span>
or the invariant <span class="math-container">$$\exp( 2 \pi i (k/8) \int_{M^2} \; \text{(abk)}),$$</span>
with <span class="math-container">$k \in \mathbb Z_8$</span></p>
<blockquote>
<ul>
<li>My question: Does there exist any surjective <span class="math-container">$$G \to Pin^-,$$</span>
such that it is a surjective map <span class="math-container">$G \to Pin^-$</span>, and the
Arf-Brown-Kervaire invariant <span class="math-container">$\exp( 2 \pi i (k/8) \int_{M^2} \; \text{(abk)})$</span> at the even integer <span class="math-container">$k$</span> of <span class="math-container">$\Omega_2^{Pin^-}(pt)$</span> becomes a trivial bordism invariant, under pullback, evaluated at <span class="math-container">$$\Omega_3^{G}?$$</span>
Namely the corresponding invariant for <span class="math-container">$\Omega_2^{G}$</span> on <span class="math-container">$\mathbb{RP}^2$</span> (at the even integer <span class="math-container">$k$</span>) becomes 1. (i.e. a trivial group element.)</li>
</ul>
</blockquote>
<p>p.s. It is obviously possible to find <span class="math-container">$G$</span>, if the <span class="math-container">$G \to Pin^-$</span> is an injective instead of surjective map. Say, we can take <span class="math-container">$G=Spin \to Pin^-,$</span> then <span class="math-container">$\Omega_2^{Spin}(pt)=\mathbb{Z}/(2\mathbb{Z})$</span> is obviously trivial, when we map back from any even integer <span class="math-container">$k$</span> in <span class="math-container">$\Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z})$</span> to <span class="math-container">$k \mod 2$</span> as <span class="math-container">$0$</span> in <span class="math-container">$$\Omega_2^{Spin}(pt)=\mathbb{Z}/(2\mathbb{Z}).$$</span></p>
http://www.2874565.com/q/3157301Complex structures on topological surfacesStudenthttp://www.2874565.com/users/1245492018-11-19T21:52:44Z2018-11-20T14:09:19Z
<ol>
<li><p>I am interested in the number of complex structures on a surface. More precisely, given a genus <span class="math-container">$g$</span> surface (topological manifold of real dimension 2) with <span class="math-container">$n$</span> punctures <span class="math-container">$X_{(g,n)}$</span>, how many complex structures (up to biholomorphic maps) are there? The usual answer I can find online are for those without punctures.</p></li>
<li><p>Also, is there any formula that describes the size of <span class="math-container">$[X_{(g,n)} , X_{(g',n')}]$</span>, which is defined to be the set of all holomorphic maps (up to biholomorphic maps)
<span class="math-container">$$
\mbox{i.e. } \mbox{Aut}(X_{(g,n)})\backslash \mbox{ Holo}(X_{(g,n)},X_{(g',n')}) \,/ \mbox{Aut}(X_{(g',n')}),
$$</span>
from the left to the right?</p></li>
</ol>
http://www.2874565.com/q/3157226Quantitatively characterizing the failure of the converse of Dirac's theoremTom Holthttp://www.2874565.com/users/885022018-11-19T20:52:10Z2018-11-20T16:37:34Z
<p>First, I am an undergraduate so I apologize if this is trivial and certainly understand if it is closed immediately. </p>
<p>I am currently in a combinatorics and graph theory class and recently we have been studying Hamiltonian graphs. We have been discussing a few theorems characterizing these graphs. I am interested in Dirac's theorem which states</p>
<blockquote>
<p><strong>Dirac (1952)</strong> Let <span class="math-container">$G$</span> be a simple graph with <span class="math-container">$n \geq 3$</span> vertices such that for any vertex <span class="math-container">$v \in G$</span> we have <span class="math-container">$\deg(v) \geq \frac{n}{2}$</span>. Then <span class="math-container">$G$</span> is Hamiltonian. </p>
</blockquote>
<p>The converse is easily seen to be false. I am interested in understanding how often the converse fails. From my view, one way to make this precise is as follows. Let <span class="math-container">$H_n$</span> denote the set of Hamiltonian graphs on <span class="math-container">$n$</span>-vertices. What can we say about the probability <span class="math-container">$$p_n = P(\deg(v) \geq \frac{n}{2},\forall v\in G \mid G \in H_n)$$</span></p>
<p>I am mainly interested in what happens as <span class="math-container">$n \to \infty$</span>. For example, I think it might be interesting if Dirac's theorem becomes necessary and sufficient if we take <span class="math-container">$n$</span> large enough. One could also investigate analogous question for other theorems that give sufficient conditions for <span class="math-container">$G$</span> to be Hamiltonian (Ore's theorem, Posa's theorem). However, Dirac's seemed the simplest to investigate. </p>
<p>Is there any literature on questions resembling this?</p>
<p>Thanks.</p>
<p><strong>Edit:</strong></p>
<p>The previous calculations for various <span class="math-container">$p_n$</span> values were clearly wrong and have been removed. </p>
http://www.2874565.com/q/3156993Can a functorial factorization be modified so that it fixes the initial object?Bruno Stonekhttp://www.2874565.com/users/62492018-11-19T15:49:52Z2018-11-20T14:32:02Z
<p>Consider a category <span class="math-container">$\mathcal C$</span> with a weak factorization system which is functorial. Let <span class="math-container">$*$</span> be an initial object. If <span class="math-container">$X\in \mathcal C$</span>, denote by <span class="math-container">$\eta_X:*\to X$</span> the unique map. Using the given wfs, it factors as as <span class="math-container">$*\stackrel{\eta_{QX}}{\to} QX \stackrel{q_X}{\to} X $</span>.</p>
<p>Consider the following assertion:</p>
<blockquote>
<p>(A) The maps <span class="math-container">$Q\eta_X:Q*\to QX$</span> and <span class="math-container">$Q*\stackrel{q_*}{\to} * \stackrel{\eta_{QX}}{\to} QX$</span> coincide.</p>
</blockquote>
<p>This is a minimalistic scenario: I'm interested in the situation where <span class="math-container">$\mathcal C$</span> is a (cofibrantly generated) model category with functorial factorizations and the wfs is (cofibrations, acyclic fibrations), so <span class="math-container">$Q$</span> is a cofibrant replacement functor. I would like to know the following:</p>
<blockquote>
<p>Is it possible to modify the functorial factorization of the wfs (cofibrations, acyclic fibrations) of <span class="math-container">$\mathcal C$</span> without changing the model category structure (i.e. keeping the same classes of cofibrations, fibrations, weak equivalences) so that assertion (A) is true?</p>
</blockquote>
<p>Manipulating some diagrams and the condition of functoriality lets us conclude that (A) is true if <span class="math-container">$q_X:QX\to X$</span> is a monomorphism, or if <span class="math-container">$\eta_{Q*}:*\to Q*$</span> is an epimorphism. </p>
<blockquote>
<p>Is it possible to modify the functorial factorization of the wfs (cofibrations, acyclic fibrations) so that <span class="math-container">$\eta_{Q*}$</span> is an epimorphism? An isomorphism? The identity <span class="math-container">$id_*$</span>?</p>
</blockquote>
<p>We could be more ambitious with this last question: it would be nice if we could change the functorial factorization so that it factors any identity map through identity maps (rather than asking that it does this just with the identity map of <span class="math-container">$*$</span>).</p>
<p>For background, I'd add that my category <span class="math-container">$\mathcal C$</span> is the category of commutative <span class="math-container">$\mathbb S$</span>-algebras of Elmendorf, Kriz, Mandell, May (EKMM), so <span class="math-container">$*$</span> is <span class="math-container">$\mathbb S$</span>.</p>
http://www.2874565.com/q/3156813Integral of product of Schur functionsMarcelhttp://www.2874565.com/users/780612018-11-19T12:13:51Z2018-11-20T12:30:41Z
<p>Schur functions are irreducible characters of the unitary group <span class="math-container">$\mathcal{U}(N)$</span>. This implies the integration formulae
<span class="math-container">$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^2}{s_\lambda(1)}$$</span>
and
<span class="math-container">$$ \int_{\mathcal{U}(N)}s_\lambda(AU)\overline{s_\mu(A U)}dU=\frac{\delta_{\lambda\mu}s_\lambda(AA^{\dagger})}{s_\lambda(1)},$$</span>
where the overline means complex conjugation.</p>
<p>My question is whether we can compute the integral
<span class="math-container">$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)\overline{s_\mu(AUA^\dagger U^\dagger)}dU$$</span></p>
http://www.2874565.com/q/3156397Index of the endomorphism ring of an abelian surfaceSun Rahttp://www.2874565.com/users/1315232018-11-19T00:01:39Z2018-11-20T15:20:03Z
