on a random walk with memory and its relation with markovian processes we study a one dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker the feedback is quite efficient and leads to a non diffusive walk the time evolution of the displacement is given by an equivalent markovian dynamical process the probability density for the position of the walker is the same at any time as for a random walk with shrinking steps although the two time correlation functions are quite different

binary n step markov chains and long range correlated systems a theory of systems with long range correlations based on the consideration of binary n step markov chains is developed in our model the conditional probability that the i th symbol in the chain equals zero or unity is a linear function of the number of unities among the preceding n symbols the correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length l are obtained analytically and numerically if the persistent correlations are not extremely strong the variance is shown to be nonlinearly dependent on l a self similarity of the studied stochastic process is revealed the applicability of the developed theory to the coarse grained written and dna texts is discussed

noncommutative standard table algebras with at most one nontrivial multiplicity abstract standard table algebras that have an irreducible character of degree two and with all other irreducible characters of multiplicity one are classified to exact isomorphism

discrete stochastic evolution rules and continuum evolution equations abstract the continuum evolution equations are derived from updating rules for three classes of stochastic models the first class corresponds to models whose stochastic continuum equations are of the langevin type obtained after carrying out a local average known as coarse graining the second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations this average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables in which case a hierarchy of ordinary differential equations is obtained the third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice this double average generates a hierarchy of deterministic ordinary differential equations the second and third classes of equations are truncated using a mean field m n closure approximation in order to obtain a finite set of equations illustrative examples of every class are given

on hypergeometric functions and function spaces the aim of this paper is to discuss the role of hypergeometric functions in function spaces and to prove some new results for these functions the first part of this paper proves results such as monotone convexity and concavity properties of sums of products of hypergeometric functions the second part of our results deals with the space a of all normalized analytic functions f f 0 0 f 0 1 in the unit disk and the subspace r f a a n r such that re ei f z 0 z for f a we consider integral transforms of the type v f 01 t f tz t dt where t is a real valued nonnegative weight function normalized so that 01 t 1 we obtain conditions on and the function such that v f takes each member of r into a starlike function of order 0 1 2 these results extend and improve the earlier known results in these directions we end the paper with an open problem

limit theorems for the laziest minimal random walk model of elephant type we consider a minimal model of one dimensional discrete time random walk with step reinforcement introduced by harbola kumar and lindenberg 2014 the walker can move forward never backward or remain at rest for each n 1 2 ldots a random time u n between 1 and n is chosen uniformly and if the walker moved forward resp remained at rest at time u n then at time n 1 it can move forward with probability p resp q or with probability 1 p resp 1 q it remains at its present position for the case q 0 several limit theorems are obtained by coletti gava and de lima 2019 in this paper we prove limit theorems for the case q 0 where the walker can exhibit all three forms of asymptotic behavior as p is varied as a byproduct we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees

on structures of quasi thin table algebras and applications to association schemes abstract in this paper we develop a structure theory for a class of table algebras i e the so called quasi thin table algebras they are direct generalizations of quasi thin association schemes although this class of table algebras is rather close to finite groups its structural properties are quite different from those of finite groups the thin radical and thin residue give rise to two group structures for every table algebra they play the central role in our discussions applications to association schemes are also discussed and some new results about quasi thin association schemes are obtained in particular we will give necessary and sufficient conditions under which a commutative quasi thin association scheme has multiplicities 1 or 2 we will need these results in the study of the characterization and classification of double quasi thin association schemes in 22

integral transforms of certain subclasses of analytic functions abstract let a be the class of normalized analytic functions in the unit disk and define the class p f a r re e i 1 f z z f z 0 z for a function f a we define the integral transform v f z 1 0 t f tz t dt where is a real valued nonnegative weight function with 10 t dt 1 for f p we will find conditions such that v f is starlike and also such that v f p we also discuss how to write the integral transform as a convolution with a 2f1 hypergeometric function

elephant random walks and their connection to polya type urns in this paper we explain the connection between the elephant random walk erw and an urn model a la polya and derive functional limit theorems for the former the erw model was introduced in phys rev e 70 045101 2004 10 1103 physreve 70 045101 to study memory effects in a highly non markovian setting more specifically the erw is a one dimensional discrete time random walk with a complete memory of its past the influence of the memory is measured in terms of a memory parameter p between zero and one in the past years a considerable effort has been undertaken to understand the large scale behavior of the erw depending on the choice of p here we use known results on urns to explicitly solve the erw in all memory regimes the method works as well for erws in higher dimensions and is widely applicable to related models

duality techniques for certain integral transforms to be starlike abstract for p denote the class of all normalized analytic functions f such that re e i 1 f z z f z 0 z for some r let s 0 s f re zf z f z z in this paper we find conditions so that the integral transform v f z 0 1 t f tz t dt carries p into s we provide a number of applications for certain special choices of t these results extend previously known results by a number of authors

the diffusion of opposite opinions in a randomly biased environment we propose a model for diffusion of two opposite opinions here the decision to be taken by each individual is a random variable which depends on the tendency of the population as well as on its own trend characteristic the influence of the population trend can be positive negative or nonexistent in a random form we prove a phase transition in the behavior of the proportion of each opinion specifically the mean square proportions are linear functions of time in the diffusive case but are given by a power law in the superdiffusive regime

current fluctuations in stochastic systems with long range memory we propose a method to calculate the large deviations of current fluctuations in a class of stochastic particle systems with history dependent rates long range temporal correlations are seen to alter the speed of the large deviation function in analogy with long range spatial correlations in equilibrium systems we give some illuminating examples and discuss the applicability of the gallavotti cohen fluctuation theorem

memory induced anomalous dynamics emergence of diffusion subdiffusion and superdiffusion from a single random walk model we present a random walk model that exhibits asymptotic subdiffusive diffusive or superdiffusive behavior in different parameter regimes this appears to be an instance of a single random walk model leading to all three forms of behavior by simply changing parameter values furthermore the model offers the great advantage of analytical tractability our model is non markovian in that the next jump of the walker is probabilistically determined by the history of past jumps it also has elements of intermittency in that one possibility at each step is that the walker does not move at all this rich encompassing scenario arising from a single model provides useful insights into the source of different types of asymptotic behavior

on the multi dimensional elephant random walk the purpose of this paper is to investigate the asymptotic behavior of the multi dimensional elephant random walk merw it is a non markovian random walk which has a complete memory of its entire history a wide range of literature is available on the one dimensional erw surprisingly no references are available on the merw the goal of this paper is to fill the gap by extending the results on the one dimensional erw to the merw in the diffusive and critical regimes we establish the almost sure convergence the law of iterated logarithm and the quadratic strong law for the merw the asymptotic normality of the merw properly normalized is also provided in the superdiffusive regime we prove the almost sure convergence as well as the mean square convergence of the merw all our analysis relies on asymptotic results for multi dimensional martingales

a martingale approach for the elephant random walk the purpose of this paper is to establish via a martingale approach some refinements on the asymptotic behavior of the one dimensional elephant random walk erw the asymptotic behavior of the erw mainly depends on a memory parameter p which lies between zero and one this behavior is totally different in the diffusive regime the critical regime and the superdiffusive regime in the diffusive and critical regimes we establish some new results on the almost sure asymptotic behavior of the erw such as the quadratic strong law and the law of the iterated logarithm in the superdiffusive regime we provide the first rigorous mathematical proof that the limiting distribution of the erw is not gaussian

n dimensional nonlinear fokker planck equation with time dependent coefficients an n dimensional nonlinear fokker planck equation is investigated here by considering the time dependence of the coefficients where drift controlled and source terms are present we exhibit the exact solution based on the generalized gaussian function related to the tsallis statistics furthermore we show that a rich class of diffusive processes including normal and anomalous ones can be obtained by changing the time dependence of the coefficients

inhomogeneous diffusion and ergodicity breaking induced by global memory effects fil budini adrian adolfo comision nacional de energia atomica gerencia del area de investigaciones y aplicaciones no nucleares gerencia de fisica centro atomico constituyentes argentina consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet patagonia norte argentina

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

nilpotent fusion categories abstract in this paper we extend categorically the notion of a finite nilpotent group to fusion categories to this end we first analyze the trivial component of the universal grading of a fusion category c and then introduce the upper central series of c for fusion categories with commutative grothendieck rings e g braided fusion categories we also introduce the lower central series we study arithmetic and structural properties of nilpotent fusion categories and apply our theory to modular categories and to semisimple hopf algebras in particular we show that in the modular case the two central series are centralizers of each other in the sense of m muger

exact moments in a continuous time random walk with complete memory of its history we present a continuous time generalization of a random walk with complete memory of its history phys rev e 70 045101 r 2004 and derive exact expressions for the first four moments of the distribution of displacement when the number of steps is poisson distributed we analyze the asymptotic behavior of the normalized third and fourth cumulants and identify new transitions in a parameter regime where the random walk exhibits superdiffusion these transitions which are also present in the discrete time case arise from the memory of the process and are not reproduced by fokker planck approximations to the evolution equation of this random walk

phase transition in random walks with long range correlations motivated by recent results in the theory of correlated sequences we analyze the dynamics of random walks with long term memory binary chains with long range correlations in our model the probability for a unit bit in a binary string depends on the fraction of unities preceding it we show that the system undergoes a dynamical phase transition from normal diffusion in which the variance d l scales as the string s length l into a superdiffusion phase d l approximately lalpha alpha 1 when the correlation strength exceeds a critical value we demonstrate the generality of our results with respect to alternative models and discuss their applicability to various data such as coarse grained dna sequences written texts and financial data

generating functions via integral transforms abstract in this paper we use some integral transforms to derive for a polynomial sequence p n x n 0 generating functions of the type g x t n 0 n p n x t n starting from a generating function of type g x t n 0 p n x t n where n n 0 is a real numbers sequence independent on x and t that allows us to unify the treatment of a generating function problem for many well known polynomial sequences in the literature

memory controlled diffusion memory effects require for their incorporation into random walk models an extension of the conventional equations the linear fokker planck equation for the probability density p stackrel p vec r t is generalized by including nonlinear and nonlocal spatial temporal memory effects the realization of the memory kernel is restricted due the conservation of the basic quantity p a general criteria is given for the existence of stationary solutions in case the memory kernel depends on p polynomially transport may be prevented owing to the delay effects a finite amount of particles remains localized and the further transport is terminated for diffusion with nonlinear memory effects we find an exact solution in the long time limit although the mean square displacement exhibits diffusive behavior higher order cumulants offer differences to diffusion and they depend on the memory strength

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

discrete time non markovian random walks the effect of memory limitations on scaling abstract an important class of random walks includes those in which the random increment at time step t depends on the complete history of the process we consider a recently proposed discrete time non markovian random walk process characterized by a memory parameter p we numerically calculate the first and second moments of the position distribution and relate our results to known analytic results for special cases we obtain data collapse for the position distribution we study the effect of reducing the memory by considering a modified model in which only a fraction f of the previous steps are remembered surprisingly the behavior becomes markovian for small f even though the correlation time diverges for f 0 we also study the transient effects near the memory edge by considering a markovian limit of the original model we observe a crossover to markovian behavior for times much larger than the range of the memory

on a comparative study between dependence scales determined by linear and non linear measures in this manuscript we present a comparative study about the determination of the relaxation i e independence time scales obtained from the correlation function the mutual information and a criterion based on the evaluation of a nonextensive generalisation of mutual entropy our results show that for systems with a small degree of complexity standard mutual information and the criterion based on its nonextensive generalisation provide the same scale whereas for systems with a higher complex dynamics the standard mutual information presents a time scale consistently smaller

