supports of derivations free factorizations and ranks of fixed subgroups in free groups for f a free group of finite rank it is shown that the fixed subgroup of any set b of endomorphisms of f has rank rank f generalizing a recent result of dicks and ventura the proof involves the combinatorics of derivations of groups some related questions are examined

generalization in a multi state neural network the generalization ability of an extremely dilute feedback neural network with multi state neurons is studied by means of a deterministic noiseless parallel dynamics the overlap with any one of a macroscopic number of binary full activity concepts is determined when the network is trained with examples of variable activity according to a hebbian learning algorithm that favours stable symmetric mixture states explicit results about the phase diagram and the generalization error are obtained for a network with three state neurons which remain inactive below a threshold it is shown that the generalization ability can be considerably enhanced either by training the network with low activity examples or by means of a moderate increase in

phase transitions for suspension flows this paper is devoted to studying the thermodynamic formalism for suspension flows defined over countable alphabets we are mostly interested in the regularity properties of the pressure function we establish conditions for the pressure function to be real analytic or to exhibit a phase transition we also construct an example of a potential for which the pressure has countably many phase transitions

regenerative representation for one dimensional gibbs states on montre que les etats de gibbs unidimensionnels peuvent etre representes comme des concatenations de listes infinies de mots i i d

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

categorization in a layered neural network abstract the layered feedforward network of domany meir and kinzel is extended to investigate the ability to recognize any one of a macroscopic number of concepts ancestors when the network is trained through a finite number of examples descendents of each concept by means of a generalized hebbian learning rule between cells on two consecutive layers learning curves describing the generalization error are obtained as well as phase diagrams and basins of attraction that exhibit the possible coexistence of a generalization phase a retrieval phase of examples and a spin glass phase

categorization ability in a biologically motivated neural network abstract in this work we study the categorization properties of an extreme and asymmetrically diluted version of the hopfield model for associative memory when the effect of the refractory period of the neurons is taken into account in the dynamics of the system the simplest way of modeling these refractory periods is by means of a time dependent threshold that acts only on those neurons that emit a signal and favors them to be at rest during a given time interval the dynamic equations are derived in the limit of extreme dilution using an approach that explicitly preserves the dependence of the system on its whole history the categorization error is analyzed for different values of the parameters in particular we confront our analytical results with numerical simulations for the noiseless case t 0 when the number of examples or their correlations increases the system always categorizes independently of the amplitude of the potential that mimics the effect of the refractory period

categorization in a hopfield network trained with weighted examples i finite number of concepts abstract we consider the categorization problem in a hopfield network with a finite number of concepts and trained with s examples of weight 1 s we find that the retrieval capacity of an example with weight 1 and the corresponding categorization error depends also on the arithmetic mean m 1 s 1 2 s of the other weights for 1 m 1 we find that the line of first order transitions between the retrieval and categorization phases ends at a critical point in the s t plane when two solutions are present the global minimum of the free energy corresponds to the solution with the highest weight

generalization in a hopfield network with noise the generalization ability of the hopfield model of neural networks trained with examples is studied in mean field theory in the presence of synaptic noise although the latter improves the generalization ability for a finite number of concepts it does not for a macroscopic number of them nevertheless the network performance is still robust against synaptic noise numerical simulations are performed to verify the mean field theory predictions

fluctuation bounds for entropy production estimators in gibbs measures we give bounds for the fluctuations of estimators of the mean entropy production in gibbsian sources these bounds are valid for every n where n denotes the size length of the sample we consider two estimators which are based on waiting and hitting times

renewal representations for markov operators abstract as is known due to the existence of an embedded renewal structure the iterates of a harris recurrent markov operator can be represented as a delayed renewal sequence we show that these kind of representations also exist for a larger class of markov operators provided only that certain filling schemes are successful as applications of the theory we study the co feller operators and markov operators which contract the variation

thermodynamics of towers of hyperbolic type we introduce a class of continuous maps f of a compact topological space x admitting inducing schemes of hyperbolic type and describe the associated tower constructions we then establish a thermodynamic formalism i e we describe a class of real valued potential functions on x such that f possess a unique equilibrium measure associated to each which minimizes the free energy among the measures that are liftable to the tower we also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem we then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable our results extend those obtained in ps05 ps08 for inducing schemes of expanding types and apply to certain multidimensional maps applications include obtaining the thermodynamic formalism for young s diffeomorphisms the henon family at the first birfucation and the katok map in particular for the katok map we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with 0 t 1

fixed subgroups of endomorphisms of free products abstract let g i 1 n g i and let be a symmetric endomorphism of g if is a monomorphism or if g is a finitely generated residually finite group then the fixed subgroup fix g g g g of has kurosh rank at most n

simulating highly diluted neural networks the asymmetric diluted hopfield model is studied through numerical simulations and reliable results are obtained even when the network sizes are not exponentially large the mean final overlap the mean convergence time and a quantity that gives information on the relative importance of spurious states are measured as a function of the load capacity alpha

