a decomposition theorem for bounded solutions and the existence of periodic solutions of periodic differential equations abstract we prove a decomposition theorem for bounded uniformly continuous mild solutions to periodic evolution equations of the form dx dt a t x f t with in general unbounded periodic a periodic f and compact monodromy operator by this theorem every bounded uniformly continuous mild solution to is a sum of a periodic solution to and a quasi periodic solution to its homogeneous equation an analog of this for bounded solutions has been proved for abstract functional differential equations dx dt ax f t x t f t with finite delay where a generates a compact semigroup as an immediate consequence the existence of such a solution implies the existence of a periodic solution to the inhomogeneous equation as well as a formula for its fourier coefficients this even for the classical case of equations improves considerably the previous results on the subject

an extension of some theorems of l barreira and c valls for the nonuniform exponential dichotomous evolution operators abstract the aim of this paper is to study the connection between the non uniform exponential dichotomy of a non uniform exponentially bounded strongly continuous evolution family and the admissibility of some function spaces thus there are extended recent results established by l barreira and c valls 2010 2011 in 10 and 11

invariant manifolds of admissible classes for semi linear evolution equations abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of invariant manifolds of this equation these manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory the existence of such manifolds is obtained in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to certain classes of admissible function spaces

using an efficient immune symbiotic evolution learning for compensatory neuro fuzzy controller this paper presents an efficient immune symbiotic evolution learning isel algorithm for the compensatory neuro fuzzy controller cnfc the proposed isel method includes three major components initial population subgroup symbiotic evolution and immune system algorithm first the self clustering algorithm that determines proper input space partitioning and finds the mean and variance of the gaussian membership functions and number of rules is applied to the initial population second the subgroup symbiotic evolution method that uses each subantibody represents a single fuzzy rule and the evolution of the rule itself third the immune system algorithm uses the clonal selection principle such that antibodies between others of high similar degree are canceled and these antibodies after processing will have higher quality accelerating the search and increasing the global search capacity finally the proposed cnfc with isel cnfc isel method is adopted to solve several nonlinear control problems the simulation results have shown that the proposed cnfc isel can outperform other methods

equivalences between nonuniform exponential dichotomy and admissibility abstract relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems it was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy in this paper we find an appropriate admissible pair of classes of lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half lines r separately on both half lines r simultaneously and on the whole line r additionally the maximal admissibility is proved in the case on both half lines r simultaneously

elicitation and fine tuning of fuzzy control rules using symbiotic evolution this paper exploits the ability of symbiotic evolution se as a generic methodology to elicit a fuzzy rule base of the mamdani type almost all fuzzy rule base generation algorithms produce rule bases with redundant and overlapped membership functions that limit their interpretability elegance in their application we address this problem by applying an algorithm to merge any similar membership functions it is shown that our proposed algorithm leads generally to a more transparent and more interpretable rule base with a minimum number of membership functions and a reduced number of rules in addition a new post processing approach is proposed for recovering any probable performance lost after membership functions merging the proposed methodology has been applied successfully for the design of an active control suspension system using a non linear bond graphs bg based half car model with parameters that relate to a ford fiesta mk2

a local property of basic rickard equivalences abstract let p be a prime and k be an algebraically closed field of characteristic p let g and g be two finite groups b and b be blocks of g and g and p e and p e be maximal brauer b g and b g pairs if kgb and k g b are basic rickard equivalent then there is a group isomorphism p p such that it induces an equivalence between the brauer categories of kgb and k g b and that for any brauer b g pair q f contained in p e and any subgroup k of the image of n g q f in aut q the block algebras k n g k q f and k n g k q f are basic rickard equivalent where q q f is the block of k c g q such that q f p e and k denotes the image of k in aut q

broue s conjecture holds for principal 3 blocks with elementary abelian defect group of order 9 abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a finite group g has an abelian sylow p subgroup p then the derived categories of the principal p blocks of g and of the normalizer ng p of p in g are equivalent we prove in this paper that broue inage kus conjecture holds for the principal 3 block of an arbitrary finite group g with an elementary abelian sylow 3 subgroup p of order 9 by using initiated works for the case where g is simple which are due to puig okuyama waki miyachi and the authors the result depends on the classification of finite simple groups

