On the exponential exponents of discrete linear systems

In this paper we introduce the concepts of exponential exponents of discrete linear time varying systems. It is shown that these exponents describe the possible changes in the Lyapunov exponents under perturbation decreasing at infinity at exponential rate. Finally we present formulas for the exponential exponents in terms of the transition matrix of the system.     

Entropy formula and continuity of entropy for piecewise expanding maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

On the Perron exponents of discrete linear systems

This note studies properties of Perron or lower Lyapunov exponents for discrete time varying system. It is shown that for diagonal system of order s there are at most 2^s-1 lower Lyapunov exponents. By example it is demonstrated that in non-diagonal case it is possible to have arbitrarily many different Perron exponents. Finally it is shown that the exponent is almost everywhere equal to the lower Lyapunov exponent of the matrices coefficient sequence.     

Dynamics of generic 2-dimensional linear differential systems

We prove that for a C⁰-generic (a dense Gδ) subset of all the 2-dimensional conservative nonautonomous linear differential systems, either Lyapunov exponents are zero or there is a dominated splitting μ almost every point.     

Relations between Bohl exponents and general exponent of discrete linear time-varying systems

In the paper, properties of the upper Bohl exponents and senior upper general exponent of discrete linear time-varying systems are investigated. The relation of these exponents to uniform exponential stability is discussed. Moreover, an example of system, which is not uniformly exponentially stable but each trajectory tends uniformly and exponentially to zero is provided.     

Piecewise linear output measures in DEA (third revision)

It is assumed in the standard DEA model that the aggregate output (input) is a pure linear function of each output (input). This means, for example, that if DMU _j1$j_1$ generates twice as much of an output as does another DMU _j2$j_2$, then the former is credited with having created twice as much value  . In many situations, however, linear pricing (_μryrj)$({\mu }_r{y}_{\mathit{rj}})$ may not adequately reflect differences in value created from one DMU to another. In this paper, a generalization of the DEA methodology is presented that incorporates piecewise linear functions of factors. We deal specifically with those situations where for certain outputs in an input-oriented model, the weight function f(_yrj)$f(y_{\mathit{rj}})$ is described by either a non-increasing or non-decreasing set of multipliers for larger amounts of the factor. We refer to such a variable r as exhibiting diminishing marginal value (DMV) or increasing marginal value   (IMV). The DMV/IMV phenomenon is common in many for-profit applications. For example, in the case that _yrj$y_{\mathit{rj}}$ is the amount of a consumer product r generated by DMU j  , and _μr${\mu }_r$ is the price of that product, it may well be that the market will force lower prices if greater amounts of that product are generated; discounts automatically lead to this DMV situation. Such a phenomenon can arise as well in not-for-profit settings, and we examine such a situation based on earlier work by Cook et al. [Cook, W.D., Roll, Y., Kazakov, A., 1990. A DEA model for measuring the relative efficiency of highway maintenance patrols. INFOR 28 (2), 113–124].     

Locally injective maps in O-minimal structures without poles are surjective

If is continuous and locally injective, then is in fact surjective and a homeomorphism, provided is definable in an o-minimal expansion without poles of the ordered additive group of real numbers; `without poles' means that every one-variable definable function is locally bounded. Some general properties of definable maps in o-minimal expansions of ordered abelian groups without poles are also established.

     

Biderivations,linear commuting maps and commutative post-Lie algebra structures on W-algebras

In this paper, the biderivations without the skew-symmetric condition of W-algebras including the Witt algebra, the algebra W(2,2) and their central extensions are characterized. Some classes of non-inner biderivations are presented. As applications, the forms of linear commuting maps and the commutative post-Lie algebra structures on aforementioned W-algebras are given.     

Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition

In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on Rm×Rk of the form (f(x),g(x,y)). Assume that f has covering relations determined by a transition matrix A. If g is locally trapping, we show that any small C⁰ perturbed system has a compact positively invariant set restricted to which the system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C¹ perturbed homeomorphism has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A. Some other results about multidimensional perturbations of f are also obtained. The strong Liapunov condition for covering relations is adapted with modification from the cone condition in Zgliczyński (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2], Li and Malkin (2006) [3], Misiurewicz and Zgliczyński (2001) [4] by considering a larger class of maps f and their multidimensional perturbations, and by concluding conjugacy rather than entropy. Our results are applicable to both the logistic and Hénon families.     

Nonconvex, lower semicontinuous piecewise linear optimization

A branch-and-cut algorithm for solving linear problems with continuous separable piecewise linear cost functions was developed in 2005 by Keha et al. This algorithm is based on valid inequalities for an SOS2 based formulation of the problem. In this paper we study the extension of the algorithm to the case where the cost function is only lower semicontinuous. We extend the SOS2 based formulation to the lower semicontinuous case and show how the inequalities introduced by Keha et al. can also be used for this new formulation. We also introduce a simple generalization of one of the inequalities introduced by Keha et al. Furthermore, we study the discontinuities caused by fixed charge jumps and introduce two new valid inequalities by extending classical results for fixed charge linear problems. Finally, we report computational results showing how the addition of the developed inequalities can significantly improve the performance of CPLEX when solving these kinds of problems.     

An improved particle swarm optimization algorithm combined with piecewise linear chaotic map

Particle swarm optimization (PSO) has gained increasing attention in tackling complex optimization problems. Its further superiority when hybridized with other search techniques is also shown. Chaos, with the properties of ergodicity and stochasticity, is definitely a good candidate, but currently only the well-known logistic map is prevalently used. In this paper, the performance and deficiencies of schemes coupling chaotic search into PSO are analyzed. Then, the piecewise linear chaotic map (PWLCM) is introduced to perform the chaotic search. An improved PSO algorithm combined with PWLCM (PWLCPSO) is proposed subsequently, and experimental results verify its great superiority.     

Indecomposable positive linear maps constructed from permutations

We generalize two classes of D-type positive linear maps by permutations. For the first class, we discuss their indecomposability and give conditions when they are indecomposable and decomposable. For the other class, we show that they are atomic.