2
votes
1answer
34 views

Can this simple integral be zero for a Jordan curve?

The following simple problem came up while doing some unrelated research. Does there exist a Jordan curve $\gamma : [0,2\pi] \to \mathbb{C}$ of positive orientation, lets say $C^1$-smooth (just to ...
0
votes
0answers
4 views

Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
1
vote
0answers
11 views

Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...
0
votes
1answer
34 views

Is there an integral point in the group generated by an rational point?

Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and $$G=\{P,2P,3P,4P,\cdots\}.$$ Is there an integral point $Q\in G?$
0
votes
0answers
9 views

On monotonicity of the roots of polynomials

I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied. Here is my ...
0
votes
0answers
16 views

Colon ideal and Artin-Rees lemma

Let $a$ and $b$ be ideals of a Noetherian integral domain $R$. Prove that there exists a natural number $r$ such that $(a^n:b) = a^{n-r}(a^r:b)$ for $n > r$.
0
votes
0answers
5 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
45
votes
20answers
4k views

What is the most useful non-existing object of your field?

When many proofs by contradiction end with "we have built an object with such, such and such properties, which does not exist", it seems relevant to give this object a name, even though (in fact ...
2
votes
1answer
245 views

Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...
-1
votes
0answers
40 views

Two Questions on $\pi(x)$

I have recently came to know about this conjecture. The questions that naturally came to my mind are, $\forall$ $x,y \geq2$ and $x,y \in \mathbb{N}$ prove that $\pi(x) \pi(y)\geq \pi(xy)$ ...
2
votes
1answer
96 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
6
votes
6answers
2k views

Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such. Yet, they could have been just ...
0
votes
0answers
34 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
8
votes
1answer
234 views

A particular specialization of symmetric polynomials: is it bijective?

Let $\Lambda^d_n$ the space of symmetric polynomials in $n$ variables, with maximum 'partial degree' of each variable $d$. A basis for this space is the set of symmetrized monomials $m_\lambda$, where ...
0
votes
0answers
14 views

When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
8
votes
2answers
672 views

(Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
3
votes
0answers
46 views

Explicit non observability example for the wave equation?

Is there a simple (one dimensional, radial, by separation of variable...) example of non observability for the linear wave for a non constant in space velocity, in a simple domain? By this I mean: ...
5
votes
2answers
249 views

Hausdorff group topologies on finitely generated groups

Suppose $G$ is a finitely generated Hausdorff topological group. Must $G$ be first countable (or perhaps a sequential space)? What if we restrict to the abelian case? I wonder if this is even true ...
-3
votes
0answers
41 views

extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question): Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...
0
votes
1answer
24 views

Maximum size of set of points with distance bounded from below

I am interesting in finding a reference for a result of the following type: Suppose $D \subset \Bbb{R}^n$ is a bounded open set and $\delta>0$. Then the size $M$ of a family of points $F = ...
0
votes
0answers
23 views

Standard form for partitions of $Z_n$

Let $A$ and $B$ be partitions of $Z_n$. Suppose we say that $A$ and $B$ are equivalent if $A=Bx+y$ for some $x\in\{1, -1\}$ and $y\in Z_n$. In other words, two partitions are equivalent if one can be ...
0
votes
0answers
12 views

Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$. Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...
2
votes
1answer
51 views

Characterization of a subset of [0,1] $III$

I have a question related to the previous one. Characterization of a subset of [0,1] $II$ Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e. $t_n$ is said to converge to ...
3
votes
2answers
146 views

Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?
0
votes
0answers
26 views

Fixed Point Algebras of Adjoint Actions of Banach Lie groups

I have the following question: Let a be an element in a connected Banach Lie group G (over K, where K is the reals or the complex numbers). We assume that G is not trivial, that has more than one ...
0
votes
0answers
15 views

Strongly regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $(v,c k, c \lambda, c \mu)$ (simple) strongly regular graph that can be given an edge coloring with $c$ ...
58
votes
2answers
2k views

Gromov's list of 7 constructions in differential topology

At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order ...
0
votes
0answers
32 views

Probability of that the sum is even [on hold]

There are $2n$ letters which are randomly placed into $2n$ envelopes (each envelope can have only one letter). The letters and the envelopes are numbered from $1$ to $2n$. What is the probability that ...
2
votes
0answers
79 views

Not too classical group characters

When teaching group theory, one has too speak of characters, which are morphisms $\chi:G\rightarrow k^\times$ into the multiplicative group of a field $k$ (mind that I don't mean representation ...
0
votes
0answers
24 views

Number of possible wall positions in the game Quoridor

I'm trying to figure out how many possible board positions there are for the game Quoridor. I think sorting out the legal positions from the illegal positions will be difficult, so to start I'm trying ...
1
vote
0answers
67 views

Examples for Markov generators with pure point spectrum

I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...
1
vote
0answers
51 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
7
votes
1answer
515 views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
1
vote
0answers
31 views

Lifting a quadratic system to a non vanishing vector field on $S^{3}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
2
votes
1answer
132 views

Does the implicit function theorem hold for discontinuously differentiable functions?

(This was posted on math.SE over 5 days ago and has not been answered, although a comment mentioned a similar question on this site.) Wikipedia's statement of the implicit function theorem requires ...
1
vote
0answers
122 views

Smooth and $GL(n)$-equivariant implies algebraic?

Context: Let $B_n$ be the space of symmetric bilinear forms on $\mathbb{R}^n$ and $L_n\subset B_n$ be the subset of non-degenerate forms of Lorentzian signature $(-,+,\ldots,+)$. Let $T$ be a finite ...
0
votes
0answers
27 views

Ordered statistics CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
2
votes
0answers
24 views

Removable sets for simply connectedness of a differentiable manifold

I am sorry that my question might be stupid for experts, but I really do not know the answer. Let $M$ be a smooth $n$-manifold. We assume that there exists a distance $d$ on $M$ such that $(M,d)$ is ...
3
votes
1answer
98 views

Existence of partitions

Good morning everybody. I would like to know if anybody is aware of nontrivial results of the following form : if a family $\mathcal I$ of subsets of $\mathbb N$ satisfies such and such assumption, ...
0
votes
1answer
223 views

A problem related to connectivity of analytic functions

Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$. Question: Is the connectivity of ...
2
votes
2answers
193 views

How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where $A_i \sim \text{exp}(\lambda), S_i \sim ...
6
votes
2answers
79 views

Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$. If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's ...
0
votes
0answers
118 views

rings and modules theory [on hold]

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-module. Now let $I$ be a proper ideal of $R$, what is the sructure of $I$. Is it true that $I$ is direct sum of cyclic ...
7
votes
1answer
260 views

Normal generators of finite index subgroups in a free group

Let $F=F(a,b)$ be the free group of rank $2$. Question 1: Given any positive integer $d$, can one always find elements $u_j,v_j,w_j \in F$, $j=1,\dots,d$, such that if $1 \le j <k \le d$ then the ...
5
votes
0answers
46 views

Can one detect smoothness of $k$-forms with $k$-dimensional manifolds

Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le ...
3
votes
1answer
107 views

A question about some notation involving the exclamation mark

What does the symbol ‘!’ signify? Is it $ \text{argmin} $? For example, $ \| A x - y \| = \min! $.
0
votes
0answers
24 views

Showing that a particular function from a locally compact Hausdorff group $ G $ to a $ C^{*} $-algebra $ A $ is Bochner-measurable

Suppose that we have the following data: A $ C^{*} $-algebra $ A $. A locally compact Hausdorff group $ G $. A strongly Borel mapping $ \alpha: G \to \text{Aut}(A) $, the automorphism group of $ A ...
2
votes
0answers
54 views

Cesaro summation of a particular Dirichlet series on the abscissae of convergence

If you've investigated the error in Perron's formula, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{n^s}$$ ...
41
votes
14answers
4k views

How to write popular mathematics well?

Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead ...
1
vote
0answers
11 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...

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