ternary composition algebras and hurwitz s theorem axioms are proposed for a certain alternative kind of ternary composition algebra termed a 3cn algebra the axioms are shown to be for n 2 in a simple correspondence with the axioms for a ternary vector cross product algebra the axioms imply that n 1 2 4 or 8 from which the usual hurwitz theorem is deduced the existence of 3c8 algebras is demonstrated by an explicit construction in four dimensional hilbert space without appeal to the properties of the algebra of octonions

a new view of d 7 clifford algebra a new view of d 7 clifford algebra is described in which starting from a septet of mutually anticommuting clifford operators one introduces a septet of mutually commuting involutions in so 8 these involutions can be seen to arise naturally from a study of ternary vector cross products in eight dimensions

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

sensitivity analysis of discrete markov chains via matrix calculus abstract markov chains are often used as mathematical models of natural phenomena with transition probabilities defined in terms of parameters that are of interest in the scientific question at hand sensitivity analysis is an important way to quantify the effects of changes in these parameters on the behavior of the chain many properties of markov chains can be written as simple matrix expressions and hence matrix calculus is a powerful approach to sensitivity analysis using matrix calculus we derive the sensitivity and elasticity of a variety of properties of absorbing and ergodic finite state chains for absorbing chains we present the sensitivities of the moments of the number of visits to each transient state the moments of the time to absorption the mean number of states visited before absorption the quasistationary distribution and the probabilities of absorption in each of several absorbing states for ergodic chains we present the sensitivity of the stationary distribution the mean first passage time matrix the fundamental matrix and the kemeny constant we include two examples of application of the results to demographic and ecological problems

generalization of a fundamental matrix abstract it is shown that for a finite ergodic markov chain basic descriptive quantities such as the stationary vector and mean first passage matrix may be calculated using any one of a class of fundamental matrices new applications of the use of these operators are discussed

ternary vector cross products ternary vector cross products are studied in their own right results include a new proof of hurwitz s theorem and a principle of duplicity upon breaking the symmetry in eight dimensions by choosing a preferred axis this last principle implies the well known triality principle for octonions and so 8 transformations in displaying canonical forms it helps to put the eight basis vectors in correspondence with the eight points of the three dimensional affine geometry over f2

on triality and automorphisms and derivations of composition algebras abstract the principle of triality for orthogonal transformations on eight dimensional spaces is used to obtain different expressions for the automorphisms derivations and orthogonal transformations of cayley dickson algebras and symmetric composition algebras over fields of arbitrary characteristic

sensitivity of the stationary distribution vector for an ergodic markov chain stationary distribution vectors p sup infinity for markov chains with associated transition matrices t are important in the analysis of many models in the mathematical sciences such as queuing networks input output economic models and compartmental tracer analysis models the purpose of this paper is to provide insight into the sensitivity of p sup infinity to perturbations in the transition probabilities of t and to understand some of the difficulties in computing an accurate p sup infinity the group inverse a sup number of i t is shown to be of fundamental importance in understanding sensitivity or conditioning of p sup infinity the main result shows that if there is a state that is accessible from every other state and the corresponding column of t has no small off diagonal elements then p sup infinity cannot be sensitive to small perturbations in t ecological examples are given a new stable algorithm for calculating a sup number is described

the weak solution of a functional differential equation in a general banach space abstract the existence of a unique weak lipschitz continuous solution of the abstract functional differential equation u t a t u t g t u t t 0 t fde u t t t r 0 is obtained by a fixed point argument here x is a general banach space a t d a t x x is m accretive t 0 t and g is lipschitz continuous in both variables stability properties of this weak solution are given and a first order hyperbolic pde to which the theory can be applied is considered

on the asymptotic behavior of a semilinear functional differential equation in banach space the family a t t 3 0 is supposed to generate a linear evolution system and f t u is a nonlinear operator continuous with respect to a fractional power of a r for some r 2 0 it is known that local solutions to 1 exist under fairly general continuity assumptions on the nonlinear term f t u see 3 4 combined with condition 5 below this guarantees the existence of a global solution u t cf 3 p 5391 our results then say that if the linearized equation u t a t u t is asymptotically stable the same holds true for 1 provided that the nonlinear perturbation f is sufficiently small the main point hereby lies in the fact that the bound forfis independent of the time lag r 0

a six variable identity for a ternary vector cross product in eight dimensional space let x a b c be a ternary vector cross product for eight dimensional euclidean space e an identity is derived which expresses x a b c x u v w in terms of the spin 7 invariant scalar quadruple product phi a b c d a x b c d the proof of the identity is coordinate free and starts out from an explicit expression for x with e viewed as complex four dimensional hilbert space

high order finite difference technique for delay pseudo parabolic equations abstract one dimensional initial boundary delay pseudo parabolic problem is being considered to solve this problem numerically we construct higher order difference method for approximation to the considered problem and obtain the error estimate for its solution based on the method of energy estimate the fully discrete scheme is shown to be convergent of order four in space and of order two in time numerical example is presented

explicit forms for ergodicity coefficients and spectrum localization explicit forms for orgodicity coefficients which bound the non unit eigenvalues of finite stochastic matrices are known in the cases p 1 when the lp norm is used 3 6 7 the purpose of this report is to generalize these results to arbitrary real matrices in particular square matrices with a real valued eigenvalue by giving a generalized and to some extent simplified synthesis of methods used hitherto alternative procedures are mentioned in the light of the framework established the results are of particular relevance for a primitive nonnegative matrix for which a bound results for the non perron frobenius eigenvalues

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

on the approximate k controllability of nonlinear systems in banach spaces abstract let x be a real banach space control problems of the type x a t x b t x u t 0 t x 0 0 are considered where for every t 0 t a t x d a x and b 0 t x 2 x are given operators the concept of approximate k controllability or controllability with preassigned responses is introduced for systems of the type it is shown that there exist lipschitz continuous approximating control functions u t t 0 t for a variety of response types evans responses and kato responses are considered for fully nonlinear problems as well as mild ones for semilinear problems it is also shown that the function r which determines the proximity of the response x t with x t a t x t b t x t u t t 0 t x 0 0 to the preassigned response f t x f r is of the type c where c is a positive constant independent of a more natural approximate k controllability concept approximate ek controllability is also introduced and a result is given about it using leray schauder theory an application is given in the field of partial differential equations

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

existence uniqueness and stability of stochastic neutral functional differential equations of sobolev type in this paper we are mainly concerned with a class of stochastic neutral functional differential equations of sobolev type with poisson jumps under two different sets of conditions we establish the existence of the mild solution by applying the leray schauder alternative theory and the sadakovskii s fixed point theorem respectively furthermore we use the bihari s inequality to prove the osgood type uniqueness also the mean square exponential stability is investigated by applying the gronwall inequality finally two examples are given to illustrate the theory results

existence and asymptotic behavior of a functional differential equation in banach space abstract existence and asymptotic behavior of solutions are given for the equation u t a t u t f t u t t 0 and u 0 c r 0 x c the space x is a banach space the family a t 0 t t of unbounded linear operators defined on d a x x generates a linear evolution system and f c x is continuous with respect to a fractional power of a t 0 for some t 0 0 t

sobolev type fractional stochastic differential equations with non lipschitz coefficients this paper investigates the existence and uniqueness of mild solutions for a class of nonlinear fractional sobolev type stochastic differential equations in hilbert spaces in this work we used the fractional calculus semigroup theory and stochastic analysis techniques for obtaining the required result a new set of sufficient condition is established with the coefficients in the equations satisfying some non lipschitz conditions which include classical lipschitz conditions as special cases more precisely the results are obtained by means of standard picards iteration finally an example is given to illustrate the obtained theory

degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators in this paper we mainly investigate controlled fractional evolution hemivariational inequalities of degenerate type in caputo and riemann liouville fractional derivatives of order in 1 2 respect

functional evolution equations abstract of concern is the functional evolution equation du dt bu f t u t t 0 u 0 s c 1 r 0 x where ut s u t s r 0 s r 0 and x is a real banach space it is shown by the method of lines combined with the crandall pazy theorem that for the initial data u0 in a generalized domain this equation has a limit solution which is lipschitz continuous in t and that this limit solution is a unique strong one if further assumptions on b are imposed and that the zero solution is asymptotically stable an application is given to a class of partial functional differential boundary value problems

