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most recent 30 from mathoverflow.net 2019-05-17T22:14:05Z

Differentiability (Hessian) of $\int \log F$ when $\int \log f$ is differentiable?

Wed, 05/08/2019 - 03:54

For a specific probability density function $f$ with support on ${\mathbb R}$, which is not differentiable everywhere, I have proven that the Hessian matrix of $$g(\theta) = \int \log f(x;\theta)d H(x),$$ exists for all $\theta \in {\mathbb R}^p$, where $H$ is another distribution function. Let $F(x;\theta) = \int_{-\infty}^x f(t;\theta)dt$ be the distribution function. I want to check if the Hessian of $$G(\theta) = \int \log F(x;\theta)dH(x),$$ also exists.

Is there a direct method of showing this? This is, some general result I can appeal to? If it wasn't for the logarithm, I could use exchange the integral and derivative symbols, for instance.

Eigenvalue-taking operator?

Wed, 05/08/2019 - 01:35

$\newcommand{Tr}{\operatorname{Tr}}$ Is there a continuous map $(p,t) \mapsto \lambda(p,t)$ which, given a path $p: [0,1] \to M(2,\mathbb R)$ and a $t \in \mathbb [0,1]$, gives back an eigenvalue of $p(t)$? Additionally, I'd like it that if $p(a) = -p(b)$ for some $a, b \in [0,1]$, then $\lambda(p,a) = -\lambda(p,b)$.

A naive possibility would be $\lambda(p, t) = \frac{\Tr(p(t)) \pm \sqrt{\Tr(p(t))^2 + 4\det(p(t))} }2$, using the quadratic formula and the characteristic polynomial. But this has got $\pm$ in it, which is not a function. Changing $\pm$ into $+$, the image of the path $$p(t) = \begin{pmatrix}0 & 1-2t \\ 2t - 1 & 0 \end{pmatrix}$$ is seen to violate the second condition: namely, $\lambda(p,0)=\lambda(p,1)=i$ when what's needed is $\lambda(p,0)=-\lambda(p,1)$.

This problem is unsolvable when $M(2, \mathbb R)$ is changed to $M(2, \mathbb C)$, as the existence of $\lambda(-,-)$ would violate the simple-connectivity of $SL(2,\mathbb C)$.

Countable union of Menger spaces

Tue, 05/07/2019 - 23:03

A topological space $X$ has Menger's property $\textsf{S}_{\mbox{fin}}(\mathcal{O}, \mathcal{O})$ if, for each sequence of open covers, $\mathcal{U}_1, \mathcal{U}_2, \cdots $, we can select finite sets $\mathcal{F}_1\subseteq\mathcal{U}_1, \mathcal{F}_2\subseteq\mathcal{U}_2, \cdots $ whose union $\bigcup_{n}\mathcal{F}_n$ covers the space.

My question is if the Menger's property is preserved by countable unions, that is, if for each $n\in\omega$, $X_n$ is a Menger space, then $\bigcup_{n\in\omega}X_n$, with the disjoint union topology is a Menger space.

Thanks

Equivalent forms of the P vs. NP problem

Tue, 05/07/2019 - 16:51

Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an intergral equality, etc.

I wonder about equivalent formulations of the N vs. NP problem. Formulations that are very much different from the questions such "Is TSP in P?", formulation that may seem unrelated to complexity theory.

Number of integers for which $np(n)$ is a perfect square

Tue, 05/07/2019 - 14:22

Let $p(n)$ be the partition function. Are $n=1,2,3$ the only cases for which $np(n)$ is a perfect square?

Nonnegativity of an integral over the unitary group

Tue, 05/07/2019 - 12:23

For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let $$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$ Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\sigma_2})dU\ge 0$ for all $\sigma_1,\sigma_2\in S_n$?

The integral can be expanded to $$\frac{1}{4}\int_{U(n)} (w_{\sigma_1}+\bar w_{\sigma_1})(w_{\sigma_2}+\bar w_{\sigma_2})dU=\frac{1}{2}\int_{U(n)} w_{\sigma_1}w_{\sigma_2}+w_{\sigma_1}\bar w_{\sigma_2}dU.$$ Once the $\det(U^*)$ factors have been expanded as sums over permutations, this can be evaluated using Weingarten functions (see Collins 2003). The latter term is \begin{align}\int_{U(n)} w_{\sigma_1}\bar w_{\sigma_2}dU&=(-1)^{\sigma_1\sigma_2}\int_{U(n)}U_{1\sigma_1(1)}\cdots U_{n\sigma_1(n)}U^*_{\sigma_2(1)1}\cdots U^*_{\sigma_2(n)n}dU\\ &=(-1)^{\sigma_1\sigma_2}Wg(\sigma_2^{-1}\sigma_1,n).\end{align} By Novak 2010, the element $\sum_\sigma Wg(\sigma,n)\sigma$ of the group algebra $\mathbb{C}[S_n]$ can be written as a product of elements $(n+J_k)^{-1}$ of the form $\sum_\sigma (-1)^\sigma|a_\sigma|\sigma$, hence $(-1)^\sigma Wg(\sigma,n)\ge 0$. Computer experiments suggest the former term is nonnegative as well, but I haven't been able to prove it.

Note that $\int_{U(n)}\mathrm{Re}(w_\sigma)dU=1/n!$ by a symmetry argument (see here).

Power series ring and monomials

Tue, 05/07/2019 - 10:34

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables.

For a given positive number $\epsilon > 0$ we call a monomial $X_{i_1}^{e_{i_1}} \cdots X_{i_n}^{e_{i_n}}$ an $\epsilon$-monomial if it satisfies the following conditions simultaneously$\colon$ \begin{align*} & e_{i_1}/p^{i_1} < \epsilon \\ & e_{i_2}/p^{i_2} < \epsilon \\ & \cdots \\ & e_{i_n}/p^{i_n} < \epsilon. \end{align*} Let $\alpha$, $\beta$ be two $\epsilon$-monomials. Then for two elements $x, y \in (X_1,\ldots,X_n)$, i.e., the unique maximal ideal of $A_n$, we consider the product defined by \begin{equation*} P_{\alpha,\beta}(x,y) \colon= (\alpha + x)(\beta + y). \end{equation*}

Q. Suppose that $x$ (resp. $y$) comprises monomials different from $\alpha$ (resp. $\beta$). That is, $x$ (resp. $y$) is composed of monomials neither of which is equal to $\alpha$ (resp. $\beta$). Then does the product $P_{\alpha, \beta}(x, y)$ always contain a non-zero $2\epsilon$-monomial?

Formalization of mathematics [on hold]

Tue, 05/07/2019 - 01:17
The first book of Bourbaki

In the first book of the Elements, Bourbaki describes a formalization of mathematics: what are valid "formulas", what are the rules that one can use to build new "formulas" from other "formulas", etc.

To be more precise, by "formalization of mathematics":

  • I mean: formalization of the activity of doing mathematics

  • I don't mean: formalization of the various branches of mathematics. Besides, it is what Bourbaki does in the following books: he formalizes different areas of mathematics (Topology, Differential geometry, etc.)

Then only, he defines sets and set theory.

Then, he formalizes different areas of mathematics in the setting of set theory.

Hence, we have this movement:

  1. Formal description of mathematics: "formulas" and rules to create new "formulas"

  2. Description of set theory within this framework

  3. Description of other branches of mathematics within the framework of set theory

Other similar works?

I would like to have other references (if possible as many) of similar works by other mathematicians/logicians/philosophers.

More precisely, I'm only interested on works where point 1. is achieved, or works where points 1. and 2. are achieved.

With type theory?

If it exists (but I believe it does not exist), I would like to find works where the following is achieved:

  1. Formal description of mathematics

  2. Description of type theory within this framework

PS
  • I believe (but I have never read it) that the Principia of Russel and Whitehead is such a similar work, isn't it?

  • Does Frege has also done such a work?

  • This question should be a community wiki I believe.

Regularity result for the boundary value problem for the heat equation

Mon, 05/06/2019 - 13:16

Let $\Omega$ be an open bounded subset of $\mathbb R^N$.

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$

Consider the following boundary value problem for the heat equation: $$ \begin{cases} u_t - \Delta u = f \\ u|_{\partial\Omega} = 0\\ u(0) = u_0 \end{cases} $$

Questions:

  • Let $k \ge 2$. Assume $u_0 \in C^k(\bar \Omega)$, $f \in C^k([0,T) \times\bar \Omega)$ such that $u_0 = \Delta u_0 = 0$ on $\partial \Omega$ and assume that $\Omega$ is of class $C^k$. Is it true that there exists a unique solution $u \in L^\infty((0,T)\times\bar\Omega) \cap C^k([0,T)\times \bar\Omega)$? How can one prove it? Do we need some additional assumptions on $f$?

  • Fix $U$ a neighborhood of $x_0 \in \Omega$, and assume that $u_0 \in C^k(U)$, and $f \in C^k([0,T) \times U)$. Is it true that there exists a unique (weak) solution of the heat equation that is regular in $U$, that is $u \in C^k([0,T)\times U) \cap L^\infty$?

  • Are the results in the first two questions true even if we assume $\Omega$ Lipschitz? An are they true with less regularity assumptions on $f$?

Motives and homotopy theories of algebraic varieties

Mon, 05/06/2019 - 09:23

The theory of motives is an attempt to cope with the fact that there are many reasonable cohomology theories of algebraic varieties. Now, sometimes your cohomology theory does not just give you a bunch of groups/vector spaces; it gives you a full-fledged (pro-)homotopy type (though based on limited responses to this question, I think there is no formal way to functorially produce homotopy types given a cohomology theory). The question is: what should be the extension of motives to homotopy types? What are some notable works in this direction? I have heard there is something called motivic homotopy theory, is it really relevant here or it just happens to have a similar name?

More specific questions:

  • First off, we need to show that usual motives are not good enough. This is probably obvious to anyone working in the field, but not everybody on this site is an expert on motives so an explicit example would be great. I think there is a rigorously constructed triangulated category of motives which is supposed but not known to be the derived category of some abelian category of motives. There is also a somewhat more pedestrian Grothendieck ring of varieties. I think the class in the Grothendieck ring does not determine the motive nor does the motive determine the class in the Grothendieck ring. Is there an example of two varieties with the same class in the Grothendieck ring which have different etale fundamental groups? An example of two varieties with the same motive in the triangulated category which have different etale fundamental groups?
  • Before you can say "every Weil cohomology theory factors through motives", you have to define a Weil cohomology theory. What should the definition be for homotopy types? Has somebody written down a manageable list of axioms definining exactly what we are interested in?
  • Before we can seriously talk about motivic homotopy types, we should understand on the categorical level what we expect the relevant category to be. In the case of motives, the relevant piece of category theory is Tannakian formalism, I believe (then we can say smart words like "a Weil cohomology theory is just a fiber functor blah blah"). What should the category theory look like for motivic homotopy types? A kind of non-abelian Tannakian formalism?

P.S.: yeah, I know the question is super naive, you are free to call me an idiot in the comments.

The expectation of binary logistics regression with respect to Gaussian distribution

Mon, 05/06/2019 - 09:10

I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If we define $$\langle g(s,x)\rangle_q=\int\mathcal{N}(x;m,v)g(s,x)\mathrm{d}x$$

I would like to re-derive the formula which is given in section 5.1, paragraph 3 of this paper $$\frac{\mathrm{d}\langle g(s,x)\rangle_q}{\mathrm{d}v}=\frac{-1}{2v}(\big(\langle x\sigma(x)\rangle_q-m\langle\sigma(x)\rangle_q\big)$$ $$\frac{\mathrm{d}\langle g(s,x)\rangle_q}{\mathrm{d}m}=s-\langle\sigma(x)\rangle_q.$$ where $q$ is the normal distribution. Does this derivation come from the direct partial differentiation of integrand with respect to $v$ and $m$? Could anybody suggest a way to re-derive these two equations?

Homology of solvable (nilpotent) Lie algebras

Mon, 05/06/2019 - 07:40

Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for $\mathbb{C}_\lambda$, the one-dimensional $\mathfrak{g}$-module with character $\lambda$.

I have calculated homology manually for the 2-dimensional algebra $\mathfrak{g}=\langle x,y \rangle$ with relation $[x,y]=y$. The thing that surprised me is that homology is nontrivial only for $\lambda(x)=0$ or $1$.

In general, I conjecture is that the homology is nontrivial, iff $\lambda$ is a weight of the adjoint representation. I can't prove it (or find counterexample). I've tried to find the answer in books, but there is a lack on literature in homology theory for solvable lie algebras.

How do I get 1/15 of something, only by divide with 2 or 3 and add the result back together?

Sun, 05/05/2019 - 14:27

I'm currently playing the game Satisfactory, where I need to balance the conveyor belts to ensure a 100% efficient factory.

To help me in this job I have Merger and Splitter. The Splitter can split belts into 2 or 3 conveyor belts and the Merger can join 2 or 3 belts into one.

Now I have a certain Input of N Ressources and want 1/15 of N Ressources at the End. Which is the amount of Splitter and Merger I need for this problem and how can I calculate, if it is even possible to achieve 1/15 or other fractures.

Hope somebody can help me with this problem.

Involutions in $\mathbb{F}_p[[x]]$

Sun, 05/05/2019 - 14:15

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$?

Here involution in $A[[x]]$ means $f\in A[[x]]$ such that $f\circ f=x$.

Obviously, a reduction mod $p$ of an involution in $\mathbb{Z}[[x]]$ yields an involution in $\mathbb{F}_p[[x]]$. Moreover, at least one involution in at least one $\mathbb{F}_p[[x]]$ is not obtainable by reducing mod $p$, namely, $1+x\in\mathbb{F}_2[[x]]$, exactly because its constant term is nonzero.

But do we get all involutions in $\mathbb{F}_p[[x]]$ with the zero constant term this way?

If there is a reference where this is discussed, that would be very helpful, too.

Multiplication in Deligne cohomology: explicit formula for $p=q=1$

Sun, 05/05/2019 - 14:13

[This is a double of my question of math.stackexchange https://math.stackexchange.com/questions/3214962/multiplication-in-deligne-cohomology-explicit-formula-for-p-q-1]

In the very beginning of [1] the geometric meaning of Deligne cohomology $H^q(X, \mathbb{Z}(p))_D$ and multiplicative structure on it is being discussed. In particular, it is not hard to see that $H^q(X, \mathbb{Z}(1))$ can be canonically identified with $H^{q-1}(X, \mathcal{O}^{\times}_X)$.

The group $H^2(X, \mathbb{Z}(2))_D$ is identified with the group of holomorphic rank $1$ bundles with holomorphic connection (group structure is given by tensor product)

The $\cup$-multiplication gives us a map $$ H^1(X, \mathbb{Z}(1))_D \otimes H^1(X, \mathbb{Z}(1))_D = H^0(X, \mathcal{O}^{\times}_X) \otimes H^0(X, \mathcal{O}^{\times}_X) \to H^2(X, \mathbb{Z}(2))_D $$ In other words, given two nowhere vanishing holomorphic functions $f$ and $g$ on $X$ we obtain a holomorphic line bundle with holomorphic connection on $X$.

Though in [1] the explicit formula for this in terms of Čhech cocycles is given, I am looking for another description of the same operation.

First of all, observe that each pair of functions $f, g \in H^0(X, \mathcal{O}^{\times}_X)$ define a holomorphic map $F_{f,g} \colon X \to (\mathbb{C}^{\times})^2$. Following Esnault and Viehweg, denote the resulting line bundle with holomorphic connection $f \cup g$ by $r(f, g)$. Then it seems clear from functoriality reasons that $$r(f, g) = F_{f,g}^*r(z,w),$$ where $z$ and $w$ are coordinate functions on $\mathbb{C}^{\times}\times \mathbb{C}^{\times}$. Thus, I'd be happy to understand, what $r(z, w)$ is.

Since $(\mathbb{C}^{\times})^2$ is a product of two Stein manifolds, there are no non-trivial holomorphic line bundles. Therefore, the only ''interesting'' part of $r(z,w)$ is the holomorphic connection. Any holomorphic connection on trivial bundle is given by $\nabla = d + \eta$, where $\eta$ is a holomorphic $1$-form. So my questions are:

  • What is this $1$-form $\eta$ on $\mathbb{C}^{\times} \times \mathbb{C}^{\times}$? It seems to me, that $\frac{dz}{z} - \frac{dw}{w}$ would be nice (at least, if this is the case, it satisfies the properties of $r(f, g)$ given in [1]), however I'm not able do deduce this explicitly form Esnault-Viehweg formulae.
  • From my speculations it follows that the underlying line bundle for any $r(f,g)$ is trivial. Is this at least true? If not, than where is my mistake?

Thank you for any comments!