central limit theorem and related results for the elephant random walk we study the so called elephant random walk erw which is a non markovian discrete time random walk on with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior we prove a law of large numbers and a central limit theorem remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive inside the superdiffusive regime the erw converges to a non degenerate random variable which is not normal we also obtain explicit expressions for the correlations of increments of the erw

quotient structures homomorphisms and partial wreath products in reality based algebras abstract homomorphisms and quotient structures of reality based algebras are presented and described related central idempotents and quotient sets are characterized and the extent to which a homomorphism preserves the given anti automorphisms of the algebras is determined the notion of a partial wreath product an algebraic abstraction of the wedge product of association schemes is developed prototypical results previously obtained for various special cases including table algebras and commutative c algebras are thus uniformly generalized

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

memory effects in recurrent and extreme events a dynamical approach to recurrent and extreme events is developed focusing on the role of correlations and memory in the structure of the probability distributions and their low order moments the procedure is illustrated on homogeneous first and second order markov chains non markovian and nonhomogeneous processes and deterministic dynamical systems substantial differences with classical statistical theory as applied to independent identically distributed random variables are identified

diffusion and memory effects for stochastic processes and fractional langevin equations we consider the diffusion processes defined by stochastic differential equations when the noise is correlated a functional method based on the dyson expansion for the evolution operator associated to the stochastic continuity equation is proposed to obtain the fokker planck equation after averaging over the stochastic process in the white noise limit the standard result corresponding to the stratonovich interpretation of the non linear langevin equation is recovered when the noise is correlated the averaged operator series cannot be summed unless a family of time dependent operators commutes in the case of a linear equation the constraints are easily worked out the process defined by a linear langevin equation with additive noise is gaussian and the probability density function of its fluctuating component satisfies a fokker planck equation with a time dependent diffusion coefficient the same result holds for a linear langevin equation with a fractional time derivative defined according to caputo 1 in the generic linear or nonlinear case approximate equations for small noise amplitude are obtained for small correlation time the evolution equations further simplify in agreement with some previous alternative derivations the results are illustrated by the linear oscillator with coloured noise and the fractional wiener process where the numerical simulation for the probability density and its moments is compared with the analytical solution

elephants can always remember exact long range memory effects in a non markovian random walk we consider a discrete time random walk where the random increment at time step t depends on the full history of the process we calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p at a critical value p 1 c 1 2 where memory effects vanish there is a transition from a weakly localized regime where the walker elephant returns to its starting point to an escape regime inside the escape regime there is a second critical value where the random walk becomes superdiffusive the probability distribution is shown to be governed by a non markovian fokker planck equation with hopping rates that depend both on time and on the starting position of the walk on large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time dependent spring constant k 2p 1 t the solution of this problem is a gaussian distribution with time dependent mean and variance which both depend on the initiation of the process

fusion rings with few degrees abstract the main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p a well known theorem of isaacs and passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary numerically compatible nc fusion rings are defined as a potential generalization of the grothendieck rings of fusion categories and the nc fusion rings of order p p 2 p 3 pq and p q 2 where p q are primes are classified in terms of certain standard table algebras as another application of the main theorem

a formalism for studying long range correlations in many alphabets sequences abstract we formulate a mean field like theory of long range correlated l alphabets sequences which are actually systems with l 1 independent parameters depending on the values of these parameters the variance on the average number of any given symbol in the sequence shows a linear or a superlinear dependence on the total length of the sequence we present exact solution to the four alphabets and three alphabets sequences we also demonstrate that a mapping of the given sequence into a smaller alphabets sequence namely a coarse graining process does not necessarily imply that long range correlations found in the latter would correspond to those of the former

a two dimensional non markovian random walk leading to anomalous diffusion abstract exact solutions are rare for non markovian random walk models even in 1d and much more so in 2d here we propose a 2d genuinely non markovian random walk model with a very rich phase diagram such that the motion in each dimension can belong to one of 3 categories i subdiffusive ii superdiffusive or iii normally diffusive the main advance reported here is a different method and the consequent physical insight for analytically solving this model simpler non markovian models such as levy walks have been solved in 2d but it is not clear if the method of solution could be made to work for more complicated models such as the one studied here we also report the exact solutions for the first two moments of the random walk propagator along with the complete phase diagram the latter is surprisingly rich and admits diverse diffusion regimes finally we discuss these results in the context of theoretical underpinnings as well as possible applications

