solvability of the quaternionic monge ampere equation on compact manifolds with a flat hyperkahler metric abstract a quaternionic version of the calabi problem was formulated by alesker and verbitsky 2010 6 it conjectures a solvability of a quaternionic monge ampere equation on a compact hkt manifold hkt stays for hyperkahler with torsion in this paper this problem is solved under the extra assumption that the manifold admits a flat hyperkahler metric compatible with the underlying hypercomplex structure the proof uses the continuity method and a priori estimates

dirichlet duality and the nonlinear dirichlet problem we study the dirichlet problem for fully nonlinear degenerate elliptic equations of the form f hess u 0 on a smoothly bounded domain n in our approach the equation is replaced by a subset f sym2 n of the symmetric n n matrices with f f 0 we establish the existence and uniqueness of continuous solutions under an explicit geometric f convexity assumption on the boundary we also study the topological structure of f convex domains and prove a theorem of andreotti frankel type two key ingredients in the analysis are the use of subaffine functions and dirichlet duality associated to f is a dirichlet dual set f that gives a dual dirichlet problem this pairing is a true duality in that the dual of f is f and in the analysis the roles of f and f are interchangeable the duality also clarifies many features of the problem including the appropriate conditions on the boundary many interesting examples are covered by these results including all branches of the homogeneous monge ampere equation over and equations appearing naturally in calibrated geometry lagrangian geometry and p convex riemannian geometry and all branches of the special lagrangian potential equation 2008 wiley periodicals inc

dynamic behavior for a system of neutral functional differential equations consider the system of neutral functional differential equations x 1 t x 2 t r f x 1 t g x 2 t r x 2 t x 1 t r f x 2 t g x 1 t r where r 0 f g c r it is shown that if f is nondecreasing on r and some additional assumptions hold then the w limit set of every bounded solution of such a system with some initial conditions is composed of 2r periodic solutions our results are new and complement some corresponding ones already known

asymptotic constancy for pseudo monotone dynamical systems on function spaces abstract a pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points the purpose of this paper is to present some convergence oscillation and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point some applications to neutral functional differential equations and semilinear parabolic partial differential equations with neumann boundary condition are given

discrete order preserving semigroups and stability for periodic parabolic differential equations on etudie la stabilite globale des solutions des equations paraboliques dans une forme non auto adjointe avec des coefficients periodiques dans le temps la structure mathematique est celle des semigroupes preservant l ordre a temps discret s n n e n sur un reseau de banach

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

a generalization of the three dimensional bernfeld haddock conjecture and its proof consider the following system of delay differential equations x 1 t f x 1 t g x 2 t r 2 x 2 t f x 2 t g x 3 t r 3 x 3 t f x 3 t g x 1 t r 1 where r 1 r 2 and r 3 are positive constants f email protected c r 1 and f is nondecreasing on r 1 these systems have important practical applications and also are a three dimensional generalization of the bernfeld haddock conjecture in this paper by using comparative technique we obtain the asymptotic behavior of solutions that each bounded solution of the systems tends to a constant vector under a desirable condition

retracted asymptotic constancy for a differential equation with multiple state dependent delays in this paper we investigate the asymptotic behavior of solutions to a differential equation with multiple state dependent delays it is shown that every bounded solution of such an equation tends to a constant as t our results improve and extend some corresponding ones already known

on the computational complexity of problems related to distinguishability sets we study the computational complexity of problems related to distinguishability sets for regular languages roughly speaking the distinguishability set mathsf d l for a not necessarily regular language l consists of all words w that are a common suffix of a word xw in l and of a word yw that does not belong to l in particular we investigate the complexity of the representation problem i e deciding for given automata a and b whether b accepts the distinguishability set of l a it is shown that this problem and some of its variants are highly intractable namely pspace complete in fact determining the size of an automaton for mathsf d l a is already pspace complete on the other hand questions related to the hierarchy induced by iterated application of the mathsf d operator turn out to be much easier for instance the question whether for a given automaton a the accepted language is equal to its own distinguishability set i e whether l a mathsf d l a holds is shown to be nl complete as a byproduct of our investigations we found a compelling characterization of synchronizing automata namely that a minimal automaton a is synchronizing if and only if mathsf d l a mathsf d 2 l a

positive operators and elliptic eigenvalue problems on considere le probleme aux valeurs propres lu mu sur et u 0 sur dans un domaine borne lisse cr n l est une expression differentielle uniformement elliptique a coefficients a valeurs reels continus de holder le coefficient de u dans l est non negatif est nu x est une fonction continue a valeur reelle on utilise la theorie abstraite des operateurs positifs pour retrouver certains resultats