existence results for partial neutral functional differential equations with unbounded delay abstract we prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the leray schauder alternative

on a time discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space abstract in this paper we construct a semidiscrete approximate solution to a semilinear one dimensional heat equation subject to integral boundary conditions by means of the rothe discretization in time method the convergence of the approximation scheme obtained is proved yielding the well posedness of the problem considered

flow invariance for nonlinear partial differential delay equations several fundamental results on existence flow invariance regularity and linearized stability of solutions to the nonlinear partial differential delay equation u t bu t f u t t 0 uo with b c x x x w accretive are developed for a general banach space x in contrast to existing results with the history response f globally defined and at least lipschitz on bounded sets the results are tailored for situations with f defined on possibly thin subsets of the initial history space e only and are applied to place several classes of population models in their natural l 1 setting the main result solves the open problem of a subtangential condition for flow invariance of solutions in the fully nonlinear case paralleling those known for the cases of a no delay b ordinary delay equations with b 0 and c the semilinear case

existence and continuous dependence results for nonlinear differential inclusions with infinite delay this paper is concerned with nonlinear functional differential inclusions with infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory existence and continuous dependence results are obtained for integral solutions without the assumption of compactness on the associated nonlinear semigroup

oscillations of solutions of vector differential equations of parabolic type with functional arguments vector parabolic differential equations with functional arguments are studied and the oscillations of solutions of boundary value problems are investigated our approach is to reduce the oscillation problems to the nonexistence of positive solutions of scalar differential inequalities by employing the concept of h oscillation introduced by domslak see r courant d hilbert methods of mathematical physics vol i interscience new york 1996 where h denotes a unit vector

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

stationary distributions and mean first passage times of perturbed markov chains stationary distributions of perturbed finite irreducible discrete time markov chains are intimately connected with the behaviour of associated mean first passage times this interconnection is explored through the use of generalized matrix inverses some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are obtained together with some improved bounds ams classification 15a51 60j10 60j20 65f20 65f35

method of lyapunov functionals construction in stability of delay evolution equations the investigation of stability for hereditary systems is often related to the construction of lyapunov functionals the general method of lyapunov functionals construction which was proposed by v kolmanovskii and l shaikhet and successfully used already for functional differential equations for difference equations with discrete time for difference equations with continuous time is used here to investigate the stability of delay evolution equations in particular partial differential equations

convergence of the kato approximants for evolution equations involving functional perturbations the existence of a unique strong solution of the nonlinear abstract functional differential equation u t a t u t f t ut u0 ec1 r 0 x te 0 t e is established x is a banach space with uniformly convex dual space and for t 0 t a t is m accretive and satisfies a time dependence condition suitable for applications to partial differential equations the function f satisfies a lipschitz condition the novelty of the paper is that the solution u t of e is shown to be the uniform limit as n of the sequence un t where the functions un t are continuously differentiate solutions of approximating equations involving the yosida approximants thus a straightforward approximation scheme is now available for such equations in parallel with the approach involving the use of nonlinear evolution operator theory

on optimal condition numbers for markov chains let t and tilde t t e be arbitrary nonnegative irreducible stochastic matrices corresponding to two ergodic markov chains on n states a function is called a condition number for markov chains with respect to the norm pair if pi tilde pi alpha leq kappa t e beta here and tilde pi are the stationary distribution vectors of the two chains respectively various condition numbers particularly with respect to the 1 and norm pairs have been suggested in the literature they were ranked according to their size by cho and meyer in a paper from 2001 in this paper we first of all show that what we call the generalized ergodicity coefficient tau p a rm sup y t e 0 frac y ta p y 1 where e is the n vector of all 1 s and a is the group generalized inverse of a i t is the smallest condition number of markov chains with respect to the p norm pair we use this result to identify the smallest condition number of markov chains among the and 1 norm pairs these are respectively 3 and 6 in the cho meyer list of 8 condition numbers kirkland has studied 3 t he has shown that kappa 3 t geq frac n 1 2n and he has characterized transition matrices for which equality holds we prove here again that 2 3 t 6 which appears in the cho meyer paper and we characterize the transition matrices t for which kappa 6 t frac n 1 n there is actually only one such matrix t j n i n 1 where j n is the n n matrix of all 1 s

