nonconvergence to unstable points in urn models and stochastic approximations a particle in mathbf r d moves in discrete time the size of the n th step is of order 1 n and when the particle is at a position mathbf v the expectation of the next step is in the direction mathbf f mathbf v for some fixed vector function mathbf f of class c 2 it is well known that the only possible points mathbf p where mathbf v n may converge are those satisfying mathbf f mathbf p mathbf 0 this paper proves that convergence to some of these points is in fact impossible as long as the noise the difference between each step and its expectation is sufficiently omnidirectional the points where convergence is impossible are the unstable critical points for the autonomous flow d dt mathbf v t mathbf f v t this generalizes several known results that say convergence is impossible at a repelling node of the flow

a simulation based algorithm for ergodic control of markov chains conditioned on rare events we study the problem of long run average cost control of markov chains conditioned on a rare event in a related recent work a simulation based algorithm for estimating performance measures associated with a markov chain conditioned on a rare event has been developed we extend ideas from this work and develop an adaptive algorithm for obtaining online optimal control policies conditioned on a rare event our algorithm uses three timescales or step size schedules on the slowest timescale a gradient search algorithm for policy updates that is based on one simulation simultaneous perturbation stochastic approximation spsa type estimates is used deterministic perturbation sequences obtained from appropriate normalized hadamard matrices are used here the fast timescale recursions compute the conditional transition probabilities of an associated chain by obtaining solutions to the multiplicative poisson equation for a given policy estimate further the risk parameter associated with the value function for a given policy estimate is updated on a timescale that lies in between the two scales above we briefly sketch the convergence analysis of our algorithm and present a numerical application in the setting of routing multiple flows in communication networks

classifying spaces of kac moody groups we study the structure of classifying spaces of kac moody groups from a homotopy theoretic point of view they behave in many respects as in the compact lie group case the mod p cohomology algebra is noetherian and lannes t functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p subgroups also spaces of maps from classifying spaces of finite p groups to classifying spaces of kac moody groups are described in terms of classifying spaces of centralizers while the classifying space of a kac moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p subgroups up to p completion we show that these properties are common to a larger class of groups also including parabolic subgroups of kac moody groups and centralizers of finite p subgroups

normal subsystems of fusion systems the notion of a fusion system was first defined and explored by puig in the context of modular representation theory later broto levi and oliver significantly extended the theory of fusion systems as a tool in homotopy theory in this paper we begin a program to establish a local theory of fusion systems similar to the local theory of finite groups in particular we define the notion of a normal subsystem of a saturated fusion system and prove some basic results about normal subsystems and factor systems

extensions of linking systems and fusion systems we correct two errors in the statement and proof of a theorem in an earlier paper 2007 and at the same time extend that result to a more general theorem about extensions of p local finite groups other special cases of this theorem have already been shown in two later papers so we feel it will be useful to have this more general result in the literature

subcentric linking systems for part of this research the author was supported by the danish national research foundation through the centre for symmetry and deformation dnrf92

existence of invariant tori in three dimensional maps with degeneracy abstract we prove a kam type result for the persistence of two dimensional invariant tori in perturbations of integrable action angle angle maps with degeneracy satisfying the intersection property such degenerate action angle angle maps arise upon generic perturbation of three dimensional volume preserving vector fields which are invariant under volume preserving action of s 1 when there is no motion in the group action direction for the unperturbed map this situation is analogous to degeneracy in hamiltonian systems the degenerate nature of the map and the unequal number of action and angle variables make the persistence proof non standard the persistence of the invariant tori as predicted by our result has implications for the existence of barriers to transport in three dimensional incompressible fluid flows simulation results indicating existence of two dimensional tori in a perturbation of swirling hill s spherical vortex flow are presented

saturated fusion systems over 2 groups we develop methods for listing for a given 2 group s all nonconstrained centerfree saturated fusion systems over s these are the saturated fusion systems which could potentially include minimal examples of exotic fusion systems fusion systems not arising from any finite group to test our methods we carry out this program over four concrete examples two of order 2 7 and two of order 2 10 our long term goal is to make a wider more systematic search for exotic fusion systems over 2 groups of small order

barriers to transport and mixing in volume preserving maps with nonzero flux abstract we identify some geometric structures secondary tori that restrict transport and prevent mixing in perturbations of integrable volume preserving systems with nonzero net flux unlike the customary kam tori secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction we also note that secondary tori persist under the addition of a net flux which destroys all customary kam tori we introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori we then implement the algorithm and after running it present assertions regarding the distribution of the escape times of the unbounded orbits the abundance of secondary tori the size of the resonant regions and the robustness of the tori under the addition of a mean flux

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

saturated fusion systems with parabolic families abstract let g be a group a finite p subgroup s of g is a sylow p subgroup if every finite p subgroup of g is conjugate to a subgroup of s in this paper we examine the relations between the fusion system over s which is given by conjugation in g and a certain chamber system c on which g acts chamber transitively with chamber stabilizer n g s next we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system we determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system we give an application to fusion systems with parabolic families of classical type

the o d e method for convergence of stochastic approximation and reinforcement learning it is shown here that stability of the stochastic approximation algorithm is implied by the asymptotic stability of the origin for an associated ode this in turn implies convergence of the algorithm several specific classes of algorithms are considered as applications it is found that the results provide i a simpler derivation of known results for reinforcement learning algorithms ii a proof for the first time that a class of asynchronous stochastic approximation algorithms are convergent without using any a priori assumption of stability iii a proof for the first time that asynchronous adaptive critic and q learning algorithms are convergent for the average cost optimal control problem

some pathological traps for stochastic approximation we consider different kinds of pathological traps for stochastic algorithms thus extending a previous study on regular traps an illustration is given by the complete proof of the convergence of a principal component analysis pca algorithm when the eigenvalues are multiple

extensions of p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we study and classify extensions of p local finite groups and also compute the fundamental group of the classifying space of a p local finite group

avoidance of traps in stochastic approximation abstract an estimate for the lock in probability of stochastic approximation i e the probability of convergence to an attractor given that the algorithm is in its domain of attraction after a sufficiently long time is derived and is used for proving a s avoidance of traps i e convergence only to the stable attractors

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

wreath products and representations of p local finite groups abstract given two finite p local finite groups and a fusion preserving morphism between their sylow subgroups we study the question of extending it to a continuous map between their classifying spaces the results depend on the construction of the wreath product of p local finite groups which is also used to study p local permutation representations

existence of kam tori in degenerate hamiltonian systems abstract this article is devoted to the proof that invariant tori still exist in perturbed integrable hamiltonian systems with n degrees of freedom h p q n p p p q if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping n p can be any positive integer smaller than n provided the image of any open set under the mapping includes a cn 2 curved one dimensional submanifold

extensions of linking systems with p group kernel we study extensions of p local finite groups where the kernel is a p group in particular we construct examples of saturated fusion systems mathcal f which do not come from finite groups but which have normal p subgroups a vartriangleleft mathcal f such that mathcal f a is the fusion system of a finite group one of the tools used to do this is the concept of a transporter system which is modelled on the transporter category of a finite group and is more general than a linking system

break up of invariant surfaces in action angle angle maps and flows abstract an important class of three dimensional volume preserving maps and flows arises as a perturbation from integrable action action angle maps and flows the properties of this class of maps and flows are discussed while action angle angle volume preserving maps admit an analog of the kam theorem general results on nonexistence of two dimensional invariant manifolds of action action angle maps are proven here nonexistence of such two dimensional invariant manifolds means possibility of global transport and a mechanism for such transport the local mechanism of resonance induced dispersion phys rev lett 75 1995 3669 is studied perturbatively resonance induced dispersion is shown to arise from the existence of periodic orbits that survive perturbation at places where two dimensional invariant manifolds break down

shearless transport barriers in unsteady two dimensional flows and maps abstract we develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two dimensional flows and maps defined over a finite time interval this principle reveals that hyperbolic lagrangian coherent structures lcss and parabolic lcss or jet cores are the two main types of shearless barriers in unsteady flows based on the boundary conditions they satisfy parabolic barriers are found to be more observable and robust than hyperbolic barriers confirming widespread numerical observations both types of barriers are special null geodesics of an appropriate lorentzian metric derived from the cauchy green strain tensor using this fact we devise an algorithm for the automated computation of parabolic barriers we illustrate our detection method on steady and unsteady non twist maps and on the aperiodically forced bickley jet

homotopy equivalences of p completed classifying spaces of finite groups we study homotopy equivalences of p completions of classifying spaces of finite groups to each finite group g and each prime p we associate a finite category p c g with the following properties two p completed classifying spaces bg p and bg p have the same homotopy type if and only if the associated categories p c g and p c g are equivalent furthermore the topological monoid aut bg p of self equivalences is determined by the self equivalences of the associated category p c g

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated