Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

We consider some reaction–diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.     

On some auxiliary mappings generated by nonexpansive and strictly pseudo-contractive mappings

Starting by a finite family of mappings, we define the concept of procedure with Lipschitzian dependence of the coefficients. We give seven concrete examples of such procedures and prove the strong convergence of two viscosity methods.     

Dynamical Analysis of a New Chaotic Fractional Discrete-Time System and Its Control

This article proposes a new fractional-order discrete-time chaotic system, without equilibria, included two quadratic nonlinearities terms. The dynamics of this system were experimentally investigated via bifurcation diagrams and largest Lyapunov exponent. Besides, some chaotic tests such as the 0–1 test and approximate entropy (ApEn) were included to detect the performance of our numerical results. Furthermore, a valid control method of stabilization is introduced to regulate the proposed system in such a way as to force all its states to adaptively tend toward the equilibrium point at zero. All theoretical findings in this work have been verified numerically using MATLAB software package.     

Regularity for minimizers of functionals with p – q growth

We prove higher integrability for the gradient of vector-valued minimizers of some integral functionals with p – q growth. Received February 9, 1998     

The set union problem with dynamic weighted backtracking

We consider an extension of the set union problem, in which dynamic weighted backtracking over sequences of unions is permitted. We present a new data structure which can support each operation inO(logn) time in the worst case. We prove that this bound is tight for pointer based algorithms. Furthermore, we design a different data structure to achieve better amortized bounds. The space complexity of both our data structures isO(n). Motivations for studying this problem arise in logic programming memory management.Work partially supported by NSF Grant CCR-88-14977, by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (Project ALCOM), and by the Italian MURST Project Algoritmi e Strutture di Calcolo.Partially supported by an IBM Graduate Fellowship.     

A θ-stable approximations of abstract Cauchy problems

Summary We study the approximation of linear parabolic Cauchy problems by means of Galerkin methods in space andA -stable multistep schemes of arbitrary order in time. The error is evaluated in the norm ofL _t ² (H _x ¹ ) L _t (L _x ² ).     

Properties of Integers and Finiteness Conditions for Semigroups

Let h and k be integers greater than 1; we prove that the following statements are equivalent: 1) the direct product of h copies of the additive semigroup of non-negative integers is not k-repetitive; 2) if the direct product of h finitely generated semigroups is k-repetitive, then one of them is finite. Using this and some results of Dekking and Pleasants on infinite words, we prove that certain repetitivity properties are finiteness conditions for finitely generated semigroups.     

Small values of signed harmonic sums

For every $\tau \in \mathbb{R}$ and every integer N, let ${\mathfrak{m}}_N(\tau )$ be the minimum of the distance of τ from the sums ${\sum }_{n=1}^N{s}_n/n$, where $s_1,\dots ,s_n\in \{?1,+1\}$. We prove that ${\mathfrak{m}}_N(\tau )<\exp ?(?C{(\log ?N)}^2)$, for all sufficiently large positive integers N (depending on C and τ), where C is any positive constant less than $1/\log ?4$.     

A conjectural extension of Hecke’s converse theorem

By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions.