3
votes
2answers
260 views

the space of continuous maps between 3-manifolds

Let $X$ be a connected hyperbolic 3-manifold (without boundary), $S^3$ the 3-sphere and $Map(X,S^3)$ the space of continuous maps between $X$ and $S^3$. Question: Is the space $Map(X,S^3)$ connected ...
0
votes
0answers
39 views

References on hyperinvariant

Reading Dubrovin & Novikov text: Modern Geometry, "hyperinvariant" is mentioned. Can someone give me references on the concepts and notations ?
-2
votes
1answer
93 views

Recursion, Common Term, Combinatorics [closed]

May we find the common term for recursive sequence? if yes that how to find the common term of recursive sequence such: 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6 ... in a ...
3
votes
1answer
186 views

Löwenheim-Skolem for many-sorted theories

Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete ...
3
votes
3answers
109 views

Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by \begin{eqnarray*} s_0 &=& a, \\ s_1 &=& b, \text{and} \\ s_{n+2} &=& s_{n+1} ...
2
votes
0answers
171 views

Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
1
vote
0answers
34 views

Basic results for chi square processes

I could not find any introductory material with basic results regarding chi-square processes. Their definition from The Supremum of Chi-Square Processes is as a sum of $d$ squares of independent ...
0
votes
1answer
167 views

Minimum number of people such that 2 can be expected to sit next to each other [closed]

We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here ...
12
votes
2answers
460 views

Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?
1
vote
0answers
100 views

Finding a lower bound in terms of given integers

Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks
3
votes
1answer
93 views

almost diagonal Positive semidefinite Matrix

Consider the set $\mathcal{D}_n$ of $n$-dimensional positive semidefinite matrices. A matrix $M\in \mathcal{D}_n$ is called $\epsilon$ diagonal in trace distance if there is a diagonal matrix $D\in ...
1
vote
1answer
120 views

A question on $J(f)$ and $J(f')$

I was confused by the following question for a long time: Does there exists a transcendental entire function $f$ such that $J(f)\cap J(f')=\emptyset$ ? where $J(f)$, ($J(f')$) is the Julia set of ...
2
votes
1answer
82 views

How to associate a proper parabolic subgroup of a real s.s Lie group $G$ to a non-trivial unipotent element in a non uniform lattice in $G$?

Let $G$ be a real semi-simple Lie group. Let $\Gamma$ be a non-uniform lattice in $G$. Then it is known that $\Gamma$ contains a non-trivial unipotent element. When $\mathbb{R}$-rank of $G$ is 1, it ...
0
votes
1answer
21 views

Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...
4
votes
3answers
630 views

Looking for techniques of How to measure the Similarity/Dissimilarity between two images?

I would like to compute the similarity/dissimilarity between two images L and R. I know one way which is : computing the histogram of blocks of each image, and then using Bhattacharyya measure I ...
5
votes
2answers
175 views

If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable?

In "A domain theoretic account of Picard's theorem" (http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf), the authors assert the following. Let $\mathbb{IR}$ be the interval domain $\lbrace ...
5
votes
0answers
105 views

Minimal possible cardinality of a $(a_1, …, a_k)$-distributable multiset

Suppose we have a multiset $M$ of positive rational numbers. Sum of $M$ equals $1$. We'll call this multiset $n$-distributable for some $n\in \mathbb{N}$, if there exists a partition $M_1 \sqcup ... ...
3
votes
0answers
213 views

When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...
1
vote
0answers
69 views

Poisson algebra automorphisms of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $V=\mathcal{C}^\infty(M)$ be the Lie algebra of smooth real valued functions. Suppose $f:\rightarrow V$ be an Lie-algebra isomorphism (an algebra ...
2
votes
1answer
65 views

Generating function of alternating even terms in the Vandermonde Convolution

I have \begin{equation} G(x) = \sum_{i = 0}^{\infty} \sum_{r = 0}^{\infty} (-1)^i \binom{k}{2i} \binom{n-k}{r} x^{2i} x^{r} = \frac{1}{2} \left( (1 + x{\iota})^{k} + (1 - x \iota)^{k} \right)(1 + ...
1
vote
0answers
97 views

Oracle queries asked in parallel

Definition: Assume that $\phi(q)$ is of the form $\exists y \leq 2^{p(n)} \varphi(q,y)$, where $p$ is a polynomial and $n = |q|$ (i.e. $n$ is the length of the binary representation of $q$). Then a ...
47
votes
3answers
5k views

Is this differential identity known?

Recently I discovered the differential identity $$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$ valid for any odd natural number $k$; for ...
0
votes
0answers
15 views

Strong solution and measurability on Ikeda and Watanabe context

This question is from Chap 4 of Ikeda and Watanabe - Stochastic differential equations and Diffusion processes pg 149 and on page 152 I believe that in the context of the last paragraph ...
1
vote
0answers
81 views

When Max(R) is Hausdorff space? [duplicate]

Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...
1
vote
0answers
11 views

Is this function $u\in SBV(\Omega)$ also belongs to $L^\infty(\Omega)$?

Some early discussion can be found here. My question: Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Also $\mathcal H^{N-1}(\partial\Omega))<\infty$. Let $u\in ...
1
vote
0answers
35 views

Characterizations of cycloid

There are several constructions of a cycloid. I have some examples below. Are there any others? Trace of a fixed point on a rolling circle Evolute of another cycloid (the locus of all its centers of ...
3
votes
3answers
329 views

Genus of a plane curve of the form $\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$

Does anybody know the genus of the following (projective) plane curve?: $$\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$$ where the $a_i$'s and the $b_i$'s are complex numbers with $a_j \ne a_i\ne b_i \ne b_j$ ...
0
votes
1answer
59 views

Ergodic automorphisms of a compact metric abelian group are Bernoulli

In the literature, such as in this article, it is proved that every ergodic automorphism of a compact metric abelian group is Bernoulli. A rotation is not isomorphic to a Bernoulli shift because it ...
167
votes
109answers
44k views

What are some examples of colorful language in serious mathematics papers? [closed]

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ...
3
votes
1answer
119 views
1
vote
1answer
117 views

A proof of the Ibragimov et al. commutation relation

Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime $x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein ...
3
votes
0answers
49 views

Derived Deformations of associative algebras

Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows: ...
3
votes
1answer
84 views

Any formula for the partial sum of a remainder series?

Let $N \ge 1$ be an integer, and there is a series $ \{ N \mod 1, N \mod 2, ... , N \mod i, ... \}$. Obviously when $i \gt N+1$, the series will become $\{N, N, N, ..., \}$. So only take $i \le N$ ...
0
votes
0answers
20 views

One parameter family differentiable dependence for linear parabolic pde's

Consider for example, the Black Schole's equation $\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0$ on $[0,T]\times[0,\infty)$ Subject to boundary conditions $u(s,T)=f(s)$. The ...
10
votes
2answers
256 views

Posets isomorphic to their endomorphism poset

Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise. Is there a poset with more than 1 point such that $P\cong ...
5
votes
1answer
235 views

Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
12
votes
2answers
446 views

Number of nonzero terms in polynomial expansion (lower bounds)

Let $f(x) = a_1x^{z_1} + a_2x^{z_2} + \cdots + a_kx^{z_k}$ be a polynomial with coefficients $(a_1, \ldots, a_k) \in \mathbb{F}_q^*$ and $z_i$ are distinct positive integers. If I need to compute the ...
1
vote
1answer
143 views

Continuous maps on compact topological spaces which induce compact (Fredholm) operators

Let $X$ be a compact Hausdorff space. A continuous map $f:X \to X$ defines a bounded linear operator $T_{f}$ on the Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi\; \text{is continuous}\}$ with ...
85
votes
7answers
4k views

Is the set $ AA+A $ always at least as large as $ A+A $?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...
7
votes
1answer
416 views

Do modular forms show up in the cohomology of moduli spaces of unmarked curves?

Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...
8
votes
1answer
172 views

Automorphisms of del Pezzo surfaces

Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p_1,p_2,p_3$. Is it true that its automorphism group is ...
0
votes
2answers
241 views

Intersection of two lattices

Suppose that $\Lambda_1, \Lambda_2$ are two sub-lattices of $\mathbb{Z}^n$ of full rank, defined by congruence modulo a prime $p$. That is, there exist two vectors with integer entries $\mathbf{a}, ...
2
votes
1answer
102 views

Does the following characterize local presentability?

Let $\mathcal C$ be a cocomplete category. Consider the following two conditions: $\mathcal C$ is locally presentable. The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors ...
7
votes
1answer
358 views

What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?

This weird problem popped up in my research: Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$? Is there a ...
1
vote
0answers
40 views

Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...
1
vote
1answer
62 views

Categorical product of graphs and chromatic number

Let $(G_i)_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod_{i\in I}G_i$ denote their categorical product. Why do we have the inequality $$\chi(\prod_{i\in ...
6
votes
0answers
201 views

Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...
2
votes
1answer
212 views

Training towards research on k3 surfaces

I am a graduate student learning basic algebraic geometry (from Hartshorne, Shafarevich). I'm planning to work in k3 surfaces (arithmetic and geometric properties, in my guide's words). I came to know ...
5
votes
0answers
129 views

Congruences involving binary forms and primes of the form $x^2+y^2$

Let $a_s$ be \begin{align*} a_s=\sum_{k=0}^s{s+k\choose k}2^k, \end{align*} which is the coefficient of $x^s$ in \begin{align*} \frac{3-\sqrt{1-8x}}{2(x+1)\sqrt{1-8x}}. \end{align*} ( see ...
5
votes
2answers
429 views

Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...

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