equations of motion for variational electrodynamics abstract we extend the variational problem of wheeler feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative velocities of bounded variation we show here that the gateaux derivative of the generalized functional defines two partial lagrangians for variations in our generalized local space one for each particle we prove that the critical point conditions of the generalized variational problem are i the euler lagrange equations must hold lebesgue almost everywhere and ii the momentum of each partial lagrangian and the legendre transform of each partial lagrangian must be absolutely continuous functions generalizing the weierstrass erdmann conditions

parabolic partial differential equations with discrete state dependent delay classical solutions and solution manifold abstract classical solutions to pdes with discrete state dependent delay are studied we prove the well posedness in a set x f which is analogous to the solution manifold used for ordinary differential equations with state dependent delay we prove that the evolution operators are c 1 smooth on the solution manifold

primitive free cubics with specified norm and trace the existence of a primitive free normal cubic x 3 ax 2 cx b over a finite field f with arbitrary specified values of a 0 and b primitive is guaranteed this is the most delicate case of a general existence theorem whose proof is thereby completed

superstability and rigorous asymptotics in singularly perturbed state dependent delay differential equations abstract we study the singularly perturbed state dependent delay differential equation e x t x t k x t r r r x t 1 x t which is a special case of the equation e x t g x t x t r r r x t one knows that for every sufficiently small e 0 eq possesses at least one so called slowly oscillating periodic solution and moreover the graph of every such solution approaches a specific sawtooth like shape as e 0 in this paper we obtain higher order asymptotics of the sawtooth including the location of the minimum and maximum of the solution with the form of the solution near these turning points and as well an asymptotic formula for the period using these and other asymptotic formulas we further show that the solution enjoys the property of superstability namely the nontrivial characteristic multipliers are of size o e for small e this stability property implies that this solution is unique among all slowly oscillating periodic solutions again for small e

normal basis of the finite field f sub 2 p 1 p sup m over f sub 2 the determination of normal bases for finite fields particularly over the finite field f sub 2 is of importance in applications such as coding and cryptography this correspondence gives an explicit normal basis for a finite field f sub 2 p 1 p sup m over f sub 2 when 2 is a primitive element module p sup 2 in the case when p 3 an explicit dual basis is also obtained

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

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

merging homoclinic solutions due to state dependent delay abstract for 2 5 2 the linear differential equation x t x t 1 with constant time lag is unstable and hyperbolic for sufficiently close to 5 2 we construct a state dependent delay d u 0 2 on an open set in the space c 1 2 0 with d u 1 for close to 0 so that the nonlinear equation x t x t d u x t has a one parameter family of entire solutions c r r r r r u which are homoclinic to zero and merge in finite time at some t y 0 c r t c 0 t for r r u and t t y this complements earlier results on complicated solution behaviour caused by state dependent delay and shows that there exist delay functionals for which certain transversality and regularity conditions along homoclinic solutions are violated

dirichlet problem for the diffusive nicholson s blowflies equation abstract the object of this paper is to consider the asymptotic behavior of solutions of the diffusive nicholson s blowflies equation under dirichlet boundary condition a new approach is developed to study the global attractivity of the positive steady state for a reaction diffusion equation with time delay when monotone or quasi monotone conditions fail to apply this approach should be applicable to other dirichlet problems as well even though our analysis is tailored to the diffusive nicholson s blowflies equation

partial differential equations with discrete and distributed state dependent delays this work is an attempt to treat partial differential equations with discrete concentrated state dependent delay the main idea is to approximate the discrete delay term by a sequence of distributed delay terms all with state dependent delays we study local existence and long time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor

compact kernel sections for lattice systems with delays abstract in this paper we consider the existence upper semi continuity and singular limiting behavior of compact kernel sections for the process associated to non autonomous lattice systems with delays

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

non linear partial differential equations with discrete state dependent delays in a metric space we investigate a class of non linear partial differential equations with discrete state dependent delays the existence and uniqueness of strong solutions for initial functions from a banach space are proved to get the well posed initial value problem we restrict our study to a smaller metric space construct the dynamical system and prove the existence of a compact global attractor

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

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

low complexity erasure insertion in rs coded sfh spread spectrum communications with partial band interference and nakagami m fading we propose two novel low complexity low delay erasure insertion schemes namely the output threshold test ott and joint maximum output and ratio threshold test mo rtt the employment of the ott and mo rtt is beneficial in the context of the errors and erasures reed solomon rs decoding in a slow frequency hopping spread spectrum sfh ss system using m ary frequency shift keying mfsk the statistics of the erasure insertion related decision variables associated with the ott mo rtt as well as with the ratio threshold test rtt are investigated when the channel of each frequency hopping fh slot is modeled as flat nakagami m fading the transmitted signals also experience both additive white gaussian noise awgn as well as partial band gaussian interference pbgi the properties of these erasure insertion schemes are investigated with the aid of their statistics the performance of the proposed erasure insertion schemes and that of the erasure insertion scheme using the rtt is investigated and compared in the context of rs coded sfh ss systems using mfsk furthermore the performance of the rs coded sfh ss systems is also compared both with and without side information concerning the pbgi

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

wavefronts and global stability in a time delayed population model with stage structure we formulate and study a one dimensional single species diffusive delay population model the time delay is the time taken from birth to maturity without diffusion the delay differential model extends the well known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes this delayed model is known to have simple global dynamics similar to that of the logistic equation through the use of a sub supersolution pair method we show that the diffusive delay model continues to generate simple global dynamics this has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics we also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population connecting the zero solution of that equation with the positive steady state our main finding here is that our fronts appear to be all monotone regardless of the size of the delay this is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

the solution manifold and c1 smoothness for differential equations with state dependent delay abstract let h 0 u c 1 h 0 r n open and f u r n continuously differentiable if f satisfies two mild additional smoothness conditions then the set x u 0 f is a c1 submanifold of codimension n in c 1 h 0 r n the maximal solutions x of the initial value problems x t f x t x 0 x define a continuous semiflow f on x and all operators f t are continuously differentiable their derivatives d2f t are given in the usual way by solutions v to the variational equation along x with segments vt in the tangent spaces tf t x the additional conditions on f are motivated by properties of differential equations with state dependent delay and are verified for an example

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

periodic attractor for reactiondiffusion high order hopfield neural networks with time varying delays this paper is concerned with a class of reactiondiffusion high order hopfield neural networks with time varying delays subject to the dirichlet boundary condition in a bounded domain easily verifiable delay independent criteria are established to ensure the existence of periodic mild solutions and the global exponential stability of the periodic mild solutions is also discussed by using the exponential dissipation property of semigroup of operators the obtained results are easy to check and they effectually complement previously known results a numerical example is given to show the effectiveness of theoretical results

a survey on normal bases over finite fields normal bases are important in efficient implementation of finite field arithmetic for many applications in coding theory and cryptography this paper gives a survey on various topics of normal bases including basic properties of normal elements primitive normal bases dual and self dual normal bases fast arithmetic under normal bases and constructions for low complexity normal bases

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

smoothness of semiflows for parabolic partial differential equations with state dependent delay in this paper the smoothness properties of semiflows on c1 solution submanifold of a parabolic partial differential equations with state dependent delay are investigated the problem is formulated as an abstract ordinary retarded functional differential equation of the form du t dt au t f ut with a continuously differentiable map g from an open subset u of the space c1 h 0 l2 where a is the infinitesimal generator of a compact c0 semigroup the present study is continuation of a previous work 14 that highlights the classical solutions and c1 smoothness of solution manifold here we further prove the continuous differentiability of the semiflow we finally verify all hypotheses by a biological example which describes a stage structured diffusive model where the delay which is the time taken from birth to maturity is assumed as a function of a immature species population

reduced gmd decoding a framework is presented for generalized minimum distance gmd decoding with a limited number of decoding trials and a restricted set of reliability values in gmd decoding symbols received from the channel may be erased before being fed into an algebraic error erasure decoder for error correction in subsequent or simultaneous trials with different erasing patterns the decision whether or not to erase a symbol in a certain trial is taken by an erasure choosing algorithm which takes into account reliability information from the channel the final gmd decoder output is a codeword which results from a decoding trial and satisfies a certain distance criterion for various erasing strategies and reliability sets the guaranteed error correction radius and the unsuccessful decoding probability of this technique are studied both known and new results with applications to concatenated coding follow from the unified approach presented in this correspondence

