Related papers: On the variance of the error term in the hyperboli…

The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The…

This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order $\alpha$ ($0 < \alpha < 1$). The solution regularity in the Sobolev space is revisited, and new regularity results in the…

The number of lattice points in $d$-dimensional hyperbolic or elliptic shells $\{m : a<Q[m]<b\}$, which are restricted to rescaled and growing domains $r\;\Omega$, is approximated by the volume. An effective error bound of order…

Let $f$ be a $C^{2+\epsilon}$ expanding map of the circle and $v$ be a $C^{1+\epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = \alpha (f(x)) - Df(x) \alpha (x)$ which has a unique bounded solution…

The hyperbolic lattice point problem asks to estimate the size of the orbit $\Gamma z$ inside a hyperbolic disk of radius $\cosh^{-1}(X/2)$ for $\Gamma$ a discrete subgroup of $\hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for…

In this paper, we prove a conditional limit theorem for independent not necessarily identically distributed random variables. Namely, we obtain the asymptotic distribution of a large number of them given the sum.

We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial…

We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the…

The \emph{Orbit Problem} consists of determining, given a linear transformation $A$ on $\mathbb{Q}^d$, together with vectors $x$ and $y$, whether the orbit of $x$ under repeated applications of $A$ can ever reach $y$. This problem was…

Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected…

We investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound, and lying in some arithmetic progression a (mod q). In particular, we prove an upper bound for its variance as a varies over…

We study the asymptotic behavior as $p\to\infty$ of the Gelfand problem \[ -\Delta_{p} u=\lambda\,e^{u}\ \textrm{in}\ \Omega\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partial\Omega. \] Under an appropriate rescaling on $u$ and $\lambda$,…

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, \Omega\\ u=g…

Let $\gamma_n $ denote the length of the $n$-th zone of instability of the Hill operator $Ly= -y^{\prime \prime} - [4t\alpha \cos2x + 2 \alpha^2 \cos 4x ] y,$ where $\alpha \neq 0, $ and either both $\alpha, t $ are real, or both are pure…

We investigate the inverse problem consisting in the identification of constant coefficients for a fractional-in-time partial differential equation governed by a finite sum of positive self-adjoint operators on a Hilbert space under…

We obtain error rates for large deviations of sums of i.i.d. random variables in, a particular case, of the domain of a non-symmetric infinite mean $\alpha=1$-stable law. The focus of this work is on the method of proof via analytic…

There is a growing body of evidence that the running of $\alpha_s$ predicted by perturbation (PT) theory is not correctly describing the accelerator experiments at the highest energies. A natural explanation is provided by the authors' 1992…

We prove a law of large numbers and functional central limit theorem for a class of multivariate Hawkes processes with time-dependent reproduction rate. We address the difficulties induced by the use of non-convolutive Volterra processes by…

We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved…

The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose…