harnack inequality and exponential separation for oblique derivative problems on lipschitz domains abstract we consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded lipschitz cylinders we derive an optimal elliptic type harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times we show several nontrivial applications of both the exponential estimate and the derived harnack inequality

selecting fuzzy if then rules for classification problems using genetic algorithms this paper proposes a genetic algorithm based method for selecting a small number of significant fuzzy if then rules to construct a compact fuzzy classification system with high classification power the rule selection problem is formulated as a combinatorial optimization problem with two objectives to maximize the number of correctly classified patterns and to minimize the number of fuzzy if then rules genetic algorithms are applied to this problem a set of fuzzy if then rules is coded into a string and treated as an individual in genetic algorithms the fitness of each individual is specified by the two objectives in the combinatorial optimization problem the performance of the proposed method for training data and test data is examined by computer simulations on the iris data of fisher

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

on growth rates of sub additive functions for semi flows determined and random cases abstract let r m m be a measurable random dynamical systems on the compact metric space m over f p t t r with time r let m p and e p denote the set of all invariant measures on m and the set of all ergodic invariant measures whose marginal on coincide with p respectively a function f r m r is sub additive with respect to if f t s x f t x f s t t x we define the maximal growth rate of f to be lim sup t 1 t sup x m f t x for p a e it is shown that it is equal to max m p where lim t 1 t m f t x d and there exists e p such that max m p the result may have some applications in the study of the dynamical spectrum of infinite dimension random dynamical systems and robust permanence for differential equations

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

the principal spectrum for linear nonautonomous parabolic pdes of second order basic properties abstract we put forward a theory extending the notion of principal eigenfunction and principal eigenvalue to the case of linear nonautonomous parabolic pdes of second order u t i j 1 n a ij t x 2 u x i x j i 1 n a i t x u x i a 0 t x u x on a bounded domain r n with dirichlet or robin boundary conditions a canonically defined one dimensional subbundle s corresponding to the solutions that are globally defined and of the same sign serves as an analog of principal eigenfunction the principal spectrum is defined to be the dynamical sacker sell spectrum of the linear skew product flow on s characterizations of principal spectrum in terms of logarithmic growth rates of positive solutions are given finally monotonicity of the principal spectrum with respect to zero order terms is proved

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

globally positive solutions of linear parabolic pdes of second order with robin boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with robin boundary conditions are unique up to multiplication by a positive constant

spectral theory for general nonautonomous random dispersal evolution operators abstract we investigate the spectral theory of the following general nonautonomous evolution equation t u t x a u t x h t x u t x x d where d is a bounded subset of r n which can be a smooth domain or a discrete set a is a general linear dispersal operator for example a laplacian operator an integral operator with positive kernel or a cooperative discrete operator and h t x is a smooth function on r d we first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal lyapunov exponent of a time dependent dispersal equation is always greater than or equal to that of the time averaged one several results about the principal eigenvalue of time periodic parabolic equations are extended to general time periodic dispersal ones finally the investigation is generalized to random time dependent dispersal equations

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

strange eigenmodes and decay of variance in the mixing of diffusive tracers we prove the existence of asymptotic spatial patterns for diffusive tracers advected by unsteady velocity fields the asymptotic patterns arise from convergence to a time dependent inertial manifold in the underlying advection diffusion equation for time periodic velocity fields we find that the inertial manifold is spanned by a finite number of floquet solutions the strange eigenmodes observed first numerically by pierrehumbert these strange eigenmodes only admit a regular asymptotic expansion in the diffusivity if the velocity field is completely integrable

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

exponential separation and principal floquet bundles for linear parabolic equations on general bounded domains nondivergence case we consider the dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients under such minimal regularity assumptions we establish the existence of a principal floquet bundle exponentially separated from a complementary invariant bundle as a special case of our main theorem assuming the coefficients are time periodic we obtain a new result on the existence of a principal eigenvalue of an associated time periodic parabolic eigenvalue problem we also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time periodic uniformly parabolic operators finally we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity

redundant fuzzy rules exclusion by genetic algorithms abstract a genetic algorithm based method for exclusion of the potential redundant if then fuzzy rules that have been extracted from numerical input output data is proposed the main idea is the input space separation into activation rectangles corresponding to certain output intervals the generation of fuzzy rules and the membership functions are based on these activation rectangles and appropriate fuzzy rules inference mechanism is proposed as the method usually produces too many rules it is necessary to exclude the potential redundant if then rules the concept for varying the family of sensitivity parameters defining the overlapping of the fuzzy regions is proposed the genetic algorithms are used to resolve the following combinatorial optimization problem the generation of families of sensitivity parameters in this way the potential redundant if then fuzzy rules are excluded the method formalizes the synthesis of the fuzzy system and could be used for function approximation classification and control purposes an illustrative example for implementation of the method for traffic fuzzy control is given

