# Math Overflow Recent Questions

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### Is there a fiber bundle for manifolds with $sec\geqslant k$ collapsing to a manifold?

Thu, 03/01/2018 - 03:21

Let $\Psi(i)\to 0$ as $i\to \infty$. Let $A_i$ be a sequence of n dimensional Alexandrov spaces with curvature $\geqslant k$, that Gromov Hausdorff converge to an m dimensional closed Riemannian manifold $A$. Here we consider the collapsing case, i.e. $n>m$. By Yamaguchi's almost Lipschitz submersion theorem, there is a $\Psi(i)$-almost Lipschitz submersion $f_i:A_i\to A$.

Next we discuss whether $f_i$ is a locally fiber bundle. For any point $p\in A$, there exist $a_1,...,a_m\in A$ near p, such that $$\tilde{\angle}_k a_j p a_l >\frac{\pi}{2} \text{ for all } j\neq l,$$ and there is a neighborhood $U\ni p$ such that $G(x)=(d(a_1,x),...,d(a_m,x)): U \to \mathbb{R}^m$ is a homeomorphism. For $x_i\in f_i^{-1}(U)$, we know $$d(f_i^{-1}(a_j),x_i)=d(a_j, f_i(x_i))+\Psi(i), \quad d(f_i^{-1}(a_j), f_i^{-1}(a_l))=d(a_j,a_l)+\Psi(i).$$ So $$G_i(x_i)=(d(f_i^{-1}(a_1),x_i),...,d(f_i^{-1}(a_m),x_i))$$ have no critical point in $f_i^{-1}(U)$.

By the Morse theorem for Alexandrov spaces proved by Perelman, $G_i:f_i^{-1}(U)\to \mathbb{R}^m$ is a locally trivial fiber bundle. We don't know whether its fiber coincide with $f_i^{-1}(p)$ since for $x,y\in f_i^{-1}(p)$, we don't have $G_i(x)=G_i(y)$. Is there a locally trivial fiber bundle from $A_i\to A$? Are there counter examples or illuminating examples?

### Is the algebra $\mathcal{C}^{\infty}(M,\mathbb{R})$ a smooth algebra in the sense of algebraic geometry?

Thu, 03/01/2018 - 02:59

First of all, let me fix some terminology: I will follow the definitions that can be found in the book "Cyclic Homology" of J.L. Loday (second edition) page 102 in the special case of $K$ a field.

Let $S$ be a commutative algebra with unit element. A sequence $(x_{1}, ...,x_{n})$ of elements of $S$ is called regular if multiplication by $x_{i}$ in $S/(x_{1}S+...+x_{i-1}S)$ is injective for every $i=1,...,n$.

A commutative unital algebra $A$ is smooth over a field $K$ if for every maximal ideal $M$ of $A$, the kernel of the localized map $$\mu_{M}\colon (A\otimes_{K} A)_{\mu^{-1}(M)}\longrightarrow A_{M}$$ is generated by a regular sequence in $(A\otimes_{K} A)_{\mu^{-1}(M)}$. Here the map $\mu\colon A\otimes A\rightarrow A$ is given by the multiplication: $\mu(a\otimes b)=ab$.

$\mathbf{Question}$: Is the algebra $\mathcal{C}^{\infty}(M,\mathbb{R})$ a smooth algebra in the previous sense ? (Here $M$ is a smooth compact manifold).

I have tried to understand what the kernel is: recalling that every maximal ideal of $\mathcal{C}^{\infty}(M,\mathbb{R})$ when $M$ is a compact manifold is of the form $$M=\{f\in \mathcal{C}^{\infty}(M,\mathbb{R}) : f(p)= 0\}$$ for a fixed $p\in M$, I have deduced that the kernel is the space generated by the elements of this form: $$\frac{f\otimes g}{\sum_\alpha r^{\alpha}\otimes q^{\alpha}} \quad \text{such that} \quad f g \equiv 0 \quad \sum_\alpha r^{\alpha}(p)q^{\alpha}(p)\neq 0.$$

I think that this kernel is even not finitely generated but I can't prove it.

### a linear programming problem

Wed, 02/28/2018 - 23:13

Recently I have a conjecture on decomposing a linear program into smaller ones. I have tested it in Mathmatica by a lot of examples. However, I cannot prove it. I will appreciate if someone can give some ideas.

Let $f_1,\dotsc,f_r \in \mathbb{R}[x_1,...,x_n]$ ($r>n+1$) be linear functions. Then $\{f_i \ge 0|\ 1 \le i \le r \}$ has a solution if and only if $\{f_i \ge 0|\ 1 \le i \le r, i \ne j\}$ has a solution for each $j=1,\dotsc,r$.

### Stochastic Extension to Fubini's Theorem

Wed, 02/28/2018 - 22:02

Fubini's theorem tells us when we can exchange the order of integration - however, does this apply in a stochastic setting?

What are the rules for changing the order of integration in stochastic calculus?

### math formulas for pi knowing all six derivatives [on hold]

Wed, 02/28/2018 - 21:45

If I have all 6 theta how do I find $\pi$ or how do I convert them into radian?are there any formulas?

For consecutive number or non consecutive numbers $x<y<z$

$(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times((1-\frac{x}{z})\times\sqrt\frac{(x+z)}{(z-x)})=\sin A$

$(\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})-(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times(\frac{x}{z})=\cos A$

$\sqrt\frac{(z-y)}{z}=\cos B$

$\sqrt\frac{y}{z}=\sin B$

$\frac{x}{z}=\cos C$

$((1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\sin C$

The following variables a,b,c represent the length of the sides of the triangles. $\frac{\sin A}{\sin C}=a$

$\frac{\sin B}{\sin C}=b$

$\frac{\sin C}{\sin C}=c$

$\frac{h_c}{h_a}=a$

$\frac{h_c}{h_b}=b$

$\frac{h_c}{h_c}=c$

### Prove that a limit about a converge series becomes $+\infty$ [on hold]

Wed, 02/28/2018 - 21:43

Assume that the series $\sum_{n=1}^{\infty} a_n$ converges, where $a_n>0$. And the sequence $\{a_n-a_{n+1}\}$ strictly monotonously decreases. Prove that $\lim_{n\to\infty}(\frac{1}{a_{n+1}}-\frac{1}{a_n})=+\infty$.

I tried to use some inequality to change the form, but failed. Then I change the form into $\frac{1}{a_n}(\frac{a_n}{a_{n+1}}-1)$ to go further, but failed again. So what's the correct way and are there some common techniques to prove similar propositions?

### Find the number of complementary subspaces of a $1$ dimensional subspace [on hold]

Wed, 02/28/2018 - 21:16

Let $V$ be a finite dimensional vector space of dimension $n$ over a finite field of order $p$ where $p=q^t$,$q$ being a prime.

If $V_1$ is a vector subspace of dimension $1$ show that the number of $n-1$ dimensional vector spaces $V_2$ such that $V_1+V_2=V$ is $p^{n-1}$.

Now it is easy to find the number of $k$ dimensional subspaces of an $n$ dimensional vector space by the formula

$\dfrac{(p^n-1)\times (p^n-1)\times \ldots \times (p^n-p^{k-1})}{(p^k-1)\times (p^k-1)\times \ldots \times (p^k-p^{k-1})}$.

So the number of $1$ dimensional subspaces is $\dfrac{p^n-1}{p-1}$.

But how to find the number of those $1$ dimensional subspaces $V_1$ such that $V_1+V_2=V$.

### Rectangular Newton polygon of a Jacobian pair

Wed, 02/28/2018 - 17:27

Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero. By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an automorphism pair if $(x,y) \mapsto (p,q)$ is an automorphism of $k[x,y]$.

There is a known result (based on S. S. Abhyankar results), Corollary 10.2.21, saying that if $p,q$ is a Jacobian pair, then there exists an automorphism $g$ of $k[x,y]$ such that $g(p)=x$ (in that case clearly $p,q$ is an automorphism pair) or the Newton polygon of $g(p)$ is contained in a rectangular $\{(i,j)|0 \leq i \leq a, 0 \leq j \leq b \}$, $1 \leq a \leq b$, with $(a,b)$ belonging to the support of $g(p)$.