[1] -- H. Esnault, E. Viehweg. Deligne-Beilinson cohomology. in: Beilinson's Conjectures on Special Values of L-Functions ( Ed.: Rapoport, Schappacher, Schneider ). Perspectives in Math. 4, Academic Press (1988) 43 - 91 (http://page.mi.fu-berlin.de/esnault/preprints/ec/deligne_beilinson.pdf)

Ratio of hypergeometric function

Sun, 05/05/2019 - 13:45

Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function? $$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$ for $1>x>y>0$ ideally in the form like some powers of $x/y$?

Is there an $L$-space whose square is selectively $d$-separable?

Sun, 05/05/2019 - 13:03

An $L$-space is a hereditarily Lindelof regular space which is not separable.

A space is $d$-separable if it contains a dense set which is the countable union of discrete sets.

An $L$-space can't be $d$-separable, because it has no uncountable discrete subsets, however, by a small modification of his celebrated construction of a ZFC $L$-space, Justin Moore provided a ZFC example of an $L$-space with a $d$-separable square.

Moore, Justin Tatch, An $L$ space with a $d$-separable square, Topology Appl. 155, No. 4, 304-307 (2008). ZBL1146.54015.

At first glance $d$-separability looks like the strongest "separability-type" property you could hope to get in the square of an $L$-space, or is it? Consider, for example, the following "selective version" of $d$-separability:

A space is called $D$-separable if, for every sequence $\{D_n: n < \omega \}$ of dense subsets of $X$, there are discrete sets $E_n \subset D_n$, for every $n<\omega$, such that $\bigcup \{E_n: n < \omega \}$ is dense.

The above property lies between a property that the square of an $L$-space clearly can't have (a $\sigma$-disjoint $\pi$-base) and a property that the square of an $L$-space can have ($d$-separability).

QUESTION: Is there an $L$-space with a $D$-separable square?

For more information about $D$-separability see: Bella, Angelo; Matveev, Mikhail; Spadaro, Santi, Variations of selective separability II: Discrete sets and the influence of convergence and maximality, Topology Appl. 159, No. 1, 253-271 (2012). ZBL1239.54014.).

What's this non-holomorphic modular function?

Sun, 05/05/2019 - 12:48

I have an expression which is of the following form $$ M(\tau,\bar{\tau})=E_2(\tau)f_{k,\bar{k}+2}(\tau,\bar{\tau})+ E_2(\bar\tau)g_{k+2,\bar{k}}(\tau,\bar{\tau}). $$ Here, $E_2$ is the second Eisenstein series. $f_{k,\bar{k}+2}(\tau,\bar{\tau})$ and $g_{k+2,\bar{k}}(\tau,\bar{\tau})$ are non-holomorphic Maass forms of weights $(k,\bar{k}+2)$ and $(k+2,\bar{k})$ respectively. They transform under modular transformations as $$ f_{k,\bar{k}+2}(\gamma.\tau,\gamma.\bar{\tau}) = (c\tau+d)^{k}(c\bar{\tau}+d)^{\bar k+2} f_{k,\bar{k}+2}(\tau,\bar{\tau}), $$ and $$ g_{k+2,\bar{k}}(\gamma.\tau,\gamma.\bar{\tau}) = (c\tau+d)^{k+2}(c\bar{\tau}+d)^{\bar k} g_{k+2,\bar{k}}(\tau,\bar{\tau}). $$ We know that the second Eisenstein series is quasimodular with weight 2. So it is clear how $M(\tau,\bar{\tau})$ transforms under modular transformations.

My question is does the function $M(\tau,\bar{\tau})$ fall under a special class of non-holomorphic modular functions which show quasimodular features? Have these been considered by mathematicians and do these have a special name?

I want ask little questions about 0=1 [on hold]

Sun, 05/05/2019 - 12:27

I have physics equation and it says tha A*b/c=0 so I need all the physics terms to be alive so I equal 0. To 1 is this correct

Calculate the norms of the matrix A [on hold]

Sun, 05/05/2019 - 11:49

I need to calculate the norms of the matrix A.

A=[-3 -4 -2; 5 9 -5; -3 8 -9]

a. ||A||_(1,1)

b. ||A||_(∞,∞)

c. ||A||_(1,∞)

d. ||A||_(2,∞)

e. ||A||_(1,2)

I know that ||A||1=21 and ||A||\infty=20, but I'm not sure what to do after that for a or b. I haven't done ||A||_2 yet, but I know how to get that information. I just don't know what do when I have (1,1) and things like that. Any help would be appreciated.

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