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory this series of papers is concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part of the series focuses on the development of general theory first the notions of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations for general positive random dynamical systems in ordered banach spaces are introduced which extend the classical notions of principal floquet subspaces principal lyapunov exponents and exponential separations for strongly positive deterministic systems in strongly ordered banach to general positive random dynamical systems in ordered banach spaces under some quite general assumptions it is then shown that a positive random dynamical system in an ordered banach space admits a family of generalized principal floquet subspaces a generalized principal lyapunov exponent and a generalized exponential separation we will consider in the forthcoming part s the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations including random leslie matrix models random cooperative systems of ordinary differential equations and random parabolic equations institute of mathematics and computer science wroc law university of technology wybrzeze wyspianskiego 27 pl 50 370 wroc law poland url http www im pwr wroc pl mierczyn index html e mail address mierczyn pwr wroc pl department of mathematics and statistics auburn university auburn university al 36849 usa e mail address wenxish auburn edu 2010 mathematics subject classification primary 37h15 37l55 37a30 secondary 15b52 34f05 35r60

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

aitken sequences and generalized fibonacci numbers consider the sequence t generated by t i avn bv l l n 5i 2 where vi 1 t 2 a with a and b real of which the fibonacci sequence is a special case it is shown that if aitken acceleration is used on the sequence x defined by a v 1 v the resulting sequence is a subsequence of jc second if newton s method and the secant method are used with suitable starting values to solve the equation x2 ax b 0 then the sequences obtained from both of those methods are also subsequences of the original sequence

generalized fibonacci and lucas sequences and rootfinding methods consider the sequences u and v generated by u 1 pun qun i and vn 1 pvn qvn i n 1 where uo 0 ul 1 vo 2 v p with p and q real and nonzero the fibonacci sequence and the lucas sequence are special cases of un and vn respectively define rn un d un rn vn d vn where d is a positive integer mccabe and phillips showed that for d 1 applying one step of aitken acceleration to any appropriate triple of elements of rn yields another element of rn they also proved for d 1 that if a step of the newton raphson method or the secant method is applied to elements of rn in solving the characteristic equation x2 px q 0 then the result is an element of rn the above results are obtained for d 1 it is shown that if any of the above methods is applied to elements of rn then the result is an element of rn the application of certain higher order iterative procedures such as halley s method to elements of rn and rn is also investigated fibonacci and lucas numbers appear repeatedly in the works of the father of computational number theory d h lehmer who contributed also to numerical analysis notably 5 to his memory is dedicated this extension of results of mccabe and phillips 6 and jamieson 4 about applying iterative formulas for solving nonlinear equations to ratios of generalized fibonacci numbers

globally positive solutions of linear parabolic partial differential equations of second order with dirichlet boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with dirichlet boundary conditions are unique up to multiplication by a positive constant

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

on iterates of mobius transformations on fields let p be a quadratic polynomial over a splitting field k and s be the set of zeros of p we define an associative and commutative binary relation on g k s so that every mobius transformation with fixed point set s is of the form x plus c for some c this permits an easy proof of aitken acceleration as well as generalizations of known results concerning newton s method the secant method halley s method and higher order methods if k is equipped with a norm then we give necessary and sufficient conditions for the iterates of a mobius transformation m to converge necessarily to one of its fixed points in the norm topology finally we show that if the fixed points of m are distinct and the iterates of m converge then newton s method converges with order 2 and higher order generalizations converge accordingly

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

genetic fuzzy logic controller an iterative evolution algorithm with new encoding method logic rules and membership functions are two key components of a fuzzy logic controller flc if only one component is learned the other one is often set subjectively thus can reduce the applicability of flc if both components are learned simultaneously a very long chromosome is often needed thus may deteriorate the learning performance to avoid these shortcomings this paper employs genetic algorithms to learn both logic rules and membership functions sequentially we propose a bi level iterative evolution algorithm in selecting the logic rules and tuning the membership functions for a genetic fuzzy logic controller gflc the upper level is to solve the composition of logic rules using the membership functions tuned by the lower level the lower level is to determine the shape of membership functions using the logic rules learned from the upper level we also propose a new encoding method for tuning the membership functions to overcome the problem of too many constraints our proposed gflc model is compared with other similar gflc artificial neural network and fuzzy neural network models which are trained and validated by the same examples with theoretical and field observed car following behaviors the results reveal that our proposed gflc has outperformed