Assume that $g(p)$ has degree $> 1$. By Proposition 10.2.6, there exist $1 \leq \hat{a} \leq a$ and $1 \leq \hat{b} \leq b$, such that each of $(\hat{a},0)$ and $(0,\hat{b})$ belong to the support of $g(p)$.

Is it possible that both $(a,0)$ and $(0,b)$ belong to the support of $g(p)$? (in the sub-rectangular case).

See this question.

Any help is welcome!

### Do certain maps between f.g. $\mathbb{C}$-algebras factor through a local (and f.g.) algebra?

Wed, 02/28/2018 - 16:57

(Intuition: in the category of non-empty sets, every function that coequalizes all points in the domain factors through the terminal object. I would like to know if something analogous happens in certain category of `connected algebraic spaces'. I formulate the precise question in terms of commutative algebra.)

Let $\cal A$ be the category of finitely generated $\mathbb{C}$-algebras with exactly two idempotents.

Let ${f : A \rightarrow B}$ be a map in $\cal A$ such that, for every ${g, h : B \rightarrow \mathbb{C}}$, ${g f = h f : A \rightarrow \mathbb{C}}$.

Does $f$ factor (inside $\cal A$) as ${f = k l}$ with ${l : A \rightarrow L}$, ${k : L \rightarrow B}$ and $L$ local?

### Is V, the Universe of Sets, a fixed object?

Wed, 02/28/2018 - 16:11

When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am beginning to have my doubts. We define

$V_{0}:=\emptyset$ .

${\displaystyle V_{\beta +1}:={\mathcal {P}}(V_{\beta }).}$

${\displaystyle V_{\lambda }:=\bigcup _{\beta <\lambda }V_{\beta }.}$ For any limit ordinal λ,

and finish by saying $V:=\bigcup _{\alpha \in Ord}V_{\alpha }.$

However this definition seems to create many problems. I can see at least two immediately: First of all, the Power Set operation is not absolute, that is it varies between models of ZFC. Secondly (and more importantly) this definition seems to be completely circular as we do not know a priori what the ordinals actually are. For instance, if we assume some two, mutually inaccesible large cardinals $\kappa , \kappa'$ to exist, and model ZFC as $V_\kappa, V_{\kappa'}$ respectively, then we get two completely different sets of ordinals! So we seem to be at an impasse:

In order to define the Universe of Sets we must begin with a concept of ordinals, but in order to define the ordinals we need to have a concept of the Universe of Sets!

So my question is to ask: Is this definition circular? The only solution I can think of is that when we define $V$, we implicitly assume a model of ZFC to begin with. Then after constructing the ordinals in this model, we construct $V$ off of them, so to speak. Is this what is being assumed here?

### "Optimal" local limit theorems for densities vanishing at zero

Wed, 02/28/2018 - 15:44

Consider a nonnegative stable distribution with a density that vanishes at zero, such as $$f(t)=\frac{e^{-1/2t}}{\sqrt{2\pi t^3}},\qquad t\geq0.$$

Suppose (for simplicity) that we have i.i.d copies $(X_k)$ of a random variable supported on the maximal lattice $\mathbb Z$ in the domain of attraction of $f$, so that $$\lim_{n\to\infty}\mathbb P[\frac1{a_n}\sum_{k=1}^nX_k\in dt]=\frac{e^{-1/2t}}{\sqrt{2\pi t^3}}dt$$ for some norming sequence $a_n$. Then, the local limit theorem states that $$\mathbb P[S_n=x]=a_n^{1/2}\frac{e^{-a_n/2x}}{\sqrt{2\pi x^3}}+\frac{o(1)}{a_n}\tag{1}$$ as $n\to\infty$, where $o(1)$ is uniform in $x\in\mathbb Z$.

Because $f$ vanishes at zero, we note that $(1)$ does not necessarily identify the leading order contribution to $\mathbb P[S_n=x]$ if $x=o(a_n)$, since $$a_n^{1/2}\frac{e^{-a_n/2x}}{\sqrt{2\pi x^3}}=o(a_n^{-1})$$ in that case.

