Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.     

A general formalism for synchronization in finite dimensional dynamical systems

In this paper, an attempt is made to propose a general definition of synchronization for finite dimensional dynamical systems. The synchronization is defined here for two coupled dynamical systems with control inputs. Output functions of such systems are introduced to describe the systems’ properties on which the synchronization problem focus. Exact synchronization, asymptotic synchronization, and approximate synchronization are, respectively, defined by comparing the output functions in the corresponding ways. The definition here can also include chaos control and anti-control. The definition here covers various synchronization investigated in the references.     

Polynomial entropies and integrable Hamiltonian systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.     

Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part I: L2-error estimates

Superconvergence properties in the L₂ norm are derived for the recovered gradients of piecewise quadratic finite element approximations on triangular partitions for two-dimensional elliptic problems and systems, including the case of linear elasticity. The analysis covers problems defined on polygonal domains, where the solutions have low regularity. The effects of numerical integration are treated.     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers hq,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak-Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.     

A biased random-key genetic algorithm for the Steiner triple covering problem

We present a biased random-key genetic algorithm (BRKGA) for finding small covers of computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Using a parallel implementation of the BRKGA, we compute improved covers for the two largest instances in a standard set of test problems used to evaluate solution procedures for this problem. The new covers for instances A ₄₀₅ and A ₇₂₉ have sizes 335 and 617, respectively. On all other smaller instances our algorithm consistently produces covers of optimal size.     

Adaptive control of linear time invariant systems via a wavelet network and applications to control Lorenz chaos

Compactly supported orthogonal wavelets have certain properties that are useful for controller design. In this paper, we explore the mechanism of a wavelet controller by integrating the controller with linear time-invariant systems (LTI). A necessary condition for effective control is that the compact support of the wavelet network covers the state space where the state trajectories stay. Closed-form bounds on the design parameters of the wavelet controller are derived, which guarantee asymptotic stability of wavelet-controlled LTI systems. The same wavelet controller is then applied to the Lorenz equations. The control objective is to stabilize the Lorenz system well into its normally chaotic region at one of its equilibria.     

Nonlinear impulsive systems on infinite dimensional spaces

In this paper we consider two different classes of nonlinear impulsive systems one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures. The later class is easily extended to systems driven by general vector measures. The principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of Banach spaces called Gelfand triple. The other nonlinear operators are more regular, non-monotone continuous operators with respect to suitable Banach spaces. We present here a new result on compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.     

软计算中的协作和融合技术综述

本文综述了软计算中的协作和融合技术，包括的专题是：模糊系统与遗传算子，模糊系统与神经网络，神经网络与遗传算法。最后探讨了今后的发展趋势。     

A maximal cover of hexagonal systems

In this paper we establish the validity of Ivan Gutman's conjecture concerning covers of hexagonal systems.     

On Axioms Contituting the Foundation of Hypergraph Theory

Hypergraphs are systems of finite sets, being the most general structures in discrete mathematics and powerful tools in dealing with discrete systems. In general, a branch of mathematics is built on some axioms. Informational scientists introduced the acyclic axiom for hypergraphs. In this paper, we first list several results concerning acyclic hypergraphs, in order to show that Acyclic-Axioms constitute the foundation of acyclic hypergraph theory. Then we give the basic theorem which shows that the Cycle-Axiom covers the Acyclic-Axioms and constitutes the foundation of hypergraph theory.     

Data compression with homomorphism in covering information systems

In reality we are always faced with a large number of complex massive databases. In this work we introduce the notion of a homomorphism as a kind of tool to study data compression in covering information systems. The concepts of consistent functions related to covers are first defined. Then, by classical extension principle the concepts of covering mapping and inverse covering mapping are introduced and their properties are studied. Finally, the notions of homomorphisms of information systems based on covers are proposed, and it is proved that a complex massive covering information system can be compressed into a relatively small-scale information system and its attribute reduction is invariant under the condition of homomorphism, that is, attribute reductions in the original system and image system are equivalent to each other.     

Uniqueness of transverse solutions for reaction–diffusion equations with spatially distributed hysteresis

The paper deals with reaction–diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. Such problems describe biological processes and chemical reactions in which diffusive and nondiffusive substances interact according to hysteresis law. Under the assumption that the initial data are spatially transverse, we prove a theorem on the uniqueness of solutions. The theorem covers the case of non-Lipschitz hysteresis branches arising in the theory of slow–fast systems.     

Projective covers as subquotients of enlargements

We show that not only completions of systems in various senses but also projective covers in the category of compact Hausdorff spaces may be obtained as subquotients of enlargements.     

State Dependent AIMD Algorithms and Consensus Problems

This papers analyses a class of nonlinear positive systems that model the dynamics of nonlinear additive-increase multiplicative-decrease (AIMD) protocols. The system class covers a range of protocols that are currently used in real communication networks, such as standard TCP, and recent proposals for congestion control protocols such as Scalable TCP. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CONJUGATE OPERATORS FROM DISPERSION CURVES FOR PERTURBATIONS OF FIBERED SYSTEMS IN CYLINDERS†

We study the spectral properties of elliptic systems on perturbed fibered cylinders. This general framework covers the acoustic and elastic cylinders. Although the electromagnetic case does not exactly fit into this framework, we prove similar results for it. Our approach is based on constructing an appropriate conjugate operator using general properties of the dispersion curves.

     

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

Modal and guarded characterisation theorems over finite transition systems

We explore the finite model theory of the characterisation theorems for modal and guarded fragments of first-order logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known tree-like unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraïssé game arguments, these covers allow us to upgrade finite approximations for full bisimulation equivalence towards approximations for elementary equivalence. These techniques are used to prove several ramifications of the van Benthem–Rosen characterisation theorem of basic modal logic for refinements of ordinary bisimulation equivalence, both in the sense of classical and of finite model theory.