Related papers: On algebraic values of Weierstrass $\sigma$-functi…

We investigate the continuous function $f$ defined by $$x\mapsto \sum_{\sigma\le_L x }2^{-K(\sigma)}$$ as a variant of Chaitin's Omega from the perspective of analysis, computability, and algorithmic randomness. Among other results, we…

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq…

Let $\Omega$ be a bounded centrally symmetric domain in $\mathbb{R}^2$ with analytic boundary $\partial \Omega$ and center $c$. Let $\tau = \tau(\Omega)$ be the number of points $p$ on $\partial \Omega$ such that the normal line to…

Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain in the plane and consider \begin{align*} -\Delta u &=1 \qquad \mbox{in}~\Omega \\ u &= 0 \qquad \mbox{on}~\partial \Omega. \end{align*} If $u$ assumes its maximum in $x_0 \in…

We identify the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $4$ and and element of order $p\geq 2$. Thus $\Gamma$ has a…

Let $P$ be a set of $m$ points in ${\mathbb R}^2$, let $\Sigma$ be a set of $n$ semi-algebraic sets of constant complexity in ${\mathbb R}^2$, let $(S,+)$ be a semigroup, and let $w: P \rightarrow S$ be a weight function on the points of…

We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions…

The Weierstrass curve $X$ is a smooth algebraic curve determined by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)=0$, where $r$ is a positive integer, and each $A_j$ is a…

Let $f$ be an entire function of finite exponential type less than or equal to $\sigma$ which is bounded by $1$ on the real axis and satisfies $f(0) = 1$. Under these assumptions H\"ormander showed that $f$ cannot decay faster than…

We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the…

Let $\Gamma$ be a finitely generated subgroup of the multiplicative group $\G_m^2(\bar{Q})$. Let $p(X,Y),q(X,Y)\in\bat{Q}$ be two coprime polynomials not both vanishing at $(0,0)$; let $\epsilon>0$. We prove that, for all $(u,v)\in\Gamma$…

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for "many" objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general framework under which it is possible to…

We show that for all $\gamma > 0$ and $\Delta \in \mathbb{N}$, there is some $n_0$ such that, if $n \geq n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3/8 + \gamma)n$ contains a copy of each oriented…

What is the maximum length ${\rm f}_{\rm max}(\ell, \Sigma)$ of a facial cycle of an inclusion-maximal graph with girth at least $\ell$ embedded on a given surface $\Sigma$? If $\Sigma=\mathcal{P}$ is a plane, we show that $3\ell-11\leq…

Let $S$ be a set of $n$ points in the plane, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that…

Let $\hat \Omega \subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $\hat \Omega$. Starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial \hat…

We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace…

Recall first the algebraic treatment of flows or tensions in a transportation network $N$, i.e. a connected antisymmetric 1-graph $G(X, U)$. Assume that, unusually, we take the values of flows (resp. tensions) in $\mathbb{C}$. So the…

For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero boundary values, we prove the optimal bound \[ \sup_{\Omega}|u| \leq (2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2 \,dx\right)^{1/4}.…

Prototypical rational vertex operator algebras are associated to affine Lie algebras at positive integer level k. They correspond physically to the Wess-Zumino-Witten theories, and their representation theory can be captured by quantum…