<p>For an abelian surface <span class="math-container">$A/\mathbb{Q}$</span> such that <span class="math-container">$R:=\mathrm{End}_{\mathbb{Q}}(A)$</span> is an order in a real quadratic field <span class="math-container">$K$</span> (so a <span class="math-container">$\mathrm{GL}_2$</span>-type surface), is there a bound on the index <span class="math-container">$[O_K : R]$</span> of the ring inside the maximal order <span class="math-container">$O_K$</span>?</p>
http://www.2874565.com/q/3148572Stationary Navier-Stokes solutionsJean Duchonhttp://www.2874565.com/users/754222018-11-08T17:08:17Z2018-11-20T17:02:31Z
<p>Are there known nontrivial (<span class="math-container">$u\neq0$</span>) stationary solutions to Navier-Stokes equations in <span class="math-container">$\mathbb R^3$</span> ? Not square integrable of course (that's impossible), but with self-similar amplitudes of Fourier modes such as <span class="math-container">$|\hat u(c\xi)|=c^{-2}|\hat u(\xi)|$</span> ?</p>
<p>That would be <span class="math-container">$\Delta u=B(u,u)$</span> with <span class="math-container">$\nabla\cdot u=0$</span>, where <span class="math-container">$B(u,v):=\sum_{i=1}^3\partial_i(u_iv)+\nabla p$</span> and <span class="math-container">$p$</span> is such that <span class="math-container">$\nabla\cdot B(u,v)=0$</span>.</p>
<p>Same question for the particular class of <em>cylindrically symmetric</em> solutions. In cylindrical coordinates: <span class="math-container">$(r,\theta,z)$</span> and <span class="math-container">$(u_r,u_\theta,u_z,p)$</span> not depending on <span class="math-container">$\theta$</span>, with <span class="math-container">$\frac1r\frac\partial{\partial r}(ru_r)+\frac{\partial u_z}{\partial z}=0$</span> (incompressibility).</p>
<p>I wonder if a solution with <span class="math-container">$u(c\mathbb x)=c^{-1}u(\mathbb x)$</span> is possible...</p>
http://www.2874565.com/q/3010000Minimum Distance Distribution of two Uniformly Distributed SamplesEinsteinhttp://www.2874565.com/users/1248212018-05-24T08:11:50Z2018-11-20T16:03:00Z
<p>Thanks for your attention. I am working with genome data, more precisely two different kinds of DNA motifs distributed over a large DNA sequence. What I am searching for, is the expected probability distribution of the minimum distances between the two kinds motifs under the assumption that they are distributed randomly and independently of each other.</p>
<p>The problem is analogous to the <strong>minimum distance distribution of two uniformly distributed random samples</strong> (with different sizes) distributed over a fixed interval of length L ([0, L] with L > 0).</p>
<p>My fall-back solution would be running many simulations of the problem (which is quite easy to implement) but I would guess, that this problem is simple enough, that a exact solution could exist (accordingly, I feel a little dump that I can not come up with a good solution).</p>
<p>I searched on Google a few times within the last weeks but could not find anything 100% related. I could find exact solutions for two randomly distributed points which would be the extreme case were both of my samples have size 1 but I need a more general solution.</p>
<p>Something like "<em>the problem is not exactly solvable</em>" would also help, if reliable.</p>
http://www.2874565.com/q/2523837The Kan construction, profunctors, and Kan extensionsFosco Loregianhttp://www.2874565.com/users/79522016-10-17T19:46:51Z2018-11-20T16:48:30Z
<p>It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "<a href="https://ncatlab.org/nlab/show/nerve+and+realization" rel="nofollow">nerve-realization</a>" adjunction
$$
\text{Lan}_y F \dashv N_F = \hom(F,1)
$$
that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as
$$
\text{Lan}_y F \dashv \text{Lan}_F y,
$$
and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[<a href="https://golem.ph.utexas.edu/category/2014/03/an_exegesis_of_yoneda_structur.html" rel="nofollow">here</a>, several comments in the discussion are mine]. </p>
<ol>
<li>Is there a reason why this is true?</li>
<li>What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?</li>
<li>My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints
$$
\text{Lan}_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F.
$$
What's the meaning of this extension, and its universal property, in $\bf Prof$?</li>
</ol>
http://www.2874565.com/q/2089374Distribution of dot product of two unit random vectorsAllenhttp://www.2874565.com/users/188272015-06-10T16:52:19Z2018-11-20T12:16:45Z
<p>Consider $\mathbf{u}, \mathbf{v}\in \mathcal{C}^M$ to be two independent unit norm random vectors on the $M-1$ dimensional complex sphere $\mathcal{S}^{M-1}$. In addition, $\mathbf{u}$ follows an isotropic distribution (i.e., $\mathbf{u}$ is uniformly distributed on the complex sphere $\mathcal{S}^{M-1}$. What is the distribution of $Z=|\mathbf{u}\cdot\mathbf{v}|^2$?</p>
<p>I know that if $\mathbf{v}$ is also uniformly distributed on the complex sphere $\mathcal{S}^{M-1}$, then $Z$ follows Beta$(1, M-2)$ distribution. (I don't know how to prove this!) Does the same result hold if $\mathbf{v}$ follows an arbitrary distribution?</p>
http://www.2874565.com/q/2077833Dual cone of 'positive' Bochner integrable functionsGuillaume Garrigoshttp://www.2874565.com/users/736082015-05-27T20:39:31Z2018-11-20T12:55:44Z
<p>If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the corresponding dual cone is $L^\infty ([0,1];\mathbb{R}_+)$.
I wonder if the same kind of statement holds when replacing $\mathbb{R}$ by an arbitrary Banach space $Y$? </p>
<p>More formally :</p>
<p>Let $L^1([0,1];Y)$ be the Banach space of Bochner integrable functions from $[0,1]$ to $Y$ (identifying the functions a.e. equal on $[0,1]$). Its topological dual space is known to be $L^\infty_{w^*}([0,1];Y^*)$, the space of $w^*$-measurable functions from $[0,1]$ to $Y^*$ (with also an identification there).</p>
<p>Suppose now that $Y$ is ordered by a closed convex cone, with non-empty interior, that we note $Y_+$. We note $Y_+^*$ the correspoding dual cone, defined by :</p>
<p>$$Y_+^* := \{ y^* \in Y^* \ | \ \langle y^*,y\rangle \geq 0 \ \forall y \in Y_+ \}.$$</p>
<p>Can we prove that the dual cone of $L^1([0,1];Y_+)$ is $L_{w^*}^\infty([0,1];Y^*_+)$ ?</p>
<p>By just applying the definitions, we directly see that $L_{w^*}^\infty([0,1];Y^*_+)$ is included in the dual cone of $L^1([0,1];Y_+)$. But the reverse inclusion seems tricky...</p>
http://www.2874565.com/q/13019414Least prime in an arithmetic progression and the Selberg sieveGH from MOhttp://www.2874565.com/users/119192013-05-09T19:39:52Z2018-11-20T13:11:32Z
<p>My question concerns a technical step in the proof of Linnik's theorem on the least prime in an arithmetic progression, as presented in Chapter 18 of Iwaniec-Kowalski: Analytic number theory. </p>
<p>The proof uses certain weights $\theta_b$ coming from the theory of the Selberg sieve. The sequence is supported on square-free numbers up to $y$ coprime with $q$ (the modulus of the arithmetic progression), and its main feature is, cf. (18.19)-(18.23) in the book,
$$ |\theta_b|\leq 1,\qquad \theta_1=1, $$
$$ 0<\sum_{b_1,b_2}\frac{\theta_{b_1}\theta_{b_2}}{[b_1,b_2]}\leq\frac{q}{\varphi(q)\log y}. $$
The corresponding Selberg upper sieve coefficients are defined by
$$ \sigma_m:=\sum_{[b_1,b_2]=m}\theta_{b_1}\theta_{b_2}, $$
so that with the notation
$$ \nu(n):=\sum_{b\mid n}\theta_b $$
we have for any integer $n\geq 1$
$$ \sum_{m\mid n}\sigma_m=\nu(n)^2\geq\sum_{m\mid n}\mu(m). $$</p>
<p>By (18.70) of the book we have, "applying a sieve of dimension 8 (see the Fundamental Lemma 6.3)",
$$ \sum_{n\leq x}\nu^2(n)\frac{\tau^3(n)}{n}\ll\left(\frac{\log x}{\log y}\right)^8. $$
Here $\tau(n)$ is the number of divisors of $n$.