Question. Does there exist general local limit theorems for such $f$ that correctly identify the leading order contribution to $\mathbb P[S_n=x]$ for $x=o(a_n^{-1})$, such as $$\mathbb P[S_n=x]=a_n^{1/2}\frac{e^{-a_n/2x}}{\sqrt{2\pi x^3}}+o(1)\frac{a_n}{x^{3/2}}$$ with $o(1)$ uniform in some range of $x$ that includes $x=o(a_n)$?

### When is GIT quotient of a projective variety smooth?

Wed, 02/28/2018 - 14:51

It might be very basic, but my google search could not find an answer for this, nor the search in previous related questions.

Let $X$ be a smooth projective variety, and let $G$ be a reductive group acting on $X$. Let $L$ be an ample $G$-equivariant line bundle on $X$. Define the GIT quotient of $X$ by $$X//^{L} G = Proj(\oplus_{n=0}^{\infty} \Gamma(X, L^{\otimes n})^G).$$

I would like to have some criterion for a point on this quotient to be a smooth point. I think it should be true for points corresponding to closed orbits of $G$ with free action, but I am more interested in points "on the boundary", namely points that corresponds to several orbits that are glued by the qoutient map $X \to X//^L G$.

Is there any criterion for that?

### When convolution with exponential kernel is bounded

Wed, 02/28/2018 - 13:11

Let $g(t)=e^{-\omega t}$, $\omega>0$. Find in terms of well-known function spaces, the subspace consisting of functions $f:\mathbb{R}^+\to \mathbb{R}^+$, $f\in L_{loc}^2(0,\infty)$, satisfying $$\lim_{t\to \infty}(g*f)(t)=\int_0^\infty g(t-s)f(s)ds<\infty.$$

### Illustrating mathematics with wysiwyg tools

Wed, 02/28/2018 - 10:06

What tools are out there for creating mathematical illustrations in a what-you-see-is-what-you-get mode?

Having struggled with tikz for several years, I've found creating figures in Omnigraffle (https://www.omnigroup.com/omnigraffle) to be a liberating experience. (And they have good tech support.)

The webpage http://wiki.illustrating-mathematics.org/wiki/What_tools_do_we_use%3F mentions Inkscape and Illustrator; how do they compare to Omnigraffle for flexibility and ease of use?

If other people know similar systems that they like better, I'd love to know about them.

### Steady Euler flows with compact support?

Wed, 02/28/2018 - 08:40

What is known about (3D) steady incompressible Euler flows with compact support?

(Looking for results in a field you are not familiar with sure is tough. I had a hope to find clues starting from the famous paper of V. Scheﬀer about a flow with compact support in space-time, and from works on vortex rings I could find, but in both cases ended up empty handed. This problem was considered by some, I presume? )

### Determining when two biquadratic polynomials generate the same field

Wed, 02/28/2018 - 07:12

Consider the family of monic biquadratic polynomials given by $f_{a,b}(x) = x^4 + 2ax^2 + b$ with $a,b$ integers. Let $K_{a,b}$ denote the isomorphism class of quartic fields obtained by adjoining any one of the roots of $f_{a,b}$. Is there a relatively nice list of criteria to decide whether $K_{a,b}$ is isomorphic to $K_{a',b'}$ when $(a,b) \ne (a',b')$?

### Exponential decay for wave equation in even dimensions

Wed, 02/28/2018 - 06:50

Consider the wave equation $$u_{tt} = \Delta_x u - q(x)u, \quad x \in\mathbb R^d, \; t > 0,\tag{1}\\ u(0,x) = u_0(x) \in H^1_\text{comp}(\mathbb R^d),\\ u_t(0,x) = u_1(x) \in L^2_\text{comp}(\mathbb R^d),$$ where $q \in L^\infty_\text{comp}(\mathbb R^d,\mathbb R)$. It is known that in odd dimensions $d$, if the operator $H=-\Delta+q(x)$ has no discrete spectrum and $0$ is not a resonance, then the solution $u$ decays exponentially on any compact set: $$\|u(t,\cdot)\|_{H^2(\Omega)} \leq C_\Omega e^{-\gamma t}\bigl(\|u_0\|_{H^1(\mathbb R^d)} + \|u_1\|_{L^2(\mathbb R^d)} \bigr), \quad t \geq T,$$ for any bounded domain $\Omega$, for some $C_\Omega$, $\gamma$, $T > 0$. This follows, in particular, from the resonance expansion of the solution $u$ (see, e.g., Thm. 3.9, p. 99 of these notes by Dyatlov-Zworski) and from the absense of real non-zero resonances due to Rellich's uniqueness theorem.