valuations on convex sets non commutative determinants and pluripotential theory abstract a new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space h n is presented in particular new examples of sp n sp 1 invariant translation invariant continuous valuations are constructed this method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers bull sci math 127 1 2003 1 j geom anal 13 2 2003 183

subsolution theorem and the dirichlet problem for the quaternionic monge ampere equation in this paper we obtain the existence of solution to the dirichlet problem for quaternionic monge ampere equation after showing a stability theorem we prove the subsolution theorem for quaternionic monge ampere equation by combining our stability theorem and kolodziej s approach for the complex case

retracted asymptotic behavior of solutions to a system of differential equations with state dependent delays in this paper we investigate the asymptotic behavior of solutions to a system of differential equations with state dependent delays it is shown that every bounded solution of such a system tends to a constant vector as t our results improve and extend some corresponding ones already known

an introduction to potential theory in calibrated geometry in this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry these functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties moreover they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce a number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds this paper introduces and investigates questions of pseudo convexity in the context of a general calibrated manifold x phi analogues of totally real submanifolds are introduced and used to construct enormous families of strictly phi convex spaces with every topological type allowed by morse theory specific calibrations are used as examples throughout in a sequel the duality between phi pluri sub harmonic functions and phi positive currents is investigated this study involves boundaries generalized jensen measures and other geometric objects on a calibrated manifold

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

on the degree of ambiguity of finite automata we investigate the ambiguity behavior of finite automata in connection with their inner structure we show that the degree of ambiguity of a finitely ambiguous nondeterministic finite automaton nfa with n states is at most 5n2 nn there is a simple criterion which characterizes the infinite degree of ambiguity of an nfa and which is decidable in polynomial time the degree of growth of the ambiguity of an nfa is computable in polynomial time starting from the first result we discuss the maximal finite degree of ambiguity of an nfa with n states and we present subclasses of nfas where this quantity is of order 2 n

convergence of a class of discrete time semiflows with application to neutral delay differential equations abstract in this paper we introduce a class of pseudo monotone maps on ordered topological spaces by exploiting monotonicity methods and the invariance of the omega limit set we establish a convergence principle for discrete time semiflows generated by the maps introduced the convergence principle is then applied to a class of periodic neutral delay differential equations which leads to some novel and sharper results

on the equivalence and containment problems for unambiguous regular expressions regular grammars and finite automata the known proofs that the equivalence and containment problems for regular expressions regular grammars and nondeterministic finite automata are pspace complete sm depend upon consideration of highly unambiguous expressions grammars and automata here we prove that such dependence is inherent deterministic polynomial time algorithms are presented for the equivalence and containment problems for unambiguous regular expressions unambiguous regular grammars and unambiguous finite automata the algorithms are then extended to ambiguity bounded by a fixed k our algorithms depend upon several elementary observations on the solutions of systems of homogeneous linear difference equations with constant coefficients and their relationship with the number of derivations of strings of a given length n by a regular grammar

the c0 estimate for the quaternionic calabi conjecture abstract we prove the c 0 estimate for the quaternionic monge ampere equation on compact hyperkahler with torsion manifolds our goal is to provide a simpler proof than the one presented in 3

convergence for a class of delayed recurrent neural networks without m matrix condition in this paper we investigate the asymptotic behavior of solutions to a class of recurrent neural network model with delays without assuming m matrix condition it is shown that every solution of the network tends to an equilibrium point as t our results improve and extend some corresponding ones already known

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

asymptotic behaviour of comparable skew product semiflows with applications the almost 1 cover lifting property of omega limit sets is established for nonmonotone skew product semiflows which are comparable to uniformly stable eventually strongly monotone skew product semiflows these results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of odes reaction diffusion systems differential systems with time delays and semilinear parabolic equations

geometry of hyper kahler connections with torsion the internal space of a n 4 supersymmetric model with wess zumino term has a connection with totally skew symmetric torsion and holonomy in sp n we study the mathematical background of this type of connection in particular we relate it to classical hermitian geometry construct homogeneous as well as inhomogeneous examples characterize it in terms of holomorphic data develop its potential theory and reduction theory

convergence for pseudo monotone semiflows on product ordered topological spaces in this paper we consider a class of pseudo monotone semiflows which only enjoy some weak monotonicity properties and are defined on product ordered topological spaces under certain conditions several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium which improve and extend some corresponding results already known some applications to delay differential equations are presented

convergence of solutions of a class of delay differential systems abstract this paper presents a new and concise approach for investigating the convergence of the solutions of a class of delay differential systems it is shown that each solution of the delayed system is convergent to a constant vector the result obtained extends the existing ones in the literature

