Related papers: The Differences Between Consecutive Smooth Numbers
Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…
In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and…
We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete…
When approximating the joint distribution of the component counts of a decomposable combinatorial structure that is `almost' in the logarithmic class, but nonetheless has irregular structure, it is useful to be able first to establish that…
The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems,…
We show a general lower bound for Mean-value of Dirichlet polynomials
The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for convex domains in $\mathbb R^n$ and $\mathbb S^n$ with Dirichlet…
We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to…
It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincare map. In contrast to…
Much recent work has shown how cross-linguistic variation is constrained by competing pressures from efficient communication. However, little attention has been paid to the role of the systematicity of forms (regularity), a key property of…
In this paper, we establish a new estimate (including lower and upper bounds) for an important quantity involved in the convergence analysis of smoothed aggregation algebraic multigrid methods. The new upper bound improves the existing…
In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference…
We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations…
Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes…
Using, as main tool, the convergence theorem for discrete martingales and the mean value property of harmonic functions we solve, a particular case of, Dirichlet problem.
It is a well observed phenomenon that natural images are smooth, in the sense that nearby pixels tend to have similar values. We describe a mathematical model of images that makes no assumptions on the nature of the environment that images…
The aim of this paper is to give two uniqueness results for the Dirichlet problem associated to the constant mean curvature equation. We study constant mean curvature graphs over strips of R^2. The proofs are based on height estimates and…
We introduce the discrete Fr\'echet gap and its variants as an alternative measure of similarity between polygonal curves. We believe that for some applications the new measure (and its variants) may better reflect our intuitive notion of…
This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections,…
We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a…