a group theoretic approach to a family of 2 local finite groups constructed by levi and oliver we extend the notion of a p local finite group defined in blo03 to the notion of a p local group we define morphisms of p local groups obtaining thereby a category and we introduce the notion of a representation of a p local group via signalizer functors associated with groups we construct a chain g g 0 g 1 of 2 local finite groups via a representation of a chain g g 0 g 1 of groups such that g 0 is the 2 local finite group of the third conway sporadic group co 3 and for n 0 g n is one of the 2 local finite groups constructed by levi and oliver in lo02 we show that the direct limit g of g exists in the category of 2 local groups and that it is the 2 local group of the union of the chain g the 2 completed classifying space of g is shown to be the classifying space b di 4 of the exotic 2 compact group of dwyer and wilkerson dw93

deconstructing hopf spaces we characterize hopf spaces with finitely generated cohomology as an algebra over the steenrod algebra we deconstruct the original space into an h space y with finite mod p cohomology and a finite number of p torsion eilenberg mac lane spaces we give a precise description of homotopy commutative h spaces in this setting

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

stratification and duality for homotopical groups abstract we generalize quillen s f isomorphism theorem quillen s stratification theorem the stable transfer and the finite generation of cohomology rings from finite groups to homotopical groups as a consequence we show that the category of module spectra over c b g f p is stratified and costratified for a large class of p local compact groups g including compact lie groups connected p compact groups and p local finite groups thereby giving a support theoretic classification of all localizing and colocalizing subcategories of this category moreover we prove that p compact groups admit a homotopical form of gorenstein duality

complexity reduction of c algorithm abstract the c algorithm introduced in 5 is designed to determine isochronous centers for lienard type differential systems in the general real analytic case however it has a large complexity that prevents computations even in the quartic polynomial case the main result of this paper is an efficient algorithmic implementation of c algorithm called reca reduced c algorithm moreover an adapted version of it is proposed in the rational case it is called rca rational c algorithm and is widely used in 1 2 to find many new examples of isochronous centers for the lienard type equation

linearizability conditions for a cubic system we obtain the necessary and sufficient conditions for linearizability of an eight parameter family of two dimensional system of differential equations in the form of linear canonical saddle perturbed by polynomials with four quadratic and four cubic terms

triangles of groups given a certain commutative diagram of groups and monomorphisms does there necessarily exist a group in which the given diagram is essentially a diagram of subgroups and inclusions in general the answer is negative but j corson and gersten and stallings have shown that in the case of a nonspherical triangle of groups the answer is positive this paper improves on these results by weakening the non sphericality requirement

isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials in this work we study isochronous centers of two dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree a complete classification of the necessary conditions for the time reversible systems of this class is given in order to have an isochronous center at the origin an open problem is stated for the sufficient conditions moreover we find two nonreversible isochronous families from the center cases known all the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods

homotopy nilpotency in p regular loop spaces we consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p completion of a product of finite numbers of spheres we determine the homotopy nilpotency of those loop spaces as an answer to this problem

the radial part of brownian motion ii its life and times on the cut locus this paper is a sequel to kendall 1987 which explained how the ito formula for the radial part of brownian motionx on a riemannian manifold can be extended to hold for all time including those times a whichx visits the cut locus this extension consists of the subtraction of a correction term a continuous predictable non decreasing processl which changes only whenx visits the cut locus in this paper we derive a representation onl in terms of measures of local time ofx on the cut locus in analytic terms we compute an expression for the singular part of the laplacian of the riemannian distance function the work uses a relationship of the riemannian distance function to convexity first described by wu 1979 and applied to radial parts of martingales in kendall 1993

linearizability conditions for lotka volterra planar complex cubic systems in this paper we investigate the linearizability problem for the two dimensional planar complex system the necessary and sufficient conditions for the linearizability of this system are found from them the conditions for isochronicity of the corresponding real system can be derived

the centre and isochronicity problems for some cubic systems we present an efficient method for computing focus and linearizability quantities of polynomial differential equation systems we apply the method to computing these quantities for ten eight parametric cubic systems and obtain the necessary and sufficient conditions of linearizability isochronicity of these systems we also show that there is a kind of duality between the problem of constructing algebraic invariant curves first integrals and linearizing transformations on one side and the problem of solving some first order linear partial differential equations on the other side

a new finite loop space at the prime two we construct a space bdi 4 whose mod 2 cohomology ring is the ring of rank 4 mod 2 dickson invariants the loop space on bdi 4 is the first example of an exotic finite loop space at 2 we conjecture that it is also the last one

a finite loop space not rationally equivalent to a compact lie group we construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact lie group hence providing a counterexample to a classical conjecture aided by machine calculation we verify that our counterexample is minimal i e that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact lie group extending the classical known bound of 5

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

ito correction terms for the radial parts of semimartingales on manifolds we consider functions f of a semimartingale x on a complete manifold which fail to bec2 only on and are sufficiently well behaved near a codimension 1 subset we obtain an extension of the ito formula which is valid for all time by adding a continuous predictable process given explicitly in terms of two geometric local times ofx on and the g teaux derivative off we then examine the cut locus of a point of the manifold in sufficient detail to show that this result applies to give a corresponding expression for the radial part of the semimartingale

limiting angles of γ martingales suppose that m is a complete simply connected riemannian manifold of non positive sectional curvature with dimension m 3 and that outside a fixed compact set the sectional curvatures are bounded above by c 1 r 2 ln r and below by c 2 r 2 where c 1 and c 2 are two positive constants and r is the geodesic distance from a fixed point we show that when 1 satisfies certain conditions the angular part of a quasi conformal martingale on m tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity this improves both a result of le for brownian motion and also results concerning the non existence of quasi conformal harmonic maps from certain types of riemannian manifolds into m

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

linearizability conditions of time reversible quartic systems having homogeneous nonlinearities abstract in this paper we investigate the linearizability problem for the two dimensional time reversible complex system x x p x y y y q x y where p and q are homogeneous quartic polynomials the necessary and sufficient conditions for the linearizability of this system are found as a corollary the necessary and sufficient conditions for the origin to be an isochronous center of the planar time reversible real system u v f u v v u g u v where f and g are homogeneous quartic polynomials are obtained

linearizability and local bifurcation of critical periods in a cubic kolmogorov system since chicone and jacobs investigated local bifurcation of critical periods for quadratic systems and newtonian systems in 1989 great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation this paper is devoted to the linearizability and local bifurcation of critical periods for a cubic kolmogorov system we use the darboux method to give explicit linearizing transformations for isochronous centers investigating the finite generation for the ideal of all period constants which are of the polynomial form in six parameters we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center moreover we prove that the maximum number of critical periods is reachable