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curious little things - aggregated feedsenMath Overflow Recent Questions: Topological similarity of solutions to Dirichlet problem
https://mathoverflow.net/questions/288360/topological-similarity-of-solutions-to-dirichlet-problem
<p>Let $\varphi_{1},\varphi_{2}:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be two
smooth general position (Morse) functions having the same set of critical
points $\left\{ p_{1},...,p_{n}\right\} \subset\mathbb{S}^{1}$ ($n$ is even)
and both $\varphi_{1}$ and $\varphi_{2}$ have a local maximum at $p_{1}$.
Suppose that $\varphi_{1}$ and $\varphi_{2}$ are <em>similar</em> in the following sense:</p>
<p>$\left( \varphi_{1}(p_{i})-\varphi_{1}(p_{j})\right) \left( \varphi
_{2}(p_{i})-\varphi_{2}(p_{j})\right) >0$ for any $i\neq j$,</p>
<p>i.e. the critical level sets of $\varphi_{1}$ and $\varphi_{2}$ are in some
sense similar. </p>
<p>Consider the corresponding two Dirichlet problems:</p>
<p>$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta u=0$</p>
<p>$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u|_{\mathbb{S}^{1}}=\varphi_{i}$ , $i=1,2$,</p>
<p>getting in such a way two harmonic solutions $u_{1},u_{2}:\mathbb{B}%
^{2}\rightarrow\mathbb{R}$.</p>
<p>Then is it true that the level lines portraits of $u_{1}$ and $u_{2}$ are the
same up to topological equivalence, i.e. there is a homeomorphism
$h:\mathbb{B}^{2}\rightarrow\mathbb{B}^{2}$ fixing all $p_{i}$ and sending the
level lines of $u_{1}$ onto the level lines of $u_{2}$? Then, of course, $h$
is sending the critical set of $u_{1}$ onto the critical set of $u_{2}$.</p>
<p>In brief: does the similarity of the boundary conditions implies similarity
between the solutions of the corresponding Dirichlet problems?</p>
<p>Note that we don't assume $\varphi_{1}$ and $\varphi_{2}$ to be close in any sense.</p>Tue, 12 Dec 2017 10:11:38 -0700Math Overflow Recent Questions: Reference request: maximal ratio of different norms of polynomials
https://mathoverflow.net/questions/288358/reference-request-maximal-ratio-of-different-norms-of-polynomials
<p>Let us consider polynomials as functions on $[0,1]$, and so define
\begin{align*}
\|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\
\|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}.
\end{align*}
I am interested in the ratio of these norms. It is easy to see that $\|f\|_2\leq\|f\|_\infty$, with equality only for constant polynomials. In the opposite direction, put
$$ f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(d-i)!i!(i+1)!}(-x)^i. $$
Experiments make it clear that $\|f_d\|_2=1$ and $\|f_d\|_\infty=(d+1)$ and that $f_d$ maximises the ratio $\|f\|_\infty/\|f\|_2$ among polynomials of degree $d$. These facts must surely be known. Can anyone point me to a reference? Do the polynomials $f_d(x)$ have a standard name?</p>Tue, 12 Dec 2017 09:39:33 -0700Math Overflow Recent Questions: Number of binary arrays of length n with k consecutive 1's
https://mathoverflow.net/questions/288357/number-of-binary-arrays-of-length-n-with-k-consecutive-1s
<p>What is the number of binary arrays of length $n$ with at least $k$ consecutive $1$'s?
For example, for $n=4$ and $k=2$ we have $0011, 0110, 1100, 0111, 1110, 1111$ so the the number is $6$.</p>Tue, 12 Dec 2017 09:36:01 -0700Math Overflow Recent Questions: Proof of $\sum_{k=1}^n \binom{n}{k} \binom{n}{n+1-k} = \binom{2n}{n+1}$ via Induction [migrated]
https://mathoverflow.net/questions/288355/proof-of-sum-k-1n-binomnk-binomnn1-k-binom2nn1-via-indu
<p>I'm having trouble to prove the following formula using Induction on $n \in \mathbb{N}$:
$$\sum_{k=1}^n \binom{n}{k} \binom{n}{n+1-k} = \binom{2n}{n+1}.$$
I've tried all the usual identities, but they seem to lead nowhere. Is there any trick to this, or is it just not possible to prove this using induction?</p>
<p>I'm thankful for any tip or advice on how to approach this :)</p>Tue, 12 Dec 2017 09:11:30 -0700Math Overflow Recent Questions: Finding binary quadratic forms which parametrize a conic
https://mathoverflow.net/questions/288353/finding-binary-quadratic-forms-which-parametrize-a-conic
<p>For a pair of integers $(a,b)$, consider the conic in $\mathbb{P}^2$ given by</p>
<p>$$C_{a,b} : z^2 = ax^2 + by^2.$$</p>
