2
votes
1answer
28 views

Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
0
votes
0answers
21 views

“Dimension” of ideals in $F_q[x]/\langle x^n-1\rangle$?

I'm very much confused by algebra. Hoping to get a bit more comfortable I tried to compute different things and see what happens... Let $F_q$ be the finite field with $q$ elements and ...
-1
votes
0answers
16 views

Is a n-th derivative of elliptic modular function $j(z)$ modular form?

Let $j(z)$ be a elliptic modular function. Then $j(\gamma z) = j(z)$ for all $\gamma \in SL(2,\mathbb{Z})$. I would like to see whether $\frac{d^n}{dz^n}j(z)$ be a modular form for $n = 1,2,3$. ...
1
vote
0answers
87 views

research statement for assistant professorship [migrated]

What should a research statement for an application for an assistant professorship contain (pure mathematics in Germany)? How long should it be?
0
votes
0answers
9 views

A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...
-2
votes
0answers
38 views

Lifting of a Diffeomorphism to an Orientable Double Cover [on hold]

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
0
votes
0answers
26 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best $k$ possible such that ...
1
vote
1answer
51 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
0
votes
0answers
36 views

$L^1$ convergence to equilibrium of solutions of heat equation

Let $u$ and $v$ be the weak solutions of $$u_t - \Delta u = f$$ $$u(0)=u_0$$ and $$-\Delta v = f$$ $$|\Omega|^{-1}\int_\Omega v =0$$ on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...
0
votes
1answer
28 views

A question on decreasing function [on hold]

Let $t\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} + f(t) = 0$ $f(t) $ is continuous decreasing function of $t$. $a_i\ge0$ for all $i$. $y(t)$ is positive real zero of the first equition. Can ...
4
votes
2answers
176 views

Collection of dense subsets as a “fingerprint” for Hausdorff topologies?

Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau_1 \neq \tau$ on $X$ such that ...
0
votes
0answers
10 views

Circle actions on graph C*-algebras

Do all graph C*-algebras admit actions of the circle? Suppose we have a graph C*-algebra which we know is the quotient of a graph algebra by a circle action. Is it possible to read off the original ...
2
votes
0answers
86 views

splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...
-3
votes
0answers
55 views

Differential geometry [migrated]

If we have integrable distribution D of rank k on a manifold, and we have k functions which are constants on the associated integral manifolds. can we glue together these functions to obtain global ...
0
votes
0answers
74 views

Advice on dealing with the gap [on hold]

A young mathematician AA is writing a paper proving a property X for a certain model. There have been quite a few articles proving the property X for various models. One of the first ones, let's call ...
6
votes
2answers
126 views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
0
votes
0answers
35 views

Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...
-2
votes
0answers
30 views

Probability Inequality when X > Y > 0 [on hold]

I want to know whether the following statement is true or not, and the proof. Let X, Y be random variable, satisfying X > Y > 0, and have finite variance, $Var(X) < \infty$ and $Var(Y) < ...
7
votes
0answers
62 views

A seemingly simple combinatorial object that must have an easy generating function

One more question related to my earlier "Special" meanders. I am trying to isolate simplest problems related to it. Here is one. For a composition (i. e. a tuple of natural numbers) ...
0
votes
0answers
10 views

Numerical evaluation of orthogonal polynomials [on hold]

I've written some Matlab procedures that evaluate orthogonal polynomials, and as a sanity check I was trying to ensure that their dot product would be zero. But, while I'm fairly sure there's not ...
-2
votes
0answers
58 views

a naive question: is the category of moniods cartesian closed? Why? [on hold]

I read steve awodey's "category theory" and could't solve the exercise in chapter6 above. Here i speak the "category of moniods" the category with objects moniods and arrows homomophisms between ...
0
votes
0answers
22 views

measure of the distance between two joint distributions

I am trying to measure the 'closeness' of two joint distributions. Specifically, I have I have economic supply curves for different time periods. Each curve is made up of a vector of prices and ...
1
vote
1answer
56 views

Absoluteness and Tree Representations

Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional ...
0
votes
0answers
32 views

How do I evaluate this integral [on hold]

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
3
votes
0answers
89 views

Flat + locally of finite presentation + monomorphism = open immersion

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$: Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then ...
5
votes
3answers
304 views

Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
25
votes
3answers
2k views

What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...
2
votes
1answer
179 views

Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...
4
votes
1answer
161 views

The fibration map $Diff(M) \rightarrow Emb(N,M)$

Let $M$ be a non-compact manifold, equipped with a (closed?) submanifold $N\subset M$. The action of $Diff(M)$ on the set of embeddings $N\hookrightarrow M$ induces a map $$ Diff(M) \rightarrow ...
1
vote
1answer
233 views

research articles in topology/geometry [on hold]

There is a saying "Do you read the masters?" I want to read some basic papers in Topology/geometry... I can not clearly state what is basic as of now... My back ground includes course in ...
4
votes
0answers
92 views

Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected. If there exists ...
13
votes
6answers
452 views

What (fun) results in graph theory should undergraduates learn?

I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...
2
votes
1answer
53 views

Classifying Low Dimensional Solutions of the Yang--Baxter Equation

What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions? To make my question more specific, have all solutions for dimension $2$ and $3$ been ...
6
votes
1answer
217 views

Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
16
votes
1answer
253 views

How big are the prime factors of $2^kp - 1$?

I have already asked this question here. No answers despite the bounty (which has now ended) Let $p$ be a prime number, $p > 3$. Does there always exist $k \in \mathbb N_{\ge 1}$ such that the ...
2
votes
0answers
60 views

Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations: $[U_i, U_j] = 0$ for $|i-j|>1$ ...
-3
votes
0answers
29 views

Riemannian metric on an open dense subset [on hold]

If we have a description of the riemannian metric $g$ on an open dense subset $U\subset M$, then can we say that $M$ should have the metric $g$ on whole $M$? For example, on some open dense subset ...
6
votes
1answer
65 views

When do powers and ends in functor categories act pointwise?

$\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$Let $(\C, \otimes, I, \multimap)$ be a complete closed monoidal category and $\I$ a small ...
-2
votes
0answers
8 views

A specific question regarding a proof in Hassan Khalil's book, Nonlinear Systems [migrated]

I am trying to understand the proof of a Lemma in the book 'Nonlinear Systems' by Hasaan Khalil (3rd edition). In the Proof of Lemma 3.1, about Lipschitz continuity of vector valued functions, I am ...
3
votes
0answers
295 views

Grothendieck, A Place to Begin [on hold]

I'm finishing up an undergrad degree in mathematics and am beginning to think about areas of research. I know that the work of Grothendieck is considered the cornerstone of modern algebraic geometry, ...
-4
votes
0answers
25 views

Vector application [on hold]

A rope is hung at both ends from a horizontal beam, and a weight m is suspended from it. The left part of the rope exerts a force G at P, while the right part of the rope exerts a force H. Find the ...
1
vote
0answers
53 views

Perturbating the boundary of a helicoid

I prepare a long helix with many periods, so that I can obtain a helicoid with soap film, i.e. a minimal surface whose boundary is the helix. The helix is not perfect, it is unavoidable that some ...
-4
votes
0answers
36 views

calculus integral with logs [on hold]

Why the solution of this integral $\displaystyle \int \frac{dx}{15-3x}$ is... $-\frac{1}{3} \ln \mid15-3x\mid$. I can't understand where $-\frac{1}{3}$ comes from, if the integral has not been ...
5
votes
2answers
146 views

When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of ...
0
votes
0answers
170 views

Faltings height on pair $(\mathcal X,\mathcal D)$

Let $(\mathcal X,\mathcal L)$ be a semi-stable Abelian variety over number field $K$ and possessing a Neron differential $\omega\in \operatorname{H}^0(X,\Omega_X^{\text{dim}X})$, then the Faltings ...
-1
votes
0answers
68 views

A question about Kähler Einstein metric [on hold]

Let $X$, and $Y$ are Kähler manifolds and $f:X\to Y$ is birational and let on $(Y,\omega)$ we have $\text{Ric}(\omega)=-\omega$, then Kähler Einstein metric on $X$ can be of which form? can we say it ...
-5
votes
0answers
35 views

Maple: In Matrices A x B = C, how do I find matrix A given B and C [on hold]

I have matrix A, B, and C which are all 8x8 matrices in Maple. in the equation A x B = C, when B and C are known, how do I find matrix A? I know how to do it by hand, but I don't know the maple ...
2
votes
0answers
62 views

Lie Symmetries of the Bessel Differential Equation

The Bessel differential equation has an arbitrary looking form, but a lot is known about it. $$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$ Is there a way to derive the Bessel ...
-5
votes
0answers
29 views

How to calculate how much more wins to get a certain winrate? [on hold]

I have two values, wins and total games played. To calculate the win rate I use the normal «formula»: wins/totalGamesPlayed*100; But let's say I have 21 wins ...
1
vote
0answers
46 views

Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...

15 30 50 per page