Recent MathOverflow Questions

The universal property of the unseparated derived category

Math Overflow Recent Questions - Fri, 10/06/2017 - 08:28

In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a Grothendieck abelian category. The unseparated derived category $\check{{\cal D}}({\cal A})$ is a stable presentable $\infty$-category equipped with a natural t-structure which is compatible with filtered colimits and whose heart is identified with ${\cal A}$. It is related to the usual derived $\infty$-category ${\cal D}({\cal A})$ by a t-exact functor $\check{{\cal D}}({\cal A}) \to {\cal D}({\cal A})$ which exhibits ${\cal D}({\cal A})$ as the "left separation" of $\check{{\cal D}}({\cal A})$. In addition, $\check{{\cal D}}({\cal A})$ itself enjoys the following universal property (see Theorem C.5.8.8 and Corollary C.5.8.9 of loc.cit): if ${\cal C}$ is any other stable presentable $\infty$-category equipped with a $t$-structure which is compatible with filtered colimits, then restriction along the inclusion of the heart $A \to \check{{\cal D}}({\cal A})$ induces an equivalence $$ {\rm LFun^{{\rm t-ex}}}(\check{{\cal D}}({\cal A}),{\cal C}) \stackrel{\simeq}{\to} {\rm LFun^{{\rm ex}}}({\cal A},{\cal C}^{\heartsuit}) $$ where on the left hand side we have colimit preserving t-exact functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ and on the right hand side we have colimit preserving exact functors ${\cal A} \to {\cal C}^{\heartsuit}$. This is indeed a very satisfying universal characterization of $\check{{\cal D}}({\cal A})$ together with its t-structure. However, for various reasons it can be useful to have a universal characterization of $\check{{\cal D}}({\cal A})$ without the t-structure. To see how this might go, note that t-exact colimit preserving functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ send short exact sequences in ${\cal A}$ to cofiber sequences in ${\cal C}$ and filtered colimits in ${\cal A}$ to filtered colimits in ${\cal C}$. One may hence consider the possibility that restriction along ${\cal A} \to \check{{\cal D}}({\cal A})$ induces an equivalences between all colimit preserving functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ on the one hand and and all functors ${\cal A} \to {\cal C}$ which preserve filtered colimits and send short exact sequences in ${\cal A}$ to cofiber sequences in ${\cal C}$ on the other.

Question 1: Is this true?

A positive answer to Question 1 would imply that $\check{{\cal D}}({\cal A})$ admits the following explicit description: we would be able to identify it with the $\infty$-category of presheaves of spectra ${\cal A}^{{\rm op}} \to {\rm Sp}$ which send filtered colimits in ${\cal A}$ to filtered limits and send short exact sequences in ${\cal A}$ to fiber sequences of spectra.

As a variant to Question 1, one may hope that the connective part $\check{{\cal D}}({\cal A})_{\geq 0}$ enjoys the same universal characterization when ${\cal C}$ is now replaced with a Grothendieck prestable $\infty$-category, or maybe even any presentable $\infty$-category. Such a characterization would imply that $\check{{\cal D}}({\cal A})_{\geq 0}$ can be identified with the $\infty$-category of presheaces of spaces ${\cal A}^{op} \to {\cal S}$ which send filtered colimits to cofiltered limits and short exact sequences to fiber sequences of spaces.

Question 2: Is this true?

Remarks: 1) A positive answer to Question 2 would imply a positive answer to Question 1 since $\check{\cal D}({\cal A}) \simeq {\rm Sp}(\check{\cal D}({\cal A})_{\geq 0})$ is the stabilization of $\check{\cal D}({\cal A})_{\geq 0}$.

2) If ${\cal A} = {\rm Ind}(A_0)$ with $A_0$ an abelian category with enough projectives in which every object has finite projective dimension then $\check{\cal D}({\cal A}) \simeq {\cal D}({\cal A})$ (Proposition C.5.8.12 in loc.cit) and can also be described using complexes of projective objects. In this case $\check{\cal D}({\cal A})_{\geq 0} \simeq {\cal P}_{\Sigma}((A_0)_{{\rm proj}})$ is the $\infty$-category obtained from $(A_0)_{{\rm proj}}$ by freely adding sifted colimits. In particular, $\check{\cal D}({\cal A})_{\geq 0}$ admits a universal characterization as a presentable $\infty$-category and $\check{\cal D}({\cal A}) \simeq {\rm Sp}(\check{\cal D}({\cal A})_{\geq 0})$ admits a universal characterizations as a stable presentable $\infty$-category (without the t-structure). However, even in this case, I don't know how to deduce from this particular universal characterization the other universal characterization I'm interested in (the missing part is to show that a coproduct preserving functors $(A_0)_{{\rm proj}} \to {\cal C}$ extends in an essentially unique way to a functor ${\cal A}_0 \to {\cal C}$ which sends short exact sequence to cofiber sequences).

Universal covering of a 2-sphere without $n$ points

Math Overflow Recent Questions - Fri, 10/06/2017 - 07:18

Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic space.

Q. How one can describe the group of deck transformations of the universal covering as a subgroup of $\mathrm{PGL}(2,\mathbb{R})$?

I expect this should be a standard material. A reference would be helpful.

Action on a normal subgroup where each coset acts freely

Math Overflow Recent Questions - Thu, 10/05/2017 - 22:13

Given a group G and a normal subgroup N of G, is there an action of G on N such that, whenever g,h are distinct members of the same N-coset, we have g•n≠h•n? If not, then can this be done in the case G is abelian?

Take for granted that we may select a set of coset representatives for the N-cosets to use as "origins" for each N-coset. I'm working on some abstract analysis/descriptive set theory ideas, and got stuck on this thought because the algebra got a little too far out of my element. If it can't be done, a counterexample would be GREATLY appreciated.

Okay. Gotta update this question: In this setting, the groups are all either countably infinite or of size continuum.

Also it's important to actually be able to describe the action.

Are polynomials of operators injective? [on hold]

Math Overflow Recent Questions - Thu, 10/05/2017 - 15:06

I am simply asking if it is possible to have $S^k=T^k$ for $S\neq T$, where $S,T$ are operators on analytic functions and $k$ is positive integer.

If a polynomial ring is finite free over a subring, is the subring polynomial?

Math Overflow Recent Questions - Thu, 10/05/2017 - 12:11

Let $R = k[x_1, \ldots, x_n]$ for $k$ a field of characteristic zero and let $S \subset R$ be a graded sub-$k$-algebra (for the standard grading: $\deg x_i = 1$) such that $R$ is a free $S$-module of finite rank. Does this imply $S \cong k[y_1,\ldots,y_n]$?

Examples of representations of quantum groups

Math Overflow Recent Questions - Thu, 10/05/2017 - 11:54

I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional representation theory of them is well understood.

I would be interested to see examples of representations of $U_q(\mathfrak{g})$ which come not from the problem of classification of representations, but rather are either "natural" or typical from the point of view of quantum groups, or appear in applications or situations unrelated to the classification problem.

To illustrate what I mean let me give examples of each kind of represenations of Lie algebras since this situation is more familiar to me.

1) Let $\mathfrak{g}$ be a classical complex Lie algebra such as $sl_n,so(n), sp(2n)$. Then one has the standard representation of it, its dual, and tensor products of arbitrary tensor powers of them.

2) Example of very different nature comes from complex geometry. The complex Lie algebra $sl_2$ acts on the cohomology of any compact Kahler manifold (hard Lefschetz theorem). Analogously $so(5)$ acts on the cohomology of any compact hyperKahler manifold (this was shown by M. Verbitsky).

The Hierarchy of “collections” for Categories

Math Overflow Recent Questions - Thu, 10/05/2017 - 09:23

When we deal with usual category $\mathbf A$ we require that Ob$(\mathbf A)$ (i.e., collection of objects of category $\mathbf A$) and Mor$(\mathbf A)$ (i.e., collection of morphisms between objects in category $\mathbf A$) form a class (proper or no).

Then if we deal with metacategory $\mathbf A$ we say that Ob$(\mathbf A)$ is a conglomerate and Mor$(\mathbf A)$ so is.

To form a conglomerate we require:

  1. every class is a conglomerate,
  2. for every “property” P, one can form the conglomerate of all classes with property P,
  3. conglomerates are closed under analogues of the usual set-theoretic constructions, and
  4. the Axiom of Choice for Conglomerates; namely for each surjection between conglomerates $f:X\rightarrow Y$, there is an injection $g:Y\rightarrow X$ with $f\circ g=id_Y$.

So the question is: can we continue this hierarchy further to form the metacategory of all metacategories and so on?

Complexity of finding semi-ordered Eulerian tours in a 4-regular graph

Math Overflow Recent Questions - Thu, 10/05/2017 - 05:32

I'm trying to figure out the time-complexity of the problem I describe below, which I call the semi-ordered Eulerian tour problem or the SOET problem. Either finding an efficient algorithm for this problem or proving that this is actually in NP-hard would settle this question for me. Below I describe the problem, give two examples of instances to the problem and explain what I have realized so far. To be precise, my questions are:

  • What is the time-complexity of this problem?

  • Is this problem NP-hard?

  • If it exists, what is an algorithm that solves this problem in polynomial time in terms of both $n$ and $k$? (see below)

The SOET problem

Given a 4-regular graph $G=(V,E)$, i.e. a graph where each vertex have degree 4, the problem is to find a semi-ordered Eulerian tour (defined below). Note that the graph is not assumed to be simple, so multi-edges and self-loops are allowed.

Definition (semi-ordered Eulerian tour): Let $S\subseteq V$ be a subset of the vertices in $G$ and $s=s_1s_2\dots s_k$ be a permutation of $S$. The Eulerian tour $U$ is a semi-ordered Eulerian tour with respect to $S$ if, $U=As_1Bs_2\dots s_kXs_1Ys_2\dots s_kZ$ for some $s$ and where $A,B\dots,X,Y,\dots,Z$ are words with letters in $V\setminus S$. That is, $U$ traverses $s$ in some order once and then again in the same order. To be clear, which specific $s$ that satisfies this is unimportant.

Note that since $G$ is 4-regular an Eulerian tour always exists and furthermore it will pass each vertex exactly twice.

Examples

Here I give two examples of 4-regular graphs. The first one, G_1, there is a SOET with respect to $\{a,b,c,d\}$, since there is a Eulerian tour $U=abcdaebced$. In the second example, G_2, there is none.

Current status

I have realized that this problem is at least fixed-parameter tractable in terms of $k$, i.e. there is an algorithm with time-complexity $\mathcal{O}(f(k)p(n))$, where $k=|S|$, $n=|V|$ and $p$ is some polynomial. This can be seen by mapping the problem to $k!k^3$ edge-disjoint paths problems (DPP) where you try to find paths between pairs of vertices in $S$. This has to be done for all permutations of the vertices in $S$, the reason for the factor $k!$ in the number of DPP:s. Furthermore each vertex has to be split into two vertices and connected by edges in one out of three ways (vertically, horizontally or diagonally), to actually give a Eulerian tour. This are in total another $3^k$ DPP:s. Note that this does not necessarily exclude the possibility of an algorithm which is also polynomial in $k$.

Note regarding duplicate:

I have now also asked the same question on Theoretical Computer Science since after two weeks of no response here I thought that this might be the better place for this question.

Jaffard's theorem - finite matrices

Math Overflow Recent Questions - Thu, 10/05/2017 - 00:19

For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+|k-l|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,l) \leq C' (1+|k-l|)^{-r}, $$ for all $k,l\in \mathbb{Z}$ and for some $C'>0$. That is if the initial matrix exhibits polynomial off-diagonal decay, so does its inverse.

I am wondering if there is a complementary version for finite matrices. Note that for finite matrices such a result would be interesting if $C'$ can be bounded in terms of $C$ and $r$.

This paper (https://www.math.umd.edu/~jjb/general_all_pq_revised.pdf) claims that $C'$ depends only on $r$, $C$, and some parameter $b$ such that $1>b\geq \|I-A\|_2$. However, there seem to be some critical typos in the surrounding discussion, and they refer (cf. p. 5 of the preprint linked) to Jaffard's paper for a proof that $C'$ depends only on certain constants (see. loc. cit. for which). The latter paper is in French, so I couldn't verify this.

So I have three questions:

  1. Does any one have a pointer to a version of Jaffard's theorem for finite matrices?
  2. Or alternatively, is there a clear argument to bound $C'$ in terms of $C,r,$ and possibly $b$?
  3. Supposing that there is indeed such a bound, can it be generalized to settings where $\|I-A\|_2 \geq 1$?

Has a discrete/quantum theory of probability based on the Cournot-Borel principle been developed? [on hold]

Math Overflow Recent Questions - Wed, 10/04/2017 - 08:50

In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement négligeables (On universally negligible probabilities) in Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 190, pp. 537-40. Here it is:

http://gallica.bnf.fr/ark:/12148/bpt6k3143v.f539

According to this question

Have some works by Émile Borel ever been translated from French to English or another foreign language?

and to the best of my knowledge, this note has never been translated in any foreign language. I would be happy to translate it entirely upon request despite my poor English.

Borel is concerned by Cournot principle. As the bridge, the connection between the mathematical theory of probability and the real world of experience,Borel considers Cournot principle to be the most important and fundamental principle of probability theory: he used to call it the fundamental law of randomness or the unique law of randomness. Hence, Borel seeks for a quantitative version of Cournot principle. He starts like this:

We know that, in the applications of the calculus of probability, when the probability becomes extremely close to unity, it can and must be practically confounded with certainty. Carnot principle, the irreversibility of many phenomena, are well-known examples in which the theoretical probability equals practical certainty. However, we may not have, at least to the best of my knowledge, sufficiently specified from which limits a probability becomes universally negligible, that is negligible in the widest limits of time and space that we can humanly conceive, negligible in our whole universe.

and concludes:

The conclusion that must be drawn is that the probabilities that can be expressed by a number smaller than ${10^{ - 1000}}$ are not only negligible in the common practice of life, but universally negligible, that is they must be treated as rigorously equal to zero [emphasized by Borel] in every questions regarding our Universe. The fact that they are not effectively null may be of interest for the metaphysicist; for the scientist they are [emphasized by Borel] null and the phenomena to which they relate are absolutely impossible [emphasized by Borel].

This Cournot-Borel principle

$\left\{ \begin{array}{l} p \in \left[ {{{0,10}^{ - 1000}}} \right]\;\;\;\;\,\; \Rightarrow p = 0\quad \quad {\rm{Borel - supracosmic}}\;{\rm{probabilities}}\\ p \in \left[ {1 - {{10}^{ - 1000}},1} \right] \Rightarrow p = 1\quad \quad \,{\rm{Borel - supercosmic}}\;{\rm{probabilities}} \end{array} \right.$

implies that there are only discrete probability measures/distributions in every probabilistic questions regarding our universe.

Indeed, consider for instance a cumulative distribution function $F\left( x \right):\mathbb{R} \to \left[ {0,1} \right]$. Suppose $F\left( x \right)$ is left-continuous at some point ${x_0}$, that is $F\left( x \right)$ is continuous at ${x_0}$ since it is right-continuous by definition:

$\forall \varepsilon > 0\;\exists \eta > 0,\forall x,{x_0} - \eta < x < {x_0} \Rightarrow \left| {F\left( x \right) - F\left( {{x_0}} \right)} \right| = F\left( {{x_0}} \right) - F\left( x \right) = {\text{Prob}}\left( {y \in \left[ {x,{x_0}} \right]} \right) = \mu \left( {\left[ {x,{x_0}} \right]} \right) < \varepsilon $

In particular, by the Cournot-Borel principle

$\forall {10^{ - 1000}} > \varepsilon > 0\;\exists \eta > 0,\forall x,{x_0} - \eta < x < {x_0} \Rightarrow \mu \left( {\left[ {x,{x_0}} \right]} \right) = 0$

Hence, either $F\left( x \right)$ is constant or it is discontinuous: $F\left( x \right)$ is nothing but a discrete cumulative distribution function or cumulative mass function.

Hence, following Borel, at least two different mathematical theories of probability would coexist: the mathematical, metaphysical, continuous one that relies heavily on measure theory, and the scientific, physical, discrete one where measure theory is almost irrelevant.

This Borel-Cournot discrete theory of probability is not necessarily inconsistent nor trivial because continuous r.v.s have discrete probability measures. By construction and definition, it constitutes another potential answer or proposal to Hilbert sixth problem or program. We can also talk about a quantum theory of (classical and quantum?) probability (not the theory of quantum probability) with Borel probabilistic quanta $b = {10^{ - 1000}}$, analogous to the energy quanta in QM.

Has something like this theory ever been developed?

Inverse Laplace transform to get CDF

Math Overflow Recent Questions - Wed, 10/04/2017 - 02:08

I have the following problem. If i can get some help, I would greatly appreciate it. I am trying to replicate a particular research paper and came across a problem:

Suppose X is a birth death process (represents population size) that evolves by:

$X -> X+1 $ if a birth occurs with rate $\mu$

$X -> X-1 $ if a death occurs with rate $\theta$

Suppose $T_A$ is first passage time of a BD process from state A to state 0 and suppose $T_B$ is first passage time of another BD process from state B to state 0. They are both independent.

I need to find $P(T_A < T_B)$. That is, probability that a population of size A goes to 0 before population of size B.

By definition:

$T_A = T_{A,A-1} + T_{A-1,A-2} + ... + T_{1,0}$

where $T_{i,i-1}$ represents first passage time from state i, to state i-1.

I read some articles online that mentioned that if $T = T_A - T_B$, then the CDF defined by:

$G_T(t) := P(T <=t)$

is what i need. The paper suggested taking inverse laplace of a CDF to obtain the CDF and evaluate it at 0. It first suggested finding laplace transform of T, which is given by

$L[T] = E(e^{-ST}) = -\frac{L[T_A](s) L[T_B](-s)}{s}$

Then it suggested taking laplace of $G_T(t)$ i.e $L[G_T(t)]$

However, $L[G_T(t)] = \frac{L[T]}{s} = \frac{L[T_A](s) L[T_B](-s)}{s}$

Then the paper suggests taking the inverse of above evaluated at 0 to get $P(T_A < T_B)$.

Question:

1) Given the Laplace transform of CDF, which is $L[G_T(t)]$, I want to use the inverse laplace to obtain $G_T(t)$ evaluated at 0. But, by definition, inverse laplace using algorithms in python are all one sided from $[0,\infty]$ My random variable T is given by difference of two first passage times, $T = T_A - T_B $. Won't this be negative?

2) In the paper, it says they are shifting the random variable X under study by a constant c such that P(X + c > 0) is approximately 1. Then inverting the corresponding one sided Laplace transform. How would i do that in this context here?

Thanks

Upper bound of the kissing number in n dimensions

Math Overflow Recent Questions - Tue, 10/03/2017 - 23:18

In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere.

Let $\tau_n$ be the kissing number in $n$ dimension.
Kabatiansky-Levenshtein (1978) proved the following asymptotic upper bound: $$\tau_n \le 2^{0.401n(1+o(1))} = (1.32\dots)^{n(1+o(1))}$$

Question: What is the smallest $\alpha$ such that $\tau_n \le \alpha^n$, for all $n$?

By using volume, we can prove that $\tau_n \le \frac{Vol(B(3))-Vol(B(1))}{Vol(B(1))}=3^n-1$, so $\alpha \le 3$.

Now $\tau_2 = 6$, so $\alpha \ge \sqrt 6 \simeq 2.45$. Moreover, for $n \le 24$, $e^{\ln(\tau_n)/n} \le \sqrt 6$. Is it true that $\alpha = \sqrt 6$?

This post is motivated by arXiv:1710.00285, Section 5.

Is it known whether or not $\aleph_\alpha=\beth_\alpha$ can be proven by ZFC?

Math Overflow Recent Questions - Tue, 10/03/2017 - 22:51

Given a cardinal number $\aleph_\alpha$, is it known whether or not $\aleph_\alpha=\beth_\alpha$ is independent of ZFC?

One could define $\mathrm{CH}(\aleph_\alpha)$ as $\aleph_\alpha=\beth_\alpha$. For which cardinals is it true that $\mathrm{ZFC}\models\mathrm{CH(\kappa)}$?

Clearly, $\mathrm{ZFC}\models\mathrm{CH}(\aleph_0)$, and $\mathrm{CH}(\aleph_1)$ is equivalent to $\mathrm{CH}$ (and is thus independent of ZFC). Because $\mathrm{GCH}$ is indepenedent of ZFC, $\neg\mathrm{CH}(\kappa)$ cannot be proven for any cardinal $\kappa$.

If $\mathrm{ZFC}\models\mathrm{CH}(\aleph_{\alpha+1})$, then $\aleph_{\alpha+1}=\beth_{\alpha+1}$. Thus, if we assume $\aleph_\alpha<\beth_\alpha$, we get a contradiction, because $\aleph_\alpha<\beth_\alpha$ and then $\aleph_{\alpha+1}<\beth_{\alpha+1}$. So, if $\mathrm{CH}(\kappa^+)$ is provable, then $\mathrm{CH}(\kappa)$ is also provable. Thus, for cardinals $\kappa<\aleph_\omega$, $\mathrm{CH}(\kappa)$ is independent of ZFC.

It is known that $\alpha\leq\aleph_\alpha$, and thus if $\beth_\kappa=\kappa$ then $\aleph_\kappa=\kappa$ (i.e. all $\beth$-fixed points are also $\aleph$-fixed points). Therefore, $\beth_\kappa=\aleph_\kappa=\kappa$, and $\beth_\kappa=\aleph_\kappa$. So, for all $\beth$-fixed points $\kappa$, $\mathrm{CH}(\kappa)$ is provable from ZFC.

Are there any other known $\kappa$ with $\mathrm{CH}(\kappa)$ independent of ZFC? Is $\exists\kappa\neq\aleph_0(\mathrm{CH}(\kappa))$ independent of ZFC? (If so, $\beth$ fixed points are independent of ZFC as well.)

Edit: Of course $\beth$-fixed points are not independent of ZFC, they do exist by $\alpha\rightarrow\beth_\alpha$ being a normal function. So, ZFC actually proves the existence of cardinals larger than $\aleph_0$ which fulfill GCH.

Asymptotics for a $n$-dimensional Watson-like integral

Math Overflow Recent Questions - Tue, 10/03/2017 - 22:24

I'm interested in the behaviour of the following integral around $\delta=0$ which is similar to the Watson integral (the case where $\delta=0$ which diverges in n=1 and n=2 dimensions, but converges for $n\geq 3$)

$$I_n=\frac{1}{(2\pi)^n}\underbrace{\int_{-\pi}^{\pi}\ldots\int_{-\pi}^{\pi}}_{\text{n times}}\frac{d\theta_1\ldots d\theta_d}{1-n^{-1}\sum_{i=1}^{n}\left[\cos(\theta_i)+i\delta\sin(\theta_i)\right]}$$

In one dimension I expect $I_1\sim \delta^{-1}$, in two dimensions I expect $I_2\sim \log(\delta^{-1})$ (see for instance Lattice random walk under gravity) and above that I'm unsure but suspect $I_n\sim C_n + \alpha_n \delta^{n-2}$ where $C_n^{-1}$ is the $n$th Polya random walk coefficient and $\alpha_d$ a constant.

