Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2018-08-14T14:09:30Zhttps://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttps://mathoverflow.net/q/3082770Functoriality in the group $G$ of the domain of the Baum-Connes maphänselhttps://mathoverflow.net/users/888552018-08-14T14:08:41Z2018-08-14T14:08:41Z
<p>Lück claims in his preliminary book, that the left hand side of the Baum-Connes map is functorial in the group $G$. For the right hand side $K(A \rtimes G)$ this is clear for the full crossed product, as he himself points out.</p>
<p>How is this justified?</p>
https://mathoverflow.net/q/3082740Are $\varepsilon$-connected components dense?Taras Banakhhttps://mathoverflow.net/users/615362018-08-14T14:01:20Z2018-08-14T14:01:20Z
<p>Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of diameter $<\varepsilon$ in $X$ such that $x\in C_1$, $y\in C_n$ and $C_i\cap C_{i+1}\ne\emptyset$ for all $i<n$. It is clear that $\sim_\varepsilon$ is an equivalence relation on $X$. The equivalence class $[x]_\varepsilon:=\{y\in X:y\sim_\varepsilon x\}$ will be called the <em>$\varepsilon$-connected component</em> of $x$.</p>
<p><strong>Problem.</strong> What can be said about the $\varepsilon$-connected components of $X$? In particular, is each $\varepsilon$-connected component $[x]_\varepsilon$ dense in $X$? What is the Borel complexity of $\varepsilon$-connected components? Are they $\sigma$-compact? </p>
https://mathoverflow.net/q/308273-4this problem is related to algebra [on hold]user127736https://mathoverflow.net/users/1277362018-08-14T13:46:03Z2018-08-14T13:56:26Z
<p>sui mother buy some share of A on day 0. On day 7 share price of A is $\$44.6$.If she sell all share of A and buy 2000 shares of B on day 7 she would receive $\$7400$. On day 12 Share price of A is $\$4.8$ and B is $\$0.5$ less than on day 7. If she sell her all shares of A and buy 5000 shares of B on day 12 she would have to pay $\$5800$. Find share of A and share price of B on day 12</p>
https://mathoverflow.net/q/3082705Union of random intervals with total length equal to infinityMostafahttps://mathoverflow.net/users/516632018-08-14T13:18:11Z2018-08-14T14:03:04Z
<p>Let $a_1,a_2,\dots$ be a sequence of positive numbers less than 1, such that $$\sum_{n=1}^\infty a_i= \infty,$$ and $S^1 = \mathbb{R}/\mathbb{Z}$.</p>
<p>Suppose $I_1,I_2,\dots$ be random intervals with respective lengths $a_1,a_2, \dots$in $S^1$ such that the distribution of the centers of $I_n$ (for every $n$) are uniform and independent.</p>
<p>It can be shown that with probability 1, $I = \cup_{n=1}^\infty I_n$ is a full measure subset of $S^1$. Is it true that "With probability 1, $I_n = S^1$"? If this is not always true, does there exist a good characterization of the sequences $\{ a_n\}_{n=1}^{\infty}$ with this property?</p>
https://mathoverflow.net/q/3082672Closed Semi-Riemannian manifolds with non-compact isometry groupJS.https://mathoverflow.net/users/994682018-08-14T13:10:46Z2018-08-14T13:10:46Z
<p>Are there general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?</p>
<p><em>Background:</em> By a theorem of Myers and Steenrod the isometry group of a Riemannian manifold is a lie group. The same is true for Semi-Riemannian manifolds where the idea of a possible proof (which also works for Riemannian manifolds) is to embed the isometry group into the bundle of orthonormal frames as a closed submanifold. In the case of Riemannian manifolds this immediately yields that the isometry group of a compact Riemannian manifold has to be compact. But this is in general false for Semi-Riemannian manifolds since the bundle of orthonormal frames is not necessarily compact in this case even if the manifold is. So I ask myself if there are general statements about the phenomenon of a closed Semi-Riemannian manifold with non-compact isometry group. One statement that I know of is about Lorentzian manifolds and goes as follows:</p>
<p><strong>Theorem:</strong> Let $(M,g)$ be a closed, simply connected Lorentz manifold and assume $M$ and $g$ are analytic. Then the isometry group of $M$ is compact.</p>
https://mathoverflow.net/q/3082651Minimal covers in hypergraphs with finite edgesDominic van der Zypenhttps://mathoverflow.net/users/86282018-08-14T13:05:49Z2018-08-14T13:05:49Z
<p>Let $H=(V,E)$ be a <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraph</a>. We say that $C\subseteq E$ is a <em>cover</em> if $\bigcup C = E$. Let $H$ be a hypergraph with the following properties: </p>
<ol>
<li>$\bigcup E = V$,</li>
<li>all members of $E$ are finite, and</li>
<li>$d,e\in E$ with $d\subseteq e$ implies $d=e$.</li>
</ol>
<p><strong>Question.</strong> Does this imply that there is a minimal cover $C_0$ (that is, $C_0$ has the property that for all $c\in C_0$ the set $C_0\setminus \{c\}$ is no longer a cover)?</p>
https://mathoverflow.net/q/3082640Unclear inequality of L2 norms (Poisson equation for modeling flow)mueller_sebhttps://mathoverflow.net/users/1276892018-08-14T13:02:26Z2018-08-14T13:02:26Z
<p>I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your help would be greatly appreciated. I try to summarize the setting hereby:</p>
<p>Let there be a convex domain $\Omega \subset \mathbb{R}^2$ with two subdomains $\Omega_1, \Omega_2$ divided by the 1-dimensional path $\gamma$ (see picture).
<a href="https://i.stack.imgur.com/5evj9.png" rel="nofollow noreferrer">sketch of domain $\Omega$ with its subdomains</a></p>
<p>Given the Hilbert spaces
\begin{align}
L &= \{r = (r_1, r_2, r_{\tau}) \in L_2(\Omega_1) \times L_2(\Omega_2) \times L_2(\gamma)\}\\
W &= \{v = (v_1, v_2, v_f) \in H(div,\Omega_1) \times H(div,\Omega_2) \times H(div_\tau,\gamma): v_i \cdot n_i \in L_2(\gamma), i=1,2\}
\end{align}</p>
<p>we let $r = (r_1, r_2, r_{\tau}) \in M$ and $\varphi = (\varphi_1, \varphi_2, \varphi_\gamma) \in H^2(\Omega_1) \times H^2(\Omega_2) \times H^2(\gamma)$ be the solution of the PDE</p>
<p>\begin{align}
-\Delta{\varphi} = \tilde{r} \text{ on } \Omega\\
\varphi = 0 \text{ on } \Gamma
\end{align}
with $\tilde{r} \in L_2(\Omega)$ given by $\tilde{r}\vert_{\Omega_i} = r_i, \ i=1,2$ and
\begin{align}
-\Delta_\tau \varphi_\gamma = r_\tau \text{ on } \gamma\\
\varphi_\gamma = 0 \text{ on } \partial \gamma
\end{align}
We pose
\begin{align}
v_i &= -\nabla \varphi \vert_{\Omega_i}, \ i=1,2\\
v_\gamma &= -\nabla_\tau \varphi_\gamma
\end{align}
so $v=(v_1,v_2,v_\gamma) \in W$ and note that
\begin{align}
\text{div} (v_i) &= r_i, \ i=1,2\\
\text{div}_\tau (v_\gamma) &= r_\tau
\end{align}
$\Delta_\tau$ and $\nabla_\tau$ are the directional Laplacian and gradient along $\gamma$.
Note that $v_1 \cdot n_1 = -v_2 \cdot n_2 \in L_2(\gamma)$ because $v_i \in (H^1(\Omega))^2$.
Furthermore we define the following norms:
\begin{align}
||v||_{0,\Omega_i}^2 = \langle v,v \rangle_{0,\Omega} = \int_{\Omega} v \cdot v \ d\omega\\
||v||_{0,\gamma}^2 = \langle v,v \rangle_{0,\gamma} = \int_{\gamma} v \cdot v \ d \Gamma
\end{align}
with $u \cdot v$ is the dot product.
The proposed inequality is
\begin{multline}
||\tilde{r}||_{0,\Omega}^2 + ||r_\tau||_{0,\gamma}^2 + ||\nabla \varphi||_{0,\Omega}^2 + ||\nabla_\tau \varphi_\gamma||_{0,\gamma}^2 + 2||v_1 \cdot n_1 ||_{0,\gamma}^2\\
\leq (1+C(\Omega))||\tilde{r}||_{0,\Omega}^2+(1+C(\gamma))||r_\tau||_{0,\gamma}^2 + C(\Omega) ||\tilde{r}||_{0,\Omega}^2
\end{multline}
with constants $C(\Omega), C(\gamma) > 0$.</p>
<p>The authorial intention I guessed:</p>
<p>Friedrichs inequality leads to
\begin{align}
||\tilde{r}||_{0,\Omega}^2 + ||\nabla \varphi||_{0,\Omega}^2 \leq (1+C(\Omega))||\tilde{r}||_{0,\Omega}^2\\
||r_\tau||_{0,\gamma}^2 + ||\nabla_\tau \varphi_\gamma||_{0,\Omega}^2 \leq (1+C(\gamma))||r_\tau||_{0,\gamma}^2\\
\end{align}</p>
<p>so it's sufficient to proof
\begin{equation}
2 ||v_1 \cdot n_1||_{0,\gamma}^2 \leq C(\Omega) ||\tilde{r}||_{0,\Omega}^2
\end{equation}
The only way I see to get an inequality of the norms $||\cdot||_{0,\gamma}$ and $||\cdot||_{0,\Omega}$ is Gauss' divergence theorem. Apparently it holds true that:
\begin{equation}
\int_{\gamma} |v_1 \cdot n_1| d\Gamma \leq \sum_{i=1}^2 \int_{\Omega_i} |\text{div} (v_i)| d\omega = \int_{\Omega} |\tilde{r}| d\omega
\end{equation}
because $\gamma \subset \partial \Omega_i,\ i=1,2$.</p>
<p>Unfortunately this is an inequality of $L_1$ norms that does NOT imply the same inequality for $L_2$ norms $||\cdot||_{0,\gamma}$ and $||\cdot||_{0,\Omega}$.</p>
<p>Source (section 4.3, p. 12 [Existence and uniqueness of the solution]):</p>
<p>Jérôme Jaffré, Vincent Martin, Jean Roberts. Modeling Fractures and Barriers as Interfaces for Flow
in Porous Media. [Research Report] RR-4848, INRIA. 2003.
<a href="https://hal.inria.fr/inria-00071735/document" rel="nofollow noreferrer">https://hal.inria.fr/inria-00071735/document</a></p>
https://mathoverflow.net/q/3082601Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequenceNaren Manjunathhttps://mathoverflow.net/users/1259972018-08-14T12:14:51Z2018-08-14T12:14:51Z
<p>I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given a representative cocycle of $H^2(Q,N)$ corresponding to such an extension. I will use the example of $\mathbb Z_4$, which is a nontrivial central extension of $\mathbb Z_2$ by $\mathbb Z_2$. I have tried to explain my reasoning below. </p>
<p>So I start with the exact sequence</p>
<p>$0 \rightarrow \mathbb Z_2 \overset{i}{\rightarrow} \mathbb Z_4 \overset{p}{\rightarrow} \mathbb Z_2 \rightarrow 0$ </p>
<p>and compute the $E_2$-page using $E_2^{p,q} = H^p(\mathbb Z_2,H^q(\mathbb Z_2,\mathbb Z))$. In the case of the trivial central extension, where the action of every group on its respective coefficient module is trivial, we get a page which vanishes for $q$ odd and is a checkerboard of $\mathbb Z_2$'s and 0's, except for $E_2^{0,0} = \mathbb Z$, when $q$ is even. This sequence stabilizes on the $E_2$-page giving the results expected by using, say, the Kunneth formula. </p>
<p>However, this cannot be the right $E_2$-page for $\mathbb Z_4$, because the higher cohomology groups would then be too large. Therefore we must have a nontrivial action of $\mathbb Z_2$ on the coefficient modules $H^q(\mathbb Z_2,\mathbb Z)$. This is where I am stuck. I have two questions:</p>
<p>1) What is the above action, and is there a systematic way to see it for large $q$?</p>
<p>2)Is there a systematic way to compute differentials in the $E_2$-page given the maps $i,p$ and a representative cocycle for this extension?</p>
https://mathoverflow.net/q/3082585Condition on a differential form arising from the theory of elasticityRaz Kupfermanhttps://mathoverflow.net/users/987332018-08-14T11:49:17Z2018-08-14T12:54:15Z
<p>Let $D$ be the unit $n$-ball (for concreteness). Let $\beta\in\Omega^1(D;R^n)$ be an $R^n$-valued one-form, having full rank (viewed as a section of $T^*D\otimes R^n$). Under what conditions on $\beta$, does there exist a section $Q$ of $SO(n,R)$ (over $D$), such that $Q\circ\beta$ is closed (hence exact)?</p>
<p>The question is non-trivial for the following reason: if there exist such $Q$ and an $f:D\to R^n$, such that $df = Q\circ\beta$, then $\beta^T\circ\beta = df^T\circ df$, and the latter is (up to a musical isomorphism) a flat metric on $D$, whose Riemann curvature tensor vanishes. </p>
<p>So in a sense, I have an answer to my question. What I am looking for is a more explicit condition; in particular, I wonder whether there exists a condition that is linear in $\beta$.</p>
<p>For the curious, this question came up twice in two different contexts in the theory of elasticity.</p>
https://mathoverflow.net/q/3082570Differential operator of globally unbounded order on connected complex manifold?user2520938https://mathoverflow.net/users/643022018-08-14T11:42:08Z2018-08-14T11:42:08Z
<p>Let $X$ be a connected complex manifold. Consider the module $\mathcal{D}_X$. I recall hearing somewhere that one has to be careful with regards to differential operators that are not globally of finite order on $X$. I do not see how this could happen though. Intuitively I would say that constructing an operator of unbounded order would require some sort of bump function, which we do not have. An example demonstrating an operator of unbounded order would be appreciated.</p>
https://mathoverflow.net/q/3082530Classification of line bundles by second cohomology of a manifoldPraphulla Koushikhttps://mathoverflow.net/users/1186882018-08-14T11:27:14Z2018-08-14T13:49:04Z
<p>In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.</p>
<blockquote>
<p>$H^2(M,\mathbb{Z})$ is the group of isomorphism classes of line bundles over $M$.</p>
</blockquote>
<p>I guess they mean there is a natural isomorphism. </p>
<p>Can some one give a rough idea of what obvious second cohomology class we can think of given a line bundle over $M$ and what line bundle can we think of given an arbitrary second cohomology class. </p>
<p>Intuitive comments are also welcome.</p>
<p>I am familiar (not the proof details) with following result:</p>
<blockquote>
<p>If $G$ is a group and $M$ is a $G$-module, then the $H^2(G, A)$ is in one-one correspondence with the set of
equivalence classes of extensions $E$ of $M$ by $G$, in which the action of $G$ on $M$ induced by conjugation in $E$ is the same as the action defined by the $G$- module $M$.</p>
</blockquote>
<p>I am expecting some intuitive explanation that looks similar to this.</p>
https://mathoverflow.net/q/308252-2How to calculate this sum \lim_{n\rightarrow\infty}\sum_{k=0}^n \frac{1}{\sqrt{n^2 +k^3}} [on hold]user381793https://mathoverflow.net/users/1277312018-08-14T11:17:08Z2018-08-14T11:17:08Z
<p>How to calculate this sum
$$\lim_{n\rightarrow\infty}\sum_{k=0}^n \frac{1}{\sqrt{n^2 +k^3}}$$</p>
<p>I tried to prove this limit is converge but it failed</p>
https://mathoverflow.net/q/3082518Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?Pierre-Yves Gaillardhttps://mathoverflow.net/users/4612018-08-14T10:51:14Z2018-08-14T13:10:00Z
<p>I asked this question on Mathematics Stackexchange (<a href="https://math.stackexchange.com/q/2863312/660">link</a>), but got no answer.</p>
<p>Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$. </p>
<p>Recall that $A$ can be defined as the set of expressions of the form $\sum_ua_uu$, where $u$ runs over the set monomials in $x_1,x_2,\dots$, and each $a_u$ is in $K$, the addition and multiplication being the obvious ones. </p>
<p>Then $A$ is a local domain, its maximal ideal $\mathfrak m$ is defined by the condition $a_1=0$, and it seems natural to ask</p>
<blockquote>
<p>Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?</p>
</blockquote>
<p>I suspect that the answer is No, and that the series $\sum_{n\ge1}x_n^n$, which is clearly Cauchy, does <em>not</em> converge $\mathfrak m$-adically.</p>
https://mathoverflow.net/q/3082503Visualization of hidden structures in numbersHans Strickerhttps://mathoverflow.net/users/26722018-08-14T10:17:27Z2018-08-14T10:26:04Z
<p>In the general context of <strong><a href="https://www.youtube.com/watch?v=NdgQQfQLtWw" rel="nofollow noreferrer">Numbers and Geometry</a></strong> I was playing around with geometric visualizations of <strong>structures in the natural numbers</strong> and
came up with a type of function graphs I haven't seen before. I'd like to know if they have been investigated before, under which name and what can be learned from them (by seeing some geometric patterns and symmetries that were not visible otherwise).</p>
<p>I call these function graphs <em>line graphs</em>. They are defined for arbitrary functions $f:X\times X \rightarrow Y$ with $X,Y = \mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ and are created by drawing a line from each point in $X\times X$ to the points $(f(x,y),0)$ and $(0,f(x,y))$.</p>
<p>This is how the <a href="http://syspedia.de/line-graphs/" rel="nofollow noreferrer">line graph for $f(x,y) = xy$</a> looks like (in different resolutions):
<a href="https://i.stack.imgur.com/RbiCp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RbiCp.png" alt="enter image description here"></a>
<br>
<a href="https://i.stack.imgur.com/mvwR4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mvwR4.png" alt="enter image description here"></a></p>
<p>We see a maybe astonishing pattern emerge: a square grid (that might be typical for multiplication-like functions). (Note that the prime numbers are exactly those nodes on the x- resp. y-axis with degree 2.)</p>
<p>Other line graphs look quite different, of course:</p>
<p><a href="https://i.stack.imgur.com/cBbqh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cBbqh.png" alt="enter image description here"></a></p>
<p>(Can you guess at a glance which function's line graph this is?)</p>
<p>Which properties of a function can be read off the geometrical patterns (symmetries) of its line graph - and how? The other way around: Which geometrical patterns (symmetries) can be predicted just by looking at the definition of a function $f$? (First of all: The pattern is symmetric when the function is symmetric.)</p>
<p><em>[If you like to play around with line graphs you can do it <a href="http://syspedia.de/line-graphs/" rel="nofollow noreferrer">here</a>.]</em></p>
<hr>
<p>Related question: By construction of line graphs, every function $f$ has a "reverse" function $f^*$ associated to it, which is defined by the point at which the line going through $(x,y)$ and $(f(x,y),0)$ crosses the y-axis, which is by definition at $(0,f^*(x,y))$. </p>
<p>We have $f^*(x,y) = \frac{f(x,y)\times y}{f(x,y) - x}$ and $f(x,y) = \frac{f^*(x,y)\times x}{f^*(x,y) - y}$</p>
<p>For $f(x,y) = x y$, we have $f^*(x,y) = y^2/(y-1)$ (which by the way does not depend on $x$):</p>
<p><a href="https://i.stack.imgur.com/mPZ23.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mPZ23.png" alt="enter image description here"></a></p>
<p>Has this construction of an associated function $f^*$ been investigated before? Might it be interesing to investigate the relationship between $f$ and $f^*$? </p>
https://mathoverflow.net/q/3082496Partial product of Euler factorsCorbennickhttps://mathoverflow.net/users/16882018-08-14T09:51:24Z2018-08-14T09:51:24Z
<p>Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let
$$
\zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}},
$$
where $\mathrm{Re}(s)>1$.
Is there any $T$ such that $T$ and $T^c={\mathbb P}\smallsetminus T$ are both infinite and $\zeta_T$ has a meromorphic continuation to $\mathbb C$?</p>
https://mathoverflow.net/q/3082402Can we specify the value of harmonic forms at a point?Asaf Shacharhttps://mathoverflow.net/users/462902018-08-14T06:56:29Z2018-08-14T12:23:34Z
<p>Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. </p>
<p>Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$.</p>
<blockquote>
<p>Does there exist an open neighbourhood $U$ of $p$, and a closed and co-closed $k$-form $\omega \in \Omega^k(U)$ satisfying $\omega_p=\alpha_p$?</p>
</blockquote>
<hr>
<p>This question is equivalent to the following question:</p>
<blockquote>
<p>Do closed and co-closed frames for $\bigwedge^k(T^*M)$ always exist locally?</p>
</blockquote>
<p>Indeed, if we can specify the value of a form in a point, we can take a basis for $\bigwedge^k(T_pM)^*$, and so obtain forms which form a frame at $p$. Since "being a frame" is an open condition, we have a local frame. On the other hand, suppose that local closed and co-closed frames exist. Then, by choosing a linear combination with constant coefficients of that frame, we can realize any given value at $p$. </p>
<p><em>Comment:</em> In general we cannot expect such a frame to be induced from coordinates. Indeed, when we specialize to even dimension $d$, and $k=\frac{d}{2}$, then, for a generic metric $g$, <a href="https://mathoverflow.net/a/301584/46290">there are no coordinate systems where even one wedge $\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_n}
$ is co-closed</a>. </p>
https://mathoverflow.net/q/3082186On the Number of Parallel Automorphism LinesMorteza Azadhttps://mathoverflow.net/users/828432018-08-13T22:10:45Z2018-08-14T10:06:46Z
<p>Given a group $G$, one can define the transfinite <em>line</em> of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct limit is taken at the limit stages: </p>
<p>$G\rightarrow Aut(G)\rightarrow Aut(Aut(G))\rightarrow\cdots\rightarrow G_{\alpha}\rightarrow G_{\alpha+1}\rightarrow\cdots$</p>
<p>The line <em>terminates</em> when a fixed point is reached, namely one of the groups in the chain is isomorphic to its automorphism group by the natural map. According to <a href="http://www.ams.org/journals/proc/1998-126-11/S0002-9939-98-04797-2/home.html" rel="nofollow noreferrer">a result of Hamkins</a> it is known that every automorphism line terminates. So there is no automorphism line of length $\text{Ord}$. </p>
<p><strong>Definition.</strong> It is clear that the automorphism line of many non-isomorphic groups may intersect each other and so have the same terminating point. In this case, we say that two automorphism lines have <strong>converged</strong>. Otherwise, we call them <strong>parallel</strong>. Precisely, the automorphism lines of $G$ and $H$ are convergent if there are ordinals $\alpha, \beta$ such that $G_{\alpha}\cong H_{\beta}$. </p>
<blockquote>
<p><strong>Question.</strong> How many distinct parallel automorphism lines of the groups of the same cardinality do exist? Can this number vary in different forcing extensions?</p>
<p>Precisely, define the equivalence relation $\sim$ on (isomorphism type of) the groups so that $G\sim H$ if the automorphism lines of $G$ and $H$ converge. Let $\kappa$ be a (finite/infinite) cardinal and $\mathcal{C}_{\kappa}$ be the collection of all groups of size $\kappa$. What is the size of $\mathcal{C}_{\kappa}/\sim$ for different $\kappa$?</p>
</blockquote>
<p><strong>Remark.</strong> If an answer to the above question is in hand, the fact that every automorphism line eventually terminates actually gives us the number of groups which can arise as the <em>terminating point</em> of the automorphism line of a group of size $\kappa$ because two lines are parallel if and only if they have distinct terminating points. </p>
https://mathoverflow.net/q/3082177Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonicsJie Panhttps://mathoverflow.net/users/1277012018-08-13T22:09:10Z2018-08-14T10:41:20Z
<p>Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of squares of $d$ dimensional Fourier harmonics up to degree $n$. </p>
<p>My question is if $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$ and can be represented as combination of spherical harmonics dimension $d$, then does there exist some spherical harmonic polynomials $g_1,\ldots, g_k$ of degree $n$ such that $p=g_1^2+\cdots g_k^2$ is a sum of squares?</p>
<p>i edit the question after Zach Teitler's comment.</p>
<p>The interval $[-\pi,\pi]^d$ means we consern the trigonometric polynomials positive on frequency domains.</p>
<p>The optimization problems about the polynomials positive on frequency domain $[-\pi,\pi]^d$ can be implemented via SDP approach(Gram matrix Rpresentation).</p>
<p>Given a positive polynomial represented as combination of spherical harmonics dimension $d$, it is obviously that it is sum of squares of $d$ dimensional Fourier harmonics. Furthermore, it implies that symmetry relationship between $[-\pi,\pi] \times [0,\pi]$ and $[-\pi,\pi] \times [-\pi,0]$ on 2-sphere as an example. May be there is less information on sphere than cube?</p>
<p>So, is it the sum of squares of spherical harmonics? </p>
<p>Please feel free to provide any advices. Any comments and references (in English) will also be very welcome !</p>
<p>Thank you very much in advance!</p>
https://mathoverflow.net/q/3082112Generating Irreducible representations of a simple lie algebra with Schur functorsSaal Hardalihttps://mathoverflow.net/users/228102018-08-13T20:29:18Z2018-08-14T13:41:51Z
<p>Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $Rep_V(\mathfrak{g}) \subset Rep(\mathfrak{g})$ to be the smallest symmetric monoidal, idempotent complete, abelian subcategory with duals (so, closed under tensor products, retracts, direct sums and duals) which contains $V$.</p>
<blockquote>
<p><strong>Question:</strong> Does there always exist an irreducible $V$ for which $Rep_V(\mathfrak{g}) \cong Rep(\mathfrak{g})$? When it exists, is there
a unique minimal one (in terms of the order on the weights) such $V$ (up to dualizing)? If not is there a unique self-dual such representation? If it doesn't exist, what is the minimal dimensional $V$
(possibly reducible) which satisfies this condition? Is it unique in some sense?</p>
</blockquote>
<p>I'm interested in the question for all $\mathfrak{g}$ of type $A,B,C$ and $D$ (the exceptionals are a luxury). I think the standard representation $V$ in the case of type $A$ generates the entire category in this sense so that the answer is positive for this case but i'm not sure about any of the other cases.</p>
https://mathoverflow.net/q/3081949Mathematical Writing: Proof Outlines/Overview in a PaperSam Thttps://mathoverflow.net/users/592642018-08-13T17:12:25Z2018-08-14T13:38:01Z
<p>While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific.</p>
<p>I am writing a paper, in which we have a section called "Outline of Proof". (It's Section 2.)
