existence of multiple positive periodic solutions for differential equation with state dependent delays by utilizing a fixed point theorem in cones we present some sufficient conditions which guarantee the existence of multiple positive periodic solutions for a class of differential equations with state dependent delays our results extend and improve some previous results

a periodic solution of a differential equation with state dependent delay we study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future the main result is existence of a periodic solution in case the equilibrium is linearly unstable the proof employs the ejective fixed point principle on a compact convex set k0 c h 0 r of lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite dimensional submanifold of the space c1 h 0 r

orbits of primitive k homogenous groups on n k partitions with applications to semigroups the purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups automorphisms and the size of minimal generating sets in order to do this we examine the k homogeneous permutation groups those which act transitively on the subsets of size k of their domain x where x n and k n 2 in the process we obtain for k homogeneous groups results on the minimum numbers of generators the numbers of orbits on k partitions and their normalizers in the symmetric group as a sample result we show that every finite 2 homogeneous group is 2 generated underlying our investigations on automorphisms of transformation semigroups is the following conjecture if a transformation semigroup s contains singular maps and its group of units is a primitive group g of permutations then its automorphisms are all induced under conjugation by the elements in the normalizer of g in the symmetric group for the special case that s contains all constant maps this conjecture was proved correct more than 40 years ago in this paper we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less the e ort in establishing this result suggests that further improvements might be a great challenge this problem and several additional ones on permutation groups transformation semigroups and computational algebra are proposed in the end of the paper

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

determination of invariant zeros and zero directions of the system s a b c e mita 1977 presented a simple method for the invariant zeros of a linear multivariable system by constructing its maximal unobservable subspace mus however the results of this method have been restricted to a class of controllable and right invertible systems satisfying the diagonal decoupling condition and described by the triple a b c here a general method based on the theory of decoupling extends mita s results for all classes of systems described by the 4 tuples a b c e with no special requirements the mus of all classes of such systems is simply constructed and its order is given in terms of the system order n and the infinite zeros or the minimal observability indices of the system in addition the invariant zeros and zero directions are found from the eigenvalues and eigenvectors of a reduced order matrix having the dimension of the mus obtained directly from the state space parameters algorithms are presented for the invariant zeros and zero directions of invertible decouplable o

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

automorphism groups of centralizers of idempotents for a set x an equivalence relation on x and a cross section r of the partition x consider the following subsemigroup of the semigroup t x of full transformations on x t x r a t x ra rand x y xa ya the semigroup t x r is the centralizer of the idempotent transformation with kernel and image r we prove that the automorphisms of t x r are the inner automorphisms induced by the units of t x r and that the automorphism group of t x r is isomorphic to the group of units of t x r

existence of periodic solutions for delay differential equations with state dependent delay abstract in this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state a fixed point problem related to a poincare operator is constructed and solved using an ejective fixed point theorem

an exact characterization of linear systems with zeros everywhere in the complex plane in this note an exact characterization of autonomous linear multivariable systems with zeros everywhere in the complex plane is given in terms of the system matrices such systems are called degenerate an important consequence of this characterization is that nondegeneracy is equivalent to the concept of functional reproducibility introduced by brockett and mesarovic based on this equivalence a new test for nondegeneracy in terms of the functional reproducibility matrix is derived an example is included to show that controllability observability and output controllability are not sufficient to guarantee the nondegeneracy of a linear system

four notions of conjugacy for abstract semigroups there have been several attempts to extend the notion of conjugacy from groups to semigroups one notion which we will denote byp was originally introduced for free semigroups and popularized by lallement s book in a general semigroup p is reflexive and symmetric but not transitive however one may consider the transitive closure ofp which is an equivalence relation in any semigroup another notion of conjugacy which we will denote byo was introduced by otto for monoids presented by finite thue systems the relationo is an equivalence relation in any semigroup but it reduces to the universal relation if a semigroup has a zero since there is a precise sense in which almost all finite semigroups have a zero it follows thato is not useful for almost all finite semigroups three authors of the present paper introduced another notion of conjugacy denoted byc which is an equivalence relation in any semigroup in addition if a semigroup s does not have a zero thenc o in s butc does not reduce to the universal relation when a semigroup has a zero in order to decide which notion is the most satisfactory in the sense that it allows the extension of the classic conjugacy results in groups and hence leads to strong and elegant results with interconnections with other parts of mathematics it is necessary to carry out a deep study comparing and separating these three notions of conjugacy this is the aim of our paper first we study the decidability and independence of the conjugacy problems for certain classes of finitely presented monoids we show that there exists a monoid m defined by a finite complete rewriting system such that the c conjugacy problem for m is undecidable and that for finitely presented monoids the c conjugacy problem and the word problem are independent as are the c conjugacy and p conjugacy problems as an especially interesting test case we describec andp in the class of polycyclic monoids and show that in this class the p conjugacy is almost transitive and thatc is strictly included inp we also characterizec in the symmetric inverse monoid i x for a countable set x and comparec andp in that monoid we then generalize an outstanding theorem proved by kudryavtseva showing that relationp is transitive in a variety of semigroups that contains all completely regular semigroups and their variants althougho andc coincide for semigroups without zero they differ in the most extreme way in some semigroups with zero we present an infinite family of semigroups with zero divisors in whicho is universal whilec is the identity we prove that commutative semigroups are precisely those where p is the identity and commutative and cancellative semigroups are those whereo is the identity we also show that a semigroup in whichp is universal is necessarily simple moreover if a semigroup s has an idempotent we conclude that s is a rectangular band if and only ifp is the universal relation the paper ends with a fairly large section of open problems

the largest subsemilattices of the endomorphism monoid of an independence algebra abstract an algebra a is said to be an independence algebra if it is a matroid algebra and every map x a defined on a basis x of a can be extended to an endomorphism of a these algebras are particularly well behaved generalizations of vector spaces and hence they naturally appear in several branches of mathematics such as model theory group theory and semigroup theory it is well known that matroid algebras have a well defined notion of dimension let a be any independence algebra of finite dimension n with at least two elements denote by end a the monoid of endomorphisms of a we prove that a largest subsemilattice of end a has either 2 n 1 elements if the clone of a does not contain any constant operations or 2 n elements if the clone of a contains constant operations as corollaries we obtain formulas for the size of the largest subsemilattices of some variants of the monoid of linear operators of a finite dimensional vector space the monoid of full transformations on a finite set x the monoid of partial transformations on x the monoid of endomorphisms of a free g set with a finite set of free generators among others the paper ends with a relatively large number of problems that might attract attention of experts in linear algebra ring theory extremal combinatorics group theory semigroup theory universal algebraic geometry and universal algebra

boundary layer phenomena for differential delay equations with state dependent time lags i in this paper we begin a study of the differential delay equation mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0jc9yq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpeps0he9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiabew7aljqadi hagaqbaiaacicacawg0bgaaiykaiabg2da9iabgkhitiaadihacagg oagaamidaiaacmcacqghrawkcawgmbgaaiikaiaadihacaggoagaam idaiabgkhitiaadkhacaggpagaaiykaiaacycacaqggagaaeiiaiaa dkhacqgh9aqpcawgybgaaiikaiaadihacaggoagaamidaiaacmcaca ggpaaaaa 5192 varepsilon x t x t f x t r r r x t we prove the existence of periodic solutions for 0 0 where 0 is an optimal positive number we investigate regularity and monotonicity properties of solutions x t which are defined for all t and of associated functions like t t r x t we begin the development of a poincare bendixson theory and phase plane analysis for such equations in a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as approaches zero

computation of the zeros and zero directions of linear multivariable systems a new geometric method of calculating multivariable system zeros and zero directions is presented by considering a particular choice of state feedback control law this control law has been motivated by the study of variable structure systems in the sliding mode the cases of singular and non singular cb have been treated separately

on the geometry of the inverse system the geometry of a particular reduced order state space realization of the inverse of a linear continuous time invariant strictly proper system is considered this geometry is characterized by the eigenstructure of the dynamic part of the inverse which is related to the invariant zero structure of the inverted system a method for deriving explicit expressions for the zeros and the zero directions is introduced this method is demonstrated in an example of a uniform rank system

almost periodic solution for a volterra model with mutual interference and beddington deangelis functional response in this paper we consider a volterra model with mutual interference and beddington deangelis functional response by applying the comparison theorem of the differential equation and constructing a suitable lyapunov functional sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

slowly oscillating periodic solutions of autonomous state dependent delay equations more than a dozen years since the peak of activity on the question of the existence of periodic slowly oscillating solutions of autonomous delay differential equations following the early work of jones l wright 2 and grafton 3 the work of nussbaum 4 51 is to be specially noted for providing several new fixed point results and a global bifurcation theorem which are particularly useful for proving the existence of periodic solutions other important works include those of hadeler and tomiuk 6 kaplan and yorke 7 chow 8 chow and hale 9 alt lo 111 and walther 12 131 see hale s book 14 for an overview to the best of our knowledge only nussbaum 5 and alt 1 l considered the question of the existence of periodic solutions of autonomous state dependent delay differential equations nussbaum considered a special equation 0 3 below in 5 but did not prove a general result alt 1 l considered more complicated integral threshold type state dependent delays which have arisen in various models in epidemiology and structured population models alt obtained a theorem for a general class of state dependent equations on which we comment below the lack of results on periodic solutions for state dependent equations is probably explained by the fact that it is not clear what kinds of state dependent delays are interesting or natural and on a lack of compelling examples of such equations arising from plausible mathematical models however some recent papers of belair and mackey 15 and belair 16 contain some interesting state dependent delay equations arising in economics and population biology state dependent delays of threshold type arise naturally in structured population models see lo 111 the motivation for this paper stems from consideration of the innocent looking equation x t kx t r x t 0 1 where r x is bell shaped e g r x 01 emxz 1 01 0 i 01 5 1 if k 7r 2 and r o 1 then the zero solution of 0 1 is unstable a formal linearization would yield z t kz t 1 which is known to be unstable for

three periodic solutions for a nonlinear first order functional differential equation this paper is concerned with the existence of three positive t periodic solutions of the first order functional differential equations of the formx t a t x t email protected t f t x h t where a t b t and t t are positive t periodic functions and l is a positive parameter leggett williams multiple fixed point theorem has been used to prove our results our results extend and improve some results in the literature

conjugation in semigroups abstract the action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory there have been several attempts to extend the notion of conjugacy to semigroups in this paper we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero we compare the new notion of conjugacy with existing definitions characterize the conjugacy in various semigroups of transformations on a set and count the number of conjugacy classes in these semigroups when the set is infinite

multiple positive periodic solutions for a class of nonlinear functional differential equations abstract by employing avery henderson fixed point theorem in cones we obtain some sufficient conditions which guarantee the existence of multiple positive periodic solutions for a class of differential equations with state dependent delays

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

system zeros analysis via the moore penrose pseudoinverse and svd of the first nonzero markov parameter a new characterization of system zeros of an arbitrary linear system described by a state space model s a b c d is presented the transmission zeros are characterized as invariant zeros of an appropriate strictly proper system with a smaller number of inputs and outputs than the original system the approach is based on singular value decomposition svd of the first nonzero markov parameter this result together with characterization of invariant and decoupling zeros based on the moore penrose inverse of the first nonzero markov parameter and the kalman canonical decomposition theorem provided in the first part of the paper yield a complete characterization of system zeros of an arbitrary multi input multi output system

boundary layer phenomena for differential delay equations with state dependent time lags iii in this paper we begin a study of the differential delay equation varepsilon x t x t f x t r r r x t

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

properties of the semigroup and the approximation of crandall liggett associated with a class of differential equations with delays depending implicitly on the state abstract in this paper we are concerned with the semigroup properties and the crandall liggett approximation associated with a class of differential equations with delays depending implicitly on the state the existence of a strongly continuous nonlinear semigroup associated with the solutions of the equations is proved various properties of this semigroup have been established such as the continuous dependence the compactness and the infinitesimal generators an approximation of crandall liggett type has also been established