I note that the author of an answer here Lattice random walk under gravity notes such scaling for $n=2$ but doesn't mention a method.

I can't seem to rearrange it into a form where the canonical methods are helpful, but perhaps I'm missing something?

multiply matrices [on hold]

Math Overflow Recent Questions - Tue, 10/03/2017 - 22:15

Can I use the normal rules to apply here? I mean the rules about real number.
Can I apply them for matrices or not?

Show that

$$ A \begin{pmatrix} B & C \ \end{pmatrix} = \begin{pmatrix} AB & AC \\ \end{pmatrix} $$ for $A ∈ M_{m×n}$, $B ∈ M_{n×p}$, $C ∈ M_{n×q}$; $$ \begin{pmatrix} A \\ B \\ \end{pmatrix} C = \begin{pmatrix} AC \\ BC \\ \end{pmatrix} $$ for $A ∈ M_{p×n}$, $B ∈ M_{q×n}$, $C ∈ M_{n×m}$.

Identification of a PDE

Math Overflow Recent Questions - Tue, 10/03/2017 - 21:30

I cannot find the following equation $$u_t=u^2u_{xx}$$ anywhere in the literature. Could someone please point me in the direction of what's been published on this?

Open subsets of the n-torus containing no nontrivial loops

Math Overflow Recent Questions - Tue, 10/03/2017 - 21:19

Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is nullhomotopic?

Expectation of max of Gaussian multiplied by a functional of Gaussian

Math Overflow Recent Questions - Tue, 10/03/2017 - 20:23

Let $X \in \mathbb{R}^{d}$ follows the standard Gaussian distribution $N(0, I_d)$. Let $Y = \max_{j\in[d] } X_j$. It is not hard to see that \begin{align} \mathbb{E}\left [ Y \cdot X\right] = \sum_{j=1}^n \mathbb{P}\left( j = \arg\max_{i \in [d]} X_i \right) \cdot e_j, \end{align} where $e_i$ is the standard basis in $\mathbb{R}^d$. Now I was wondering how to compute $$\mathbb{E} \left[ Y\cdot (X X^\top - I_d) \right ].$$ Is there a closed form solution?

Special fibers of parahorics

Math Overflow Recent Questions - Tue, 10/03/2017 - 17:28

Tit's Corvallis article introduces a map on special fibers of group schemes associated to the elements fixing sets pointwise in a building from $\bar{\mathcal{P}_\Omega}$ to $\bar{\mathcal{P}_{\Omega'}}$ where $\Omega' \subset \Omega$, but that is all he does with this map. In particular there is nothing about injectivity. Is it injective when we look at the maximal reductive quotients of the $\mathcal{P}$? By the discussion in $3.5$ it is enough to answer for facets.

I've looked in Bruhat-Tit's 5 articles and have not found anything about this, but it is very possible I am not looking in the right places for it.

Integrability over the closure of a domain

Math Overflow Recent Questions - Tue, 10/03/2017 - 17:26

Let $D$ be a bounded domain in $\mathbb{R}^{N}$ ($N\geq2$) and $E$ a closed subset of $D$ with empty interior. Suppose $f$ is a measurable function defined on $D$ and integrable on $D\setminus E$, i.e., $$\int_{D\setminus E}|f(x)|dx<\infty.$$ Can we say that at least either its positive part $f^{+}$ or its negative part $f^{-}$ are integrable on $D$?

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