on the equilibrium state of random walkers in random environments analytical results we study equilibrium properties of random walkers in one dimensional random environments of finite length l from an exact expression for the quenched average of the free energy we derive analytical results for all cumulants and all renyi entropies of the equilibrium distribution in contrast to the finite variance of a typical non equilibrium distribution in the unbiased situation we find that in equilibrium the disorder averaged variance diverges with the size of the system as

on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

linear and sub linear growth and the clt for hitting times of a random walk in random environment on a strip the main goal of this work is to study the asymptotic behaviour of hitting times of a random walk rw in a quenched random environment re on a strip we introduce enlarged random environments in which the traditional hitting time can be presented as a sum of independent random variables whose distribution functions form a stationary random sequence this allows us to obtain conditions stated in terms of properties of random environments for a linear growth of hitting times of relevant random walks in some important cases e g independent random environments these conditions are also necessary for this type of behaviour we also prove the quenched central limit theorem clt for hitting times in the general ergodic setting a particular feature of these ballistic laws in random environment is that whenever they hold under standard normalization the convergence is a convergence with a speed the latter is due to certain properties of moments of hitting times which are also studied in this paper the asymptotic properties of the position of the walk are stated but are not proved in this work since this has been done in goldhseid probab theory relat fields 139 1 41 64 2007

generalized random walk in a random environment random walk in a random environment a process where a randomly chosen transition probability is assigned to each point has been exhaustively studied in the one dimensional nearest neighbor case but is much more difficult in any more general case this paper develops crude techniques for handling transience results and zero one laws in a generalized set up

dynamic theory of quasilinear parabolic equations i abstracts evolution equations on etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

lingering random walks in random environment on a strip we consider a recurrent random walk rw in random environment re on a strip we prove that if the re is i i d and its distribution is not supported by an algebraic subsurface in the space of parameters defining the re then the rw exhibits the log t 2 asymptotic behaviour the exceptional algebraic subsurface is described by an explicit system of algebraic equations

computable examples of the maximal lyapunov exponent some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal lyapunov exponent the examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks if the original sequence is the outcome of independent bernoulli trials with success probability p then the maximal lyapunov exponent may be expressed in terms of power series in p with explicit formulae for the coefficients the convexity of the maximal lyapunov exponent as a function of p is discussed as is an application to branching processes in a random environment

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

recurrence and transience of random walks in random environments on a strip we explain the necessary and sufficient conditions for recurrent and transient behavior of a random walk in a stationary ergodic random environment on a strip in terms of properties of a top lyapunov exponent this lyapunov exponent is defined for a product of a stationary sequence of positive matrices in the one dimensional case this approach allows us to treat wider classes of random walks than before

random walks in a random environment on a strip a renormalization group approach we present a real space renormalization group scheme for the problem of random walks in a random environment on a strip which includes the one dimensional random walk in a random environment with bounded non nearest neighbour jumps we show that the model renormalizes to an effective one dimensional random walk with nearest neighbour jumps and conclude that sinai scaling is valid in the recurrent case while in the sub linear transient phase the displacement grows as a power of the time

on product of companion matrices this paper describes an explicit combinatorial formula for the product of companion matrices the method relies on the connections between matrix algebra and associated combinatorial structures to enumerate the paths in an unweighted digraph as an application we obtain bases for the solution space of the linear difference equation with variable coefficients

a variable step size lms algorithm a least mean square lms adaptive filter with a variable step size is introduced the step size increases or decreases as the mean square error increases or decreases allowing the adaptive filter to track changes in the system as well as produce a small steady state error the convergence and steady state behavior of the algorithm are analyzed the results reduce to well known results when specialized to the constant step size case simulation results are presented to support the analysis and to compare the performance of the algorithm with the usual lms algorithm and another variable step size algorithm they show that its performance compares favorably with these existing algorithms