<p>It is known that for most pairs $(a,b)$ the curve $C_{a,b}$ is not everywhere locally soluble, and thus, does not have a rational point. Given that $C_{a,b}$ has a rational point, it is then possible to find all rational points by simply considering all lines $L$ which go through a given rational point and then finding the other intersection point of $L$ with $C_{a,b}$. </p>
<p>Since $C_{a,b}$ has genus 0, it is possible to parametrize all rational points by quadratic forms, whenever a rational point exists. That is, there exist binary quadratic forms $f,g,h$ with integer coeffcients such that the map</p>
<p>$$\displaystyle (u,v) \mapsto (f(u,v), g(u,v), h(u,v))$$</p>
<p>parametrizes the points on $C_{a,b}$; that is, we have the equality</p>
<p>$$\displaystyle h(u,v)^2 = a f(u,v)^2 + b g(u,v)^2.$$</p>
<p>Let $(f,g,h)$ be an <em>admissible</em> triple of binary quadratic forms if the above holds. Let $\delta_1 = \min\{|\Delta(f)| : (f,g,h) \text{ admissible}\}$ and $\delta_2 = \min\{|\Delta(g)| : (f,g,h) \text{ admissible}\}$. Is there a known way to compute $\delta_1, \delta_2$ from $(a,b)$?</p>Tue, 12 Dec 2017 08:46:25 -0700Math Overflow Recent Questions: Clenshaw-Curtis integration without Fourier
https://mathoverflow.net/questions/288351/clenshaw-curtis-integration-without-fourier
<p>The <a href="https://en.wikipedia.org/wiki/Clenshaw%E2%80%93Curtis_quadrature" rel="nofollow noreferrer">Clenshaw-Curtis</a> quadrature rule approximates an integral $I=\int\limits_{-1}^{1} f(x) \, dx$ by $$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$
where the $x_j$'s are the roots of the $N$-th order Chebyshev polynomial, and and $w_j$'s their respective weight. To prove the accuracy of this integration formula, one usually goes by either Fourier representation of $f(x)=f(\cos (\theta))$, or by the "Fourier" expansion of $f$ in the Chebyshev polynomials. See e.g., in the Wiki page.</p>
<p><strong>My Question:</strong> Is there a way to prove the accuracy of this formula, which does not rely on spectral/Fourier theory? Specifically, to show that it is exact ($I=I_n$) for polynomials of degree $\leq n$, and to bound its error for $f\in C^n$.</p>Tue, 12 Dec 2017 08:30:17 -0700Math Overflow Recent Questions: An upper bound for skew symmetric rank 2 matrices
https://mathoverflow.net/questions/288349/an-upper-bound-for-skew-symmetric-rank-2-matrices
<p>Earlier, I had asked a similar question but that was not the correct problem where I got stuck. After a few quick answer, I realized that and I apologize for that. </p>
<p>Let $B_m$ be the space of all skew-symmetric matrices of size $m$ over the finite field $\mathbb{F}_q$ of $q$ elements. Let $E$ be a subspace of $B_m$ of dimension $r$ containing atleast one rank $2$ matrix. Write $E$ as $E= E_1 \bigoplus E_2$ with $
\dim E_i= r_i$ for $i=1,2$ and $E_1$ is a maximal subpace of $E$ containing only rank $2$ matrices. Now for a given rank $4$ matrix $Q\in E_2$ how many matrices $P\in E_1$ exist such that $P +Q$ is again of rank $2$. My Guess is $q^2$ and also that $q^2$ is a strict upper bound.</p>Tue, 12 Dec 2017 08:23:19 -0700Math Overflow Recent Questions: Is the following function a polynomial?
https://mathoverflow.net/questions/288346/is-the-following-function-a-polynomial
<p>I am reposting the second question from <a href="https://mathoverflow.net/questions/288290/variation-of-an-old-question-are-these-functions-polynomials">here</a> (after clarifying it) on the recommendation of user "GH from MO". </p>
<p>Let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. </p>
<p><strong>Question 2:</strong> Suppose I define $$G(x,y) = a_0(y) + a_1(y)(x-b_1) + a_2(y)(x-b_1)(x-b_2) + \dots$$ where the $a_k(y)$ are polynomials in $y$ and $g(x) = G(x,x)$.</p>
<p>Suppose the $a_k(y)$ are not identically zero for $k$ large enough. Otherwise, we clearly get polynomials. Is this the only way to get a polynomial? That is, if $g(x)$ is equal to a polynomial function, then is $a_k = 0$ for $k \gg 0$?</p>Tue, 12 Dec 2017 07:31:28 -0700Math Overflow Recent Questions: Retractions for completely positive unital maps, and their effect on spectral diameter
https://mathoverflow.net/questions/288345/retractions-for-completely-positive-unital-maps-and-their-effect-on-spectral-di
<p>Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$. Does there exist a choice of retraction $\Phi$ such that, for some $n > 0$ and some $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$, we have
$$ \mathrm{sdiam}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) > 2 \lVert E \rVert$$