generalized vector products duality and octonionic identities in d 8 geometry in an explicit unified and covariant formulation we study and generalize the exceptional vector products in r8 of zvengrowski gray and kleinfeld we derive the associated general quadratic cubic and quartic g2 invariant algebraic identities and uncover an octonionic counterpart to the d 4 quaternionic duality when restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on s7 we further link up with the ogievetski tzeitlin vector product and obtain explicit tensor forms of the so 7 and g2 structure constants so 8 transformations related by triality are characterized by means of invariant tensors possible physical applications are discussed

existence of periodic solutions for abstract neutral non autonomous equations with infinite delay in this paper by using sadovskii fixed point theorem we study the existence of solutions and periodic solutions for a class of abstract neutral functional evolution equations with infinite delay an example is presented in the end to show the applications of the obtained results

existence of solutions of abstract fractional integrodifferential equations of sobolev type this paper deals with the study of existence of solutions of nonlinear fractional integrodifferential equations of sobolev type with nonlocal condition in banach spaces the results are obtained by using resolvent operators fractional calculus and fixed point technique an example is provided to illustrate the theory

controllability of sobolev type integrodifferential systems in banach spaces sufficient conditions for controllability of sobolev type integrodifferential systems in banach spaces are established the results are obtained using compact semigroups and the schauder fixed point theorem as an example is provided to illustrate the results

flow invariance for solutions to nonlinear nonautonomous partial differential delay equations we investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations u t b t u t f t ut t s us where b t is a family of nonlinear multivalued accretive operators with d b t possibly depending on t and the operators f t being defined and lipschitz continuous possibly only on thin subsets of the initial history space e the results are applied to population dynamics models we also study the asymptotic behavior of solutions to this equation our analysis will be based on the evolution operator associated to the equation in the initial history space e

a partial functional differential equation of sobolev type where r t t 0 is the semigroup of bounded linear operators generated by ab equations of type 1 1 have been considered by several authors 24 g 101 in particular our motivation comes from the recent article of brill 2 the method employed here is different from brill s however our results reduce to his when r 0 the article 111 contains results on partial functional differential equations when b i equation 1 1 can be considered as an abstraction of the partial functional differential equation

comparison of perturbation bounds for the stationary distribution of a markov chain abstract the purpose of this paper is to review and compare the existing perturbation bounds for the stationary distribution of a finite irreducible homogeneous markov chain

mixing times with applications to perturbed markov chains abstract a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain is considered the statistic i j 1 m m ij j where j is the stationary distribution and mij is the mean first passage time from state i to state j of the markov chain is shown to be independent of the initial state i so that i for all i is minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations an application considering the effects perturbations of the transition probabilities have on the stationary distributions of markov chains leads to a new bound involving for the 1 norm of the difference between the stationary probability vectors of the original and the perturbed chain when is large the stationary distribution of the markov chain is very sensitive to perturbations of the transition probabilities

nonlinear partial functional differential equations existence and stability existence and uniqueness of solutions for a class of nonlinear functional differential equations in hilbert spaces are established sufficient conditions which guarantee the transference of exponential stability from partial differential equations to partial functional differential equations are studied the stability results derived are applied also to ordinary differential equations with hereditary characteristics

approximate controllability results for impulsive neutral differential inclusions of sobolev type with infinite delay abstractthe paper is mainly concerned with the approximate controllability results for a class of impulsive neutral differential control systems first we establish a set of sufficient conditions

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

a method of lines for a nonlinear abstract functional evolution equation let x be a real banach space with x uniformly convex a method of lines is introduced and developed for the abstract functional problem lipschitzian like function in its two variables further conditions are given for the convergence of the method to a strong solution of e recent results for perturbed abstract ordinary equations are substantially improved the method applies also to large classes of functional parabolic problems as well as problems of integral perturbations the method is straightforward because it avoids the introduction of the operators a t and the corresponding use of nonlinear evolution operator theory 1 introduction preliminaries let x be a real banach space with norm 11 let