perfect simulation of a coupling achieving the bar d distance between ordered pairs of binary chains of infinite order we explicitly construct a stationary coupling attaining ornstein s bar d distance between ordered pairs of binary chains of infinite order our main tool is a representation of the transition probabilities of the coupled bivariate chain of infinite order as a countable mixture of markov transition probabilities of increasing order under suitable conditions on the loss of memory of the chains this representation implies that the coupled chain can be represented as a concatenation of i i d sequences of bivariate finite random strings of symbols the perfect simulation algorithm is based on the fact that we can identify the first regeneration point to the left of the origin almost surely

uniqueness of gibbs measures and absorption probabilities 1 introduction in studying uniqueness of gibbs measures it may be useful to make a comparison with a stochastic process we present a method connecting absorption probabilities of a simple markov chain at 0 with uniqueness our specific result theorem 1 1 is used for one dimensional systems it includes a well known result on uniqueness in ruelle 1968 and succeeds in section 6 to find a natural improvement of other results that needed refined estimates in our method the key role is played by a certain inequality comparison of processes is quite well known already in the literature duality see liggett 1985 is a nice technique based on an equality we hope that our comparison method using absorption probabilities is useful in more problems let us now describe our problem setting let s be a countable set and x a finite or countable set for a c s and a e xs write ga gj j ea we define gibbs measures on xs in terms of a given energy difference function ah a a for the energy difference between a and a in xs this function has to be properly defined only in case aa a o for the complement a of any finite set a assume that for any finite a c s this energy difference has the form

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

on rate of convergence of statistical estimation of stationary ergodic processes stationary ergodic processes with finite alphabets are approximated by finite memory processes based on an n length realization of the process under the assumptions of summable continuity rate and non nullness a rate of convergence in d distance is obtained with explicit constants asymptotically as n the result is near the optimum

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

infinite systems of interacting chains with memory of variable length a stochastic model for biological neural nets we consider a new class of non markovian processes with a countable number of interacting components at each time unit each component can take two values indicating if it has a spike or not at this precise moment the system evolves as follows for each component the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component this class of systems extends in a non trivial way both the interacting particle systems which are markovian spitzer in adv math 5 246 290 1970 and the stochastic chains with memory of variable length which have finite state space rissanen in ieee trans inf theory 29 5 656 664 1983 these features make it suitable to describe the time evolution of biological neural systems we construct a stationary version of the process by using a probabilistic tool which is a kalikow type decomposition either in random environment or in space time this construction implies uniqueness of the stationary process finally we consider the case where the interactions between components are given by a critical directed erdos renyi type random graph with a large but finite number of components in this framework we obtain an explicit upper bound for the correlation between successive inter spike intervals which is compatible with previous empirical findings

sharp error terms and neccessary conditions for exponential hitting times in mixing processes we prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in mixing processes with exponential decay mixing processes with a summable function and for general mixing processes with a finite alphabet in the first case the bound is uniform as a function of the measure of the event in the last two cases the bound depends also on the time scale t this allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time a uniform bound is a consequence we present an example that shows that this bound is sharp we also prove that second moments are not necessary for having the exponential law moreover we prove a necessary condition for having the exponential limit law

dna viewed as an out of equilibrium structure the complexity of the primary structure of human dna is explored using methods from nonequilibrium statistical mechanics dynamical systems theory and information theory a collection of statistical analyses is performed on the dna data and the results are compared with sequences derived from different stochastic processes the use of 2 tests shows that dna can not be described as a low order markov chain of order up to r 6 although detailed balance seems to hold at the level of a binary alphabet it fails when all four base pairs are considered suggesting spatial asymmetry and irreversibility furthermore the block entropy does not increase linearly with the block size reflecting the long range nature of the correlations in the human genomic sequences to probe locally the spatial structure of the chain we study the exit distances from a specific symbol the distribution of recurrence distances and the hurst exponent all of which show power law tails and long range characteristics these results suggest that human dna can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the monte carlo rejection sampling method we construct a model dna sequence this method allows us to keep both long and short range statistical characteristics of the native dna data the model sequence presents the same characteristic exponents as the natural dna but fails to capture spatial correlations and point to point details 2014 american physical society

positivity of entropy production we discuss the positivity of the mean entropy production for stochastic systems driven from equilibrium as it was defined in refs 7 and 8 non zero entropy production is closely linked with violation of the detailed balance condition this connection is rigorously obtained for spinflip dynamics we remark that the positivity of entropy production depends on the choice of time reversal transformation hence on the choice of the dynamical variables in the system of interest

chains with complete connections general theory uniqueness loss of memory and mixing properties we introduce a statistical mechanical formalism for the study of discrete time stochastic processes with which we prove i general properties of extremal chains including triviality on the tail algebra short range correlations realization via infinite volume limits and ergodicity ii two new sufficient conditions for the uniqueness of the consistent chain the first one is a transcription of a criterion due to georgii for one dimensional gibbs measures and the second one corresponds to the dobrushin s criterion in statistical mechanics iii results on loss of memory and mixing properties for chains in the dobrushin regime these results are complementary to those existing in the literature and generalize the markovian results based on the dobrushin ergodic coefficient

relative entropy dimensions and large deviations for g measures we prove large deviation results for empirical measures with respect to an arbitrary g measure on a subshift of finite type by using an ergodic theorem for relative entropy we establish extending some results of cajar on billingsley dimensions to g measures we relate the relative entropy to the dimensions of saturated sets this allows a dimensional interpretation of the large deviation results and the study of the set of non generic points which are proved to be of full dimension

stable representatives for symmetric automorphisms of groups and the general form of the scott conjecture let g be a group acting on a tree x such that all edge stabilizers are finite we extend bestvina handel s theory of train tracks for automorphisms of free groups to automorphisms of g which permute vertex stabilizers using this extension we show that there is an upper bound depending only on g for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of g

countable state shifts and uniqueness of g measures in this paper we present a new approach to studying g measures which is based upon local absolute continuity we extend an earlier result that square summability of variations of g ensures uniqueness of g measures the first extension is to the case of countably many symbols the second extension is to some cases where g 0 relaxing the earlier requirement that inf g 0

finiteness results for subgroups of finite extensions abstract we discuss in the context of finite extensions two classical theorems of takahasi and howson on subgroups of free groups we provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free groups virtually nilpotent groups or fundamental groups of finite graphs of groups with virtually polycyclic vertex groups and finite edge groups as an application of our generalization of takahasi s theorem we provide a uniform bound for the rank of the periodic subgroup of any endomorphism of the fundamental group of a given finite graph of groups with finitely generated virtually nilpotent vertex groups and finite edge groups

optimal concentration inequalities for dynamical systems for dynamical systems modeled by a young tower with exponential tails we prove an exponential concentration inequality for all separately lipschitz observables of n variables when tails are polynomial we prove polynomial concentration inequalities those inequalities are optimal we give some applications of such inequalities to specific systems and specific observables

toeplitz minimal flows which are not uniquely ergodic we study minimal symbolic dynamical systems which are orbit closures of toeplitz sequences we construct 0 1 subshifts of this type for which the set of ergodic invariant measures has any given finite cardinality is countably infinite or has cardinality of the continuum

chains with infinite connections uniqueness and markov representation if for a process xi n n infty infty the conditional distribution of n given the past does not depend on n for e g n 0 then the process may be called a chain with infinite connections under a well known continuity condition on this conditional distribution the process is shown to be distributed as an instantaneous function of a countable state markov chain also under a certain weaker continuity condition uniqueness of the distributions of the stationary chains is obtained

one dimensional infinite memory imitation models with noise in this paper we study stochastic process indexed by mathbb z constructed from certain transition kernels depending on the whole past these kernels prescribe that at any time the current state is selected by looking only at a previous random instant we characterize uniqueness in terms of simple concepts concerning families of stochastic matrices generalizing the results previously obtained in de santis and piccioni j stat phys 150 6 1017 1029 2013

regeneration for chains with infinite memory a regeneration structure is established for chains with infinite memory the memory is required to decay only along a single recurrent path when there are many recurrent paths e g under conservativity the construction yields a decomposition into regenerative recurrent classes

categorization in the symmetrically dilute hopfield network a symmetrically dilute hopfield model with a hebbian learning rule is used to study the effects of gradual dilution and of synaptic noise on the categorization ability of an attractor neural network with hierarchically correlated patterns in a two level structure of ancestors and descendants categorization consists in recognizing the ancestors when the network has been trained exclusively with the descendants we consider a macroscopic number of ancestors each with a finite number of descendants and take into account the stochastic noise produced by the former in an equilibrium study of the network by means of replica symmetric mean field theory phase diagrams are obtained that exhibit a categorization a spin glass and a paramagnetic phase as well as the dependence of the order parameters on the relevant quantities the de almeida thouless lines that limit the validity of the replica symmetric results are also obtained it is shown that gradual dilution increases considerably the region where a stable categorization phase may be found

dimensions and waiting times for gibbs measures for shifts of finite type we relate the waiting time between two different orbits one chosen according to an ergodic measure the other according to a gibbs measure to billingsley dimensions of generic sets this is achieved by computing billingsley dimensions of saturated sets in terms of a relative entropy which satisfies a pointwise ergodic result as a by product a similar result is obtained for match lengths that are dual quantities of waiting times

on the definition of entropy production via examples we present a definition of entropy production rate for classes of deterministic and stochastic dynamics the point of departure is a gibbsian representation of the steady state path space measure for which the density is determined with respect to the time reversed process the gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems our definition is motivated by recent work on the gallavotti cohen local fluctuation theorem and it is illustrated via a number of examples