But what is the situation in even dimensions? Do we still have exponential decay when zero is not a resonance and the discrete spectrum of $H$ is empty?

### convert sum binary number to power of 2 [on hold]

Tue, 02/27/2018 - 08:10

I was reading the Supporting Information of paper.

I

I was confused the part of converting sum binary number to power of 2.

The detail and partial paper as follow:

Replacing the fractions $1/\sum_{j=1}^{d_i}[e_h]_j$ by the negative powers of 2 gives

seems:

$$\sum_{j=1}^{d_i}[e_{{}_h}]_j=2^{\sum_{j=1}^{d_i}\,[e_{{}_h}]_j-1}$$

$e_h$ is a series of binary number. For example four binary number series, $d_i=4$. The 9th series is $e_{9}=[1\, 0\, 0\, 1]^T$, where $[e_9]_1 = 1$, $[e_9]_2 = 0$, $[e_9]_3 = 0$, $[e_9]_4 = 1$. So sum those binary number: 1+0+0+1=2.

$$\sum_{j=1}^4[e_{{}_9}]_j=2^{\sum_{j=1}^4\,[e_{{}_9}]_j-1}=2$$.

But what if the 11th series: $e_{11}=[1\, 0\, 1\, 1]^T$, where $[e_{11}]_1 = 1$, $[e_{11}]_2 = 0$, $[e_{11}]_3 = 1$, $[e_{11}]_4 = 1$.

$$\sum_{j=1}^4[e_{{}_{11}}]_j=1+0+1+1=3$$ $$2^{\sum_{j=1}^4\,[e_{{}_{11}}]_j-1}=2^2=4$$.

II

$$\frac{1}{2^{\sum_{j=1}^{d_i}\,[e_{{}_h}]_j}}=\prod_j(\frac{1}{2}[e_{{}_h}]_j+(1-[e_{{}_h}]_j)1)=\prod_j(1-\frac{1}{2}[e_{{}_h}]_j)$$

No clues about that the left power of 2 equal to the right product part.

Could you give me any favor? Thanks in advance.

You can help me to edit the question. Thanks.

### Kazhdan constant and finite index subgroups

Tue, 02/27/2018 - 01:18

I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups?

Let $G$ be a finitely generated group with a generating set $\Sigma$ that satisfies Kazhdan property (T) with constant $\kappa(G,\Sigma)$.

1) if $\Gamma$ is a finite index subgroup of $G$, is there a generating set $\Theta$ of $\Gamma$ for which $\kappa(\Gamma,\Theta)$ can be estimated in terms of $\kappa(G,\Sigma)$?

2) if $G$ is a finite index subgroup in $H$, is there a generating set $\Theta$ of $H$ such that $\kappa(H,\Theta)$ can be estimates in terms of $\kappa(G,\Sigma)$?

### fibrations of classyfing spaces - Leray Hirsch Theorem converse

Mon, 02/26/2018 - 16:16

Let $G$ be a topological group and let $H$ be a closed subgroup, assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG$$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ it is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence ; but I want to consider cases where $G$ is not connected.

EDIT 28/02

Looking around , I realize that maybe the Leray-Hirsch theorem might play a role here in some specific situations: If the spectral sequence collapses, and $H^*(G/H)$ is a free $R$-module, then $H^*(BH)$ is a free $H^*(BG)$-module.

Conversely, if I assume that $H^*(G/H)$ is a free $R$-module, and $H^*(BH)$ is a free $H^*(BG)$-module, does it follows that the spectral sequence collapses and $H^(BH) \cong H^(BG) \otimes H^*(G/H)$\$