convolutional codes with maximum column sum rank for network streaming the column hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels we introduce a metric known as the column sum rank that parallels the column hamming distance when streaming over a network with link failures we prove the rank analogues of several known column hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory our construction involves finding a class of super regular matrices that preserve this property after multiplication with non singular block diagonal matrices in the ground field

optimal erasure selection of m ary pam signaling for errors and erasures decoding algorithms erasure selection for errors and erasures decoding algorithms is investigated and analyzed where declaring erasures provides additional gain for hard decision decoding to derive an analytical criterion for optimal erasure selection we first investigate erasure effects on decoding performance by which a metric function for erasure selection is derived from the derived metric a suboptimal threshold is defined which can be used for errors and erasures decoding of large code distance channel codes moreover application to m ary m 2 pam constellation is also described for verification of the proposed erasure threshold simulation results with popular channel codes are included

chemical principle and pde of variational electrodynamics abstract the two body problem of variational electrodynamics possesses differential delay equations of motion with state dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets from a periodic orbit possessing some mild properties at breaking points we define a synchronization function in r r 3 which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points the oscillatory functions are associated with a natural pde identity in w 2 2 r 3 we postulate an asymptotic ordering for a generalized pde identity which defines a fredholm schroedinger operator in w 2 2 r 3 with a forcing term belonging to l 2 r 3 o 1 r 2 as an application we introduce the chemical principle criterion to select orbits with asymptotically vanishing far fields and estimate the bohr radius parameter of the fredholm schroedinger pde using the boundary layers of orbits chosen according to the chemical principle criterion last working backward we derive an orbital property of the chemical principle type from the ordering conditions

a non local pde model for population dynamics with state selective delay local theory and global attractors we propose a non local pde model for the evolution of a single species population that involves delayed feedback where the delay such as the maturation time in the delayed birth rate is selective and the selection depends on the status of the system this delay selection in contrast with the usual state dependent delay widely used in ordinary delay differential equation ensures the lipschitz continuity of the nonlinear functional in the classical phase space we also develop the local theory and the existence and upper semi continuity of the global attractor with respect to parameters

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions abstract partial differential equations with discrete concentrated state dependent delays in the space of continuous functions are investigated in general the corresponding initial value problem is not well posed so we find an additional assumption on the state dependent delay function to guarantee the well posedness for the constructed dynamical system we study the long time asymptotic behavior and prove the existence of a compact global attractor

approximations for the probability in the tails of the binomial distribution corresp upper and lower bounds for the probability in the tails of a binomial distribution are investigated new bounds are obtained which are computationally simple and in the case of the lower bound significantly better than previously known bounds

a condition on delay for differential equations with discrete state dependent delay abstract parabolic differential equations with discrete state dependent delay are studied the approach based on an additional condition on the delay function introduced in a v rezounenko differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions nonlinear anal 70 11 2009 3978 3986 is developed we propose and study an analogue of the condition which is sufficient for the well posedness of the corresponding initial value problem on the whole space of continuous functions c the dynamical system is constructed in c and the existence of a compact global attractor is proved

c1 α strict solutions and wellposedness of abstract differential equations with state dependent delay abstract we study the existence and uniqueness of c 1 strict solutions for a general class of abstract differential equations with state dependent delay we also study the local well posedness of this type of problems on subspaces of c 1 p 0 x some examples involving partial differential equations are presented

generators and irreducible polynomials over finite fields weil s character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field an application to the distribution of irreducible polynomials is given which confirms an asymptotic version of a conjecture of hansen mullen

uniform attractors for non autonomous parabolic equations with delays abstract the present paper is devoted to the asymptotic behavior of non autonomous parabolic equations with nonlinearity containing general delay and with time dependent external forces the existence uniqueness and regularity of solutions for the equation are obtained if the time dependent external forces is a translation compact function in l loc 2 r l 2 then the existences of uniform attractor for non autonomous dynamics system are provided and the structures of attractors are also studied which is the union of all bounded complete trajectory of the equation at a time moment finally the main results are applied to the population dynamics with distributed state dependent delay and with almost periodic external force and the uniform attractor and its kernel section are obtained

a fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377 the standard algorithm for testing reducibility of a trinomial of prime degree r over gf 2 requires 2r o 1 bits of memory we describe a new algorithm which requires only 3r 2 o 1 bits of memory and significantly fewer memory references and bit operations than the standard algorithm if 2r 1 is a mersenne prime then an irreducible trinomial of degree r is necessarily primitive we give primitive trinomials for the mersenne exponents r 756839 859433 and 3021377 the results for r 859433 extend and correct some computations of kumada et al the two results for r 3021377 are primitive trinomials of the highest known degree

primitive normal polynomials over finite fields in this note we significantly extend the range of published tables of primitive normal polynomials over finite fields for each pn 1050 with p 97 we provide a primitive normal polynomial of degree n over fp moreover each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over fp the roots of such a polynomial generate a primitive normal basis of fpn over fp and so are of importance in many computational problems we also raise several conjectures concerning the distribution of such primitive normal polynomials including a refinement of the primitive normal basis theorem

on primitive polynomials over finite fields abstract let f gf q and let e gf qk be the field extension of degree k of f we show that the following statement holds for all but finitely many exceptional pairs q k given any element a f there exists a primitive element of e with trace t e f a equivalently the coefficient of xk 1 in a primitive polynomial of degree k over gf q may be arbitrarily selected from gf q

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

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

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

the coefficients of primitive polynomials over finite fields for n 7 we prove that there always exists a primitive polynomial of degree n over a finite field f q q odd with the first and second coefficients prescribed in advance

dynamics of the diffusive nicholson s blowflies equation with distributed delay in this paper we study the diffusive nicholson s blowflies equation generalizing previous works we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible the linearized stability of the non zero uniform steady state is studied in detail mainly by using the principle of the argument global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations finally we consider what bifurcations are possible from the non zero uniform state in the case when it is unstable

compact θ method for the generalized delay diffusion equation abstract the generalized diffusion equation with a delay has inherent complex nature because its analytical solutions are hard to obtain therefore one has to seek numerical methods especially the high order accurate ones for their approximate solutions in this paper we have established the results of the numerical asymptotic stability of the compact method for the generalized delay diffusion equation it shows that the compact method is asymptotically stable if and only if k r t h 2 10 cos h 12 1 cos h 1 2 for 0 1 2 and is unconditionally asymptotically stable for 1 2 1 respectively the convergent results in the maximum norm are studied according to the consistency analysis and lax theorem in the end a series of numerical tests on stability and convergence are carried out to support our theoretical results

existence results for an impulsive abstract partial differential equation with state dependent delay in this paper we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state dependent delay

on abstract differential equations with state dependent non local conditions abstract in this paper we introduce the concept of state dependent nonlocal condition and study the existence and uniqueness of solution for a general class of abstract differential equations with state dependent delay subject to this class of conditions in this type of problem the nonlocal condition can be dependent on times or time intervals determined by the unknown solution some examples on partial differential equations with state dependent delay arising in population dynamics are presented

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

dynamics of projective adaptive resonance theory model the foundation of part algorithm projective adaptive resonance theory part neural network developed by cao and wu recently has been shown to be very effective in clustering data sets in high dimensional spaces the part algorithm is based on the assumptions that the model equations of part a large scale and singularly perturbed system of differential equations coupled with a reset mechanism have quite regular computational performance this paper provides a rigorous proof of these regular dynamics of the part model when the signal functions are special step functions and provides additional simulation results to illustrate the computational performance of part

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

delay partial differential equations with some large diffusion it is often the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v its eigenvalues and eigenfunctions will also depend upon v in many cases see for example l 7 there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies the remaining eigenvalues blow up at a certain rate in this case one expects the dynamics to be dictated by a system of ordinary differential equations this is indeed the case in several applications and its proof strongly uses the invariant manifold theory in the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain whereas the remaining blow up this is the case for example in parabolic equations in thin domains studied by hale and raugel s we will not be able to address the general case of thin domains but only the case of a thin straight channel however the property that allows this theory to work is present in the general case that is there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up see also arrieta 9 the theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation the main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem a physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that in that direction spatial changes will not be observed the same interpretation can be given to a damped wave problem in a thin straight channel the wave speed can be interpreted as being very large in one direction this would intuitively imply that there would be no spatial changes for the amplitude of the wave in that direction we also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances in this

