# All Questions

**2**

votes

**0**answers

68 views

### Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...

**5**

votes

**1**answer

204 views

### Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?
Given a convex body K,, such that t K=-K, is there a point; such that $diam(\partial K)=d(x,-x)$
I am ...

**2**

votes

**0**answers

57 views

### Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange.
Question: Which inequalities similar to the famous isoperimetric inequality is known?
conjectured?
I recently learned about some ...

**13**

votes

**3**answers

1k views

### Algebra and Cancer Research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research ?
It is well-known that mathematical modeling and computational biology are effective tools in cancer ...

**2**

votes

**0**answers

103 views

### Heat kernel and Wiener measure

A theorem by Barry Simon says that for arbitrary open sets $\Omega\subset \mathbb{R}^n$, we have $$[\exp(t\Delta_{\Omega}^D)](x,y) = \mu_{x,y,t}\lbrace \omega \text{ } \vert \text{ } \omega(s) \in ...

**0**

votes

**1**answer

72 views

### Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...

**3**

votes

**0**answers

124 views

### Dead Flies Problem [duplicate]

If a set of points in the plane contains one point in each convex region of
area 1, then can it have finite density?
what is the density of the points? In my understanding, it means the average ...

**-1**

votes

**0**answers

43 views

### vector-matrix notation and expectation of matrix and Hermitian product [on hold]

Let $\textbf{h} \in \mathbb{C}^{N\times 1}$, $\textbf{a} \in \mathbb{C}^{N\times 1}$, $\textbf{b} \in \mathbb{C}^{N\times 1}$ and $\textbf{c} \in \mathbb{C}^{N\times 1}$. The variable $h_i$ is defined ...

**1**

vote

**0**answers

25 views

### abstract simplicity results for anisotropic quasi-simple algebraic groups defined over a non-archimedean local field

I was wondering if anyone has some references I can look at that deal with results that identify some abstractly quasi-simple normal subgroup of the group of rational points of an anistropic ...

**4**

votes

**2**answers

224 views

### Simple groups and words

Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...

**3**

votes

**2**answers

173 views

### Schreier's index formula

A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...

**1**

vote

**0**answers

70 views

### number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...

**0**

votes

**0**answers

84 views

### Find two triangles of longest side length 25? [on hold]

I'm using the quadratic Diophantine equations to solve for two integer triangles of longest side 25. It's been shown that for $a^2 + b^2 = c^2$, which goes to $x^2 + y^2 = 1$ where $x = a/c$, $y = ...

**0**

votes

**0**answers

37 views

### Canonical relations and phase functions of a Fourier Integral Operator

I'm thinking about the (semiclassical) Fourier Integral Operator $T$ given by
$T=h^{-n}\int{e^{i\phi(x,y,\theta)/h}a(x,y,\theta,h)d\theta}$
(that is, $T$ has phase $\phi$ and amplitude $a$). ...

**4**

votes

**2**answers

310 views

### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

**8**

votes

**0**answers

100 views

+100

### $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...

**3**

votes

**1**answer

136 views

### What are the modular transformation properties of q-Pochhammer symbols?

Do q-Pochhammer symbols, defined as
$(a;q) = \prod_{k=0}^{\infty} (1- a q^k)$
have known modular transformation properties? That is, if we write $q = q[z] = e^{2\pi i z}$, is there any reasonably ...

**14**

votes

**2**answers

542 views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a ...

**0**

votes

**1**answer

70 views

### Non-hyperbolic fixed points in multidimensional systems

Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start ...

**5**

votes

**2**answers

185 views

### Limit of distance between two random points in a unit $n$-cube

What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average ...

**2**

votes

**2**answers

123 views

### Perimeter of a 'trapped' convex set

Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit ...

**0**

votes

**0**answers

40 views

### Total differential of Lipschitz submanifolds embedding

My interest is analysis on Lipschitz manifolds. I want to define traces of differential forms on a Lipschitz submanifold $N$ of a Lipschitz manifold $M$. In other words, I want to push-forward ...

**3**

votes

**1**answer

157 views

### References for von Neumann Algebras

I have some -possibly- simple but broad questions: Where to begin the study of von Neumann Algebras? Which are the important questions in the field that guide current research? I'm interested in ...

**2**

votes

**0**answers

83 views

+50

### projective modules over noncommutative tori

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...

**-2**

votes

**0**answers

55 views

### Complexity of a function [on hold]

I am looking for a natural definition of the complexity a function. If the image is discrete, i was thinking it could be: consider the preimage of an element of the image, count the number of ...

**-5**

votes

**1**answer

76 views

### computing integral of dz/(z+1) on unit circle [on hold]

I guess it must be a simple matter in complex analysis, but I would like to compute the the following integral:
$$
\oint_C\frac{dz}{z+1}
$$
and
$$
\int_C\frac{dz}{z+1}
$$
where $C=\{z\in \mathbb{C}: ...

**3**

votes

**1**answer

104 views

### In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...

**13**

votes

**0**answers

235 views

### The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on ...

**24**

votes

**1**answer

2k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**1**

vote

**0**answers

34 views

### Identification of model involving convex polynomials

I want to solve a nonlinear least squares problem on the following form
\begin{equation}
\begin{array}{l}
\min_{\theta,\phi} J(\theta,\phi) &=& \min_{\theta,\phi} \sum_{i=1}^k ...

**1**

vote

**1**answer

86 views

### infinitary logic and partial fixed point logic

Is there a property definable in finite-variable infinitary logic $L^{\omega}_{\omega\infty}$ but not definable in partial fixed point logic FO(PFP) ?

**-9**

votes

**0**answers

149 views

### Fermat and the abc conjecture [on hold]

There exist finitely many triples of coprime $a+b=c$ such that $$|\frac{b}{c}|^n+|\frac{a}{b}|^n=b^n-a^n$$ where $n>2$ and $b^n-a^n=\mathfrak{Prime}.$
We know that it maybe true in this version ...

**0**

votes

**1**answer

61 views

### reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...

**-1**

votes

**0**answers

56 views

### Rising Sun Inequality (Dunford-Schwartz maximal inequality) [migrated]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function
$$f^*(x) := ...

**0**

votes

**0**answers

30 views

### Bounding Rayleigh quotioent for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...

**8**

votes

**0**answers

189 views

### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...

**2**

votes

**3**answers

61 views

### a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...

**1**

vote

**1**answer

184 views

### Why tangent vector of statistical manifold is a function?

In differential geometry, tangent vectors are considered operators.
At point p, the local tangent space is defined as
$$
T_p(M)=\{X^i\partial_i|X\in R^n\}
$$
This is quite easy to understand for me.
...

**1**

vote

**0**answers

48 views

### About the integral of the plurisubharmonic functions on the whole manifold

Let $(M,\omega)$ be a Kahler manifold with positive first Chern class. $\omega\in c_1(M)$ is a Kahler current. $D$ is a smooth simple divisor in $-K_M$. $s$ is a determining section of $D$. The ...

**3**

votes

**0**answers

233 views

### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

**-2**

votes

**0**answers

60 views

### Books and papers on differential equation method [on hold]

I wanted to understand the differential equations method for analyzing stochastic sequences. Is there a good book/ papers that provide a gentle survey this topic with a good number of examples? A good ...

**0**

votes

**0**answers

10 views

### Equating coefficients [migrated]

Heading
Excuse me,i don't know how to deal with this problem,i try it for all time of last night,
this equation is on "Concrete Mathematics" page 200:
d(n) is the number of derangements,e^z is the ...

**-2**

votes

**0**answers

50 views

### solving an exponential inequality with parameters [on hold]

Can someone help me how I can solve this inequality:
$$b^{x/2-3/2} - b^{ax} - 2c > 0$$
where $b, a, c$ are the parameters. I want to solve it to have a range for $x.$
If I should do it with a ...

**2**

votes

**2**answers

137 views

### If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...

**5**

votes

**2**answers

327 views

### Geometric explanation of Hutton's formula?

$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$
Is there some geometric construction that explains this beautiful equation
(known as Hutton's formula)?
Perhaps a "proof without ...

**2**

votes

**1**answer

317 views

### On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

In response to a comment posted under
Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...

**2**

votes

**1**answer

41 views

### Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end.
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...

**0**

votes

**0**answers

114 views

### Conjecture relating differential equation and sum of a function over partitions

The following is an addition to A function from partitions to natural numbers - is it familiar?; the function $f(\lambda)$ as defined there, when summed over all partitions of n, gives an unexpected ...

**6**

votes

**1**answer

325 views

### The sum over zeros in the explicit formula for $\zeta(s)$

The explicit formula for $\zeta(s)$ is:
$$
\psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right),
$$
where ...

**-4**

votes

**0**answers

62 views

### Recurrence relation practice problem that I can't figure out [on hold]

thanks for taking the time to look at this problem. I'm trying to prepare for a test on Monday by doing some extra odd numbered problems from my textbook. I'm having a lot of trouble trying to solve ...