The outline is fairly informal, and we omit some technical details, making approximations.
However, among these approximations, my co-author wants to state (and label) <em>important</em> definitions and results (lemmas, equations, etc). He then wants to, later in the paper when we are doing the corresponding part carefully and rigorously, refer back to these (say, "by equation (2.4)", or "by Lemma 2.2"). Moreover, he is very against redundancies, so does not like things being stated twice precisely (including in the outline) -- once precisely in the text and once approximately in the outline is fine.</p>
<p>To me, this seems insane. (Of course, I did not use such a phrase when speaking with him!) When I read a paper, I never carefully read the outline:
I just read it, and try to get an overview (or 'outline') of the proof;
if there are parts that I don't really understand, I don't get hung up on them, trusting that with the more rigorous explanation later I'll be able to make sense of what the authors are saying.</p>
<p><em>However</em>, I'm a pretty junior author -- 2nd year of PhD -- while he is a first year postdoc. That doesn't mean that I haven't read a reasonable number of papers (and in fact my lack of experience and knowledge means that I am less able to understand poorly written papers); moreover, he has said that he feels writing papers well isn't his best attribute.</p>
<p>So my question is this:</p>
<blockquote>
<p>(a) is it standard to read an outline of a proof carefully?</p>
<p>(b) is it standard (or at least not discouraged) to state precisely important, even key, results/definitions that will be referred back to in the main body of the paper when giving proofs?</p>
</blockquote>
<hr>
<p>Just as some extra comments... I'm not here to try to get people to tell me that I'm right and my co-author is wrong and/or being silly! I know that <em>sometimes</em> some people come to Stack Exchange for such comments (see, particular, Workplace/Interpersonal Skills SEs!). I should have perhaps made the following clear: <em>if everything my co-author does is standard in the field, and I'm in the wrong, I definitely want to know that and will accept it!</em> -- I'm here to learn :-) please have no qualms about hearing criticism! (assume that it's constructive, of course)</p>
https://mathoverflow.net/q/3081311references on group representation over local fields / a question on an argument of a Ralph Greenberg's papergualteriohttps://mathoverflow.net/users/1232262018-08-13T05:01:47Z2018-08-14T14:01:57Z
<p>I'm currently studying Iwasawa theory.</p>
<p>1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act.
So I often face some facts on the group representation over local fields or p-adic integer ring. But I can't find any references yet.</p>
<p>Of course, there are articles on p-adic representation. But I want references that are not too deep.
I want references using just easy-to-follow arguments of algebra and representation theory. </p>
<p>Can you suggest any references?</p>
<p>2) Currently, I'm reading the paper "On the Iwasawa Invariants of Totally Real Number Fields" written by Ralph Greenberg. There I cannot understand a line which I have underlined with red line.</p>
<p><a href="https://i.stack.imgur.com/YlGLI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YlGLI.png" alt="enter image description here"></a></p>
<p>I'm afraid that there are many counter-examples against the line.(For example we can take cyclic group of order prime to the order of the group of units.) Can you please explain the line to me?</p>
https://mathoverflow.net/q/3078805A paradox on the deformation of singularitiesLi Yutonghttps://mathoverflow.net/users/297302018-08-09T02:33:59Z2018-08-14T11:05:21Z
<p>Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X_o$ has a certain class of singularities, one wants to study if $\mathcal X$ preserves such singularities.</p>
<p>It has been shown (see "<a href="https://doi.org/10.1090/S0894-0347-99-00285-4" rel="nofollow noreferrer">Deformations of canonical singularities</a>") that the above property holds for canonical singularities; but it fails for klt singularities (see "<a href="https://arxiv.org/abs/math/9809091v1" rel="nofollow noreferrer">On the extension problem of pluricanonical forms</a>," Example 4.3). </p>
<p>However, I found an one-line argument for both cases, could anyone point out where I was wrong?</p>
<p>Here is the argument: First, it is known that $\mathcal X$ is $\mathbb Q$-Gorenstein. Because $\{o\}$ is a divisor on $C$, $\mathcal X_o$ can also be viewed as the pull-back Cartier divisor $\pi^*{o}$, hence it is Cartier, and by adjunction (assuming $\mathcal X$ is CM)
$$(K_{\mathcal X}+\mathcal X_o)|{_{\mathcal X_o}} = K_{\mathcal X_o}.$$ Then by the precise inversion of adjunction,
$${\rm total~discrepancy}\{\mathcal X_o\} = {\rm total~discrepancy}\{(\mathcal X, \mathcal X_o){\rm~with~center~intersects~} \mathcal X_o\}.$$ Hence the minimal discrepancy of $(\mathcal X, \mathcal X_o)$ near $\mathcal X_o$ is $\geq 0$ in the canonical case and $>-1$ in the klt case. In particular, $\mathcal X$ is canonical and klt respectively.</p>
https://mathoverflow.net/q/3073562Reference request for weak solutions of an Elliptic PDERajesh Dachirajuhttps://mathoverflow.net/users/144142018-08-02T02:19:10Z2018-08-14T13:16:44Z
<blockquote>
<p><strong>Edit</strong> : I just learned that all weak solutions are $C^\infty$, so <a href="https://mathoverflow.net/q/307363/14414">this question</a>, by Willie, seems more appropriate than the current one.</p>
</blockquote>
<p>I want to find weak, non trivial, <strong>continuous</strong>, solutions of $$\Delta u - \lambda u = 0$$ for a square domain in $\mathbb{R}^N$, $N \ge 2$, under periodic boundary conditions, and under an added constraint that, the weak solutions $u$ should take given values, at a given finite set of points in the interior of the domain. $u(x_i) = d_i$, $x_i$ lie in the interior of the domain, and $d_i$ are reals.</p>
<p>Reference request, if someone already solved it, or partially solved it or any relevant work. I am trying to solve it and I want to know if it makes sense, and I am not re-inventing, or barking up the wrong tree.</p>
<p>PS : Solving, I mean, having a numerical solution that converges pointwise, to the actual solution.</p>
https://mathoverflow.net/q/3066730root of identity matrix and lexicographic orderjcdornanohttps://mathoverflow.net/users/1123822018-07-23T17:12:43Z2018-08-14T13:26:53Z
<p>I asked a question here <a href="https://mathoverflow.net/questions/306572/order-of-a-permutation-and-lexicographic-order">order of a permutation and lexicographic order</a> but it seems*** that a very powerful and rich generalization can be made!</p>
<p>Let $A$ be a finite ring together with an arbitrary <strong>total order</strong> $<^*$ and let $L: M_n(A)\to M_n(A)$ be the application that sorts the rows of a matrix according to the (increasing) lexicographic order (induced by $<^*$) and the columns of the matrice we get, according to the increasing lexicographic order (increasingly too). We then define $L_Q(M):=L(M).Q$ for any $Q\in M_n(A)$.</p>
<blockquote>
<p>Suppose that $Q^q=Id$ for some $q\in \mathbb N$ and $Q\in GL_n(A)$. Is is true that for any $M\in M_n(A)$, there exists $r\in \mathbb N$ such that for any $i\in \mathbb N$, we have $L_Q^r(M)=L_Q^{r+iq}(M)$ ?</p>
</blockquote>
<p>[edit : it is not true for $A=\mathbb Z$, but it seems asymptotically true up to a scalar multiplication of matrices, anyway I edited and ask $A$ to be finite]</p>
https://mathoverflow.net/q/3064315Model category of diagrams with the colimit detecting the weak equivalencesPhilippe Gaucherhttps://mathoverflow.net/users/245632018-07-20T07:37:40Z2018-08-14T12:53:43Z
<p>Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category. </p>
<blockquote>
<p>Is it known a model category structure on the functor category
$\mathcal{K}^I$ such that a map of diagrams $D\to E$ is a weak
equivalence if and only if $\mathrm{colim} D \to \mathrm{colim} E$ is
a weak equivalence of $\mathcal{K}$ ? (so not the objectwise weak
equivalences, and by $\mathrm{colim}$, I mean the colimit)</p>
</blockquote>
<p>I have no trace of a thing like that in the nLab or in the MathReview. I don't know what keyword to use in fact. Since there is no reason for a colimit of and objectwise weak equivalence to be a weak equivalence, it is not possible to see it as a localization of the projective or the injective model category structure.</p>
https://mathoverflow.net/q/3025474Exit time of a stochastic process defined by a SDEgerdhttps://mathoverflow.net/users/885352018-06-11T17:09:35Z2018-08-14T09:36:20Z
<p>Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\mathrm{d} x_t & = v_t \mathrm{d}t \\
m\mathrm{d} v_t& = -\nabla \Psi(x_t) \mathrm{d}t -\gamma v_t \mathrm{d}t + \sigma \mathrm{d} W_t
\end{align*}
where $x_t$ and $v_t$ are the position and velocity vectors, $\Psi(x)$ is a bell-shaped potential with a single minimum at $\Psi(0)$ and $W_t$ is a vector Wiener process. Now define the energy of the particle as
\begin{align}\label{eq:energy}
h(x_t, v_t) = \Psi(x_t) + \frac{1}{2}mv_t^2.
\end{align}
I am interested in calculating the time it takes for $\mathcal{P}$ to reach a certain energy $r$, i.e., the stopping time
$$
\tau_h = \inf\{ t \geq 0: h(x_t, v_t) = r \}.
$$
Calculating $\mathbb{E}(\tau_h)$ would be enough for now. So far I have tried calculating
\begin{align}\label{eq:en_evolution}
h(x_t, v_t) = h(x_0, v_0) + \int_0^t \mathrm{d} h(x_s, v_s),
\end{align}
and then, by taking expected values, I hoped I'd be able to solve by $\mathbb{E}(\tau_h)$. With the Wiener process I got the right and well-known result of the exit time from a ball of radius $r$ (this is, in fact, a naive way of using Dynkin's formula), but with the process I described before I didn't get too far. Assuming the initial conditions are distributed according to the invariant (Gibbs) measure, one gets, of course, that
$$
\mathbb{E}[h(x_t, v_t)] = \mathbb{E}[\Psi(x_t)] + \frac{1}{2}k_B T,
$$
which is true, and reassuring, but not very useful, since we lost $\mathbb{E}(\tau_h)$.</p>
<p>I'm not especially used to calculations like this, so any suggestions of "tricks" and/or common techniques for calculating hitting times would be appreciated. I would also be grateful for any references where similar exit time calculations are performed (for non trivial processes).</p>
https://mathoverflow.net/q/3002754Asymptotic for number of partitions of $n$ into $k$ squares, uniform in $n,k \to +\infty$Meganhttps://mathoverflow.net/users/1244852018-05-15T15:43:15Z2018-08-14T09:00:36Z
<p>Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since
$$9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2$$
$$9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2\phantom{1^2 +\;\,+ 1^2 + 1^2}$$
$$9 = 1^2 + 2^2 + 2^2\phantom{ + 1^2 + 1^2 + 1^2 + 2^2\;\,+ 1^2 + 1^2}$$
$$9 = 3^2 \phantom{+ 2^2 + 2^2 + 1^2 + 1^2 + 1^2 + 2^2\;\,+ 1^2 + 1^2}$$
and there are no other ways of writing $9$ as sum of squares.</p>
<p>It is known that
$$\log p^{(s)}(n) \sim (s+1)\left(\frac1{2}\Gamma\!\left(1+\frac1{s}\right)\zeta\!\left(1+\frac1{s}\right)\right)^{s/(s+1)} n^{1/(s+1)},$$
as $n \to +\infty$ (See <em>Hardy and Littlewood, Asymptotic formulæ in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVII, 1918, 75-115</em>).</p>
<p>My question is: If $p_k^{(s)}(n)$ is the number of ways of writing the positive integer $n$ as a sum of <em>exactly $k$</em> perfect $s$-powers, is there an asymptotic formula for $\log p_k^{(s)}(n)$ holding in a reasonable range of $n,k \to \infty$? I am particularly interested in the case of squares $s = 2$. </p>
<p>Thank you in advance for any suggestion.</p>
https://mathoverflow.net/q/1268527$a^5+b^5=c^5+d^5$ and polynomial identitiesjorohttps://mathoverflow.net/users/124812013-04-08T13:24:08Z2018-08-14T09:36:40Z
<p>No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.</p>
<p>(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from a genus 0 curve and the identity
$$ (-2 y - 1)^5 + (x + \frac{1}{3} y - 1)^5- (\frac{1}{3} y - 1)^5 - (x - 2 y - 1)^5 = $$
$$ \left(-\frac{35}{81}\right) \cdot y \cdot (-3 x + 5 y + 6) \cdot x \cdot (9 x^{2} - 15 x y + 37 y^{2} - 18 x + 30 y + 18)$$</p>
<p>I wonder if some similar genus 0 or 1 curve might have points over $\mathbb{Q}$.</p>
<p>Let $p_1,p_2,p_3,p_4 \in \mathbb{Q}[x,y]$, where $\deg(p_i)=1$ and the $p_i^2$
are distinct.</p>
<p>Let $P=p_1^5+p_2^5-p_3^5-p_4^5$.</p>
<p>Q1. Can $P$ have an irreducible factor of degree 3?</p>
<p>Q2. Is there a reason for all degree 2 factors of $P$ to not have infinitely many
rational points over $\mathbb{Q}$?</p>
<p>I couldn't solve Q1 by equating coefficients (couldn't solve the system). </p>
<p>Found a lot of genus 0 factors, but all of them didn't have rational points
and linear factors gave only trivial solutions.</p>
https://mathoverflow.net/q/453314Examples of Riemannian SubmersionsHenry Wegenerhttps://mathoverflow.net/users/97622010-11-08T17:48:11Z2018-08-14T12:40:32Z
<p>Is there any example of a Riemannian submersion, which is no fibration?</p>
<p>As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. Nevertheless, I could not think of or find an example.</p>
https://mathoverflow.net/q/1720263Sum of 'the first k' binomial coefficients for fixed nmathyhttps://mathoverflow.net/users/44052010-03-05T19:16:01Z2018-08-14T09:25:41Z
<p>I am interested in the function $\sum_{i=0}^{k} {N \choose i}$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other notable properties? Any literature references?</p>
<p>In particular, does it have a closed form or notable algorithm for computing it efficiently?</p>
<p>In case you are curious, this function comes up in information theory as the number of bit-strings of length $N$ with Hamming weight less than or equal to $k$.</p>
<p>Edit: I've come across a useful upper bound: $(N+1)^{\underline{k}}$ where the underlined $k$ denotes falling factorial. Combinatorially, this means listing the bits of $N$ which are set (in an arbitrary order) and tacking on a 'done' symbol at the end. Any better bounds?</p>