where $\mathrm{sdiam}(X) = \lambda_{\max}(X) - \lambda_{\min}(X)$ is the spectral diameter? For instance: in the case $h = k$ (in which $\Phi = \Psi^{-1}$ would be unique), are there $\Psi$, $n>0$, and $E \in \mathbf M_h(\mathbb C) \otimes \mathbf M_n(\mathbb C)$ for which
$$\begin{align}
\lambda_{\max}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &> \lVert E \rVert, \tag{1}\\ \quad\text{and}\quad \lambda_{\min}\Bigl( \bigl(\Phi \otimes \mathrm{id}_n\bigr)(E) \Bigr) &< - \!\!\;\lVert E \rVert ? \tag{2}\end{align} $$</p>
<p>(This question is a follow-up to a <a href="https://mathoverflow.net/q/285178/3723">previous question</a>, in which it was established that there are maps $\Psi$ and operators $E$ for which every retraction $\Phi$ satisfies Eqn. (1) above.)</p>Tue, 12 Dec 2017 07:10:40 -0700Math Overflow Recent Questions: General existence theorem for cup products
https://mathoverflow.net/questions/288344/general-existence-theorem-for-cup-products
<p>I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on topological spaces, quasicoherent sheaf cohomology, things like etale cohomology where we look at sheaves on more general sites, etc.</p>
<p>In the sources I've looked at (e.g. Cassels-Fröhlich, Lang's <em>Topics in Cohomology of Groups</em>), the existence of cup products is proven in terms of cochains, Čech cohomology, etc., even when more abstract definitions and uniqueness theorems are given. </p>
<p>What I'm looking for is something like this: Let $\mathscr{A}$ is an abelian category with a symmetric monoidal structure $\otimes$ such that $(\mathscr{A}, \otimes)$ satisfies certain conditions (e.g. enough injectives, existence of $\mathrm{Hom}$-objects, whatever other features are common in practice) and $\{H^i\}$ is a universal $\delta$ functor from $\mathscr{A}$ to another abelian symmetric monoidal category $\mathscr{B}$ (assuming whatever we want for $\mathscr{B}$, even $\mathscr{B} = \mathbf{Ab}$, the category of abelian groups). If $\phi \colon H^0(M) \otimes H^0(N) \rightarrow H^0(M \otimes N)$ is an additive bi-functor, then there is a unique sequence of additive bi-functors $\Phi^{p,q} \colon H^p(M) \otimes H^q(N) \rightarrow H^{p+q}(M \otimes N)$ such that:</p>
<ul>
<li>$$\Phi^{0,0} = \phi$$</li>
<li>$\Phi$ is a "map of $\delta$-functors separately in $M$ and $N$": if \begin{equation}\tag{1} \label{s1} 0 \rightarrow{A'} \rightarrow A \rightarrow A'' \rightarrow 0
\end{equation}
is an exact sequence in $\mathscr{A}$ with \begin{equation}
\tag{2}\label{s2} 0 \rightarrow A' \otimes B \rightarrow A \otimes B \rightarrow A'' \otimes B \rightarrow 0
\end{equation}
still exact, then $\Phi^{p +1 ,q} \circ (\delta_1 \otimes H^0(\mathrm{id}_B)) = \delta_2 \circ \Phi^{p+1, q}$. Here, $\delta_1 \colon H^p(A'') \rightarrow H^{p+1}(A')$ and $\delta_2 \colon H^p(A'' \otimes B) \rightarrow H^{p+1}(A' \otimes B)$ are the maps provided by the $\delta$-functor structure on $H$ via the sequences (\ref{s1}), (\ref{s2}). Similarly, if we swap the roles of $A$ and $B$, we require that $\Phi^{p +1 ,q} \circ (\delta_1 \otimes H^0(\mathrm{id}_B)) = (-1)^{p} (\delta_2 \circ \Phi^{p+1, q})$. </li>
</ul>
<p>The answers to <a href="https://mathoverflow.net/questions/674/what-is-a-cup-product-in-group-cohomology-and-how-does-it-relate-to-other-branc">this question</a> shed some light on this matter: Suppose we are in a setting where $H^0(M) = \mathrm{Hom}(O, M)$ for some object $O$ of $\mathscr{A}$ (e.g. group cohomology where we can take $O = \mathbf{Z}$, sheaf cohomology where we can take $O = \mathscr{O}_X$, etc.) Then $H^p(M) = \mathrm{Ext}^p(O, M)$, so we should get a pairing $H^p(O) \otimes H^p(O) \rightarrow H^{p+q}(O)$ induced by the 'composition' mapping $\mathrm{Hom}(O, O) \otimes \mathrm{Hom}(O, O) \rightarrow \mathrm{Hom}(O,O)$. I'm not sure exactly how to prove this part in general either, but I've at least seen it discussed in terms of classes of extensions of modules (I'm not sure how generally the result that $\mathrm{Ext}$ describes extension classes holds). This also doesn't allow general group objects, and I'm not sure how to do the extension.</p>
<p>The above question also discusses a more homotopical/$\infty$-categorical way to think about cup products, but I'm not familiar enough in that language to really get what's going on: I'd much prefer an argument working in ordinary abelian categories. </p>Tue, 12 Dec 2017 06:57:19 -0700