statistical properties of locally free groups with applications to braid groups and growth of random heaps the main statistical characteristics of locally free groups the growth the drift and the entropy are considered and relations between them are established our results assert that i the statistical properties of random walks markov chains on locally free and braid groups are not the same as the uniform statistics on these groups and ii the stabilization of the statistical characteristics exists when the number of generators of the group grows

anisotropic branching random walks on homogeneous trees symmetric branching random walk on a homogeneous tree exhibits a weak survival phase for parameter values in a certain interval the population survives forever with positive probability but with probability one eventually vacates every finite subset of the tree in this phase particle trails must converge to the geometric boundary of the tree the random subset of the boundary consisting of all ends of the tree in which the population survives called the limit set of the process is shown to have hausdorff dimension no larger than one half the hausdorff dimension of the entire geometric boundary moreover there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic it is further shown that in all cases there is a distinguished probability measure supported by such that the hausdorff dimension of where is the set of generic points of converges to one half the hausdorff dimension of at the phase separation point exact formulas are obtained for the hausdorff dimensions of and and it is shown that the log hausdorff dimension of has critical exponent 1 2 at the phase separation point

continuous time quantum walk on integer lattices and homogeneous trees this paper is concerned with the continuous time quantum walk on d and infinite homogeneous trees by using the generating function method we compute the limit of the average probability distribution for the general isotropic walk on and for nearest neighbor walks on d and infinite homogeneous trees in addition we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases

a phase transition for hyperbolic branching processes consider a time and space homogeneous random branching markov process on a d d lobachevsky space mathcal h d its asymptotic behaviour can be described in terms of the hausdorff dimension of the random set of the accumulation points on the absolute partial mathcal h d the simplest and most well known example is the laplace beltrami branching diffusion in the case d 2 the hausdorff dimension of was calculated in ls in this paper we extend the formula for the hausdorff dimension to d 3 and a larger class of branching processes it turns out that the hausdorff dimension of takes either a value from 0 d 1 2 or equals d 1 the euclidean dimension of partial mathcal h d which gives an interesting exmaple of a geometric phase transition

branching random walks on free products of groups we study certain phase transitions of branching random walks brw on cayley graphs of free products the aim of this paper is to compare the size and structural properties of the trace that is the subgraph that consists of all edges and vertices that were visited by some particle with those of the original cayley graph we investigate the phase when the growth parameter is small enough such that the process survives but the trace is not the original graph a first result is that the box counting dimension of the boundary of the trace exists is almost surely constant and equals the hausdorff dimension which we denote by the main result states that the function has only one point of discontinuity which is at c r where r is the radius of convergence of the green function of the underlying random walk furthermore r is bounded by one half the hausdorff dimension of the boundary of the original cayley graph and the behaviour of r as r is classified in the case of free products of infinite groups the end boundary can be decomposed into words of finite and words of infinite length we prove the existence of a phase transition such that if c the end boundary of the trace consists only of infinite words and if c it also contains finite words in the last case the hausdorff dimension of the set of ends of the trace and the original graph induced by finite words is strictly smaller than the one of the ends induced by infinite words

the branching random walk and contact process on galton watson and nonhomogeneous trees we show that the branching random walk on a galton watson tree may have one or two phase transitions depending on the relative sizes of the mean degree and the maximum degree we show that there are some galton watson trees on which the branching random walk has one phase transition while the contact process has two this contradicts a conjecture of madras and schinazi we show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree contradicting a conjecture of pemantle

discontinuity induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle abstract this paper applies methods of numerical continuation analysis to document characteristic bifurcation cascades of limit cycles in piecewise smooth hybrid dynamical system models of the eukaryotic cell cycle and associated period adding cascades in piecewise defined maps with gaps a general theory is formulated for the occurrence of such cascades for example given the existence of a period two orbit with one point on the system discontinuity and with appropriate constraints on the forward trajectory for nearby initial conditions in this case it is found that the bifurcation cascade for nearby parameter values exhibits a scaling relationship governed by the largest in magnitude floquet multiplier here required to be positive and real in complete agreement with the characteristic scaling observed in the numerical study a similar cascade is predicted and observed in the case of a saddle node bifurcation of a period two orbit away from the discontinuity provided that the associated center manifold is found to intersect the discontinuity transversally