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

on some random walks on z in random medium we consider random walks on z in a stationary random medium defined by an ergodic dynamical system in the case when the possible jumps are l 1 0 1 for some fixed integer l we provide a recurrence criterion expressed in terms of the sign of the maximal liapounov exponent of a certain random matrix and give an algorithm of calculation of that exponent next we characterize the existence of the absolutely continuous invariant measure for the markov chain of the environments viewed from the particle and also characterize in the transient cases the existence of a nonzero drift to study the validity of the central limit theorem we consider the notion of harmonic coordinates introduced by kozlov we characterize the existence of both the invariant measure and the harmonic coordinates and show in the recurrent case that the existence of those two objects is equivalent to the validity of an invariance principle we give sufficient conditions for the validity of the central limit theorem in the transient cases finally we consider the previous results in the context of a random medium defined by an irrational rotation on the circle and study their realization in terms of regularity and diophantine approximation

transient random walks on a strip in a random environment we consider transient random walks on a strip in a random environment the model was introduced by bolthausen and goldsheid comm math phys 214 2000 429 447 we derive a strong law of large numbers for the random walks in a general ergodic setup and obtain an annealed central limit theorem in the case of uniformly mixing environments in addition we prove that the law of the environment viewed from the position of the walker converges to a limiting distribution if the environment is an i i d sequence

global stability and periodic solution of the viral dynamics it is well known that the mathematical models provide very important information for the research of human immunodeficiency virus type 1 and hepatitis c virus hcv however the infection rate of almost all mathematical models is linear the linearity shows the simple interaction between the t cells and the viral particles in this paper we consider the classical mathematical model with saturation response of the infection rate by stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection free steady state we also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution numerical simulations are presented to illustrate the results

a variable step size diffusion lms algorithm for distributed estimation we propose a new variable step size diffusion least mean square algorithm for distributed estimation that adaptively adjusts the step size in every iteration for a network application the proposed method determines a suboptimal step size at each node to minimize the mean square deviation for the intermediate estimate the algorithm thus adapts the different node environments and profiles across the networks and requires relatively less user interaction than existing algorithms in experiments the algorithm achieves both fast convergence speed and low misadjustment by remarkable improvement in an adaptation stage we analyze the mean square performance of the proposed algorithm also the proposed algorithm works well even in non stationary environments

a new variable step size nlms algorithm and its performance analysis numerous variable step size normalized least mean square vss nlms algorithms have been derived to solve the dilemma of fast convergence rate or low excess mean square error in the past two decades this paper proposes a new easy to implement nonparametric vss nlms algorithm that employs the mean square error and the estimated system noise power to control the step size update theoretical analysis of its steady state behavior shows that when the input is zero mean gaussian distributed the misadjustment depends only on a parameter controlling the update of step size simulation experiments show that the proposed algorithm performs very well furthermore the theoretical steady state behavior is in very good agreement with the experimental results

a robust variable step size lms type algorithm analysis and simulations a number of time varying step size algorithms have been proposed to enhance the performance of the conventional lms algorithm experimentation with these algorithms indicates that their performance is highly sensitive to the noise disturbance this paper presents a robust variable step size lms type algorithm providing fast convergence at early stages of adaptation while ensuring small final misadjustment the performance of the algorithm is not affected by existing uncorrelated noise disturbances an approximate analysis of convergence and steady state performance for zero mean stationary gaussian inputs and for nonstationary optimal weight vector is provided simulation results comparing the proposed algorithm to current variable step size algorithms clearly indicate its superior performance for cases of stationary environments for nonstationary environments our algorithm performs as well as other variable step size algorithms in providing performance equivalent to that of the regular lms algorithm

lyapunov functions for random walks and strings in random environment we study two typical examples of countable markov chains in random environment using the lyapunov functions method random walk and random string in random environment in each case we construct an explicit lyapunov function investigating the behavior of this function we get the classification for recurrence transience ergodicity we obtain new results for random strings in random environment though we simply review well known results for random walks using our approach

modeling and mathematical analysis of an initial boundary value problem for hepatitis b virus infection abstract in this work we investigate the hepatitis b virus infection we first derive a nonlinear pde model for the studied biological phenomenon the obtained initial boundary value problem is completely analyzed to begin with the analysis of the model we use lou and zhao lemma concerning globally attractive steady states to prove boundedness of potential solutions then we prove global existence uniqueness and positivity of the solution by a variational method combined with semigroups theory and some other useful tools from functional analysis moreover the basic reproduction number r 0 determining the extinction or the persistence of the hbv infection is thoroughly computed via a new method based on spectral properties of differential operators the residue theorem and some arguments from numerical analysis also the global asymptotical properties of the hbv free equilibrium of the model are derived via a skillful construction of a suitable lyapunov function local stability of the endemic equilibrium of the model is studied as well finally numerical simulations are performed to support the theoretical results obtained