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

global stability of a class of delay differential systems this paper is concerned with a class of systems of delay differential equations which are defined on the nonnegative function space under proper conditions we employ a novel proof to establish several criteria of the global stability of positive equilibrium moreover we give two examples to illustrate our main results

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

on the finite valuedness problem for sequential machines abstract for a set a a 1 a m of n x n matrices with nonnegative integer entries let p a be the set of all finite products of matrices in a we show that there is a square space algorithm to decide given a whether or not p a is finite as a corollary we show that there is an exponential space algorithm to decide given a nondeterministic mealy sequential machine m with accepting states whether there exists an integer d 0 such that on every input the number of accepting computations of m producing distinct outputs is at most d if d exists the smallest such d can be computed in space exponential in size m log d the space bound reduces to polynomial for the analogous problem of ambiguity of nondeterministic finite acceptors

on the linear algebra in the quaternionic pluripotential theory abstract we clarify the linear algebra used in the quaternionic pluripotential theory so that proofs of several results there can be greatly simplified in particular we characterize and normalize real 2 forms with respect to the quaternionic structure and show that the moore determinant of a quaternionic hyperhermitian matrix is the coefficient of the exterior product of the associated real 2 form as a corollary the quaternionic monge ampere operator is the coefficient of the exterior product of the baston operator

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

the parallel complexity of finite state automata problems the goal of this paper is to study the exact complexity of several important problems concerning finite state automata and to classify the degrees of ambiguity of nondeterministic finite state automata our results are as follows 1 minimization of deterministic finite automata is nc 1 complete for nl 2 testing whether the degree of ambiguity of a nondeterministic finite automaton is exponential or polynomial or bounded is nc 1 complete for nl 3 checking whether a given nondeterministic finite automaton is unambiguous or k ambiguous is nc 1 complete for nl where k is some fixed constant 4 the bounded nonuniversality problem for nondeterministic finite automata which is the problem of deciding whether l m s n s n for a given nondeterministic finite automaton m and a unary integer n is log space complete for np 5 the bounded nonuniversality problem for unambiguous finite automata is in det the class of problems nc 1 reducible to computing the determinants of integer matrices and for deterministic finite automata it is nc 1 complete for nl 6 the inequivalence problems for unambiguous and k ambiguous finite automata are both in det where k is some fixed constant

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

asymptotic behavior of solutions to a class of non autonomous delay differential equations abstract in this paper we study the asymptotic behavior of solutions of a class of non autonomous delay differential equations these equations have important practical applications and generalize those on which bernfeld and haddock conjectured that each solution of the equations tends to a constant it is shown that every solution of the equations is bounded and tends to a constant as t which improves and extends some existing ones

convergence in smooth strongly monotone flows defined by semilinear parabolic equations on considere des equations paraboliques semi lineaires dy dt ay f y ou a est un operateur sectoriel sur un banach x ayant une resolvante compacte et f est une c 2 application de l espace des puissances fractionnaires x 0 1 dans x soit y t une solution qui est quasi convergente mais non convergente alors il existe un voisinage u de y 0 y 0 avec la propriete suivante si z u y 0 a une trajectoire bornee et soit z y 0 soit z y 0 alors la solution partant de z converge vers un equilibre

convergence of bounded solutions for a class of systems of delay differential equations abstract in this paper the convergence of a class of systems of delay differential equations is considered we show that every bounded solution of such systems tends to an equilibrium under certain hypotheses our results extend some corresponding ones already known

geometric plurisubharmonicity and convexity an introduction this is an essay on potential theory for geometric plurisubharmonic functions it begins with a given closed subset g of the grassmann bundle g p tx of tangent p planes to a riemannian manifold x this determines a nonlinear partial differential equation which is convex but never uniformly elliptic p dimx a surprising number of results in complex analysis carry over to this more general setting the notions of a g submanifold an upper semi continuous g plurisubharmonic function a g convex domain a g harmonic function and a g free submanifold are defined results include a restriction theorem as well as the existence and uniqueness of solutions to the dirichlet problem for g harmonic functions on g convex domains

on the periodic solutions of periodic multi species competitive systems with delays the paper considers general n species periodic kolmogorov competitive systems of functional differential equations with finite or infinite delays applying schauder s fixed point theorem a criterion on the existence of the positive periodic solutions is established as applications the existence and global asymptotic stability of the single species logistic periodic model and n species lotka volterra type periodic model with delays are discussed