static and dynamic phase transitions in multidimensional voting models on continua a voting model or a generalization of the glauber model at zero temperature on a multidimensional lattice is defined as a system composed of a lattice each site of which is either empty or occupied by a single particle the reactions of the system are such that two adjacent sites one empty the other occupied may evolve to a state where both of these sites are either empty or occupied the continuum version of this model in a d dimensional region with a boundary is studied and two general behaviors of such systems are investigated the stationary behavior of the system and the dominant way of relaxation of the system toward its stationary state based on the first behavior a static phase transition discontinuous changes in the stationary profiles of the system is studied based on the second behavior a dynamical phase transition discontinuous changes in the relaxation times of the system is studied it is shown that the static phase transition is induced by the bulk reactions only while the dynamical phase transition is a result of both bulk reactions and boundary conditions

sk1 like functors for division algebras abstract we investigate the group valued functor g d d f d where d is a division algebra with center f and d the commutator subgroup of d we show that g has the most important functorial properties of the reduced whitehead group sk 1 we then establish a fundamental connection between this group its residue version and relative value group when d is a henselian division algebra the structure of g d turns out to carry significant information about the arithmetic of d along these lines we employ g d to compute the group sk 1 d as an application we obtain theorems of reduced k theory which require heavy machinery as simple examples of our method

oscillation and global attractivity in a delay periodic host macroparasite model abstract in this paper we shall consider the nonlinear delay host macroparasite model with periodic coefficients with common period by means of a continuation theorem in coincidence degree theory we establish a sufficient condition for the existence of a positive periodic solution m t with strictly positive components also we establish some sufficient conditions for oscillation of all positive solutions about the positive periodic solution m t and establish a sufficient condition for the global attractivity of m t

on the existence of periodic solutions of a neutral delay model of single species population growth abstract in this paper we employ some new techniques to study the existence of positive periodic solutions of the neutral delay model n t n t a t t n t b t n t t c t n t t our result gives a correct answer to the open problem 9 2 due to y kuang 1993 delay differential equations with applications in population dynamics academic press new york

totally ramified splitting fields of central simple algebras over henselian fields abstract let d be a central division algebra over a henselian field f with algebraically closed residue field f it is proved that if the degree of d is relatively prime to the characteristic of f then every finite extension of f which splits d contains a maximal subfield of d

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

multispecies extension of the solvable partially asymmetric reaction diffusion processes by considering the master equation of the partially asymmetric diffusion process on a one dimensional lattice the most general boundary condition i e interactions for the multispecies reaction diffusion processes is considered the resulting system has various interactions including diffusion to the left and right two particle interactions a a a a and the extended n particle drop push interactions to the left and right we obtain three distinct new models the conditions on reaction rates to ensure the solvability of the resulting models are obtained the two particle conditional probabilities are calculated exactly

galois groups of maximal p extensions let p be an odd prime and f a field of characteristic different from p containing a primitive pth root of unity assume that the galois group g of the maximal p extension of f has a finite normal series with abelian factor groups then the commutator subgroup of g is abelian moreover g has a normal abelian subgroup with pro cyclic factor group if in addition f contains a primitive p 2 th root of unity then g has generators x y i i i with relations y i y j y j y i and xy i x 1 y i q 1 where q 0 or q p n for some n 1

a new construction of noncrossed product algebras new examples of noncrossed product division algebras are obtained using methods different from all previous noncrossed product constructions the examples are division algebras over intersections of henselian valued fields and they have schur index and exponent p for any prime number p and any integers m n 2 n 3if 2 the basic tools used in the construction are valuation theory and galois cohomology no generic methods are applied and there is no p i theory along the way local global principles are proved for central simple algebras over intersections of p henselian valued fields the first examples of central simple division algebras which are not crossed products were obtained by amitsur in am in 1972 amitsur thereby settled a question that had been one of the outstanding open problems in the theory of algebras for at least thirty years his examples were the generic division algebras ud q n of index n over q the rational numbers for any natural number n such that p2 n p an odd prime or 8 all subsequent constructions of noncrossed products in ss 83 883 r r an ti have heretofore been based on amitsur s specialization argument and they are all generic division algebras or extensions of generic division algebras the centers of these noncrossed product algebras are not known nor are the brauer groups nor the absolute galois groups of the centers we present here a new method of constructing noncrossed product algebras in our approach the noncrossed product is realized as the underlying division algebra d of a tensor product of suitably chosen cyclic algebras over a field f l n l2 where each l is a pth root henselian valued field we prove local global principles relating the splitting fields of d to those of d f lt i 1 2 it is shown that the part of the brauer group of f is completely determined by that of ll and l2 computations for central simple f algebras thus become very tractable for exam ple we not only show that d is not a crossed product but also calculate exactly how large r must be so that the matrix ring mr d is a crossed product and how large s must be so that ms d is a tensor product of cyclic algebras indeed the structure of the brauer group of f is so nice that we were somewhat surprised that noncrossed products could possibly exist over f

almost periodic solutions of lasota wazewska type delay differential equation sufficient conditions are derived for the existence of a globally attractive almost periodic solution of a single species model given by the nonautonomous lasota wazewska type delay differential equations x t txt ptfxt in which 0 p t 0 t are continuous almost periodic functions and f is a decreasing positive c1 function conditions of the uniform asymptotical stability for some associated nonautonomous delay differential equations x t txt ptxt are also discussed