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays

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

primitive normal bases for finite fields it is proved that any finite extension of a finite field has a normal basis consisting of primitive roots introduction let q be a prime power q 1 we denote by f9 a finite field of q elements it is well known that for every positive integer m there exists a normal basis of f m over f9 i e a basis of the form q q 2 qp aaq aq2 a9 1 with a e f m it is also well known that the multiplicative group f of fqm is cyclic i e that for some a e pq we have f an n e z such an element a is called a primitive root of fqm following davenport 4 we call qq2 rn i a normal basis a a a9 a a a9 of fq over fq a primitive normal basis if a is a primitive root of fq carlitz 2 3 proved in 1952 that for all sufficiently large qm there exists a primitive normal basis of f m over fq davenport 4 proved in 1968 that a primitive normal basis exists for all m if q is prime in the present paper this result is extended to the general case theorem for every prime power q 1 and every positive integer m there exists a primitive normal basis of fqm over fq section 1 contains an exposition of certain results due to ore 7 concerning the galois module structure of finite fields these lead to an alternative formulation of the theorem in section 2 we describe an improved version of the method of carlitz and davenport which handles all but finitely many pairs q m in section 3 we determine which are the remaining pairs and they are dealt with in section 4 we denote the cardinality of a set s by s and the group of units of a ring r with 1 by r if f g are polynomials in one variable we mean by g i f that g divides f and is monic i e has leading coefficient one the same notation for divisibility is used for positive integers received march 5 1986 1980 mathematics subject classification 1985 revision primary 11t30 12e20

principle of linearized stability and instability for parabolic partial differential equations with state dependent delay abstract in this paper the stability properties of a parabolic partial differential equation with state dependent delay are investigated by the heuristic approach the previous works 1 2 obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions and resolved the problem of linearization for this equation here we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution by a biological application we finally verify all hypotheses for an age structured diffusive model with state dependent delay and consider its stability behavior

stability and hopf bifurcation for a cell population model with state dependent delay we propose a mathematical model describing the dynamics of a hematopoietic stem cell population the method of characteristics reduces the age structured model to a system of differential equations with a state dependent delay a detailed stability analysis is performed a sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a lyapunov razumikhin function a unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state the analysis of the positive steady state behavior through the study of a first order exponential polynomial characteristic equation concludes the existence of a hopf bifurcation and gives criteria for stability switches a numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

approximate inertial manifolds for retarded semilinear parabolic equations abstract the asymptotic behavior of a system of retarded parabolic equations is considered for any given 0 we construct an approximate inertial manifold aim which contains all the steady states of the system and has an attractive neighborhood of thickness the dependence of aims on the delay time is investigated

the asymptotic behavior of the strong solutions for a non autonomous non local pde model with delay abstract in this paper we study the asymptotic behavior of the strong solutions of a non autonomous non local pde model with time delay we present the existence and structure of the uniform attractor by constructing the skew product flow of the family of processes generated by the strong solutions in order to obtain the existence of the uniform attractor we prove the family of processes satisfies uniform condition c by using some special technique of phase space decomposition additionally it is shown that all the bounded complete trajectories are globally asymptotic stable under some assumptions as the application of our result we obtain a globally asymptotic stable nontrivial strong periodic solution of a non local pde model

finding isomorphisms between finite fields we show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time every finite field has cardinality p for some prime number p and some positive integer conversely if p is a prime number and a positive integer then there exists a field of cardinality p and any two fields of cardinality p are isomorphic these results are due to e h moore 1893 10 in the present paper we are interested in an algorithmic version of his theorem in particular of the uniqueness part we say that a finite field is explicitly given if for some basis of the field over its prime field we know the product of any two basis elements expressed in the same basis let more precisely p be a prime number and a positive integer then by explicit data for a finite field of cardinality p we mean a system of elements ajjk k x of the prime field fp z pz suchthat f becomes a field with the ordinary addition and multiplication by elements of fp and the multiplication determined by n

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach