# All Questions

**2**

votes

**0**answers

47 views

### Second eigenvalue of biased reflected random walk

Let $Z_n$ be a reflected random walk on the non-negative integers with negative drift. That is,
$Z_n$ is non-negative and moves one to the right w.p. $p<1/2$. It moves one to the left w.p. $1-p$, ...

**2**

votes

**1**answer

94 views

### Positive rational numbers as sum of unit fractions [duplicate]

Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see ...

**4**

votes

**2**answers

155 views

### Find the expansion of the exact solution (beyond Taylor)

In a paper by Kitagawa & Ueda Squeezed spin states they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V_{min} \propto S^{-2/3}$. I will shortly describe ...

**7**

votes

**0**answers

111 views

### Complete list of exceptions and efficient algorithm for Waring's problem

2 weeks ago, Samir Siksek http://arxiv.org/abs/1505.00647 proved more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, ...

**2**

votes

**0**answers

45 views

### Rank of the augmented jacobian matrix

I'm struggling to understand the proof of the following theorem found in 'Solving the Likelihood Equations'.
Suppose that $V$ is a complete intersection, i.e. its defining ideal $P$ can be generated ...

**-1**

votes

**0**answers

35 views

### Which spaces admit bump functions? [migrated]

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets.
Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...

**0**

votes

**0**answers

126 views

### Are del-Pezzo surfaces complete intersections?

Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$
to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$
for some $n$.
$\textbf{Question:}$ Can $X_k$ ...

**0**

votes

**1**answer

79 views

### topology of setwise convergence of measures

It is well known that if $X$ is, say, compact and metric, then the set of probability measures on the Borel subsets of $X$ endowed with the usual topology of weak convergence of measures has as a ...

**10**

votes

**1**answer

234 views

### Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$

Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?

**4**

votes

**1**answer

112 views

### What's the relation between spin model for subfactors theory and physics?

In the sense of subfactor theory, a spin model is a commuting square of the form
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset ...

**0**

votes

**0**answers

11 views

### Copula theory on discrete random variables [on hold]

How can I find the joint pmf on two discrete random variables using the copula theory

**3**

votes

**1**answer

82 views

### Measurability and continuity for general topological spaces

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K ...

**0**

votes

**0**answers

14 views

### Approximation of a generlized hypergeometric function for large parameters

I found the following approximation in the literature:
If z is fixed and $|ph(1−z)|<π$. Assuming that none of a1,a2,…,ap is a nonpositive integer, $p<q$, $r\rightarrow+\infty$. For each ...

**2**

votes

**0**answers

54 views

### Newton polyhedron and product of ideals

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and
$J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ...

**3**

votes

**1**answer

66 views

### Hyperelliptic generalization of Euler's formula

Are there any hyperelliptic generalizations of the following formula, first proved by Euler in 1782,
...

**0**

votes

**1**answer

63 views

### What is the maximal number of solutions of $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$?

What is the maximal number of solutions of the following equation?
$\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$
where $x$ is the unknown and $n, m$, $a_i$'s, $b_i$'s are constant.
It ...

**7**

votes

**1**answer

197 views

### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...

**1**

vote

**0**answers

21 views

### Is there an equivalent form for Wishart to a power times a normal?

Lin described two equivalent characterizations of the multivariate t-distribution, viz.
As a normal vector divided by an independent chi. That is, $t = Z / \sqrt{\chi^2/v}$, where $Z$ is ...

**-1**

votes

**0**answers

26 views

### Probability of co-occurence [on hold]

Of total N people, m people are good at mathematics and c people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...

**0**

votes

**0**answers

28 views

### Functional approximation that vanishes at infinity

I have a function $f(x)$ that I wish to approximate using a linear combination of basis functions
$$
\hat{f}(x) = \sum_{i=1}^k c_i \varphi_i(x).
$$
The approximation is done via an orthogonal ...

**1**

vote

**0**answers

37 views

### Projecting a convex partition onto a convex set

Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of ...

**1**

vote

**0**answers

30 views

### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...

**0**

votes

**0**answers

47 views

### Global existence solutions NLS

Let's consider the following NLS in $\mathbb{R}^3$
$$i\partial_t\psi=-\Delta\psi+\vert\psi\vert^2\psi$$
How to prove that $H^2$-solutions are globally in time? Can someone suggest references to me?

**1**

vote

**0**answers

54 views

### Does the monodromy of such VHS have to be trivial

Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...

**3**

votes

**0**answers

56 views

### Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?
More specifically given a smooth manifold of $M$ and a ...

**0**

votes

**0**answers

60 views

### partial pullback-completion of a category

Let $\mathcal{C}$ be a (possibly enriched) category with all finite product, and $\mathbf{M}$ a class of morphisms.
Can one construct completion of $\mathcal{C}$ w.r.t. all pullbacks along morphisms ...

**0**

votes

**0**answers

48 views

### inequality involving determinants and quadratic forms [on hold]

I'm interested in comparing $\det(x'\boldsymbol{A}x)$ and $\det(\boldsymbol{A})x'x$ where $\boldsymbol{A}$ is symmetric positive semidefinite, and $x$ is a free vector of constant. My argument is: ...

**8**

votes

**1**answer

121 views

### Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?
For ...

**2**

votes

**2**answers

110 views

### Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent (“weakly”)?

It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I ...

**4**

votes

**1**answer

240 views

### Flat Riemannian manifold

Is it true that a Riemannian manifold is flat, if and only if a coordinate transformation $f$ exists, such that the geodesics after transformation is in linear form ...

**4**

votes

**2**answers

171 views

### hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...

**7**

votes

**0**answers

66 views

### Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An ...

**0**

votes

**0**answers

26 views

### Bounds on the moments of truncated sub-gaussian random variables

If $X$ is a centered sub-gaussian random variable, then there exists a constant $c$ such that
$$
\mathbb{P}[|X|>t] \leq \exp(1-ct^2)
$$
for all $t\geq 0$. Moreover, we know that the normalized ...

**7**

votes

**0**answers

214 views

### Why does $Mf$ always support an $Mf$-orientation?

Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...

**3**

votes

**1**answer

143 views

### Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...

**5**

votes

**3**answers

408 views

### Permutations with all cycles odd length and permutations with all cycles even length

If $n$ is even, then the number of permutations of $n$ in which all cycles have odd length equals the number of permutations of $n$ in which all cycles have even length. This fact is easily proved, ...

**0**

votes

**0**answers

86 views

### Cycle class map in smooth quasi-projective varieties

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ be a closed subvariety of codimension $k$.
Q1. How to define a cycle class $[Z]\in H^k(X,\Omega_X^{k})$ ?
Q2. More general, ...

**-5**

votes

**0**answers

60 views

### Integral 1/(2x) [on hold]

What is right way to calculate this integral and why?
$$
\int\frac{1}{2x}dx
$$
I thought, that substituion is right
$$
t = 2x
$$
$$
dt = 2dx
$$
$$
\frac{dt}{2} = dx
$$
$$
...

**0**

votes

**0**answers

51 views

### Sum of weighted permutations over natural numbers [on hold]

If we consider the sum $\sum\limits_{i=1}^n i\sigma(i)$ where $\sigma$ is a permutation of the first $n$ natural numbers, then this sum has a few interesting properties for various values of $n$. My ...

**0**

votes

**0**answers

56 views

### Multiple convex sets hyperplane separations

Hyperplane separation theorem states that if there is a bounded convex set $X$, a convex set $Y$, then there is a hyperplane that separates $X$ from $Y$.
That is, there is a statement that gives ...

**10**

votes

**0**answers

149 views

### Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...

**2**

votes

**0**answers

54 views

### Geometric/algebraic interpretation of quadratic points of rank r

In the paper of Eckl/Puhklikov (http://arxiv.org/abs/1210.3715) the following terminology is introduced:
"
Let $X \subset Y$ be a subvariety of codimension 1 in a smooth quasiprojective
complex ...

**2**

votes

**2**answers

58 views

### Resolvent operator of fractional Laplacian

For $0<\alpha<2$, we define the fractional Laplacian with Fourier transform
\begin{align}
\widehat{(-\Delta)^{\frac{\alpha}{2}} u}(\xi) = |\xi|^\alpha \widehat u(\xi).
\end{align}
Consider the ...

**0**

votes

**0**answers

16 views

### Binary strings and one-to-one correspondence [on hold]

Assume T be the set of binary strings of length 30 with 10 1’s and 20 0’s. Let X be the set of the first 30 positive integers {1,2,3,…,30}. Let Y be the set of all subsets of X containing 10 numbers. ...

**2**

votes

**1**answer

67 views

### Epsilon-approximations of set systems with finite VC-dimension

ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following
Theorem. For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ ...

**1**

vote

**0**answers

26 views

### Fusion category R-symbols diagonal even when multiplicity present?

Preface: Please bear my lousy ASCII :-)
I automatically thought that a twist move on a trivalent node -(/ = a* -( would be diagonal even in the case of multiplicity, for example: -1(/= a* -1( + 0* ...

**2**

votes

**3**answers

539 views

### A Diophantine equation with prime powers

Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$?
This specific equation appears when computing order components of finite groups.

**0**

votes

**0**answers

56 views

+50

### Cardinality of non-integer points in the translation of the Minkowski sum of convex hull.

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_m\}$. Let $\mathbb{Z}_+=\mathbb{N}\cup\{0\}$ and $\mathbb{Q}_+$ denotes the positive (inluding 0) rational numbers. ...

**5**

votes

**1**answer

187 views

### Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory.
The facts are stated as follows:
Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally ...

**2**

votes

**1**answer

55 views

### Dedekind-MacNeille completion of the strictly increasing members of $\omega^\omega$

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ ordered by $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \omega$.
Set $K = \{f\in \omega^\omega: m<n\in \omega \implies ...