self equivalences of stable module categories let p be an abelian p group e a cyclic p group acting freely on p and k an algebraically closed field of characteristic p 0 in this work we prove that every self equivalence of the stable module category of k p rtimes e comes from a self equivalence of the derived category of k p rtimes e work of puig and rickard allows us to deduce that if a block b with defect group p and inertial quotient e is rickard equivalent to k p rtimes e then they are splendidly rickard equivalent that is broue s original conjecture implies rickard s refinement of the conjecture in this case all of this follows from a general result concerning the self equivalences of the thick subcategory generated by the trivial module

time periodic traveling wave solutions for periodic advection reaction diffusion systems abstract we study the existence uniqueness and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection reaction diffusion systems under certain conditions we prove that there exists a maximal wave speed c such that for each wave speed c c there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system it is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co moving frame coordinate we also show that the traveling wave solutions with wave speed c c are asymptotically stable in certain sense in addition we establish the nonexistence of time periodic traveling waves with speed c c

characterizations of dichotomies of evolution equations on the half line abstract in this paper we investigate the characterization of dichotomies of an evolution family u u t s t s 0 of bounded linear operators on banach space x we introduce operators i0 and ix on subspaces of lp r x using the integral equation u t u t s u s ts u t f d the exponential and ordinary dichotomies of u are characterized by properties of i0 ix

admissibility for nonuniform exponential contractions abstract we study the relation between the notions of nonuniform exponential stability and admissibility in particular using appropriate adapted norms which can be seen as lyapunov norms we show that if any of their associated l p spaces with p 1 is admissible for a given evolution process then this process is a nonuniform exponential contraction we also provide a collection of admissible banach spaces for any given nonuniform exponential contraction

existence and uniqueness of time periodic physically reasonable navier stokes flow past a body let be a three dimensional exterior domain of class c2 0 1 assume that a navier stokes liquid is moving in under the action of a body force f that is time periodic of period t and that the velocity of the liquid is zero at spatial infinity in this paper we show that if f satisfies suitable conditions and its norm in appropriate function spaces is sufficiently small there is at least one time periodic strong solution furthermore the velocity field v of such a solution decays to zero for large x as x 1 and its spatial gradient decays as x 2 both uniformly in time in addition the pressure p decays like x 2 and its gradient like x 3 for almost all t 0 t in the special case where f is time independent these solutions are also time independent and coincide with finn s physically reasonable solutions 4 moreover we show that our time periodic solutions are unique in a very large class namely the class of time periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in 0 t

admissibly inertial manifolds for a class of semi linear evolution equations consider the semi linear evolution equation du t dt au t f t u we prove the existence of a new class of inertial manifolds called admissibly inertial manifolds for this equation these manifolds are constituted by trajectories of the solutions belonging to rescaledly admissible function spaces which contain wide classes of function spaces like weighted lp spaces the lorentz spaces lp q and many other rescaling function spaces occurring in interpolation theory the existence of these manifolds is obtained in the case that the partial differential operator a is positive definite and self adjoint with a discrete spectrum and the nonlinear forcing term f satisfies the lipschitz conditions on the domain d a 0 1 i e f t x f t y t a x y and f t x t 1 a x for being a real and positive function which belongs to certain classes of admissible function spaces

on equivalences between blocks of group algebras reduction to the simple components abstract a conjecture of michel broue states that if d is an abelian sylow p subgroup of a finite group g and h n g d then the principal blocks of g and h are rickard equivalent the structure of groups with abelian sylow p subgroups as determined by p fong and m e harris raises the following question assuming that broue s conjecture holds for the simple components of g under what conditions does it hold for g itself due to the structure of g this problem requires mainly the lifting of rickard complexes to p extensions of the simple components and the construction of complexes over wreath products we give here these reduction steps which may be regarded as a clifford theory of tilting complexes

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

existence of nonuniform exponential dichotomies and a fredholm alternative abstract we consider linear equations x a t x in a banach space with nonuniform exponential dichotomies in r and r and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations we also obtain a fredholm alternative for the existence of bounded solutions under nonlinear perturbations

trivial source bimodule rings for blocks and p permutation equivalences we associate with any p block of a finite group a grothendieck ring of certain p permutation bimodules we extend the notion of p permutation equivalences introduced by boltje and xu 4 to source algebras of p blocks of finite groups we show that a p permutation equivalence between two source algebras a b of blocks with a common defect group and same local structure induces an isotypy

exponential dichotomy of evolution equations and admissibility of function spaces on a half line consider an evolution family u u t s ts0 on a half line r and an integral equation u t u t s u s t s u t f d we characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of lp type the lorentz spaces lp q and many other function spaces occurring in interpolation theory we then apply our results to study the robustness of the exponential dichotomy of evolution families on a half line under small perturbations

boundedness almost periodicity and stability of certain navier stokes flows in unbounded domains abstract we investigate the existence uniqueness and stability of bounded and almost periodic mild solutions to several navier stokes flow problems in our strategy we propose a general framework for studying the semi linear evolution equations with certain smoothing properties of the linear part and with the local lipschitz continuity of the nonlinear operator our method is based on interpolation functors combined with differential inequalities our abstract results are applied to navier stokes oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to navier stokes equations on aperture domains and or in besov spaces

on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domains we present a uniqueness theorem for time periodic solutions to the navier stokes equations in unbounded domains thus far results on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domain roughly speaking have only found that a small time periodic ln solution is unique within the class of solutions which have sufficiently small l ln norm in this paper we show that a small time periodic ln solution is unique within the class of all time periodic ln solutions which contains large solutions we also consider the uniqueness of solutions in weak ln space the proof of the present uniqueness theorem is based on the method of dual equations

l p theory of the navier stokes flow in the exterior of a moving or rotating obstacle in this paper we describe two recent approaches for the lp theory of the navier stokes flow in the exterior of a moving or rotating obstacle

the principal 3 blocks of four and five dimensional projective special linear groups in non defining characteristic abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he has conjectured that for any prime p if a finite group g has an abelian sylow p subgroup p then the principal p blocks of g and the normalizer ng p of p in g are derived equivalent let q be a power of a prime such that q 2 or 5 mod 9 in this paper we show that broue s conjecture is true for p 3 and for g psl4 q and g psl5 q in these cases g has elementary abelian sylow 3 subgroups of order 9 what we prove here is the following in the case g psl4 q all the principal 3 blocks of g are morita even puig equivalent independently of infinitely many q in the case g psl5 q all the principal 3 blocks of g are morita even puig equivalent to the principal 3 block of ng p independently of infinitely many q

lp lq estimate of the stokes operator and navier stokes flows in the exterior of a rotating obstacle we consider the motion of a viscous fluid filling the whole three dimensional space exterior to a rotating obstacle with constant angular velocity we develop the l p l q estimates and the similar estimates in the lorentz spaces of the stokes semigroup with rotation effect we next apply them to the navier stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in l3 weak l3 space

exponential dichotomy of difference equations and applications to evolution equations on the half line abstract for a sequence of bounded linear operator a n n 0 on a banach space x we investigate the characterization of exponential dichotomy of the difference equations v n 1 a n v n we characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations v n 1 a n v n f n in l space then we apply the results to study the exponential dichotomy of evolution families generated by evolution equations

periodic motions of stokes and navier stokes flows around a rotating obstacle we prove the existence and uniqueness of periodic motions to stokes and navier stokes flows around a rotating obstacle d subset mathbb r 3 with the complement omega mathbb r 3 backslash d being an exterior domain in our strategy we show the cb regularity in time for the mild solutions to linearized equations in the lorentz space l 3 infty omega known as weak l3 spaces and prove a massera typed theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak l3 spaces we then use the obtained results for such equations and the fixed point argument to prove such results for navier stokes equations around a rotating obstacle we also show the stability of such periodic solutions

asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition diffusion system abstract this paper is concerned with a time periodic competition diffusion system u t u x x u r 1 t a 1 t u b 1 t v t 0 x r v t d v x x v r 2 t a 2 t u b 2 t v t 0 x r where u t x and v t x denote the densities of two competing species d 0 is some constant r i t a i t and b i t are t periodic continuous functions under suitable conditions it has been confirmed by bao and wang 2013 2 that this system admits periodic traveling fronts connecting two stable semi trivial t periodic solutions p t 0 and 0 q t associated to the corresponding kinetic system in the present work we first investigate the asymptotic behavior of periodic bistable traveling fronts with non zero speeds at infinity by a dynamical approach combined with the two sided laplace transform method with these asymptotic properties we then obtain some key estimates as a result by applying the super and subsolutions techniques as well as the comparison principle we establish the existence and various qualitative properties of the so called entire solutions defined for all time and the whole space which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment

existence uniqueness and conditional stability of periodic solutions to evolution equations abstract using an ergodic approach we investigate the condition for existence and uniqueness of periodic solutions to linear evolution equation u a t u f t t 0 and to semi linear evolution equations of the form u a t u g u t where the operator valued function t a t and the vector valued function f t are t periodic and nemytskii s operator g is locally lipschitz and maps t periodic functions to t periodic functions we then apply the results to study the existence uniqueness and conditional stability of periodic solutions to the above semi linear equation in the case that the family a t t 0 generates an evolution family having an exponential dichotomy

circular spectrum and bounded solutions of periodic evolution equations abstract we consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u a t u h t u f t where a t is in general an unbounded operator depending 1 periodically on t h is 1 periodic in t is small and f is a bounded and continuous function that is not necessarily uniformly continuous we propose a new approach to the spectral theory of functions via the concept of circular spectrum and then apply it to study the linear equations u a t u f t with general conditions on f for small we show that the perturbed equation inherits some properties of the linear unperturbed one the main results extend recent results in the direction saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f

resolvents of operators and partial functional differential equations with nonautonomous past abstract to a backward evolution family u u t s t s 0 on a banach space x we associate an abstract differential operator g through the integral equation u t u t s u s t s u t f d on a banach space of x valued functions on r we compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator we then apply the results to prove existence exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in s brendle r nagel dist contin dynam systems 8 2002 953 966 our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

broue s conjecture for the nonprincipal block of sl 2 q with full defect abstract m broue gives an important conjecture which is called broue s abelian defect group conjecture this conjecture says that a p block where p is a prime number of a finite group with an abelian defect group is derived equivalent to its brauer correspondent in the normalizer of the defect group in this paper we prove that this conjecture is true for the nonprincipal block of sl 2 p n for a positive integer n

nonuniform mean square exponential dichotomies and mean square exponential stability abstract in this paper the existence conditions of nonuniform mean square exponential dichotomy nms ed for a linear stochastic differential equation sde are established the difference of the conditions for the existence of a nonuniform dichotomy between an sde and an ordinary differential equation ode is that the first one needs an additional assumption nonuniform lyapunov matrix to guarantee that the linear sde can be transformed into a decoupled one while the second does not therefore the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity this paper is also concerned with the mean square exponential stability of nonlinear perturbation of a linear sde under the condition of nonuniform mean square exponential contraction nms ec for this purpose the concept of second moment regularity coefficient is introduced this concept is essential in determining the stability of the perturbed equation and hence we deduce the lower and upper bounds of this coefficient our results imply that the lower and upper bounds of the second moment regularity coefficient can be expressed solely by the drift term of the linear sde

inertial manifolds for semi linear parabolic equations in admissible spaces we prove the existence of inertial manifolds for the solutions to the semi linear parabolic equation du t dt au t f t u when the partial differential operator a is positive definite and self adjoint with a discrete spectrum having a sufficiently large distance between some two successive points of the spectrum and the nonlinear forcing term f satisfies the lipschitz conditions on the domain d a 0 1 i e f t x f t y t a x y and f t x t 1 a x where t belongs to one of admissible function spaces containing wide classes of function spaces like lp spaces the lorentz spaces lp q and many other function spaces occurring in interpolation theory

a general approach to time periodic incompressible viscous fluid flow problems this article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations it yields on the one hand a unified approach to various classical problems in incompressible fluid flow and on the other hand gives new results for periodic solutions to the navier stokes oseen flow the navier stokes flow past rotating obstacles and in the geophysical setting for ornstein uhlenbeck and various diffusion equations with rough coefficients the method is based on a combination of interpolation and topological arguments as well as on the smoothing properties of the linearized equation

strong fusion control and stable equivalences abstract this article is dedicated to the proof of the following theorem let g be a finite group p be a prime number and e be a p block of g assume that the centraliser c g p of an e subpair p e p strongly controls the fusion of the block e and that a defect group of e is either abelian or for odd p has a non cyclic centre then there exists a stable equivalence of morita type between the block algebras o g e and o c g p e p where o is a complete discrete valuation ring of residual characteristic p this stable equivalence is constructed by gluing together a family of local morita equivalences which are induced by bimodules with fusion stable endo permutation sources broue had previously obtained a similar result for principal blocks in relation with the search for a modular proof of the odd z p theorem thus our theorem points towards a block theoretic analogue of the z p theorem which we state in terms of fusion control and morita equivalences

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

the derived categories of some blocks of symmetric groups and a conjecture of broue abstract a well known conjecture of broue in the representation theory of finite groups involves equivalences of derived categories of blocks the aim of this paper is to verify this conjecture for defect 2 blocks of symmetric groups actually we prove for these blocks a refinement of broue s conjecture due to rickard

rule base self generation and simplification for data driven fuzzy models data driven fuzzy modeling has been used in a wide variety of applications however in fuzzy rule based models acquired from numerical data redundancy often exists in the form of redundant rules or similar fuzzy sets this results in unnecessary structural complexity and decreases the interpretability of the system in this paper a rule base self extraction and simplification method is proposed to establish interpretable fuzzy models from numerical data a fuzzy clustering technique associated with the proposed fuzzy partition validity index is used to extract the initial fuzzy rule base and find out the optimal number of fuzzy rules to reduce the complexity of fuzzy models while keeping good model accuracy some approximate similarity measures are presented and a parameter fine tuning mechanism is introduced to improve the accuracy of the simplified model using the proposed similarity measures the redundant fuzzy rules are removed and similar fuzzy sets are merged to create a common fuzzy set in the rule base the simplified rule base is computationally efficient and linguistically interpretable the approach has been successfully applied to fuzzy models of non linear function approximation dynamical system modeling and mechanical property prediction for hot rolled steels

simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms this paper examines the applicability of genetic algorithms ga s in the simultaneous design of membership functions and rule sets for fuzzy logic controllers previous work using genetic algorithms has focused on the development of rule sets or high performance membership functions however the interdependence between these two components suggests a simultaneous design procedure would be a more appropriate methodology when ga s have been used to develop both it has been done serially e g design the membership functions and then use them in the design of the rule set this however means that the membership functions were optimized for the initial rule set and not the rule set designed subsequently ga s are fully capable of creating complete fuzzy controllers given the equations of motion of the system eliminating the need for human input in the design loop this new method has been applied to two problems a cart controller and a truck controller beyond the development of these controllers we also examine the design of a robust controller for the cart problem and its ability to overcome faulty rules

existence of periodic solutions in abstract semilinear equations and applications to biological models abstract in this paper we study the existence of mild periodic solutions of abstract semilinear equations in a setting that includes several other types of equations such as delay differential equations first order hyperbolic partial differential equations and reaction diffusion equations under different assumptions on the linear operator and the nonhomogeneous function sufficient conditions are derived to ensure the existence of mild periodic solutions in the abstract semilinear equations when the semigroup generated by the linear operator is not compact banach fixed point theorem is used whereas when the semigroup generated by the linear operator is compact schauder fixed point theorem is employed in applications we apply the main results to establish the existence of periodic solutions in delayed red blood cell models age structured models with periodic harvesting and the diffusive logistic equation with periodic coefficients

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

projectors on the generalized eigenspaces for functional differential equations using integrated semigroups the aim of this article is to derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues for linear functional differential equations fde by using integrated semigroup theory the idea is to formulate the fde as a non densely defined cauchy problem and obtain an explicit formula for the integrated solutions of the non densely defined cauchy problem from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues the results are useful in studying bifurcations in some semi linear problems

the navier stokes equations in the weak l 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 n ge 3 we give conditions on the external force sufficient for the unique existence of small solutions in the weak l n 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 l n space with time independent external force

input selection and partition validation for fuzzy modelling using neural network a simple and effective method for selecting significant input variables and determining optimal number of fuzzy rules when building a fuzzy model from data is proposed in contrast to the existing clustering based methods in this approach both input selecting and partition validating are determined on the basis of a class of sub clusters created by a self organising network instead of on the data the important input variables which independently and significantly influence the system output can be extracted by a fuzzy neural network on the other hand the optimal number of fuzzy rules can be determined separately via the fuzzy c means algorithm with a modified fuzzy entropy as the criterion of cluster validation the simulation results show that the proposed method can provide good model structures for fuzzy modelling and has high computing efficiency

the p adic group ring of sl2 pf in the present article we show that the zp pf 1 order zp pf 1 sl2 pf can be recognized among those orders whose reduction modulo p is isomorphic to fpfsl2 pf using only ring theoretic properties in other words we show that fpfsl2 pf lifts uniquely to a zp pf 1 order provided certain reasonable conditions are imposed on the lift this proves a conjecture made by nebe in 8 concerning the basic order of z2 2f 1 sl2 2f

characterization of nonuniform exponential trichotomies for flows abstract for an evolution family we characterize completely the notion of a nonuniform exponential trichotomy in terms of admissibility properties as a nontrivial application we establish the robustness of the notion in a very simple manner we also obtain corresponding results for a strong nonuniform exponential trichotomy we emphasize that both notions are ubiquitous in the context of ergodic theory moreover we develop a corresponding theory for the notion of a nonuniformly partially hyperbolic set in which case one considers simultaneously a collection of trajectories instead of a single one in particular this required first developing an appropriate theory for nonuniformly hyperbolic sets which is of independent interest

stable manifolds for semi linear evolution equations and admissibility of function spaces on a half line abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of stable manifolds of solutions to this equation in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory

optimal harvesting for periodic age dependent population dynamics here we investigate the optimal harvesting problem for some periodic age dependent population dynamics namely we consider the linear lotka mckendrick model with periodic vital rates and a periodic forcing term that sustains oscillations existence and uniqueness of a positive periodic solution are demonstrated and the existence and uniqueness of the optimal control are established we also state necessary optimality conditions a numerical algorithm is developed to approximate the optimal control and the optimal harvest some numerical results are presented

integral manifolds for partial functional differential equations in admissible spaces on a half line abstract in this paper we investigate the existence of stable and center stable manifolds for solutions to partial functional differential equations of the form u t a t u t f t u t t 0 when its linear part the family of operators a t t 0 generates the evolution family u t s t s 0 having an exponential dichotomy or trichotomy on the half line and the nonlinear forcing term f satisfies the lipschitz condition i e f t u t f t v t t u t v t c where u t v t c c r 0 x and t belongs to some admissible function space on the half line our main methods invoke lyapunov perron methods and the use of admissible function spaces

periodicity of inhomogeneous trajectories and applications abstract we prove the existence and uniqueness of periodic solutions to inhomogeneous linear evolution equations and then give applications to stokes equations in the space of bounded functions and to the inhomogeneous trajectories corresponding to semigroups which admit gaussian estimates