Can anyone help me understand why this is true? The quoted lemma is about Brun's sieve (where $\sigma_m$ would be $\mu(m)$ restricted to certain integers), but even if I accept it for the Selberg sieve, I do not see the stated bound. Similarly, I do not understand why (18.75) in the book is true.</p>
<p><strong>Added.</strong> Based on the expert response of Dimitris Koukoulopoulos I think that the proof of Linnik's theorem, as presented in Iwaniec-Kowalski's book, works fine if we replace the Selberg sieve $\sigma_m$ with an upper $\beta$-sieve $\beta_m$. More precisely, we redefine $\nu(n)$ so that
$$\nu(n)^2=\sum_{m\mid n}\beta_m,$$
which makes sense as the right hand side is nonnegative.</p>
http://www.2874565.com/q/937787Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)?tzhanghttp://www.2874565.com/users/228292012-04-11T15:51:32Z2018-11-20T15:13:43Z
<p>We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra <span class="math-container">$U(\mathfrak{g})$</span>, my question is: is there any "quantum Lie algebra" embeded into the quantum enveloping algebra <span class="math-container">$U_q(\mathfrak{g})$</span>?</p>
<p>The related question is, take <span class="math-container">$sl(2)$</span> generated by <span class="math-container">$\{X,Y,H|[XY]=H, [HX]=2X, [HY]=-2Y\}$</span> for example, consider the representation on polynomial <span class="math-container">$K[x,y]$</span>, <span class="math-container">$K[x,y]$</span> is in fact a module-algebra over <span class="math-container">$U(sl(2))$</span>, the elment of <span class="math-container">$sl(2)$</span> can be represented by <span class="math-container">$X=x\frac{\partial}{\partial y}, Y=y\frac{\partial}{\partial x}, H=x\frac{\partial_q}{\partial x}-y\frac{\partial_q}{\partial y}$</span> (see Kassel "Quantum groups" (GTM155), pp. 109). In fact, <span class="math-container">$\{x\frac{\partial}{\partial y}, y\frac{\partial}{\partial x}, x\frac{\partial_q}{\partial x}-y\frac{\partial_q}{\partial y}\}$</span> generated a three dim Lie subalgbebra (isomorphic to <span class="math-container">$sl(2)$</span> under the above correspondence) of derivation algebra of <span class="math-container">$K[x,y]$</span>.</p>
<p>Similariy, Is there quantum Lie algebra contained in <span class="math-container">$U_q(sl(2))$</span>? In fact, by Kassel "Quantum groups" (GTM155), pp. 146-149, there is an action of <span class="math-container">$U_q(sl(2))$</span> on quantum plane <span class="math-container">$K_q[x,y], E=x\frac{\partial_q}{\partial y}, E=y\frac{\partial_q}{\partial x}, K=\sigma_x\sigma_y^{-1}, K^{-1}=\sigma_y\sigma_x^{-1}$</span> , so is there any finite dim quantum Lie algebra generated by <span class="math-container">$E,F,K,K^{-1}$</span>, or does the operators <span class="math-container">$x\frac{\partial_q}{\partial y}, y\frac{\partial_q}{\partial x}, \sigma_x, \sigma_y^{-1}, \sigma_y, \sigma_x^{-1}$</span> generate a Lie subalgebra of of derivation algebra of <span class="math-container">$K_q[x,y]$</span>?</p>
http://www.2874565.com/q/882770Sigma algebra generatedSantoshttp://www.2874565.com/users/211172012-02-12T13:52:20Z2018-11-20T11:17:40Z
<p>Let <span class="math-container">$\mathcal{L} \subset \mathbb{R}$</span> the Lebesgue sigma algebra and <span class="math-container">$\mathcal{B} \subset \mathbb{R}^{n}$</span> the Borel sigma algebra. I'll denotes by <span class="math-container">$\mathcal{L} \times \mathcal{B}$</span> the smallest sigma algebra containing products <span class="math-container">$A \times B$</span>, where <span class="math-container">$A \in \mathcal{L}$</span> and <span class="math-container">$B \in \mathcal{B}$</span>.</p>
<p>Let <span class="math-container">$\mathbb{T} \subset \mathbb{R}$</span> a compact set and <span class="math-container">$\Delta$</span> a sigma algebra of subsets of <span class="math-container">$\mathbb{T}$</span>. Furthermore, if <span class="math-container">$E \in \Delta$</span> then <span class="math-container">$E \in \mathcal{L}$</span>. If <span class="math-container">$E \subset \mathbb{T}$</span> and <span class="math-container">$E \in \mathcal{L}$</span> then <span class="math-container">$E \in \Delta$</span>. </p>
<p>If <span class="math-container">$D \subset \mathbb{T} \times \mathbb{R}^{n}$</span> and <span class="math-container">$D \in \mathcal{L} \times \mathcal{B}$</span> then <span class="math-container">$D \in \Delta \times \mathcal{B}$</span> ?</p>
<p>Context: I'm finishing a work on the control on time scales. The sigma algebra product is common in control theory. But I have some doubts about it.</p>
山西福彩快乐十分钟