Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2016-07-28T02:58:48Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2452790Random variable of random variableJimSDhttp://mathoverflow.net/users/955712016-07-28T02:56:40Z2016-07-28T02:56:40Z
<p>This is confusing and difficut, but I hope it makes a sence.
I am interested in kind of like Random variable of Random variable.
This issue might've been mentioned below before.</p>
<p><a href="http://mathoverflow.net/questions/107364/the-probability-distribution-of-random-variable-of-random-variable">The Probability distribution of Random variable of Random variable</a></p>
<p>Let me clarify.<br>
Random variable of Random variable (RVoRV) here is <strong>not</strong><br>
add/subtract operation of two random variables.<br>
It is about a sigle measurement of random variables, which actually came from another random variables.</p>
<p>Is there any material or anything that mentioned this issue?
Thank you.</p>
<p>As an example
I made matlab code below.<br>
It creates N different gaussian distributions with their N standard deviations chosen uniform randomly with some variance.
Then it draw a single value from each of N different gaussians as random variable, making a list of N values.<br>
now what is the standard deviation of this N value list?<br>
As i run this code, sd stays near certain value.
Any equation for that?<br>
Thank you.<br><br>--------------------------<br></p>
<p>clear all;
close all;
clc;</p>
<p>N=100000;% number of gaussian distributions</p>
<p>% N means for gaussian -------------<br>
%gauss_list_mean=<strong>rand</strong>(1,N).*20+10;%average<br>
gauss_list_mean=repmat(0,1,N);%average same value</p>
<p>% N sigmas for gaussian -------------<br>
gauss_list_sd=<strong>rand</strong>(1,N).*20;%standard deviation<br>
%gauss_list_sd=repmat(15,1,N);%standard deviation same value</p>
<p>% draw a value from each of N different gaussian distributions -----<br>
if 0T% this is fast<br>
 list_drawn_value=<strong>randn</strong>(1,N).*gauss_list_sd+gauss_list_mean;<br>
else% slow<br>
 list_drawn_value=zeros(1,N);<br>
 for i=1:N<br>
  list_drawn_value(i) = normrnd(gauss_list_mean(i),gauss_list_sd(i));<br> end<br>
end</p>
<p>sd_value=std(list_drawn_value);%Get standard deviation</p>
<p>fprintf(' sd of drawn values from %d Gaussians =%g \n',sd_value,N);</p>
<p>fprintf('mean(gauss_list_sd)=%g std(gauss_list_sd)=%g\n',mean(gauss_list_sd),std(gauss_list_sd));</p>
http://mathoverflow.net/q/2452780Generalizing approximate $\mathbb{Z}$-equivariance of a simple functionSteve Huntsmanhttp://mathoverflow.net/users/18472016-07-28T02:46:28Z2016-07-28T02:46:28Z
<p>Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. <a href="http://math.stackexchange.com/questions/1865370/">http://math.stackexchange.com/questions/1865370/</a>) that $F$ is approximately equivariant w/r/t the $\mathbb{Z}$-actions corresponding to iteration $f^{\circ N}(x)$ and addition $x+N$. That is, we have that $F(f^{\circ N}(x)) \approx F(x)+N$. This is shown in the figure below: blue curves are $f^{\circ N}(x)$ and red curves are $F^{-1}(F(x)+N)$, both for $0 \le N \le 10$.</p>
<p><a href="http://i.stack.imgur.com/YGs8d.png" rel="nofollow"><img src="http://i.stack.imgur.com/YGs8d.png" alt="enter image description here"></a></p>
<blockquote>
<p>Is this approximate equivariance a manifestation of some more general phenomenon?</p>
</blockquote>
http://mathoverflow.net/q/2452770Importance of $E_n$-algebras over ring structures on $\pi_*(E)$Pax Kivimaehttp://mathoverflow.net/users/411032016-07-28T02:33:31Z2016-07-28T02:33:31Z
<p>Hopefully this question is not too vague to be closed. I am looking for examples of when a construction/theorem that involves $E$-(co)homology or even simply the ring $E_*$ requires an understanding of an some sort of $E_n$-structure on $E$ or its module category. For example:</p>
<p>When $n=\infty$, $E_\infty$-rings possess the power operations $P_n:E^*(X)\to E^*_{\Sigma_n}(X)$.</p>
<p>When $n=3$, and $X$ is a quasi-projective scheme, $Br(X)\to H^2(X; \mathbb{G}_m)$ is an isomoprhism, but not in general. However, if $X$ is quasi-compact and quasi-seperated (or more generally, a derived scheme based on non-connected ring spectra with the same conditions, then $Br_{der}(X)=\pi_0(\mathfrak{br}(X))\to H^2(X; \mathbb{G}_m)$ is an isomorphism.</p>
<p>The construction of $Tmf$ naively is only an inverse limit over the category of the Landweber spectrum coming from elliptic curves. This limit though cannot be computed until coherence conditions are fixed, and this is only possible by noting that the subset of $E_\infty$-maps between these lifts is far more manageable.</p>
<p>Similar statements for Picard groups and other invariants exist. I am looking for more examples of situations where the normal algebraic category of $E_*$ ($E^*$)-modules doesn't cut it, but interesting results can be obtained only by working in a structured category.</p>
http://mathoverflow.net/q/2452760An asymptotic formula involving the $2$-torsion subgroup of the class group of real quadratic fieldsStanley Yao Xiaohttp://mathoverflow.net/users/108982016-07-28T01:27:45Z2016-07-28T02:43:36Z
<p>Let $R$ be an order in some number field $K$ (not necessarily maximal). Then the class number $\text{Cl}(R)$ is equal to the cardinality of the Picard group of $R$, which is the group of equivalence classes of invertible fractional ideals of $R$. There is a unique quadratic order $R_D$ for each discriminant $D > 0$, and we put $h(D)$ for the class number of $R_D$. It is well-known from Gauss that $h(D)$ is also the number of $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of binary quadratic forms with integer coefficients of discriminant $D$. Let $\epsilon_D = u + v \sqrt{D}$, where $(u,v)$ is the smallest positive pair which solves the equation</p>
<p>$$\displaystyle u^2 - Dv^2 = \pm 4.$$</p>
<p>Moreover, it is a famous theorem of Siegel that the asymptotic formula</p>
<p>$$\displaystyle \sum_{D < x} h(D) \log \epsilon_D = \frac{\pi^2}{36} x^{3/2} + O(x).$$</p>
<p>Let $h_2(D)$ denote the size of the 2-torsion subgroup of $\text{Cl}(R_D)$. Is there an analogous asymptotic formula for</p>
<p>$$\displaystyle \sum_{D < x} h_2(D) \log \epsilon_D ?$$</p>
http://mathoverflow.net/q/245273-1An Example to Marsden-Weinstein TheoremDLINhttp://mathoverflow.net/users/952962016-07-28T00:58:25Z2016-07-28T02:54:28Z
<p>Suppose that the action of a compact Lie group $G$ on the <strong>closed</strong> symplectic manifold $(M,\omega)$ is Hamiltonian, with moment map $\mu : M\to \mathfrak{g}^*$. From the Hamiltonian condition it follows that $ \mu^{-1}(0)$ is invariant under $G$.</p>
<p>Assume now that 0 is a regular value of $\mu$ and that $G$ acts freely and properly on $ \mu^{-1}(0)$. Thus, $\mu ^{-1}(0)$and its quotient $\mu^{-1}(0) / G$ are both manifolds. By <strong>Marsden-Weinstein Theorem</strong>, the quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to $\mu^{-1}(0)$ equals the restriction of $\omega$ to $ \mu^{-1}(0)$. </p>
<p>Q: Is there any example to satisfy the <strong>Marsden-Weinstein</strong>?</p>
<p>PS: I consider the $M:=\mathbb P^3$ , $G:=S^1$, but I do not know how to give the moment map. Sorry.</p>
http://mathoverflow.net/q/2452721Bounding a distance function on polynomials with real rootsTurbohttp://mathoverflow.net/users/100352016-07-27T23:49:09Z2016-07-28T02:52:21Z
<p>Consider univariate polynomials $f(x)=\sum_{i=0}^{r_f}f_ix^i$ and $g(x)=\sum_{i=0}^{r_g}g_ix^i$ in $\Bbb Z[x]$ with only distinct real roots $\alpha_1,\dots,\alpha_{r_f}\in\Bbb R$ and $\beta_1,\dots,\beta_{r_g}\in\Bbb R$ respectively where $r_p$ refers to number of roots of polynomial $p(x)$.</p>
<p>Consider the function $$d(f,g)=\sum_{i\in\{1,\dots,r_f\}}\min_{j\in\{1,\dots,r_g\}}(\alpha_i-\beta_j)^2+\sum_{j\in\{1,\dots,r_g\}}\min_{i\in\{1,\dots,r_f\}}(\alpha_i-\beta_j)^2$$</p>
<p>Can we bound $d(f,g)$ by
\begin{equation}
c_1\cdot F_1(f_0,\dots,f_{r_f},g_0,\dots,g_{r_g})\leq d(f,g)\leq c_2\cdot \tag{*} F_2(f_0,\dots,f_{r_f},g_0,\dots,g_{r_g})
\end{equation}
for some constants $c_1,c_2>0$ and ${real}$ functions $F_1,F_2$?</p>
<hr>
<p>The reason I posted this problem is because it is not clear how to explicitly describe $d(f,g)$ in terms of coefficients of $f,g$ since Galois theory forbids such expressions using ring operations. I would like to have explicit description of $F_1,F_2$ since lower and upper bounds may be easy to express in terms of coefficients.</p>
http://mathoverflow.net/q/245267-1MDAS solution glitch [on hold]Jhomar Maravillashttp://mathoverflow.net/users/955682016-07-27T22:20:10Z2016-07-27T22:20:10Z
<p>I always get confused with this kind of math problem, How do we solve this problem? </p>
<p>Example:</p>
<pre><code>1 * 5 - 10 + 2 = ?
