two cocycle of a kac moody group the structure of the group corresponding to a kac moody algebra is analyzed the two cocycle for the group is constructed in terms of local coordinates and the transition functions between different coordinate systems are computed the results are extended to the group mathrm map s 3 mathfrak g which arises in a 3 1 dimensional yang mills theory

on the hopf index and the conley index the following generalization of the poincare hopf index theorem is proved if s is an isolated invariant set of a flow on a manifold m then the sum of the hopf indices on s is equal up to a sign to the euler characteristic of the homology conley index of 5 one of the classical results in dynamical systems relating local and global information is the poincare hopf index theorem if sf is a vectorfield on a compact manifold m with isolated zeros then the intersection number of with the zero section of tm is defined to be the sum of the hopf indices of the zeros i ist f yhlx s if is without boundary or if af is outward on dm the poincare hopf theorem asserts that the intersection number of sf is equal to the homological euler characteristic of m i see 4 7 this has the restriction that the local information the hopf indices is only related to the global information the betti numbers when the vectorfield is outward on the boundary pugh 8 removes this restriction in the case of generic contact of with m by relating the intersection number to a sum of euler characteristics dm j where r is the codimensional exit region and is its boundary in this paper we further develop this result by relating the hopf index and intersection numbers to the conley index for dynamical systems suppose that x is a smooth vectorfield on m with s c m an isolated invariant set of the induced flow then the intersection number of a on a neighborhood of s is up to a sign the euler characteristic of the homology conley index of s in m the conley euler number 5 in particular if all zeros on af on s are nondegenerate then the conley index and the hopf index are related by the formula nx m s 2ix there are two aspects to this result first it gives the hopf index and intersection numbers a homological interpretation in a broad setting second it shows that the properties of the hopf index such as additivity stability etc are also properties of conley euler numbers to develop the theorem we first received by the editors february 8 1988 1980 mathematics subject classification 1985 revision primary 34c35 55m20 secondary 54h20 58f25 work supported in part by oarpa applied and computational mathematics program 1989 american mathematical society 0002 9947 89 1 00 5 25 per page

isolating segments fixed point index and symbolic dynamics ii homoclinic solutions we present a topological method for detecting homoclinic solutions in nonautonomous ode s as an application we show that there exist infinitely many geometrically distinct homoclinic solutions to the trivial 0 solution in the planar systemz z 1 ei t z 2 for 0 0 495

a cohomological conley index for maps on metric spaces we define a cohomological conley index of an isolated invariant set of a time discrete semidynamical system on a metric space we prove the homotopy invariance and the additivity properties of the index for applications to time periodic parabolic partial differential equations pdes the reader is referred to the authors paper nontrivial full bounded solutions of time periodic semilinear parabolic pdes proc roy soc edinburgh to appear

periodic solutions in a given set of differential systems the existence of a periodic solution in a given set of caratheodory differential systems is studied the proof of the main statement is based on the wazewski type approach jointly with the degree arguments an illustrating example is supplied and a comparison with the analogous results of other authors is indicated

the classification of p compact groups for p odd a p compact group as defined by dwyer and wilkerson is a purely homotopically defined p local analog of a compact lie group it has long been the hope and later the conjecture that these objects should have a classification similar to the classification of compact lie groups in this paper we finish the proof of this conjecture for p an odd prime proving that there is a one to one correspondence between connected p compact groups and finite reflection groups over the p adic integers we do this by providing the last and rather intricate piece namely that the exceptional compact lie groups are uniquely determined as p compact groups by their weyl groups seen as finite reflection groups over the p adic integers our approach in fact gives a largely self contained proof of the entire classification theorem for p odd

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

isolating segments fixed point index and symbolic dynamics iii applications abstract we prove the existence of complicated dynamics in some class of periodic systems on the plane the proof is based on the continuation of the considered system to the model one

wazewski type theorem for non autonomous systems of equations with a disconnected set of egress points in this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form y t f t y t where f t r n r n and t is a time scale for a given set t r n we formulate conditions for function f which guarantee that at least one solution y of the above system stays in unlike previous papers the set is considered in more general form i e the time section t is an arbitrary closed bounded set homeomorphic to the disk for every t t and the boundary t does not contain only egress points thanks to this we can investigate a substantially wider range of equations with various types of bounded solutions a relevant example is considered the results are new also for non autonomous systems of difference equations and the systems of impulsive differential equations

remark on complicated dynamics of some planar system abstract in this paper we study the complicated dynamics generated by the planar periodic system z z 1 e i t z 2 for 0 0 495 we prove the existence of infinitely many geometrically distinct periodic solutions with some special properties

formal group laws for affine kac moody groups and group quantization we describe a method for obtaining formal group laws from the structure constants of affine kac moody groups and then apply a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group as an intermediate step we get an explicit expression for two cocycles on loop groups the programme is applied to the affinesu 2 group

a geometric method for detecting chaotic dynamics a new method of detection of chaos in dynamical systems generated by time periodic nonautonomous differential equations is presented it is based on the existence of some sets called periodic isolating segments in the extended phase space satisfying some topological conditions by chaos we mean the existence of a compact invariant set such that the poincare map is semiconjugated to the shift on two symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the poincare map as an application we prove that the planar equationz 1 ei t z 2 zgenerates chaotic dynamics provided 0 1 288

a topological method for bounded solutions of nonautonomous ordinary differential equations the existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied this is done by associating the given equation to nonlinear autonomous ones by means of a family of skew product flows related by homotopy the existence of a bounded solution to the original differential equation is then related to the nontriviality of a certain conley index associated with the autonomous differential equations the existence of nontrivial bounded solutions is also considered the differential equations studied are perturbations of homogeneous ones

isolating segments fixed point index and symbolic dynamics abstract an extension of the recently introduced srzednicki wojcik method for detecting chaotic dynamics in periodically forced ordinary differential equations is presented as an application of the method we construct a topological model for the planar equation z 1 e i t z 2 z z c 1 and we show by a continuation argument that the symbolic dynamics on three symbols for the topological model continues to eq 1 for 0

conley index of poincare maps in isolating segments abstract we calculate the discrete time conley index of the poincare map of a time periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment as an application we present results on the existence of bounded solutions of some planar equations

group cocycles line bundles and anomalies the relation between complex line bundles and certain group cocycles is explored in general to obtain explicit formulae for the transition functions and curvature of the determinant line bundle det of a family of dirac operators coupled to yang mills fields a covariant derivative on sections of det is constructed which realizes the curvature and minimally couples to the integrated anomaly which thus appears as a functional magnetic field on gauge orbit space the transcription of group cohomological cocycles into geometrical line bundles information is refined in such a way that the relevant cohomology groups can be computed in many cases giving insight into the classification of lifts of principal group actions

on periodic solutions of certain nth order differential equations we present two theorems on periodic solutions of nth order ordinary differential equations with the right hand side t periodic with respect to the time variable we apply a topological method based on the lefschetz fixed point theorem and the wazewski principle

field models from two cocycles on infinite dimensional lie groups and symplectic structures 2d gravity and chern simons theory in this paper the connection between field models and infinite dimensional lie groups is widely analysed on the bases of a new group quantization approach we also relate the poincare cartan form of variational calculus to the symplectic current structure of the covariant phase space formulation of higher derivative field theory the virasoro and kac moody groups are considered in the first case the action functional of the 2d induced gravity in the light cone formulation is derived the hidden sl 2 r simply appears as generated by the kernel of the lie algebra two cocycle and plays the role of a gauge type symmetry nevertheless it is shown that a proper space like formulation is out of reach of the virasoro group the corresponding symplectic structure of the nonlocal action functional is determined showing that it is related to the symplectic structure associated with the sl 2 r kac moody group this unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint orbit approach the relation between the kac moody groups and the chern simons gauge theory on a disc in the presence of a source is considered using the new approach

nontrivial full bounded solutions of time periodic semilinear parabolic pdes consider the semilinear evolution equation p u a u f t u where a is a sectorial operator on a banach space and f is periodic in t using a time discrete conley index developed in a previous paper 6 we prove a few existence results on bounded solutions of p defined for all t r more specific results are given for time periodic scalar parabolic equations

periodic points of the planar area preserving poincare map inside isolating segment abstract the aim of this note is to study the structure of the set of periodic points of the area preserving poincare map p associated to t periodic local planar process we will use the approach based on the method of periodic isolating segments isolating chains and the fixed point index theory the number of the applications to the processes generated by periodic in time non autonomous planar ode s are given

on periodic solutions of planar polynomial differential equations with periodic coefficients abstract we consider the planar equation z ak l t zk z l where ak l is a t periodic complex valued continuous function equal to 0 for almost all k l n we present sufficient conditions imposed on ak l which guarantee the existence of its t periodic solutions and in the case a0 0 0 the conditions for the existence of nonzero ones we use a method which computes the fixed point index of the poincare andronov operator in isolated sets of fixed points generated by so called periodic blocks the method is based on the lefschetz fixed point theorem and the topological principle of wazewski

remark on krasnosielski s guiding functions and srzednicki s periodic segments abstract we show that the existence of the complete set of guiding functions guarantees the existence of periodic isolating segment that carries the same information concerning periodic solutions of the non autonomous periodic equations x f x t

a new attempt towards the unification of space time and internal gauge symmetries the neat formulation that describes the gauge interactions associated with internal symmetries is extended to the case of a simple yet non trivial symmetry group structure which mixes gravity and electromagnetism by associating a gauge symmetry with a central extension of the poincare group

hyperbolicity and averaging for the srzednicki wojcik equation for the srzednicki wojcik equation the planar nonautonomous ode parameterized by r z z 1 z 2exp i t z t c using averaging we show how the region of hyperbolicity grows with based on this we give bounds on the sizes of bounded orbits

homotopy fixed point methods for lie groups and finite loop spaces a loop space x is by definition a triple x bx e in which x is a space bx is a connected pointed space and e x qbx is a homotopy equivalence from x to the space qbx of based loops in bx we will say that a loop space x is finite if the integral homology h x z is finitely generated as a graded abelian group i e if x appears at least homologically to be a finite complex in this paper we prove the following theorem

another construction of the virasoro group in this paper the group corresponding to the virasoro algebra is constructed the so called descent equation is used to derive the 2 cocycle for the group

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

periodic solutions of complex valued differential equations and systems with periodic coefficients has been considered by srzednicki 11 13 using a combination of generalized conley s isolating blocks the index of singular points of autonomous systems and the lefschetz fixed point theorem recently it has been shown that slight extensions of srzednicki s results could be obtained in a simpler and more elementary way using continuation theorems of the leray schauder type see 6 9 the aim of the present paper is to show that the same methodology can be adapted to prove the existence of periodic solutions for more general classes of equations and systems article no 0054

kac moody groups topology of the dirac determinant bundle and fermionization the relation between kac moody groups and algebras and the determinant line bundle of the massless dirac operator in two dimensions is clarified analogous objects are studied in four space time dimensions and a generalization of witten s fermionization mechanism is presented in terms of the topology of the dirac determinant bundle

on periodic solutions inside isolating chains abstract we develop a geometric approach to problems concerning the existence of t periodic solutions of a non autonomous time t periodic ordinary differential equation we consider isolating segments subsets of the extended phase space of the equation which in some ways resemble isolating blocks from the theory of isolated invariant sets the union of several contiguous isolating segments is called an isolating chain isolating segments determine some homomorphisms in reduced singular homologies the main theorem asserts that the lefschetz number of the composition of the homomorphisms determined by segments such that their union is a periodic isolating chain is equal to the fixed point index of the poincare map of the equation in the set of initial values of t periodic solutions contained inside the chain we give some applications of the theorem to planar polynomial equations in particular we prove that the equation z z 5 sin 2 t z has four nonzero periodic solutions provided 0 336

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

factor maps and invariant distributional chaos abstract the main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite to one factor maps this solves some open questions in the literature of the topic we also show how our method can be applied in practice taking as example poincare map of time periodic nonautonomous planar differential equation z v t z 1 e i t z 2 z 3 n where and n are sufficiently small positive real numbers

on the p compact groups corresponding to the p adic reflection groups g q r n there exists an infinite family of p compact groups whose weyl groups correspond to the finite p adic pseudoreflection groups g q r n of family 2a in the clark ewing list in this paper we study these p compact groups in particular we construct an analog of the classical whitney sum map a family of monomorphisms and a spherical fibration which produces an analog of the classical j homomorphism finally we also describe a faithful complexification homomorphism from these p compact groups to the p completion of unitary compact lie groups