2
votes
0answers
69 views

(Reference) Asymptotics of hitting probability by Brownian motion

The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" ...
1
vote
1answer
135 views

Is the Cassels-Tate pairing defined for elliptic curves over function fields?

The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...
1
vote
0answers
49 views

Is a rigid cycle a chordal graph?

There are two relevant questions: (1) We know an edge set $C$ is a rigid cycle in $\mathcal{G}_2(n)$ if and only if $|E(C)|=2|V(C)|−2$ and $|F|≤2|V(F)|−3$ for every proper subset $F$ of $E(C)$. Thus, ...
-2
votes
0answers
15 views

Population Variance PDF given Sample Variance [on hold]

I require the pdf of the population variance (v2) conditioned on the sample variance (s2). I know that (n-1)s2/p2 follows a chi-squared distribution but can it be applied in the other direction? ...
1
vote
0answers
19 views

Perturbing moments of multivariable distributions

Let $P$ be a multivariate probability distribution on $\mathbb R^n$ which is moment-determinate and let $\{m_k : k \in \mathbb N_0^n\}$ be the sequence of moments $P$. Fix an order $p$ and consider ...
2
votes
2answers
192 views

Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that If $K$ is a virtual knot whose underlying Gauss ...
3
votes
1answer
56 views

Extreme unit linear functional not norming a vector

If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only ...
8
votes
0answers
261 views

Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...
4
votes
1answer
548 views

About the hypothesis of Zorn's lemma

The proofs I know of Zorn's lemma give the following refinement: Let $(X,<)$ be a partially ordered set such that every well-ordered subset of $X$ has an upperbound. Then $X$ has a maximal ...
-4
votes
0answers
86 views

Löwenheim–Skolem as an argument for discrete mathematics? [on hold]

At least as far as first-order theories go, one could construe the (downward) Löwenheim–Skolem theorem as an incentive to invest more in discrete models rather than in continuous ones. This would ...
7
votes
1answer
264 views
+100

What is the decomposition group at $p$ in the Galois group unramified outside $\ell?$

Let $p\ne\ell$ be two prime numbers, and let $K_{\ell}$ be the maximal extension of $\mathbb Q$ ramified only at $\ell$ and $\infty$ (i.e. it is the fixed subfield of $\overline{\mathbb Q}$ by all the ...
0
votes
0answers
57 views

What is the proper Zariski-closed subset in these examples for Vojta's more general abc conjecture?

In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$ $$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...
0
votes
0answers
73 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
0
votes
2answers
57 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
4
votes
3answers
244 views

Poincare duality for (co)homology of Lie algebras?

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200 it ...
1
vote
0answers
43 views

some computational result on Hurwitz zeta function

According to Wolfram Alpha (an online computation tool), one representation of the Hurwitz zeta-function is that \begin{equation} \zeta(s, a) = \frac{1}{2a^{s}} + \frac{a^{1 - s}}{s - 1} + 2 ...
2
votes
0answers
51 views

Regularity of the Minkowski functionnal of a convex

Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is: \begin{align*} \varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K ...
0
votes
0answers
65 views

Normal Form of Homotopy Pure Braids?

It is well known that a pure braid has a normal form (also called the combed form). Namely, let $P_n$ be the set of pure braids of $n$ strands and let $d_i:P_n\to P_{n-1}$ be the $i$th "forgetting ...
1
vote
0answers
48 views

relations between nori motives and pure motives

The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) ...
0
votes
0answers
96 views

Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$ $$f(\alpha x+(1-\alpha)y)\leq ...
7
votes
1answer
397 views

Higher vector spaces

As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at ...
-7
votes
0answers
53 views

Please help me to judge a proposition about fixed point [on hold]

We make the following two assumptions: Assume that the vector set Ω is a closed convex set, and is bounded as well (We can simply consider that the vector set is a n-dimensional Euclid space.). We ...
8
votes
1answer
203 views

Nim and the Sierpinski Gasket

(I discovered this in high school, sent it off to a journal, never heard back, and forgot about it. I've never found anyone else who appeared to know about it; the combinatorial game theorists I've ...
2
votes
1answer
128 views

Rational functions and polynomials evaluated on a set of points

Let $S$ be a collection of points on the real line. Let $\{x_i\}_{i=1}^n$ take values in $S$. Consider a polynomial $p(x_1,x_2,\dots,x_n)$ over $\mathbb R[x_1,x_2,\dots,x_n]$ of degree $d$ which ...
9
votes
3answers
290 views

The continuum hypothesis for packing shapes without overlapping

Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...
4
votes
0answers
52 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
2
votes
0answers
66 views

Tracy-Widom distribution - Phase transitions - catastrophe/chaos - 'surface-fit'/'curve-fit' software [on hold]

There's an article that interested me about asymmetric distributions: http://www.simonsfoundation.org/quanta/20141015-at-the-far-ends-of-a-new-universal-law/ This mentions the finding that ...
0
votes
1answer
45 views

Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain. Definition of Newtonian ...
13
votes
1answer
248 views

Is the regularity of finitely generated rings decidable?

Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular? I mean by regular that the localization at every prime ideal is a regular local ...
0
votes
2answers
76 views

Form of the Shannon capacity for Heptagon?

Is the $0$-error capacity of $7$-cycle: $(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?
-4
votes
0answers
32 views

Floating point evaluation with taylor series and matlab [on hold]

I know how to do part (a) and I know that f2 is better at avoiding the pit falls, but I'm not entirely sure why, I know it has to do with catastrophic cancelation. I also don't know how to relate the ...
1
vote
1answer
26 views

Average entropy of quantum system in bipartite pure state for finite temperature

[I got halfway through writing this when I found the paper that answers the question in (essentially) the affirmative. I'll post it anyways in case anyone is interested.] Background: If a random ...
1
vote
2answers
226 views

A question from the proof of affine algebraic group is a linear

In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", ...
7
votes
1answer
161 views

On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F. The following are equivalent (and groups satisfying these conditions are called unramified): (a) G is quasisplit ...
2
votes
1answer
68 views

Linearly constrained eigenvalue problem

Suppose I'd like to: \begin{align} \mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\ && ...
-4
votes
0answers
26 views

How to prove a relation holds almost surely? [closed]

Let's assume x(t) and y(t) are to random processes. How should we prove a relation say x(t)>y(t) holds almost surely? Thanks in advance.
5
votes
0answers
78 views

Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
7
votes
2answers
830 views

Does the Gamma function preserve integers?

Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the ...
3
votes
1answer
60 views

Conditions conformal mapping to be expansive

Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that ...
0
votes
0answers
25 views

Coding for channels with concentrated error

A note: I accidentally over-simplified my first attempt at this question to the point of triviality - unfortunately, I didn't recognize this until after I had already placed a bounty to draw ...
3
votes
1answer
82 views

Points with pairwise integer distances in the plane

Consider $n>3$ points with pairwise integer distances in the plane! What is the relationship between these $n(n-1)/2$ integers? Do we have a theorem or result about these points? Does there exist a ...
10
votes
1answer
345 views

Is there any relationship between the Euler class and the Vandermonde determinant?

Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the ...
5
votes
1answer
317 views

Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?

$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$ $\zeta(-2n) = 0$ $\zeta(-1) = - \dfrac{1}{12}$ $\zeta(-3) = \dfrac{1}{120}$ $\zeta(-5) = - \dfrac{1}{252}$ $\zeta(-7) = \dfrac{1}{240}$ $\zeta(-9) = - ...
6
votes
1answer
157 views

Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...
0
votes
0answers
76 views

What are some continued fractions for $2\pi$ and for $\dfrac{\pi}{2}$? [closed]

I've found on Wikipedia three simple and beautiful continued fractions for $\pi$ : I would like to see some continued fractions for $2\pi$ and for $\dfrac{\pi}{2}$.
-1
votes
1answer
79 views

terminology: “complex” and “sequence” in homological algebra

It appears that the terms "complex" and "sequence" are used synonymously in homological algebra. But there seem to be collocations (in the linguistic sense) that prefer one of those words. For ...
2
votes
1answer
62 views

Triangulations of translation surfaces whose edges are shorter than the diameter

Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that ...
-1
votes
0answers
24 views

convex analysis [closed]

So I read this theorem in a convex analysis book saying Let f: R^N --> R(Bar) be convex with x(bar) in dom f. the following are equivalent. (i) f is continuous at x(bar) (ii) x(bar) in int(dom f) ...
0
votes
0answers
39 views

Semicubical parabola homeomorphic to C^2 [closed]

a semicubical parabola $L$ in $\mathbb C^2$ is given by $y^2=x^3$. I showed that a bijective function $f\colon\mathbb C \to L$ defined by $t \mapsto (t^2, t^3)$ becomes a homeomorphism regarding the ...
-1
votes
0answers
43 views

Characterization of a family of interval graphs

Let $G=(V,E)$ be a graph, where $V$ is a set of integral intervals from $[1,n]$ and $\left \{i,j \right \} \in E$ if $i \cap j \neq \emptyset $. Is the family of these graphs a proper subset of the ...

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