Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

Attraction and Stochastic Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Respect to Semimartingales

Abstract

In this paper, we will establish new results on the attraction for solutions to stochastic functional differential equations with respect to semimartingale. Most of the existing results stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in the study of attraction. Moreover, from our results on the attraction follow several new criteria on almost surely asymptotic stability and boundedness of the solutions.     

An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for L ₁-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in L ₁. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.     

Robustness of Pathwise Stability of Semilinear Perturbed Stochastic Evolution Equations

Abstract

The purpose of this paper is to investigate pathwise stability for certain Hilbert space-valued stochastic evolution equations. We are especially interested in the robustness analysis of perturbed stochastic differential equations in infinite dimensions. Sufficient conditions are established to ensure the almost surely stable decay of the given stochastic systems. Lastly, a corollary and corresponding example are studied to illustrate our theory.     

Maximally homogeneous para-CR manifolds

We define the notion of a (weak) almost para-CR structure on a manifold M as a distribution HM ⊂ TM together with a field K ∈ Γ(End(HM)) of involutive endomorphisms of HM. If K satisfies integrability conditions, then (HM, K) is called a (weak) para-CR structure. Under some regularity conditions, an almost para-CR structure can be identified with a Tanaka structure. The notion of maximally homogeneous almost para-CR structure of a semisimple type is defined. A classification of such maximally homogeneous almost para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras. All such maximally homogeneous structures of depth two (which correspond to depth two gradations) are listed and the integrability conditions are verified.Mathematics Subject Classifications (1991): 53C15, 53D99, 58A14     

On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type

Abstract

We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.     

On the class of weak almost limited operators

Abstract

We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T : E→F is almost limited (resp. almost Dunford- Pettis). As consequences, we will give some interesting results.     

Fast Algorithms for Nonparametric Curve Estimation

Abstract

Naive implementations of local polynomial fits and kernel estimators require almost O(n ²) operations. In this article two fast O(n) algorithms for nonparametric local polynomial fitting are presented. They are based on updating normal equations. Numerical stability is guaranteed by controlling ill-conditioned situations for small bandwidths and data-tuned restarting of the updating procedure. Restarting at every output point results in a moderately fast but highly stable O(n ^7/5) algorithm. Applicability of algorithms is evaluated for estimation of regression curves and their derivatives. The idea is also applied to kernel estimators of regression curves and densities.     

Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence,uniqueness, exponential stability and invariant measures

Abstract

In this work, we consider the two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows. We investigate the well-posedness of such models in two-dimensional bounded and unbounded (Poincaré domains) domains, both in deterministic and stochastic settings. The existence and uniqueness of weak solution in the deterministic case is proved via a local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique. Some results on the exponential stability of stationary solutions are also established. The global solvability results for the stochastic counterpart are obtained by a stochastic generalization of the Minty-Browder technique. The exponential stability results in the mean square as well as in the pathwise (almost sure) sense are also discussed. Using the exponential stability results, we finally prove the existence of a unique invariant measure, which is ergodic and strongly mixing.     

The Navier-Stokes equations in the weak- LnL^n space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension . We give conditions on the external force sufficient for the unique existence of small solutions in the weak- space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak- space with time-independent external force. Received: 30 March 1999 / Accepted: 21 September 1999 / Published online: 28 June 2000     

Existence of an Almost Periodic Solution in a Difference Equation with Infinite Delay

In order to obtain the existence of an almost periodic functional difference equation x(n + 1) = ?(n,x_n ),n ∈ Z ⁺ and where x_n is defined by x_n (s) = x(n + s) for s ∈ Z ^?, on an axiomatic phase space B, we consider a certain stability property, which is referred to as BS-stable under disturbances from Ω(?) with respect to K, this stability implies ρ-stable under disturbances from Ω(?) with respect to compact set K.     

A Generalization of Herrlich's Question on Almost Reflective and Coreflective Subclasses of Top and Unif

H. Herrlich asked in Topology Appl. 49 (1993), 251–264, whether there are nontrivial classes of topological spaces that are almost reflective and almost coreflective at the same time. This question was dealt with (in Huek and Tozzi, Appl. Categ. Structures 4 (1996), 57–68) in a more general setting than almost reflective and almost coreflective classes. The present paper investigates a modified question: when a nontrivial generalized reflective class of topological or uniform spaces is equivalent to a generalized coreflective class of spaces.     

Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.     

Equimorphy in varieties of distributive double p-algebras

Any finitely generated regular variety V of distributive double p-algebras is finitely determined, meaning that for some finite cardinal n(V), any subclass S V of algebras with isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double p-algebras must be almost regular.     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

Simultaneous inhomogeneous diophantine approximation on manifolds

In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine exponent w ₀(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of \mathbbRⁿ {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any “inhomogeneous” vector θ ∈ \mathbbRⁿ {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w ₀(x , θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w ₀(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈ \mathbbRⁿ {\mathbb{R}^n} , the inhomogeneous exponent w ₀(x , θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.     

Stability Analysis of nth Order Nonlinear Impulsive Differential Equations in Quasi–Banach Space

Abstract

This article is about Ulam’s type stability of n^th order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam’s type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder’s inequality and the beta integral inequality.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.