non uniform specification and large deviations for weak gibbs measures we establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak gibbs measures under some mild assumptions we obtain upper and lower bounds for the measure of deviation sets of some non uniformly expanding maps including quadratic maps and robust multidimensional non uniformly expanding local diffeomorphisms for that purpose a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional non uniformly expanding local diffeomorphisms and viana maps

analyticity of smooth eigenfunctions and spectral analysis of the gauss map we provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form uf x a x y f b x y dy acting on functions f u user1 v to mathbb c evolution operators of one dimensional dynamical systems and markov processes have this form we estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the frobenius perron operator of the continuous fraction gauss map we prove that any infinitely differentiable eigenfunction f of this frobenius perron operator corresponding to a non zero eigenvalue admits a unique analytic extension to the set mathbb c backslash infty 1 analyzing the spectrum of the frobenius perron operator in spaces of smooth functions we extend significantly the domain of validity of the mayer and ropstorff asymptotic formula for the decay of correlations of the gauss map

invariant measures exist under a summability condition for unimodal maps for unimodal maps with negative schwarzian derivative a sufficient condition for the existence of an invariant measure absolutely continuous with respect to lebesgue measure is given namely the derivatives of the iterations of the map in the unique critical value must be so large that the sum of some root of the inverses is finite

large deviation principle for benedicks carleson quadratic maps since the pioneering works of jakobson and benedicks carleson and others it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to lebesgue these measures allow one to statistically predict the asymptotic fate of lebesgue almost every initial condition estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space we use the sub exponential slow recurrence condition of benedicks carleson to build induced markov maps of arbitrarily small scale and associated towers to which the absolutely continuous measures can be lifted these various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle for the absolutely continuous measures this result encompasses dynamics far from equilibrium and thus significantly extends presently known local large deviations results for quadratic maps

on spurious detection of linear response and misuse of the fluctuation dissipation theorem in finite time series abstract using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems a goodness of fit statistics is developed for a linear statistical model of the observations based on results for central limit theorems for deterministic dynamical systems and used to detect linear response breakdown we apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series the magnitude of the perturbation and on the choice of the observable we find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed even for simple systems such as the logistic map one needs of the order of 10 6 observations to reliably detect the breakdown with a confidence level of 95 if less observations are available one may be falsely led to conclude that linear response theory is valid the amount of data required is larger the smaller the applied perturbation for judiciously chosen observables the necessary amount of data can be drastically reduced but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems furthermore we explore the use of the fluctuation dissipation theorem fdt in cases with limited data length or coarse graining of observations the fdt if applied naively to a system without linear response is shown to be very sensitive to the details of the sampling method resulting in erroneous predictions of the response

normal forms for spiking neural p systems the spiking neural p systems are a class of computing devices recently introduced as a bridge between spiking neural nets and membrane computing in this paper we prove a series of normal forms for spiking neural p systems concerning the regular expressions used in the firing rules the delay between firing and spiking the forgetting rules used and the outdegree of the graph of synapses in all cases surprising simplifications are found without losing the computational completeness sometimes at the price of slightly increasing other parameters which describe the complexity of these systems

the extremal index hitting time statistics and periodicity abstract the extremal index appears as a parameter in extreme value laws for stochastic processes characterising the clustering of extreme events we apply this idea in a dynamical systems context to analyse the possible extreme value laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures we derive new easily checkable conditions which identify extreme value laws with particular extremal indices in the dynamical context we prove that the extremal index is associated with periodic behaviour the analogy of these laws in the context of hitting time statistics as studied in the authors previous works on this topic is explained and exploited extensively allowing us to prove for the first time the existence of hitting time statistics for balls around periodic points moreover for very well behaved systems uniformly expanding we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point this theory then also applies directly to general stochastic processes adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests

non uniform hyperbolicity and universal bounds for s unimodal maps an s unimodal map f is said to satisfy the collet eckmann condition if the lower lyapunov exponent at the critical value is positive if the infimum of the lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure we prove that an s unimodal map satisfies the collet eckmann condition if and only if it has a uniform hyperbolic structure the equivalence of several non uniform hyperbolicity conditions follows one consequence is that some renormalization of an s unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the collet eckmann condition is satisfied the proof uses new universal bounds that hold for any s unimodal map without periodic attractors

harmonic gibbs and hausdorff measures on repellers for holomorphic maps i we prove that for a simply connected domain q c c whose boundary a q is self similar there is the following dichotomy concerning the harmonic measure x on a q viewed from 0 either a q is piecewise real analytic or else o is singular with respect to the hausdorff measure a notation o i a using makarov s function 4dc t t exp c log 1 t log log log 1 t for some c c x 0 and x is absolutely continuous with respect to a notation x c so if q has fractal boundary then the boundary compression and the radial growth of j logir l j for a riemann mapping r d are as strong respectively as fast as permitted by makarov s theory we prove that c x 2a 2 x for some asymptotic variance a 2 for a sequence of weakly dependent random variables and a lyapunov characteristic exponent x this includes the case where a q is a mixing repeller in ruelle s sense for a holomorphic map f defined on its neighbourhood the case a q is a quasi circle invariant under the action of a quasi fuchsian group for a pair of isomorphic compact surface fuchsian groups and the cases of the boundary of the snowflake and more generally of carleson s fractal jordan curves this dichotomy is partially deduced from the dichotomy concerning gibbs measures for holder continuous functions on an arbitrary mixing repeller x c c for a holomorphic map either i i ak where k is the hausdorff dimension of y

on the universality of axon p systems axon p systems are computing models with a linear structure in the sense that all nodes i e computing units are arranged one by one along the axon such models have a good biological motivation an axon in a nervous system is a complex information processor of impulse signals because the structure of axon p systems is linear the computational power of such systems has been proved to be greatly restricted in particular axon p systems are not universal as language generators it remains open whether axon p systems are universal as number generators in this paper we prove that axon p systems are universal as both number generators and function computing devices and investigate the number of nodes needed to construct a universal axon p system it is proved that four nodes respectively nine nodes are enough for axon p systems to achieve universality as number generators respectively function computing devices these results illustrate that the simple linear structure is enough for axon p systems to achieve a desired computational power

distribution of the periodic points of the farey map we expand the cross section of the geodesic flow in the tangent bundle of the modular surface given by series to produce another section whose return map under the geodesic flow is a double cover of the natural extension of the farey map we use this cross section to extend the correspondence between the closed geodesics on the modular surface and the periodic points of the gauss map to include the periodic points of the farey map then analogous to the work of pollicott we prove an equidistribution result for the periodic points of the farey map when they are ordered according to the length of their corresponding closed geodesics

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

spiking neural p systems with polarizations spiking neural p sn p systems are a class of parallel computation models inspired by neurons where the firing condition of a neuron is described by a regular expression associated with spiking rules however it is np complete to decide whether the number of spikes is in the length set of the language associated with the regular expression in this paper in order to avoid using regular expressions two major and rather natural modifications in their form and functioning are proposed the spiking rules no longer check the number of spikes in a neuron but in exchange a polarization is associated with neurons and rules one of the three electrical charges 0 surprisingly enough the computing devices obtained are still computationally complete which are able to compute all turing computable sets of natural numbers on this basis the number of neurons in a universal sn p system with polarizations is estimated several research directions are mentioned at the end of this paper

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

deterministic solutions to qsat and q3sat by spiking neural p systems with pre computed resources in this paper we continue previous studies on the computational efficiency of spiking neural p systems under the assumption that some pre computed resources of exponential size are given in advance specifically we give a deterministic solution for each of two well known pspace complete problems qsat and q3sat in the case of qsat the answer to any instance of the problem is computed in a time which is linear with respect to both the number n of boolean variables and the number m of clauses that compose the instance as for q3sat the answer is computed in a time which is at most cubic in the number n of boolean variables

small universal turing machines abstract let utm m n be the class of universal turing machine with m states and n symbols universal turing machines are proved to exist in the following classes utm 24 2 utm 10 3 utm 7 4 utm 5 5 utm 4 6 utm 3 10 and utm 2 18

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

whitney holder continuity of the srb measure for transversal families of smooth unimodal maps we consider c 2 families t mapsto f t of c 4 nondegenerate unimodal maps we study the absolutely continuous invariant probability srb measure mu t of f t as a function of t on the set of collet eckmann ce parameters upper bounds assuming existence of a transversal ce parameter we find a positive measure set of ce parameters delta and for each t 0 in delta a set delta 0 subset delta of polynomially recurrent parameters containing t 0 as a lebesgue density point and constants c ge 1 gamma 4 so that for every 1 2 holder function a begin aligned big int a d mu t int a d mu t 0 big le c vert a vert c 1 2 t t 0 1 2 log t t 0 gamma forall t in delta 0 end aligned in addition for all t in delta 0 the renormalisation period p t of f t satisfies p t le p t 0 and there are uniform bounds on the rates of mixing of f t p t for all t with p t p t 0 if f t x tx 1 x the set delta contains almost all ce parameters lower bounds assuming existence of a transversal mixing misiurewicz thurston parameter t 0 we find a set of ce parameters delta mt accumulating at t 0 a constant c ge 1 and a c infty function a 0 so that begin aligned c t t 0 1 2 ge big int a 0 d mu t int a 0 d mu t 0 big ge c 1 t t 0 1 2 forall t in delta mt end aligned

hitting time statistics and extreme value theory we consider discrete time dynamical systems and show the link between hitting time statistics the distribution of the first time points land in asymptotically small sets and extreme value theory distribution properties of the partial maximum of stochastic processes this relation allows to study hitting time statistics with tools from extreme value theory and vice versa we apply these results to non uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing for example we also give applications of our theory to higher dimensional examples for which we also obtain classical extreme value laws and exponential hitting time statistics for balls we extend these ideas to the subsequent returns to asymptotically small sets linking the poisson statistics of both processes

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

large deviations for poisson systems of independent random walks we prove large deviation theorems for occupation time functionals of independent random walks started from a poisson field on zd in dimensions 1 and 2 the large deviation tails are larger than exponential exact asymptotics are derived

piecewise invertible dynamical systems the aim of the paper is the investigation of piecewise monotonic maps t of an interval x the main tool is an isomorphism of x t with a topological markov chain with countable state space which is described by a 0 1 transition matrix m the behavior of the orbits of points in x under t is very similar to the behavior of the paths of the markov chain every irreducible submatrix of m gives rise to a t invariant subset l of x such that l is the set x of all limit points of the orbit of an x x the topological entropy of l is the logarithm of the spectral radius of the irreducible submatrix which is a l 1 operator besides these sets l there are two t invariant sets y and p such that for all x x the set x is either contained in one of the sets l or in y or in p the set p is a union of periodic orbits and y is contained in a finite union of sets y with y x and has topological entropy zero this isomorphism of x t with a topological markov chain is also an important tool for the investigation of t invariant measures on x results in this direction which are published elsewhere are described at the end of the paper furthermore a part of the proofs in the paper is purely topological without using the order relation of the interval x so that some results hold for more general dynamical systems x t

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

a spectral approach for quenched limit theorems for random hyperbolic dynamical systems we extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving to quenched random piecewise hyperbolic dynamics for general ergodic sequences of maps in a neighborhood of a hyperbolic map we prove a quenched large deviations principle ldp central limit theorem clt and local central limit theorem lclt

spiking neural p systems with rules on synapses spiking neural p systems sn p systems for short are a class of membrane systems inspired from the way the neurons process information and communicate by means of spikes in this paper we introduce and investigate a new class of sn p systems with spiking rules placed on synapses the computational completeness is first proved then two small universal sn p systems with rules on synapses for computing functions are constructed specifically when using standard spiking rules we obtain a universal system with 39 neurons while when using extended spiking rules on synapses a universal sn p system with 30 neurons is constructed

large deviation principle in one dimensional dynamics we study the dynamics of smooth interval maps with non flat critical points for every such a map that is topologically exact we establish the full level 2 large deviation principle for empirical means in particular the large deviation principle holds for every non renormalizable quadratic map this includes the maps without physical measure found by hofbauer and keller and challenges the widely shared view of the large deviation principle as a refinement of laws of large numbers

higher string topology on general spaces in this paper i give a generalized analogue of the string topology results of chas and sullivan and of cohen and jones for a finite simplicial complex x and k geq 1 i construct a spectrum maps s k x s x which is obtained by taking a generalization of the spivak bundle on x which however is not a stable sphere bundle unless x is a poincare space pulling back to maps s k x and quotienting out the section at infinity i show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the k 1 dimensional unframed little disk operad mathcal c k 1 i also prove a conjecture of kontsevich which states that the quillen cohomology of a based mathcal c k algebra in the category of chain complexes is equivalent to a shift of its hochschild cohomology as well as prove that the operad c ast mathcal c k is koszul dual to itself up to a shift in the derived category this gives one a natural notion of derived koszul dual c ast mathcal c k algebras i show that the cochain complex of x and the chain complex of omega k x are koszul dual to each other as c ast mathcal c k algebras and that the chain complex of maps s k x s x is naturally equivalent to their equivalent hochschild cohomology in the category of c ast mathcal c k algebras

decay of correlations for certain quadratic maps we prove exponential decay of correlations for f wheref belongs in a positive measure set of quadratic maps of the interval and is its absolutely continuous invariant measure these results generalize to other interval maps

structure and f dependence of the a c i m for a unimodal map f of misiurewicz type by using a suitable banach space on which we let the transfer operator act we make a detailed study of the ergodic theory of a unimodal map f of the interval in the misiurewicz case we show in particular that the absolutely continuous invariant measure can be written as the sum of 1 square root spikes along the critical orbit plus a continuous background we conclude by a discussion of the sense in which the map f mapsto rho may be differentiable

statistical properties of unimodal maps the quadratic family we prove that almost every nonregular real quadratic map is colleteckmann and has polynomial recurrence of the critical orbit proving a conjecture by sinai it follows that typical quadratic maps have excellent ergodic properties as exponential decay of correlations keller and nowicki young and stochastic stability in the strong sense baladi and viana this is an important step in achieving the same results for more general families of unimodal maps

binary tree approach to scaling in unimodal maps ge rusjan and zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one dimensional unimodal maps we consider the scaling behavior in a modified tree which takes into account the self similarity of the window structure a nonuniversal geometric convergence of the associated superstable parameter values towards a misiurewicz point is observed for almost all binary sequences with periodic tails for these sequences the window period grows arithmetically down the binary tree there are an infinite number of exceptional sequences however for which the growth of the window period is faster numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences

low temperature phase transitions in the quadratic family abstract we give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function this implies that several dimension spectra and large deviation rate functions associated to this map are not expected to be real analytic in contrast to the uniformly hyperbolic case the quadratic map we study has a non recurrent critical point so it is non uniformly hyperbolic in a strong sense

limit theorems and markov approximations for chaotic dynamical systems we prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients we also develop techniques of markov approximations for dynamical systems we apply our results to expanding interval maps axiom a diffeomorphisms chaotic billiards and hyperbolic attractors

on small universal antiport p systems it is known that p systems with antiport rules simulate register machines i e they are computationally complete hence due to the existence of universal register machines there exist computationally complete subclasses of antiport p systems with bounded size i e systems where each size parameter is limited by some constant however so far there has been no estimation of these numbers given in the literature in this article three universal antiport p systems of bounded size are demonstrated different from each other in their size parameters we present universal antiport p systems with 73 43 and 30 rules where the maximum of the weight of the rules is 4 5 and 6 respectively

derived representation theory of lie algebras and stable homotopy categorification of slk abstract we set up foundations of representation theory over s the sphere spectrum which is the initial ring of stable homotopy theory in particular we treat s lie algebras and their representations characters g l n s verma modules and their duals harish chandra pairs and zuckermann functors as an application we construct a khovanov s l k stable homotopy type with a large prime hypothesis which is a new link invariant using a stable homotopy analogue of the method of j sussan

small universal register machines abstract several small universal register machines are constructed the number of instructions vary from 32 to 14 depending on the chosen instruction base and the chosen notion of universality the proof uses a special coding function for finite sequences of positive integers and some strong classical results concerning distribution of primes

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

sharp estimates in ruelle theorems for matrix transfer operators a matrix coefficient transfer operator on the space of sections of an m dimensional vector bundle over n dimensional compact manifold is considered the spectral radius of is estimated by para and the essential spectral radius by here is the set of ergodic f invariant measures and for is the measure theoretic entropy of f is the largest lyapunov exponent of the cocycle over f generated by and is the smallest lyapunov exponent of the differential of f

minimization strategies for maximally parallel multiset rewriting systems maximally parallel multiset rewriting systems mpmrs give a convenient way to express relations between unstructured objects the functioning of various computational devices may be expressed in terms of mpmrs e g register machines and many variants of p systems in particular this means that mpmrs are turing universal however a direct translation leads to quite a large number of rules like for other classes of computationally complete devices there is a challenge to find a universal system having the smallest number of rules in this article we present different rule minimization strategies for mpmrs based on encodings and structural transformations we apply these strategies to the translation of a small universal register machine korec 1996 9 and we show that there exists a universal mpmrs with 23 rules since mpmrs are identical to a restricted variant of p systems with antiport rules the results we obtained improve previously known results on the number of rules for those systems

hyperbolicity properties of multi modal collet eckmann maps without schwarzian derivative assumptions in this paper we study the dynamical properties of general c2 maps f 0 1 0 1 with quadratic critical points and not necessarily unimodal we will show that if such maps satisfy the well known collet eckmann conditions then one has a hyperbolicity on the set of periodic points b nonexistence of wandering intervals c sensitivity on initial conditions and d exponential decay of branches intervals of monotonicity of fn as n oo for these results we will not make any assumptions on the schwarzian derivative f we will also give an estimate of the return time of points that start near critical points

three small universal spiking neural p systems in this work we give three small spiking neural p systems we begin by constructing a universal spiking neural p system with extended rules and only 4 neurons this is the smallest possible number of neurons for a universal system of its kind we prove this by showing that the set of problems solved by spiking neural p systems with 3 neurons is bounded above by nlnl and so there exists no such universal system with 3 neurons if we generalise the output technique we immediately find a universal spiking neural p system with extended rules that has only 3 neurons this is also the smallest possible number of neurons for a universal system of its kind finally we give a universal spiking neural p system with standard rules and only 7 neurons in addition to giving a significant improvement in terms of reducing the number of neurons our systems also offer an exponential improvement on the time and space overheads of the small universal spiking neural p systems of other authors

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

design of logic gates using spiking neural p systems with homogeneous neurons and astrocytes like control in biological nervous systems the operation of interacting neurons depends largely on the regulation from astrocytes inspired by this biological phenomenon spiking neural p systems i e sn p systems with astrocyte like control were proposed and were proven to have turing completeness as computing models in this work the application of such systems for creating logical operators is investigated specifically it is obtained in a constructive way that sn p systems with astrocyte like control can synthesize the operations of boolean logic gates i e and or not nor xor and nand gates the resulting systems are simple and homogeneous which means only one type of neuron with a unique spiking rule is used with these neural like logic gates more complex boolean circuits with cascade connections can be constructed as such they can be used to implement finite computing devices such as the finite transducers these results demonstrate a novel method of constructing logic circuits that work in a neural like manner as well as shed some lights on potential directions of designing neural circuits theoretically

a bernstein type inequality for some mixing processes and dynamical systems with an application to learning we establish a bernstein type inequality for a class of stochastic processes that include the classical geometrically mixing processes rio s generalization of these processes as well as many time discrete dynamical systems modulo a logarithmic factor and some constants our bernsteintype inequality coincides with the classical bernstein inequality for i i d data we further use this new bernstein type inequality to derive an oracle inequality for generic regularized empirical risk minimization algorithms and data generated by such processes applying this oracle inequality to support vector machines using the gaussian kernels for both least squares and quantile regression it turns out that the resulting learning rates match up to some arbitrarily small extra term in the exponent the optimal rates for i i d processes

spiking neural p systems an improved normal form spiking neural p systems in short sn p systems are computing devices based on the way the neurons communicate through electrical impulses spikes these systems involve various ingredients among them we mention forgetting rules and the delay in firing rules however it is known that the universality can be obtained without using these two features in this paper we improve this result in two respects i each neuron contains at most two rules which is optimal for systems used in the generative mode and ii the rules in the neurons using two rules have the same regular expression which controls their firing this result answers a problem left open in the literature and in this context an incompleteness in some previous proofs related to the elimination of forgetting rules is removed moreover this result shows a somewhat surprising uniformity of the neurons in the sn p systems able to simulate turing machines which is both of a theoretical interest and it seems to correspond to a biological reality when a bound is imposed on the number of spikes present in a neuron at any step of a computation such sn p systems are called finite two surprising results are obtained first a characterization of finite sets of numbers is obtained in the generative case this contrasts the case of other classes of sn p systems where characterizations of semilinear sets of numbers are obtained for finite sn p systems second the accepting case is strictly more powerful than the generative one all finite sets and also certain arithmetical progressions can be accepted a precise characterization of the power of accepting finite sn p systems without forgetting rules and delay remains to be found