# All Questions

**2**

votes

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**6**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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 ...