2
votes
0answers
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
1answer
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
0answers
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
3answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
2answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
2answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
3answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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 ...

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