totally ramified valuations on finite dimensional division algebras division algebras d with valuation v are studied where d is lsnite dimensional and totally ramified over its center f i e the ramification index of v over wlf equals d f such division algebras have appeared in some important constructions but the structure of these algebras has not been systematically analyzed before when vlf is henselian a full classification of the f subalgebras of d is given when f has a henselian valuation v with separably closed residue field and a is any tame central simple f algebra an algorithm is given for computing the underlying division algebra of a from a suitable subgroup of a f some examples are constructed using this valuation theory including the first example of finitbdimensional f central division algebras d1 and d2 with d1 xf d2 not a division ring but d1 and d2 having no common subfield k d f valuation theory long a basic tool in commutative algebra has been relatively neglected in the study of division algebras until quite recently nontheless valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties particularly algebras over iterated laurent power series fields for example such division algebras have been key ingredients in amitsur s noncrossed product construction am and in platonov s construction p of division algebras d with sk1 d 7 1 valuations are not so prevalent on division algebras as on fields but if a division algebra d does have a valuation this structure contains a substantial amount of information about d which would scarcely be accessible otherwise we consider here valued divisioll algebras d for which d is totqlly ramified and tame over its center f i e for which the ramification index ird rfi equals the dimension d f of d over f and the characteristic char d does not divide d f here rd is the value group of the valuation on d and d is the residue division algebra we assume throughout that d f oo valued division algebras of this type appear e g in amitsur s noncrossed product paper am 2 in saltman s work on indecomposable division algebras sa in certain of the malcevneumann division algebras considered by the first author and amitsur ta2 4 etc however the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before this may be because most past work on valued division algebras has concentrated on discrete valuations when rd z for such a valuation d is never totally ramified over its center f unless d f cf 3 2 below received by the editors june 26 1986 1980 mathematics subject clmsification 1985 reon primary 16a39 2supported in part by f n r s 2supported in part by the national science foundation 1987 american mathematlcal society 0002 9947 87 00 25 per page

global attractivity in population dynamics we established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation n t mu n t i 1m pi exp n t i t 0 m 1 for m 1 equation 1 was used by wazewska czyzewska and lasota as a model for the survival of red blood cells in an animal

a note on valued division algebras let d be a finite dimensional division algebra with center f and let b be a valuation ring of d i e xeb or x 1 e b holds for all nonzero x in d then b is called an extension of v f b let s be the set of all extensions of v to d and let p b be the group of all bijections of s which are induced by the inner automorphisms of d in this note we investigate p b it is shown that p b is solvable and an upper bound for p b is given finally theorem 4 5 gives a necessary and sufficient condition for b such that p b is equal to this bound

new oscillation criteria for linear delay differential equations abstract we obtain new sufficient conditions for the oscillation of all solutions of the first order linear delay differential equation x t p t x t 0 with p t 0 t t our results improve the known results in the literature which include those obtained recently in papers by elbert and stavroulakis li and yu et al

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

on the residue fields of henselian valued stable fields abstract let k v be a henselian valued field satisfying the following conditions for a given prime number p i central division k algebras of finite p primary dimensions have schur indices equal to their exponents ii the value group v k properly includes its subgroup p v k the paper shows that if k is the residue field of k v and r is an intermediate field of the maximal p extension k p k then the natural homomorphism br k br r of brauer groups maps surjectively the p component br k p on br r p it proves that br k p is divisible if p 2 or k is a nonreal field and that br k 2 is of order 2 when k is formally real we also obtain that r embeds as a k subalgebra in a central division k algebra if and only if the degree r k divides the index of

on the behaviour of brauer p dimensions under finitely generated field extensions abstract the present paper shows that if q p or q 0 where p is the set of prime numbers then there exist characteristic q fields e q k k n of brauer dimension brd e q k k and infinite absolute brauer p dimensions abrd p e q k for all p p not dividing q 2 q this ensures that brd p f q k p q 2 q for every finitely generated transcendental extension f q k e q k we also prove that each sequence a p b p p p satisfying the conditions a 2 b 2 and 0 b p a p equals the sequence abrd p e brd p e p p for a field e of characteristic zero

environmental periodicity and time delays in a food limited population model abstract sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the food limited population system modelled by the equation n t r t k t n t m k t c t r t n t m where m is a nonnegative integer and k r c are continuous positive periodic functions of period

oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics abstract in this paper we shall consider the nonlinear delay differential equation where m and n are positive integers and v t and t are positive periodic functions of period in the nondelay case we shall show that has a unique positive periodic solution x t and provides sufficient conditions for the global attractivity of x t in the delay case we shall present sufficient conditions for the oscillation of all positive solutions of about x t and establish sufficient conditions for the global attractivity of x t

oscillations of delay differential equations with variable coefficients consider the delay differential equation t p t x t 0 where p t c t0 r and is a positive constant we show that every solution of this equation oscillates if tt p t dt 1 e for sufficiently large t and t0 p t exp tt p s ds 1 e 1 dt

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

solvable multispecies reaction diffusion processes a family of one dimensional multispecies reaction diffusion processes on a lattice is introduced it is shown that these processes are exactly solvable provided a nonspectral matrix equation is satisfied some general remarks on the solutions to this equation and some special solutions are given the large time behavior of the conditional probabilities of such systems is also investigated

global linearized stability theory for delay differential equations abstract for nonlinear delay differential equations where nonlinearities are approximated by linear functions we discuss a rather general global stability principle several examples which extend and improve known global stability results are presented

global dynamics of one class of nonlinear nonautonomous systems with time varying delays abstract a class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques the results for the nonautonomous systems with time varying delays are novel e g the global stability of differential equations with nonlinear casual volterra operators is considered for the first time in the literature criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying verifying in practice we illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous nicholson type models