on a class of random walks with reinforced memory this paper deals with different models of random walks with a reinforced memory of preferential attachment type we consider extensions of the elephant random walk introduced by schutz and trimper phys rev e 70 044510 r 2004 with stronger reinforcement mechanisms where roughly speaking a step from the past is remembered proportional to some weight and then repeated with probability p with probability 1 p the random walk performs a step independent of the past the weight of the remembered step is increased by an additive factor b ge 0 making it likelier to repeat the step again in the future a combination of techniques from the theory of urns branching processes and alpha stable processes enables us to discuss the limit behavior of reinforced versions of both the elephant random walk and its alpha stable counterpart the so called shark random swim introduced by businger j stat phys 172 3 701 717 2004 we establish phase transitions separating subcritical from supercritical regimes

some properties of group theoretical categories abstract we first show that every group theoretical category is graded by a certain double coset ring as a consequence we obtain a necessary and sufficient condition for a group theoretical category to be nilpotent we then give an explicit description of the simple objects in a group theoretical category following o2 and of the group of invertible objects of a group theoretical category in group theoretical terms finally under certain restrictive conditions we describe the universal grading group of a group theoretical category

close to convexity properties of gaussian hypergeometric functions abstract let a be the class of normalized analytic functions in the unit disk let z be either zf a b c z or c ab f a b c z 1 where f a b c z denotes the classical hypergeometric function the purpose of this paper is to study close to convexity and hence univalency of z in the unit disc more generally we find conditions on a b c and such that satisfies re e i 1 z z 0 for all z and for some real 1 2 1 2

univalency of weighted integral transforms of certain functions for 0 z for some r for f p we consider the integral transform v f z 01 t f tz t dt where t is a real valued nonnegative weight function so that 01 t dt 1 the main aim of this paper is to find conditions such that v f p1 whenever f p for 1 we also obtain conditions such that v f p1 whenever f p0 for various choices of t as a useful consequence we find conditions for certain hypergeometric functions to be univalent

interacting elephant random walks the elephant random walk is a history dependent random walk we study a class of interacting elephant random walks our model includes the exclusion process as a special case by means of monte carlo simulations and mean field arguments we find that it exhibits a condensation phenomenon with a phase transition that is manifestly of first order the transition point depends on the initial configuration

tensor categories with fusion rules of self duality for finite abelian groups abstract semisimple tensor categories with fusion rules of self duality for finite abelian groups are classified as an application we prove that the tannaka duals of the dihedral and the quaternion groups of order 8 and the eight dimensional hopf algebra of kac and paljutkin are not isomorphic as abstract tensor categories whereas they have the same fusion rule

on extremal problems related to integral transforms of a class of analytic functions abstract for 0 and 1 given let p denote the class of all analytic functions f in the unit disk with the normalization f 0 f 0 1 0 and satisfying the condition re e i f z z f z 0 z d for some r we shall investigate the integral transform v f z 0 1 t f t z t d t where is a nonnegative real valued function normalized by 0 1 t d t 1 from our main results we get conditions on the number and the function such that v f is starlike of order 0 1 2 when f p as applications we study various choices of t related to classical integral transforms

spontaneous symmetry breaking in amnestically induced persistence we investigate a recently proposed non markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence we report numerical and analytical results showing the complete phase diagram consisting of four phases for this system i classical nonpersistence ii classical persistence iii log periodic nonpersistence and iv log periodic persistence driven by negative feedback the first two phases possess continuous scale invariance symmetry however log periodicity breaks this symmetry instead log periodic motion satisfies discrete scale invariance symmetry with complex rather than real fractal dimensions we find for log periodic persistence evidence not only of statistical but also of geometric self similarity