</code></pre>
<p>Solution 1 [My solution]:</p>
<p>Of couse I use MDAS rule </p>
<p>(1 * 5) - 10 + 2</p>
<pre><code>5 - (10 + 2)
5 - 12
-7
</code></pre>
<p>Solution 2 [My prof]:</p>
<pre><code>(1 * 5) - 10 + 2
5 (-10 + 2)
5 - 8
-3
</code></pre>
<p>The difference in our soution is I did not include the subtraction sign and my prof include it what is the rules in solving that kind of problem? </p>
http://mathoverflow.net/q/2452640Sobolev regularity for systems of elliptic boundary value problemsIdempotenthttp://mathoverflow.net/users/254902016-07-27T22:01:22Z2016-07-27T22:58:51Z
<p>My question is about Sobolev estimates near the boundary for elliptic systems (equivalently, elliptic boundary-value problems for vector-valued functions). </p>
<p>Note, results for the scalar case are easier to find, but it seems more difficult to find ones for the case when the solution is a vector-valued function.</p>
<p>I am interested in results that go something like this:</p>
<p>Suppose we have a second-order linear elliptic system on some domain in $\mathbb{R}^n$ with a smooth boundary. Suppose also that we have a solution with some degree of Sobolev regularity (i.e. the solution belongs to $H^s$ for some $s$). If the nonhomogeneous part of the equation and the boundary data also have some given levels of Sobolev regularity, then we can conclude that the solution actually has a higher level of Sobolev regularity. Not just in the interior (i.e. not on sets that are relatively compact in a domain which we assume is open), but actually up to the boundary.</p>
<p>If anyone can point me toward a reference, that would be great! Thank you!</p>
http://mathoverflow.net/q/2452630Riemannian manifolds conformally diffemorphic to sphereayubhttp://mathoverflow.net/users/955672016-07-27T21:55:09Z2016-07-27T21:55:09Z
<p>let a Riemannian manifold be conformally diffeomorphic to a sphere. Is its scalar curvature constant?</p>
http://mathoverflow.net/q/2452611Counting tournaments with tiesBernardo Recamán Santoshttp://mathoverflow.net/users/607322016-07-27T21:16:32Z2016-07-27T23:33:19Z
<p>An improper tournament, or tournament with ties, is a graph in which every pair of nodes is connected by a single uniquely directed edge or by a single undirected edge.
There are 1, 2, and 7 improper tournaments of orders 1, 2, and 3, respectively. How many are there of order 4? Of order n?</p>
http://mathoverflow.net/q/2452603Uncountable divisible groups and the existence of order-preserving ismorphisms of their subsetssN.W.http://mathoverflow.net/users/955632016-07-27T21:09:16Z2016-07-27T22:11:53Z
<p>Let $(G,+,0,<)$ be an ordered divisible group of uncountable dimension. Consider the subset $G^{<0}$ of $G$. </p>
<p>Question: Are $G$ and $G^{<0}$ isomorphic as ordered sets? Does there exists an order-preserving ismorphism of $G^{<0}$ onto $G$?
Intuitively I would say, this is true.</p>
<p>My ideas:
1. In the case where $(K,+, \cdot, 0,1,<)$ is an divisible ordered field we can give an explicit order-preserving isomorphism of $K$ onto $K^{<0}$, e.g. $f(x):= \begin{cases}
x-1 \quad \text{if} \; x \leq0, \\
-\frac{1}{x+1} \quad \text{else}.
\end{cases}
$</p>
<ol start="2">
<li>In the case where $G$ is in ordered set the statement is false.</li>
</ol>
<p>How can we use the group structure and the divisibility of $G$ to construct an isomorphism or does anybody know a counterexample?</p>
http://mathoverflow.net/q/2452592Whether a given algebra is the algebra of endomorphisms for a vector spaceBedovlathttp://mathoverflow.net/users/893132016-07-27T21:00:35Z2016-07-27T21:00:35Z
<p>Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms for some $\mathbb{F}$-vector space $\mathbf{V}$? Generalizations to $A$ being an associative unital ring and $\mathbf{V}$ an Abelian group or similar are welcome. Answers for particular cases $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$ are also appreciated. Thank you.</p>
<p>This is a repetition of the <a href="http://math.stackexchange.com/questions/1869442/whether-a-given-algebra-is-the-algebra-of-endomorphisms-for-a-vector-space">following</a> question I posed on MathSE days ago which has no answers so far. As I am no expert in algebra, so I thought maybe it is a research level question.</p>
http://mathoverflow.net/q/2452580Sub-matrices with a real spectrumLior Eldarhttp://mathoverflow.net/users/362722016-07-27T20:27:40Z2016-07-27T23:45:28Z
<p>This question arises from the study of PT-symmetric quantum mechanics.</p>
<p>Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum.</p>
<p>If $A$ is Hermitian, then any sub-matrix corresponding to to the
Cartesian product $I \times I$, where $I \subseteq [n]$ is also Hermitian
and hence has a real spectrum.</p>
<p>But what if $A$ is a non-Hermitian matrix (but with a real spectrum)?
Under what conditions do all principal sub-matrices of $A$ also have a real spectrum?
[edited following the first two comments.]</p>
http://mathoverflow.net/q/2452552Elementary question: Curvature change under Complexified Gauge TransformationHLChttp://mathoverflow.net/users/837862016-07-27T20:01:23Z2016-07-27T20:01:23Z
<p>Forgive me for this elementary question.</p>
<p>Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge transformation. Denote $\partial=\nabla^{1,0}$ and $\overline{\partial}=\nabla^{0,1}$. Then</p>
<p>$g.\nabla=\nabla-(\overline{\partial}g)g^{-1}+((\overline{\partial}g)g^{-1})^\dagger=\nabla-(\overline{\partial}g)g^{-1}+(g^{-1}\partial g)$</p>
<p>since $g$ is self adjoint.</p>
<p>Let $F_\nabla$ be the curvature of $\nabla$. We have the formula</p>
<p>$F_{\nabla+A}=F_\nabla+\nabla A+\frac{1}{2}[A,A].$</p>
<blockquote>
<p>How to show</p>
<p>$F_{g.\nabla}=F_{\nabla}-\partial((\overline{\partial}g)g^{-1})+\overline{\partial}(g^{-1}(\partial g))-(\overline{\partial}g)g^{-2}(\partial g)+g^{-1}(\partial g)(\overline{\partial}g)g^{-1}$?</p>
</blockquote>
<p>(Let $A=-(\overline{\partial}g)g^{-1}+(g^{-1}\partial g)$. How to compute $[A,A]$? How to compute even just $[(\overline{\partial}g)g^{-1},(\overline{\partial}g)g^{-1}]$?)</p>
<p>This question is taken from page 8 of Donaldson's <a href="https://projecteuclid.org/download/pdf_1/euclid.jdg/1214437664" rel="nofollow">proof</a> of Narasimhan-Seshadri theorem.</p>
<p>Thank you.</p>
http://mathoverflow.net/q/2452524Section of ellipsoidsJ. Dianoushttp://mathoverflow.net/users/913312016-07-27T19:36:28Z2016-07-27T20:17:30Z
<p>I am reading some survey about euclidean sections of convex bodies(<a href="http://arxiv.org/abs/1110.6401" rel="nofollow">http://arxiv.org/abs/1110.6401</a>). It is written that
any $k$-dimensional ellipsoid easily seen to have a $k/2$-dimensional
section which is a multiple of an Euclidean ball.
Can anyone tell me the reference of this fact or how to prove it?</p>
http://mathoverflow.net/q/2452491About consecutive integers covered by arithmetic progressionsuser1726559http://mathoverflow.net/users/372892016-07-27T18:42:06Z2016-07-27T22:58:59Z
<p>Help me please to solve the following problem.</p>
<p>There are $n$ arithmetic progressions of the form:</p>
<p>$$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$</p>
<p>Initial integer terms $x_i \geq 0$ are varying.</p>
<p>The problem is to cover $m(n)$ - the maximum possible number of consecutive integers starting with $ 1 $ (the number is covered, if it belongs to at least one of these progressions).</p>
<p>For example, is it true, that:
$$m(n) \approx \operatorname*{LCM}\limits_{i=1,\,\ldots\,,\,n}(\{2i+1\}) \text{?}$$</p>
<p>But it is so much and I am in search of the best solutions for this problem.</p>
<p>Numerical simulation shows something like this:
$m(n) \approx Cn$</p>
http://mathoverflow.net/q/245242-1Clarification of the proof of the main theorem of the paper of Hulse et alMedhttp://mathoverflow.net/users/443192016-07-27T16:58:15Z2016-07-27T20:06:40Z
<p>I am trying to understand some open steps in the following article <a href="http://www.worldscientific.com/doi/abs/10.1142/S179304211250042X" rel="nofollow">The Sign of Fourier coefficients of Half-integral Weight Cusp Form</a> by Hulse, Kiral, Kuan, and Lim, I find the following : </p>
<p>Let $f\in S_{\frac{k}{2}}(\Gamma_0(4))$ be an eigenform of all Hecke operators $T_{\frac{k}{2}}(p^2)$ for $p$ prime, where $k$ is an odd integer. Take the Dirichlet series : $$M(s)=\sum_{t\geq 1, \; t\;\text{square-free}}\frac{a(t)}{t}$$ With the inverse Mellin transform, we get : $$\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}M(s)\Gamma(s)x^sds=\sum_ta(t)e^{-t/x}$$ They assert that the integral on the left-hand side above is $O(x^{3/4+\varepsilon})$ for any $\varepsilon>0.$ I don't understand why ? </p>
<p>I have another questions :</p>
<p>Considering the inverse Mellin transform
$$I=\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}L^{(2)}(s)\Gamma(s)x^sds=\sum_na(n)^2e^{-n/x}$$
and shifting the line of integration to $\Re(s)=\frac{1}{2}$
past the pole at $s=1,$ we get
$$I=(\mathrm{Res}_{s=1}L^{(2)}(f,s))x+\frac{1}{2\pi i}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}L^{(2)}(s)\Gamma(s)x^sds\;\;\;(*)$$ They assert that :<br>
$$\frac{1}{2\pi i}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}L^{(2)} (s)\Gamma(s)x^sds=O(x^{\frac{1}{2}})\;\;\;\;(1)$$
and $(1)$ combined with $(*)$ implies, that
$$x\ll\sum_na(n)^2e^{-n/x}.\;\;\;\;\;\;(2)$$
I don't see why $(1)$ is true ? and why $(1)\Rightarrow (2).$</p>
<p>Can someone clarify to me it ? Thanks in advance.</p>
http://mathoverflow.net/q/2451900An inequality in product space $V$ [on hold]Oai Thanh Đàohttp://mathoverflow.net/users/766982016-07-27T08:17:57Z2016-07-28T01:04:47Z
<p><strong>I found an inequality as following:</strong> Let $x, y, z$ be three complex numbers then:</p>
<p>\begin{equation*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation*} (1)</p>
<p>The inequality holds with equality if and only if $x+y+z=0$</p>
<p><strong>Note that:</strong> <em>I have a proof of the inequality (1).</em></p>
<blockquote>
<p><strong>My question:</strong> I am looking for a proof of conjecture as following:</p>
<p>Let $x, y, z$ in an inner product space $V$ then</p>
<p>\begin{equation*}\frac{1}{2}(\|y+z-x\|+\|x+z-y\| + \|y+x-z\|) \le \|x\| + \|y\|+\|z\|+\frac{1}{2}\|x+y+z\|\end{equation*}</p>
<p>where the norm ||z|| denotes the norm induced by the inner product</p>
</blockquote>
<p><strong>See also</strong> </p>
<ul>
<li><a href="http://mathworld.wolfram.com/HlawkasInequality.html" rel="nofollow">Hlawka's inequality</a> </li>
<li><a href="http://mathoverflow.net/questions/167685/absolute-value-inequality-for-complex-numbers/167741#167741">Absolute value inequality for complex numbers</a></li>
</ul>
http://mathoverflow.net/q/2451806A $n$-gon is isospectral to a regular $n$-gon (Isospectral $\implies$ isometry ?)David Labrecquehttp://mathoverflow.net/users/953522016-07-27T06:11:33Z2016-07-28T02:31:05Z
<p>If an $n$-gon $P$ is isospectral to a regular $n$-gon $Q$, what could we say about the shape of the $P$. Otherwise, what could we say about $Q$? In fact, some hints or simply some ideas should be appreciated.</p>
<p><strong>Clarification :</strong> I talk about the spectrum of the Laplacian on the interior of the polygon, acting on the space of functions vanishing on the boundary.</p>
http://mathoverflow.net/q/2450344Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvaturesvahttp://mathoverflow.net/users/161832016-07-25T10:06:05Z2016-07-27T19:40:58Z
<p>Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov compactness theorem says that there exists a subsequence which converges in the Gromov-Hausdorff sense to a compact metric space $X$.</p>
<p><strong>Questions. (1) Is the Hausdorff dimension of $X$ necessarily integer?</strong></p>
<p><strong>(2) Is it at most $n$?</strong></p>
<p><strong>Remark.</strong> If one assumes a stronger condition that the sectional (rather than Ricci) curvature of $M_i$ is uniformly bounded below then the answers to both questions are positive as it is shown in the theory of Alexandrov spaces.</p>
http://mathoverflow.net/q/2449165Stopping times for Brownian motionJames Martinhttp://mathoverflow.net/users/57842016-07-23T11:15:12Z2016-07-27T21:00:10Z
<p>Let $B_t, t\geq 0$ be standard Brownian motion. </p>
<p>Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$.</p>
<p>Define also a filtration $\big(\mathcal{F}_t, t\geq 0\big)$ by $\mathcal{F}_t=\bigcap_{\epsilon>0} \mathcal{G}_{t+\epsilon}$.</p>
<blockquote>
<p>Let $\tau$ be a stopping time with respect to the filtration $(\mathcal{F}_t)$. Does there always exist $\tau'$ which is a stopping time with respect to the filtration $(\mathcal{G}_t)$ such that $\tau=\tau'$ with probability 1?</p>
</blockquote>
<p>Related: Blumenthal's 0-1 law says that, for fixed $t$, for any event $A\in\mathcal{F}_t$, there is an event $\tilde{A}\in\mathcal{G}_t$ such that the symmetric difference of $A$ and $\tilde{A}$ has probability 0. </p>
<p>However, this on its own is not enough. For example, let $U$ be a uniform random variable on $[0,1]$, and define a process $C_t, t\geq 0$ by $C_t=0$ for $t\leq U$ and $C_t=t-U$ for $t\geq U$. Then a similar 0-1 law holds, and $U$ itself is a stopping time for the filtration $\mathcal{F}_t=\bigcap_{\epsilon>0}\sigma(C_s, 0\leq s\leq t+\epsilon)$, but there is no stopping time $V$ for the filtration $\mathcal{G}_t=\sigma(C_s, 0\leq s\leq t)$ such that $U=V$ with probability 1.</p>
http://mathoverflow.net/q/2447587Hierarchical (Recursive) Random Walk (also known as Hierarchical Hidden Markov Model)Moskowitzhttp://mathoverflow.net/users/823582016-07-21T07:48:19Z2016-07-27T21:55:18Z
<p>Consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (<a href="https://en.wikipedia.org/wiki/Hierarchical_hidden_Markov_model" rel="nofollow">https://en.wikipedia.org/wiki/Hierarchical_hidden_Markov_model</a>). </p>
<ol>
<li><p>For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete-time) random walk. </p></li>
<li><p>For the next level $\ell=2$, we consider a set of $T$ random walks.
In particular, the $t$-th $(t \in \{1,\ldots, T\})$ random walk in this set has parameters (especially transition probabilities) depending on the realization of $X_t^{(1)}$ in the first level. </p></li>
<li><p>In the same way we recursively generate the next level $\ell = 3$, which contains $T^2$ random walks, and so on to the $L$-th level. </p></li>
</ol>
<p>It seems that results for this model is rather sparse in existing probability or statistics literature, or perhaps it has a different name. I would much appreciate any pointers to existing results. </p>
<p>A further question: Consider the concatenation of the $T^L$ random walks in the $L$-th level. Does this sequence have stronger long range correlation in comparison with canonical random walks?</p>
http://mathoverflow.net/q/2442403Integrable systems and Arnol'd - Liouville theoremPPeghttp://mathoverflow.net/users/950252016-07-13T13:57:51Z2016-07-27T20:36:34Z
<p>A system with a $2n$-dimensional phase space is Liouville-integrable if it admits $n$ independent first intgrals in involution.</p>
<p>Here integrable means that you can, in some way, solve the equations of motion by quadratures. </p>
<p>The Liouville-Arnol'd theorem states that a Liouville-integrable system admits a canonical transformation to action-angle coordinates, provided that it respects some other topological conditions.</p>
<p>These are that the level set of the first integrals must be compact and connected.
My question is: is this condition very restrictive in the usual case? And does it imply that the orbit is quasi-periodic under those conditions?</p>
<p>I wonder if a problem like the Kepler problem with an open orbit (when the energy is greater than zero) is treatable with the Arnold-Liouville method.</p>
http://mathoverflow.net/q/2397511Differential inequalities for a strictly diagonal dominant system of linear ODEsAntonyhttp://mathoverflow.net/users/383222016-05-25T16:54:22Z2016-07-27T23:42:33Z
<p>Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column diagonally dominant ($\forall j, |a_{jj}|>\sum_{i\neq j}|a_{ij}|$).</p>
<p>Consider the system of differential equations given by ${\bf \dot x}(t)=A {\bf x}(t)$ and suppose that the set of inequalities $\dot y_i(t)\leq (A {\bf y}(t))_i$ with $i=1,\ldots,d$ holds for all $t$. Given the initial conditions ${\bf x}(0)={\bf y}(0)$ with $x_i(0)\geq 0 \,\forall i$, do we necessarily have $x_i(t)\geq y_i(t)\,\forall i$ at every $t>0$?</p>
http://mathoverflow.net/q/23868610Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geodesic flow is recurrent?italo lirahttp://mathoverflow.net/users/856812016-05-12T19:10:51Z2016-07-27T20:49:26Z
<p>Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = \mu(A)$ for every $t \in \Bbb{R}$ and for all borelian set A. In particular, the time-one map $\varphi^1$ also preserves the measure $\mu$ because $\mu(\varphi^{-1}(A)) = \mu(A)$.</p>
<p>We say that the map $\varphi^1$ is recurrent (or conservative) if all wandering set W for $\varphi^1$( meaning that $W \cap \varphi^{-n}(W) = \varnothing $ for $n \geq 1$) has necessarily $\mu(W) = 0$.</p>
<p>An another equivalent definition is,
if A is borelian set with $\mu(A) > 0$ then
$$ A \subset \displaystyle\bigcup_{n \geq 1} \varphi^{-n}(A) \ \ (mod \ \ \mu) $$ </p>
<p>Question: Is there a complete Riemannian manifold M with infinite volume whose the time-one map of the geodesic flow is recurrent ?</p>
http://mathoverflow.net/q/2005331Q re Kaprekar's fixed mapping pointsAlexhttp://mathoverflow.net/users/558532015-03-20T14:07:23Z2016-07-27T19:34:26Z
<p>Jens Kruse Andersen in his comment in <a href="https://oeis.org/A099009" rel="nofollow">OEIS's A099009</a> noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine): </p>
<p>"Let $d(n)$ denote $n$ repetitions of the digit $d$. The sequence includes the following for all $n\ge0$: $5(n)499(n)4(n)5, 63(n)176(n)4, 8643(n)1976(n)532$."</p>
<p>The above comment, made by Jens Kruse Andersen, is missing one more family of terms (which starts with one or more digits "$9$" and ends with the digit "$1$"): 97508421, 9753086421, 9975084201, 975330866421, 997530864201, 999750842001, ... . </p>
<p>This family could be generalized (using the same method as in Andersen's comment) and it is actually covered by Syed Iddi Hasan in <a href="https://oeis.org/A214559" rel="nofollow">A214559</a>:
$9(x_1+1)//8(x_2)//7(x_3+1)//6(x_2)//5(x_3+1)//4(x_2)//3(x_4)//2(x_2)//1(x_3)//0//9(x_2)//8(x_3+1)//7(x_2)//6(x_4)//5(x_2)//4(x_3+1)//3(x_2)//2(x_3+1)//1(x_2)//0(x_1)//1$
where the sign // denotes concatenation of digits in the definition, $d(x)$ denotes $x$ repetitions of $d$, $x\ge0$.</p>
<p>NB - in his OEIS wiki page Syed Iddi Hasan wrote: "I narrowed it down to four parameters. I ordered the digits from largest to smallest and smallest to largest, and by comparing them I was able to find the interdependent pairs of numbers. However, these four parameters seem to be independent of each other."</p>
<p>Also A214557 and A214558 (both by Syed Iddi Hasan) are two variants relevant to Andersen's 8643(n)1976(n)532 - those two should be somehow combined, in my opinion, for the purpose of identifying unique families of Kaprekar mapping fixed points.</p>
<p>Could someone finalize classification of distinct families for Kaprekar's fixed mapping points and prove that each of Kaprekar's fixed mapping points belong ONLY to the one of the above mentioned families ?</p>
http://mathoverflow.net/q/1843253Self-adjointness of the components of the magnetic derivativeGeno Whirlhttp://mathoverflow.net/users/338042014-10-13T16:07:47Z2016-07-27T21:34:29Z
<p>On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic potential on $\mathbb{R}^{n}$ and $u \in Dom\ \Pi_{j} := \{ u \in L^{2}(\mathbb{R}^{n}) \ \vert\ -i\partial u/\partial x_{j} - A_{j}u \in L^{2}(\mathbb{R}^{2}) \}$, where $\partial u/\partial x_{j}$ is the weak derivative of $u$.</p>
<p>I wish to understand what sort of conditions on $A_{j}$ ensure that this operator is self-adjoint.</p>
http://mathoverflow.net/q/1723182$L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equationthomashttp://mathoverflow.net/users/531632014-06-20T19:47:22Z2016-07-27T22:57:41Z
<p><strong>In short:</strong></p>
<p>In P. Gérard's paper on the existence of global solutions to the Boltzmann equation from 1988 (or equivalently Cercignani's book), why are the stated assumptions (especially $A_n \in L^\infty_{loc}$) sufficient for proving velocity averaged compactness of the linearized collision kernels in Proposition 8 iii) ?</p>
<p>In particular, why is $\psi_n = \frac{A_n \ast f_n}{1 + \int f_n dv} \phi$ uniformly bounded in $L^\infty$ (here, $\phi \in L^\infty$ with compact support)?</p>
<p><strong>Background information:</strong></p>
<p>In addition to the famous paper by Lions and DiPerna "On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability" (Annals, 1989) there is the paper by P. Gérard "Solutions globales du problème de Cauchy pour l'équation de Boltzmann" (Séminaire Bourbaki, 1987-1988) which shows existence under the hypothesis that the kernel satisfies an $L^\infty_\mathrm{loc}$ assumption. This version of the proof of existence is also formulated in the book "The Mathematical Theory of Dilute Gases" (Cercignani, Illner, Pulvirenti, 1994).</p>
<p>The original DiPerna-Lions proof starts with the assumption that $A_n \in L^\infty$ then gets rid of this assumption completely in a second step. The Gérard approach tries to modify the first step in that it does only assume $A_n \in L^\infty_\mathrm{loc}$. This is already appropriate for most collision kernels, hence it's reasonable to stick to this assumption for the sake of a less technical proof (and skip the second part of DiPerna-Lions).</p>
<p>Incomprehensibly, it seems to me like the crucial step where DiPerna and Lions use $A_n \in L^\infty$ is not changed at all in the $L^\infty_\mathrm{loc}$ case (it's the step described above)! There is no comment from Cercignani or Gérard why the same argument should still work even without $L^\infty$.</p>
http://mathoverflow.net/q/12171931Richness of the subgroup structure of p-groupsStefan Kohlhttp://mathoverflow.net/users/281042013-02-13T13:44:22Z2016-07-27T21:13:50Z
<p>Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest
number such that there is a group of order $p^{f_p(n)}$ which all groups of
order $p^n$ embed into. What is the asymptotic growth of $f_p(n)$ when
$n$ tends to infinity?</p>
<p>The question asks in a certain sense for how dense $p$-groups can be "packed together"
as subgroups of a larger group.</p>
<p>Let's give an example for illustration: By the bound by Francois Brunault, all groups of
order $2^{20}$ embed into a group of order $2^{2^{20}-1}$, which is a number with
315653 decimal digits. On the other hand, by Nick Gill's bound, they do not embed
into a group of order $2^{66}$, which is a 20-digit number.
Can these bounds be refined?</p>
<p><b>Added on Feb 21, 2013:</b> Even if finding precise asymptotics for $f_p(n)$ turns out to be
delicate, isn't it at least possible to decide whether $f_p(n)$ grows polynomially or
exponentially, or whether its growth rate lies somewhere in between?
Or alternatively, are there reasons to believe that this is a difficult problem?</p>
<p><b>Added on Dec 4, 2013:</b> The question whether it is true that $f_p(n)$ grows faster than polynomially but slower than exponentially when $n$ tends to infinity will appear as Problem 18.51 in:</p>
<p>Kourovka Notebook: <em>Unsolved Problems in Group Theory</em>. Editors V. D.
Mazurov, E. I. Khukhro. 18th Edition, Novosibirsk 2014.</p>
http://mathoverflow.net/q/88922spectral radius of a matrix as one element changeshal iiihttp://mathoverflow.net/users/25692009-12-14T17:39:35Z2016-07-27T22:09:59Z
<p>Here's my question --</p>
<p>Let $A$ be an $n \times n$ real matrix, and suppose that the spectral radius $\rho(A)$ is less than one (spectral radius = max eigenvalue). Let's choose some $1 \leq i \leq N$ and look at $A_{N,i}$. Namely, let's replace $A_{N,i}$ with some new value, $a$, to give us a new matrix $\hat A$. I want to characterize the set $\lbrace a : \rho(\hat A) < 1 \rbrace$. It pretty clear that this set is of the form $[0, a_{max})$, but I want to be able to compute $a_{max}$ analytically, given $A$ and $i$. (Also clearly $a_{max} \geq A_{N,i}$, since $\rho(A) < 1$ by assumption.)</p>
<p>This seems like it should be a fairly easy exercise but I haven't been able to make any useful progress on it.</p>
<p>Thanks!</p>
<p>-h</p>