multifractality of entangled random walks and non uniform hyperbolic spaces multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a non symmetric lattice of obstacles are investigated using the equivalence of the model to random walks on a locally non symmetric tree statistical properties of topological invariants such as drift and return probabilities have been studied by means of a renormalization group rg technique the comparison of the analytical rg results with numerical simulations as well as with the rigorous results of gerl and woess demonstrates clearly the validity of our approach it is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space which has an exponentially large number of states has been broken

hyperbolic branching brownian motion hyperbolic branching brownian motion is a branching diffusion process in which individual particles follow independent brownian paths in the hyperbolic plane 2 and undergo binary fission s at rate 0 it is shown that there is a phase transition in for 1 8 the number of particles in any compact region of 2 is eventually 0 w p 1 but for 1 8 the number of particles in any open set grows to w p 1 in the subcritical case 1 8 the set of all limit points in 2 the boundary circle at of particle trails is a cantor set while in the supercritical case 1 8 the set has full lebesgue measure for 1 8 it is shown that w p 1 the hausdorff dimension of is 1 1 8 2

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

the similarity problem for fourier algebras and corepresentations of group von neumann algebras let g be a locally compact group and let a g and vn g be its fourier algebra and group von neumann algebra respectively in this paper we consider the similarity problem for a g is every bounded representation of a g on a hilbert space h similar to a representation we show that the similarity problem for a g has a negative answer if and only if there is a bounded representation of a g which is not completely bounded for groups with small invariant neighborhoods i e sin groups we show that a representation a g b h is similar to a representation if and only if it is completely bounded this in particular implies that corepresentations of vn g associated to non degenerate completely bounded representations of a g are similar to unitary corepresentations we also show that if g is a sin maximally almost periodic or totally disconnected group then a representation of a g is a representation if and only if it is a complete contraction these results partially answer questions posed in effros and ruan 2003 7 and spronk 2002 25

the full martin boundary of the bi tree are interested in the martin compactification of the bi tree with respect to each eigenvalue t p of the transition operator acting on nonnegative functions the minimal martin boundary i e set of minimal t harmonic functions is well understood see 20 and also 22 for cartesian products of markov chains in general on the other hand the description of the full martin compactification has remained an open problem some evidence has been obtained by picardello and sjogren 21 who studied boundary behaviour of harmonic functions on the bi tree the situation is similar for the hyperbolic bi disk equipped with the laplace beltrami operator in this case the minimal martin boundary is known via the work of karpelevic 13 and guivarc h 10 see also the general result of freire 7 and taylor 26 concerning cartesian products of manifolds the problem of determining the full martin boundary was proposed by guivarc h and taylor 11 who gave the answer only for the critical eigenvalue where the behaviour is different in the present paper we describe the full martin compactification of the bitree with respect to all eigenvalues t p this is possible thanks to recent work of lalley 14 who has given a local limit theorem for aperiodic nearestneighbour random walks on homogeneous trees that is an asymptotic estimate of the n step transition probabilities which is uniform both in time and space a considerable improvement of previous work of sawyer 24 gerl and woess

exact learning of discretized geometric concepts we first present an algorithm that uses membership and equivalence queries to exactly identify a discretized geometric concept defined by the union of m axis parallel boxes in d dimensional discretized euclidean space where each coordinate can have n discrete values this algorithm receives at most md counterexamples and uses time and membership queries polynomial in m and log n for any constant d furthermore all equivalence queries can be formulated as the union of o md log m axis parallel boxes next we show how to extend our algorithm to efficiently learn from only equivalence queries any discretized geometric concept generated from any number of halfspaces with any number of known to the learner slopes in a constant dimensional space in particular our algorithm exactly learns from equivalence queries only unions of discretized axis parallel boxes in constant dimensional space in polynomial time furthermore this equivalence query only algorithm can be modified to handle a polynomial number of lies in the counterexamples provided by the environment finally we introduce a new complexity measure that better captures the complexity of the union of m boxes than simply the number of boxes and the dimension our new measure sigma is the number of segments in the target where a segment is a maximum portion of one of the sides of the target that lies entirely inside or entirely outside each of the other halfspaces defining the target we present a modification of our first algorithm that uses time and queries polynomial in sigma and log n in fact the time and queries both membership and equivalence used by this single algorithm are polynomial for either m or d constant

discontinuity induced bifurcations in models of mechanical contact capillary adhesion and cell division a common framework abstract this paper collects four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and establishes selected general properties of the associated near grazing dynamics in particular it is shown that for critical choices of parameter values commonly used physical models of rigid or compliant mechanical contact capillary adhesion and cell division satisfy the conditions required by the general framework the paper relies on the well known discontinuity mapping formalism in contrast to previous treatments the proposed abstract framework more clearly establishes the origin of the large state space stretching in the initial and possibly only step of the construction of a discontinuity mapping it further highlights the nonuniqueness in the formulation of the discontinuity mapping and its connection to the choice of a locally smooth map with which the discontinuity mapping is composed to describe the near grazing dynamics the analysis is illustrated with examples from tapping mode atomic force microscopy in the presence of thin fluid layers on the sample and the probe tip and from the study of protein activity during a eukaryotic cell cycle

the poisson transform and representations of a free group abstract let g be a free group with r generators 1 r g which plays the same role of the laplace beltrami operator on semisimple lie groups are characterized furthermore an analytic family of representations z of g on functions on the boundary is considered defined by z x p z x x 1 where p x is the poisson kernel relative to the action of g on it is proved that for 0 s re z z is uniformly bounded on an appropriate hilbert space h s finally the uniform boundedness of other special representations of g obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of gl 2 q p is proved

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

corner bifurcations in non smoothly forced impact oscillators abstract we consider an impacting mechanical system in which a particle at position u t impacts with a periodically moving obstacle at position z t the motion of which is non smooth in particular we look at corner events when u impacts with z very close to a point where z loses smoothness we show that this leads through a corner bifurcation to complex dynamics in u which can include periodic orbits of arbitrary period and period adding cascades by analysing associated maps close to the corner event we show that this dynamics can be understood in terms of the iterations of a two dimensional piecewise linear discontinuous map we also show some links between this analysis and the difficult problem of understanding the motion of three objects which may have simultaneous impacts

the contact process on trees the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter is varied for small values of a single infection eventually dies out for larger the infection lives forever with positive probability but eventually leaves any finite set the survival probability is a continuous function of and the proof of this is much easier than it is for the contact process on d dimensional integer lattices for still larger the infection converges in distribution to a nontrivial invariant measure for an n ary tree with n large the first of these transitions occurs when 1 n and the second occurs when 1 2 n e n nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n ary trees provided that vertices of degree n are not too rare in particular letting n go to galton watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at 0 the nature of the second phase transition is not yet clear and several problems are mentioned in this regard

on the proper learning of axis parallel concepts we study the proper learnability of axis parallel concept classes in the pac learning and exact learning models these classes include union of boxes dnf decision trees and multivariate polynomials for constant dimensional axis parallel concepts c we show that the following problems have time complexities that are within a polynomial factor of each other c is properly exactly learnable with hypotheses of size at most times the target size from membership and equivalence queries c is properly pac learnable without membership queries under any product distribution there is an approximation algorithm for the minequic problem given a g c find a minimal size f c that is logically equivalent to g in particular if one has polynomial time complexity they all do using this we give the first proper learning algorithm of constant dimensional decision trees and the first negative results in proper learning from membership and equivalence queries for many classes for axis parallel concepts over a nonconstant dimension we show that with the equivalence oracle 1 3 we use this to show that binary decision trees are not properly learnable in polynomial time assuming p np and dnf is not se properly learnable e 1 in polynomial time even with an np oracle assuming 2p pnp

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

cascades of particles moving at finite velocity in hyperbolic spaces a branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space poincare half plane and poincare disk is examined each particle can split into two particles only once at poisson spaced times and deviates orthogonally when splitted at time t after n t poisson events there are n t 1 particles moving along different geodesic lines we are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles we derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as t increases and for different values of the parameters c hyperbolic velocity of motion and rate of reproduction the mean hyperbolic distance of each moving particle is also examined and a useful representation as the distance of a randomly stopped particle moving over the main geodesic line is presented

limit set of a weakly supercritical contact process on a homogeneous tree a conjecture of liggett 9 concerning the regime of weak survival for the contact process on a homogeneous tree is proved the conjecture is shown to imply that the hausdor dimension of the limit set of such a contact process is no larger than half the hausdor dimension of the space of ends of the tree the conjecture is also shown to imply that at the boundary between weak survival and strong survival the contact process survives only weakly a theorem previously proved by zhang 14 finally a stronger form of a theorem of hawkes and lyons concerning the hausdor dimension of a galton watson tree is proved

survival extinction and approximation of discrete time branching random walks we consider a general discrete time branching random walk on a countable set x we relate local strong local and global survival with suitable inequalities involving the first moment matrix m of the process in particular we prove that while the local behavior is characterized by m the global behavior cannot be completely described in terms of properties involving m alone moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the possibly finite state space is large enough an analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site we compare these results with the ones obtained in the continuous time case and we give some examples and counterexamples

computing norms of free operators with matrix coefficients let g 1 g 2 g n be the generators of the free group f n and denote by its left regular representation a formula for norms of operators of the form a i g i where a i are complex matrices or operators is derived similarly to the formula of akemann and ostrand for the scalar case this formula can be used to compute norms of arbitrary finitely supported convolution operators on the free group essentially the same technique yields a formula for the norms of sums of matrix valued creation and annihilation operators on full fock space finally some results about the numerical evaluation of the formula are proved

spherical functions and harmonic analysis on free groups let f r 1 be a free group with r generators in this paper we study a principal series and a complementary series of irreducible unitary representations of f which are defined through the action of f on its poisson boundary relative to a simple random walk we show that the regular representation of f can be written as a direct integral of the representations of the principal series and that the resulting harmonic analysis on the free group bears a close resemblance with the harmonic analysis of sl 2 ir

the resolvent for simple random walks on the free product of two discrete groups on montre que si g est le produit libre de 2 groupes discrets g 1 g 2 et si est une mesure sur g qui est une combinaison convexe de 2 mesures de probabilite 1 et 2 sur g 1 et g 2 alors la resolvante zse 1 est proportionnelle au produit libre des resolvantes i se i 1 ou e est l element identite et i depend analytiquement de z

exact learning boolean functions via the monotone theory we study the learnability of boolean functions from membership and equivalence queries we develop the monotone theory that proves 1 any boolean function is learnable in polynomial time in its minimal disjunctive normal form size its minimal conjunctive normal form size and the number of variables n in particular 2 decision trees are learnable our algorithms are in the model of exact learning with membership queries and unrestricted equivalence queries the hypotheses to the equivalence queries and the output hypotheses are depth 3 formulas

statistics of a polymer chain in an array of obstacles abstract exact relations for the statistical characteristics of a polymer chain in a regular lattice of obstacles are obtained the edges of this lattice cannot be crossed by the chain in the course of its motion such a model is now frequently used for the description of the dynamics of a polymer chain which is strongly entangled with other chains numerical calculations of exact characteristics for the simplest realization of the model under consideration are performed results of these calculations are compared with the results of approximate analytic methods which were proposed earlier it is shown that the partition function of a polymer chain in a lattice of obstacles depends not only on the integral characteristics of the so called primitive path which defines the topological state of the chain with respect to the obstacles but on the detailed trajectory of the primitive path as well however if the trajectory of the primitive path is a random walk it is this case that corresponds to the real situation most frequently the dependence on the details of the primitive path is very weak due to this fact it is possible to develop analytical approaches to the solution of the problems under study

characterization of critical values of branching random walks on weighted graphs through infinite type branching processes we study the branching random walk on weighted graphs site breeding and edge breeding branching random walks on graphs are seen as particular cases two kinds of survival can be identified a weak survival with positive probability there is at least one particle alive somewhere at any time and a strong survival with positive probability the colony survives by returning infinitely often to a fixed site the behavior of the process depends on the value of a certain parameter which controls the birth rates the threshold between survival and almost sure extinction is called critical value we describe the strong critical value in terms of a geometrical parameter of the graph we characterize the weak critical value and relate it to another geometrical parameter we prove that at the strong critical value the process dies out locally almost surely while at the weak critical value global survival and global extinction are both possible

the rate of escape for anisotropic random walks in a tree let g be the group generated by l free involutions whose cayley graph t is the infinite homogeneous tree with l edges at every node a general central limit theorem and law of the iterated logarithm is proven for left invariant random walks z n on g or t which applies to the distance of z n from a fixed point as well as to the distribution of the last r letters in z n for nearest neighbor random walks we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point a generalization for z n with a finitely supported step distribution is derived and discussed

recurrence and transience of branching diffusion processes on riemannian manifolds we relate the recurrence and transience of a branching diffusion process on a riemannian manifold m to some properties of a linear elliptic operator onm including spectral properties there is a trade off between the tendency of the transient brownian motion to escape and the birth process of the new particles if the latter has a high enough intensity then it may override the transience of the brownian motion leading to the recurrence of the branching process and vice versa in the case of a spherically symmetric manifold the critical intensity of the population growth can be found explicitly

branching random walks and contact processes on homogeneous trees branching random walks and contact processes on the homogeneous tree in which each site has d 1 neighbors have three possible types of behavior for d 2 local survival local extinction with global survival and global extinction for branching random walks we show that if there is local extinction then the probability that an individual ever has a descendent at a site n units away from that individual s location is at most d n 2 while if there is global extinction this probability is at most d n next we consider the structure of the set of invariant measures with finite intensity for the system and see how this structure depends on whether or not there is local and or global survival these results suggest some problems and conjectures for contact processes on trees we prove some and leave others open in particular we prove that for some values of the infection parameter there are nontrivial invariant measures which have a density tending to zero in all directions and hence are different from those constructed by durrett and schinazi in a recent paper

equivalence of models for polynomial learnability abstract in this paper we consider several variants of valiant s learnability model that have appeared in the literature we give conditions under which these models are equivalent in terms of the polynomially learnable concept classes they define these equivalences allow comparisons of most of the existing theorems in valiant style learnability and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality we also give a useful reduction of learning problems to the problem of finding consistent hypotheses and give comparisons and equivalences between valiant s model and the prediction learning models of haussler littlestone and warmuth in 29th annual ieee symposium on foundations of computer science 1988

statistics of reduced words in locally free and braid groups abstract studies and applications to ballistic growth model we study numerically and analytically the average length of reduced primitive words in the so called locally free and braid groups we consider the situations when the letters in the initial words are drawn either without or with correlations in the latter case we show that the average length of the reduced word can be increased or lowered depending on the type of correlation the ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth model

random walks on three strand braids and on related hyperbolic groups we investigate the statistical properties of random walks on the simplest nontrivial braid group b3 and on related hyperbolic groups we provide a method using cayley graphs of groups allowing us to compute explicitly the probability distribution of the basic statistical characteristics of random trajectories the drift and the return probability the action of the groups under consideration in the hyperbolic plane is investigated and the distribution of a geometric invariant the hyperbolic distance is analysed it is shown that a random walk on b3 can be viewed as a magnetic random walk on the group psl 2

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

an analytic family of uniformly bounded representations of free groups harmonic analysis on a free group f has attracted considerable attention in the last ten years or so there seem to be two reason for that one is the discovery of deep analogy of certain aspects of harmonic analysis on a free group and harmonic analysis on sl 2 r cf e g fundamental works of p cartier 4 a figa talamanca and m a picardello 8 the other being the interest in the c algebra generated by the regular representation of f cf a connes 5 j cuntz 6 u haagerup 10 m pimsner and d voiculescu 14

the existence of an intermediate phase for the contact process on trees let mathbb t d be a homogeneous tree in which every vertex has d neighbors a new proof is given that the contact process on mathbb t d exhibits two phase transitions when d geq 3 a behavior which distinguishes it from the contact process on mathbb z n this is the first proof which does not involve calculation of bounds on critical values and it is much shorter than the previous proof for the binary tree mathbb t 3 the method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties for which no such result was previously known

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

how many queries are needed to learn we investigate the query complexity of exact learning in the membership and proper equivalence query model we give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model we give applications of this characterization including results on learning a natural subclass of dnf formulas and on learning with membership queries alone query complexity has previously been used to prove lower bounds on the time complexity of exact learning we show a new relationship between query complexity and time complexity in exact learning if any honest class is exactly and properly learnable with polynomial query complexity but not learnable in polynomial time then p np in particular we show that an honest class is exactly polynomial query learnable if and only if it is learnable using an oracle for p 4

multiple transition points for the contact process on the binary tree the contact process on z d is known to have only two fundamental types of behavior survival and extinction recently pemantle discovered that the phase structure for the contact process on a tree can be more complex there are three possible types of behavior strong survival weak survival and extinction he proved that all three occur on homogeneous trees in which each vertex has d 1 neighbors provided that d 3 but he left open the case d 2 since d 1 corresponds to z 1 in which weak survival does not occur d 2 is the boundary case in this paper we complete this picture by showing that weak survival does occur on the binary tree for appropriate parameter values in doing so we extend and develop techniques for obtaining upper and lower bounds for the critical values associated with strong and weak survival of the contact process on more general graphs

an extension of hawkes theorem on the hausdorff dimension of a galton watson tree let be the genealogical tree of a supercritical multitype galton watson process and let be the limit set of i e the set of all infinite self avoiding paths called ends through that begin at a vertex of the first generation the limit set is endowed with the metric d 2 n where n n is the index of the first generation where and differ to each end is associated the infinite sequence of types of the vertices of let be the space of all such sequences for any ergodic shift invariant probability measure on define to be the set of all generic sequences i e the set of all sequences such that each finite sequence v occurs in with limiting frequency v where v is the set of all that begin with the word v then the hausdorff dimension of 1 in the metric d is

non intersection of transient branching random walks let g be a cayley graph of a nonamenable group with spectral radius rho 1 it is known that branching random walk on g with offspring distribution mu is transient i e visits the origin at most finitely often almost surely if and only if the expected number of offspring overline mu satisfies overline mu le rho 1 benjamini and muller groups geom dyn 6 231 247 2012 conjectured that throughout the transient supercritical phase 1 overline mu le rho 1 and in particular at the recurrence threshold overline mu rho 1 the trace of the branching random walk is tree like in the sense that it is infinitely ended almost surely on the event that the walk survives forever this is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely we prove this conjecture along with several other related conjectures made by the same authors a central contribution of this work is the introduction of the notion of local unimodularity which we expect to have several further applications in the future

dynamics of a piecewise linear map with a gap in this paper we consider periodic solutions of discontinuous non smooth maps we show how the fixed points of a general piecewise linear map with a discontinuity a map with a gap behave under parameter variation we show in detail all the possible behaviours of period 1 and period 2 solutions for positive gaps we find that period 2 solutions can exist independently of period 1 solutions conversely for negative gaps period 1 and period 2 solutions can coexist higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation finally we compare our results with those of jain banerjee jain banerjee 2003 int j bifurcat chaos 13 3341 3351 and banerjee et al banerjee et al 2004 ieee trans circ syst ii 51 649 654 and explain their numerical simulations

on the computation of spectra in free probability abstract we use free probability techniques to compute borders of spectra of non hermitian operators in finite von neumann algebras which arise as free sums of simple operators to this end the resolvent is analyzed with the aid of the haagerup inequality concrete examples coming from reduced c algebras of free product groups and leading to systems of polynomial equations illustrate the approach

learning functions represented as multiplicity automata we study the learnability of multiplicity automata in angluin s exact learning model and we investigate its applications our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its hankel matrix with this theorem in hand we present a new simple algorithm for learning multiplicity automata with improved time and query complexity and we prove the learnability of various concept classes these include among others the class of disjoint dnf and more generally satisfy o 1 dnf the class of polynomials over finite fields the class of bounded degree polynomials over infinite fields the class of xor of terms certain classes of boxes in high dimensions in addition we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over gf 2 while multiplicity automata are shown to be useful to prove the learnability of some subclasses of dnf formulae and various other classes we study the limitations of this method we prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general dnf formulas or even k term dnf for k log n or satisfy s dnf formulas for s 1 these results are proven by exhibiting functions in the above classes that require multiplicity automata with super polynomial number of states

codimension three bifurcations explanation of the complex one two and three dimensional bifurcation structures in nonsmooth maps many physical and engineering systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied such systems have been found to yield piecewise smooth maps and in some cases the obtained map is discontinuous by investigating the normal form of such maps we have detected a type of codimension three bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space the results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering

equidistribution of random walks on spheres we study the equidistribution on spheres of the n step transition probabilities of random walks on graphs we give sufficient conditions for this property being satisfied and for the weaker property of asymptotical equidistribution we analyze the asymptotical behaviour of the green function of the simple random walk on 2 and we provide a class of random walks on cayley graphs of groups whose transition probabilities are not even asymptotically equidistributed

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

local limits and harmonic functions for nonisotropic random walks on free groups nearest neighbour random walks on the homogeneous tree representing a free group withs generators 2 s are investigated by use of generating functions and their analytic properties a local limit theorem is derived a study of the harmonic functions corresponding to the random walk leads to properties that characterize ther harmonic function connected with the local limits

efficient learning with virtual threshold gates abstract we reduce learning simple geometric concept classes to learning disjunctions over exponentially many variables we then apply an online algorithm called winnow whose number of prediction mistakes grows only logarithmically with the number of variables the hypotheses of winnow are linear threshold functions with one weight per variable we find ways to keep the exponentially many weights of winnow implicitly so that the time for the algorithm to compute a prediction and update its virtual weights is polynomial our method can be used to learnd dimensional axis parallel boxes whendis variable and unions ofd dimensional axis parallel boxes whendis constant the worst case number of mistakes of our algorithms for the above classes is optimal to within a constant factor and our algorithms inherit the noise robustness of winnow we think that other online algorithms with multiplicative weight updates whose loss bounds grow logarithmically with the dimension are amenable to our methods