Related papers: Bounds for Smooth Theta Sums with Rational Paramet…

In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set…

Let $\Theta=(\theta_{j,k})_{3\times 3}$ be a non-degenerate real skew-symmetric $3\times 3$ matrix, where $\theta_{j,k}\in [0,1).$ For any $\varepsilon>0$, we prove that there exists $\delta>0$ satisfying the following: if $v_1,v_2,v_3$ are…

The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbol{\beta}=(\beta_0,\dots,\beta_{p-1})$ determining sofic systems. We show that a necessary condition is that the product…

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable…

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other…

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^*$. More precisely, we show that solutions are…

Let F_n be the free group of rank n, with generating set S=\{x_1,...,x_n\}. An automorphism \phi of F_n is called symmetric if for each 1\leq i\leq n, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let \Sigma Aut(F_n) be…

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the…

In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $f$ be a nonnegative multiplicative function. We prove that if there exists a $\beta<1$ such that…

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if…

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves…

We define a variant of Hochster's theta pairing and prove that it is constant in flat families of modules over hypersurfaces with isolated singularities. As a consequence, we show that the theta pairing factors through the Grothendieck…

In this paper we construct a family of holomorphic functions $\beta_\lambda (s)$ which are solutions to the asymptotic tetration equation. Each $\beta_\lambda$ satisfies the functional relationship ${\displaystyle \beta_\lambda(s+1) =…

Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: \alpha$ and $\dim_{\mathrm{H}} B =: \beta$. I prove that $$\dim_{\mathrm{H}} [A + \theta B] \geq \alpha + \beta \cdot \tfrac{1 - \alpha}{2 -…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$…

In the first paper of this series we established new upper bounds for multi-variable exponential sums associated with a quadratic form. The present study shows that if one adds a linear term in the exponent, the estimates can be further…

For an open set $V\subset\mathbb{C}^n$, denote by $\mathscr{M}_{\alpha}(V)$ the family of $\alpha$-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded domain $\Omega\subset \mathbb{C}^n$, with…

Given a finite alphabet $\mathbb{A}$ and a primitive substitution $\theta:\mathbb{A}\to\mathbb{A}^\lambda$ (of constant length $\lambda$), let $(X_\theta,S)$ denote the corresponding dynamical system, where $X_{\theta}$ is the closure of…

This short note studies $C^{\infty}$-smooth cocycles in $\mathbb{T}\times SO(3)$ that have $0$ degree and are non-homotopic to constants. The study picks up from where the author's PhD thesis left the subject, and shows that, under a…