# All Questions

**-1**

votes

**0**answers

11 views

### calculus integral with logs

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 ...

**2**

votes

**1**answer

61 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: ...

**-3**

votes

**0**answers

20 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 ...

**-3**

votes

**0**answers

26 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

**1**answer

132 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
...

**-2**

votes

**0**answers

23 views

### Arrange numbers? [on hold]

hello the question i would like to ask is very difficult as english is not my native language so here it goes...I need a program or exel chart that arrange numbers in a group in order to get the ...

**1**

vote

**0**answers

25 views

### Perturbating the boundary of a helicoid

I prepare a long helix (with many periods) as the boundary of a long helicoid. I unavoidably made some mistake and the helix is not perfect, some perturbation or even defect is happening somewhere. ...

**1**

vote

**0**answers

48 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 ...

**0**

votes

**0**answers

37 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 ...

**5**

votes

**0**answers

100 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 ...

**1**

vote

**0**answers

26 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 ...

**-1**

votes

**0**answers

40 views

### A question about Kahler Einstein metric

Let $X$, and $Y$ are Kahler manifolds and $f:X-->Y$ is birational and let on $(Y,\omega)$ we have $\text{Ric}(\omega)=-\omega$, then Kahler Einstein metric on $X$ can be of which form?
can we say ...

**1**

vote

**0**answers

20 views

### Common Point of Intersection of n-dimensional ellipsoids [on hold]

Suppose we have two ellipses in 2-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have 4 points of intersection. Can we say that in ...

**-4**

votes

**0**answers

28 views

### A set containing more than half elements of a group [on hold]

I wish to prove the exercise which states that for a set $A$ containing more than half elements of a group $G$, every element of $G$ is a product of two elements of $A$.
My attempt:
By Lagrange ...

**-2**

votes

**0**answers

19 views

### AQA A Level Normal Distribution [on hold]

The question goes like:
A wholesaler decides to grade such oranges by weight. He decided that the smallest 30% should be graded as small, largest 20% as large and in between as medium.
The ...

**3**

votes

**0**answers

73 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 ...

**3**

votes

**0**answers

19 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 ...

**-4**

votes

**0**answers

21 views

### Trouble with an assignment [on hold]

Can anyone please be kind enough to help me with this.
On one shelf there are 5 hardcover books and 6 paperbacks and on the other shelf there are 7 hardcover and 4 paperback. From the first shelf ...

**0**

votes

**0**answers

13 views

### Markov Modulated Markov Chain

Consider a discrete time Markov chain $X_t$ on some finite state space $\mathcal{S}$ with transition matrix $P$. Now consider a process $Y_t$ also on $\mathcal{S}$, which conditioned on $X_{t}=s$ ...

**4**

votes

**0**answers

74 views

### Moduli space of complex Tori

Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?

**10**

votes

**0**answers

125 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 ...

**4**

votes

**0**answers

73 views

### Fibers of a morphism

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.
If there exists a point $z_0\in ...

**3**

votes

**1**answer

80 views

### Least ordinal not embedded in a total order

If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$.
I am trying to prove the following:
If $(M,+,.,0,1)$ is a model of open induction, (or ...

**3**

votes

**0**answers

34 views

### Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...

**2**

votes

**0**answers

47 views

### Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?

**0**

votes

**0**answers

18 views

### Inequality for coefficient of ergodicity

Let $Α$, $B$, $C$ stochastic matrices and $τ(Α)= \max(A^T(e^i - e^j) )$, coefficient of ergodicity. We know that $τ(ΑΒ)\le τ(Α)τ(Β)$. Is true that $τ(ΑΒC)\le τ(ΑC)$
if $B$ has positive digonal ...

**4**

votes

**0**answers

36 views

### Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...

**2**

votes

**2**answers

50 views

### Symplectic manifolds with dense group of periods

Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H_2(M) \to \mathbb{R}$, whose image $\Gamma_{\omega} \subseteq \mathbb{R}$ is called the ...

**0**

votes

**0**answers

17 views

### How do we prove that a specific kernel is positive definite (case of logarithm)? [on hold]

I have a problem proving that some specific kernels are positive definite. In general, I can find the answer quickly enough but here I have a specific case involving a logartihm :
$K : ...

**0**

votes

**0**answers

19 views

### The derivatives of Riemann xi function [migrated]

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...

**0**

votes

**0**answers

29 views

### clustering permutations by shared subsequences [on hold]

I have a question, stimulated by some biology, about comparing sets of permutations.
The problem
Let's think of genes on a bacterial chromosome as beads on a string - atomic, unique objects, with ...

**2**

votes

**1**answer

65 views

### How many times does a simple symmetric random walk of length n return to the origin?

Consider the simple symmetric random walk on the integers starting from
the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as
$$ w:= \omega_0 \omega_1 \ldots ...

**0**

votes

**0**answers

31 views

### Context Free Languages closed under Kleene Star? [on hold]

I'm looking at the proof for the closure property of CFL under kleene star and I'm having a little trouble understand what it means. From what I saw this is the proof:
...

**1**

vote

**0**answers

47 views

### Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$.
So $G\in\mathcal{G}$ can be written ...

**0**

votes

**0**answers

45 views

### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...

**2**

votes

**0**answers

62 views

### extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$.
Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$.
If we have a $\mathbb{Z}_{\ell}$ local ...

**1**

vote

**0**answers

34 views

### moduli space of curves under prescribed tangency conditons

We consider an irreducible component of the Hilbert Scheme of curves in $\mathbb P^2$. Denote it as $\mathcal D.$ We fix a line $L$ and a point $A\in L.$ Denote $\mathcal D_0$ as the subscheme of ...

**2**

votes

**0**answers

41 views

### Factoring a multiset into a product of two multisets

Given a multiset $S$ of $mn$ numbers, how hard is to find multisets $A$ and $B$, of $m$ and $n$ numbers respectively, such that
$$ S = \{ xy \mid x\in A, y\in B \}~~~~\text{(multiset sense)}, $$
or ...

**21**

votes

**3**answers

999 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 ...

**1**

vote

**1**answer

110 views

### Direct image for crystals?

If we get a morphism $f : X \to Y$ of schemes over $k$, how should I define the direct image functor
$$
f_* : Crys(X) \to Crys(Y)?
$$
By a crystal I mean a quasi-coherent sheaf $M$ on $X$ with a ...

**3**

votes

**2**answers

148 views

### Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...

**9**

votes

**1**answer

141 views

### Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...

**2**

votes

**0**answers

62 views

### A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.
The Künneth-Theorem which I ...

**0**

votes

**0**answers

16 views

### Connected sum of two “same” Kleins bottles [migrated]

If I have two surfaces of Klein's bottle, K, given by edge words [a b- a- b-] and [a- b a b] what space do I get when I identify same edges. In other words, i have two same Klein's bottles that are ...

**5**

votes

**3**answers

154 views

### Numerical invariants for a graph or its complement that are bounded by some constant

I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but ...

**1**

vote

**0**answers

18 views

### Equivalence of first order quasilinear PDE to linear PDE [migrated]

Given a system of nonlinear PDE of the special form:
$\sum_{i=1}^n A_i(x, \phi) - \frac{\partial \phi_j}{\partial x_i} = B_j(x,\phi) $ $(1)$
with $(j=1,...,m)$ and $x \in R^n,\phi \in R^m$. If we ...

**2**

votes

**1**answer

41 views

### Properties of bipartite graphs

For a connected bipartite graph $G$ are the two following properties equivalent:
1)Every minimal cycle in $G$ has length 4, that is every cycle of length strictly greater than 4 can be divided in ...

**4**

votes

**2**answers

106 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

122 views

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

Let $\mathcal X$ 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 height can be ...

**3**

votes

**0**answers

126 views

### A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form
$$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...