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6
votes
0answers
246 views
+50

Is it always optimal to evenly split search space in Rényi-Ulam game?

Suppose that we want to find one of finitely many elements (called pivot) by asking binary questions (one after the other) of which for at most $e$ we might get an incorrect answer and our goal is to ...
3
votes
0answers
169 views
+100

A density criterion and a submersion map of a Hodge bundle

In Voisin's excellent book 《Hodge theory and complex algebraic geometry II》5.3.4 - a density criterion, there is a important theorem: Let $X$ be a compact Kähler manifold, $\pi:\mathcal X \rightarrow ...
4
votes
0answers
102 views
+100

When is the optimum of an optimization problem convex in the constraint parameter?

Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing on $[1,\infty)$ and strictly decreasing on $(0,1]$, and that $\lim_{x\to 0} f(x)$ ...
11
votes
0answers
194 views
+100

Identity involving zonal polynomials and $\operatorname O(N)$ irrep dimensions

$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Schur functions $s_\lambda(x)$ with $\lambda\vdash n$ are simultaneously the irreducible characters of the unitary group $\U(N)$ and proportional to ...
1
vote
0answers
71 views
+50

Stochastic integral with respect to a random field

I came across a generalized Black-Scholes equation formulation in this paper. Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
2
votes
0answers
185 views
+50

When is the exterior derivation $d$ a Lie algebra morphism?

In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We ...
2
votes
0answers
110 views
+50

Anti-self-dual Ricci-flat and Kähler Ricci-flat manifolds

Let $(M^4,g)$ be a complete Riemannian $4$-manifold with anti-self-dual (i.e. $W_+=0$) and Ricci-flat metric $g$. Can we find a finite cover $(\tilde M, \tilde g)$ of $(M,g)$ such that $(\tilde M, \...
4
votes
1answer
238 views
+50

Does every smooth function agree with some formal power series at uncountably many points?

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function. Does there exist an uncountable set $X\subset \mathbb{R}$ and real numbers $a_0, a_1, \dots$ such that for any $x\in X$ the sum $\sum_{i=0}^{\...
0
votes
0answers
99 views
+50

A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
6
votes
1answer
225 views
+100

Sum involving determinants of binomial coefficients, indexed by partitions

I would appreciate some help proving a conjecture related to combinatorics and representation theory. Given an integer partition $\lambda\vdash n$, define a polynomial in $N$ whose roots are the ...
1
vote
0answers
73 views
+50

On variations of a claim due to Kaneko in terms of Lehmer means

This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...
2
votes
0answers
82 views
+50

P-adic distance between solutions to S-unit equation

Let $p$ be a fixed prime number and $S$ is a finite set of prime numbers which does not contain $p$. A theorem of Siegel asserts that the number of solutions to the $S$-unit equations are finite; that ...
3
votes
0answers
201 views
+50

Can an orthogonal matrix move monotonically toward a signed permutation matrix?

The question is motivated by this question on Mathematics SE. Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
8
votes
0answers
502 views
+100

Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$? (was checked up to $10^{18}$)

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$ ...
1
vote
0answers
109 views
+50

Integer partitions into restricted parts

Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $...
4
votes
0answers
74 views
+100

Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. $\textbf{Theorem}$ Let $1\...
7
votes
0answers
170 views
+100

Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset ...