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

exactly soluble random field ising models in one dimension the authors solve exactly the one dimensional random field ising model for two classes of magnetic field distributions symmetric exponential model i and non symmetric exponential model ii for both models expressions for the free energy at all finite temperatures are presented the low temperature region is examined in more detail they obtain the zero temperature energy and entropy in closed form it is shown that the free energy of both models has an expansion in integer powers of temperature model i has a non vanishing zero point entropy for all values of the parameters as soon as randomness is diluted in model ii the zero point entropy is zero except for a discrete sequence of values of one parameter in some cases the zero temperature magnetisation is positive whereas the average magnetic field is negative the magnetisation may also change sign as a function of temperature

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

large deviations and nontrivial exponents in coarsening systems we investigate the statistics of the mean magnetization of its large deviations and persistent large deviations in simple coarsening systems in particular we consider more specifically the case of the diffusion equation of the ising chain at zero temperature and of the two dimensional voter model for the diffusion equation at large times the mean magnetization has a limit law which is studied analytically using the independent interval approximation the probability of persistent large deviations defined as the probability that the mean magnetization was for all previous times greater than some level x decays algebraically at large times with an exponent continuously varying with x when x 1 is the usual persistence exponent similar behaviour is found for the glauber ising chain at zero temperature for the two dimensional voter model large deviations of the mean magnetization are algebraic while the probability of persistent large deviations seem to behave as the usual persistence probability

a crossover in the scaling law of the lyapunov exponent the authors consider products of random matrices appearing in the study of schrodinger operators or dynamical systems they show how a crossover in the scaling law of the lyapunov exponent takes place when the standard deviation of the process increases

fluctuations of correlation functions in disordered spin systems the authors study the fluctuations of the two point correlation function in one dimensional disordered spin models these survive even in the thermodynamic limit and in order to reconstruct their probability distribution from the moments they study a set of generalised correlation lengths zeta q these moments may also be calculated within the transfer matrix formalism and provide insight on disorder induced fluctuations they show that the zeta q can be computed in monte carlo simulations they discuss the crossover of the correlation decay rate at large distances to dominance by the most probable value given by zeta 0 and the relation with the finite volume fluctuations of the free energy finally they sketch how to extend their arguments to dimensions two and three

exact generalized lyapounov exponents for one dimensional disordered tight binding models a conceptually simple procedure is given for computing the moments of the distribution of states of one dimensional disordered tight binding models from which generalized lyapounov exponents are defined

statistics of the occupation time for a class of gaussian markov processes we revisit the work of dhar and majumdar 1999 phys rev e 59 6413 on the limiting distribution of the temporal mean mt t 1 0tdu sign yu for a gaussian markovian process yt depending on a parameter which can be interpreted as brownian motion in the time scale t t2 this quantity the mean magnetization is simply related to the occupation time of the process that is the length of time spent on one side of the origin up to time t using the fact that the intervals between sign changes of the process form a renewal process on the time scale t we determine recursively the moments of the mean magnetization we also find an integral equation for the distribution of mt this allows a local analysis of this distribution in the persistence region mt 1 as well as its asymptotic analysis in the regime where is large finally we put the results thus found in perspective with those obtained by dhar and majumdar by another method based on a formalism due to kac

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

on the compatibility of lorentz metrics with linear connections on four dimensional manifolds this paper considers four dimensional manifolds upon which there is a lorentz metric h and a symmetric connection which are originally assumed unrelated it then derives sufficient conditions on h and expressed through the curvature tensor of for to be the levi civita connection of some local lorentz metric g and calculates the relationship between g and h some examples are provided which help to assess the strength of the sufficient conditions derived

scaling law and asymptotic distribution of lyapunov exponents in conservative dynamical systems with many degrees of freedom the authors study by numerical means the infinite product of 2n 2n conservative random matrices which mimics the chaotic behaviour of hamiltonian systems with n 1 degrees of freedom made of weakly nearest neighbour coupled oscillators the maximum lyapunov exponent lambda 1 exhibits a power law behaviour as a function of the coupling constant epsilon lambda 1 approximately epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements these power laws do not depend on n and moreover increasing n lambda 1 rapidly tends to an asymptotic value lambda 1 which only depends on epsilon and on the kind of probability distribution chosen for building up the matrices they also compute the spectrum of the lyapunov exponents and show that is has a thermodynamic limit of large n this suggests the existence of a kolmogorov entropy per degree of freedom proportional to lambda 1

classification and conformal symmetry in three dimensional space times three dimensional manifolds admitting lorentz metrics are studied the first part of the paper gives a classification of the ricci and curvature tensors and also of the conformal schouten cotton york tensor the second part of the paper investigates killing and conformal symmetry and also the nature of the zeros of the associated vector fields the maximum dimension of the killing and conformal algebras is calculated a theorem regarding the reduction of the conformal algebra to a killing algebra of a conformally related metric is given

affine collineations in space time the existence of affine collineations in space time is discussed and the types of space time admitting proper affine collineations is displayed the close connection between such space times and their holonomy structure and local decomposability is established affine collineations with fixed points are also considered as is the problem of extending local affine collineations to the whole of space time

a mean field approximation for the maximum lyapunov exponent of bernoulli trials of matrices the author considers the infinite products of d d random matrices xi which are taken according to a bernoulli distribution i e xi a with probability 1 p and xi b with probability the maximum lyapunov exponent is estimated by selecting a suitable periodic sequence one need only then compute the eigenvalues of one particular matrix which is obtained as a finite product of matrices a and b he also discusses the cases in which the method fails because of fluctuation effects

estimating generalized lyapunov exponents for products of random matrices we discuss several techniques for the evaluation of the generalized lyapunov exponents which characterize the growth of products of random matrices in the large deviation regime a monte carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general purpose numerical method which is both efficient and easy to implement alternative techniques complementing this method are presented these include the computation of the generalized lyapunov exponents by solving numerically an eigenvalue problem and some asymptotic results corresponding to high order moments of the matrix products taken together the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized lyapunov exponents in a broad range of applications their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows

analytical results for generalized persistence properties of smooth processes we present a general scheme to calculate within the independent interval approximation generalized level dependent persistence properties for processes having a finite density of zero crossings our results are especially relevant for the diffusion equation evolving from random initial conditions one of the simplest coarsening systems exact results are obtained in certain limits and rely on a new method to deal with constrained multiplicative processes an excellent agreement of our analytical predictions with direct numerical simulations of the diffusion equation is found

determination of the metric tensor from components of the riemann tensor an algebraic method is presented which shows how to determine the components of the metric tensor gmu nu up to an arbitrary conformal factor from a given set of components of its riemann tensor rmu nu alpha beta in some coordinate frame this procedure follows and generalises one given by ihrig 1975 since the computations are purely algebraic and are carried out at a point in the manifold no differentiability or continuity conditions are assumed a number of examples are given to illustrate the technique although the method determines the metric up to one arbitrary scalar in a number of cases either one or three other arbitrary scalars arise these latter cases are rare and the form of the riemann tensor for such cases have been listed elsewhere

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

singular behavior of the density of states and the lyapunov coefficient in binary random harmonic chains we study the integrated density of statesh 2 of a chain of harmonic oscillators with a binary random distribution of the masses we show in particular that there is a dense set of values of the squared frequency for which the differenceh 2 h 2 has a singularity of the type 2 multiplied by a periodic function of ln where the exponent and the period depend continuously on 2 in the region where 1 2 h is not differentiate on a dense set of points the same type of singularities is also present in the lyapunov coefficient

from rational bubbles to crashes we study and generalize in various ways the model of rational expectation re bubbles introduced by blanchard and watson in the economic literature bubbles are argued to be the equivalent of goldstone modes of the fundamental rational pricing equation associated with the symmetry breaking introduced by non vanishing dividends generalizing bubbles in terms of multiplicative stochastic maps we summarize the result of lux and sornette that the no arbitrage condition imposes that the tail of the return distribution is hyperbolic with an exponent 1 we then outline the main results of malevergne and sornette who extend the re bubble model to arbitrary dimensions d a number d of market time series are made linearly interdependent via d d stochastic coupling coefficients we derive the no arbitrage condition in this context and with the renewal theory for products of random matrices applied to stochastic recurrence equations we extend the theorem of lux and sornette to demonstrate that the tails of the unconditional distributions associated with such d dimensional bubble processes follow power laws with the same asymptotic tail exponent 1 for all assets the distribution of price differences and of returns is dominated by the same power law over an extended range of large returns although power law tails are a pervasive feature of empirical data the numerical value 1 is in disagreement with the usual empirical estimates 3 we then discuss two extensions the crash hazard rate model and the non stationary growth rate model of the re bubble model that provide two ways of reconciliation with the stylized facts of financial data

small strain accompanied by moderate rotation this paper is mainly concerned with the construction of a theory of material behavior with infinitesimal strain accompanied by moderate rotation after introducing a definition for moderate rotation and establishing a number of theorems pertaining to its properties precise estimates are obtained for the local moderate rotation and related kinematical results in terms of infinitesimal strain for motions which result in small strain accompanied by moderate rotation the invariance of constitutive equations under arbitrary superposed rigid body motions is discussed with particular reference to an elastic material

scaling law for the maximum lyapunov characteristic exponent of infinite product of random matrices the authors present a simple derivation for the scaling behaviour of the maximum lyapunov characteristic exponent lambda of infinite product of symplectic random matrices the considered random matrices depend on a parameter epsilon and lambda 0 for epsilon 0 they obtain lambda varies as epsilon beta with either beta 1 2 or beta 2 3 depending on the probability distribution of the matrix elements the results are in agreement with a previous numerical simulation

the riemann tensor the metric tensor and curvature collineations in general relativity the equation x r x r 0 where x and r are the components of an arbitrary symmetric tensor and of the riemann tensor formed from the metric tensor g is trivially satisfied by x g nontrivial solutions are important in various areas of general relativity such as in the study of curvature collineations and also in the study of algebraic methods given by hlavat and ihrig for the determination of g from a given set of r we have found all r for which there exist nontrivial solutions of the above equation and we have given the form of the x in each case various examples of space times for explicit nontrivial solutions are discussed

approximate kinematical relations in plasticity abstract within the framework of finite plasticity it is shown how a number of kinematical approximations can be systematically derived from the multiplicative decomposition of the deformation gradient particular attention is devoted to two types of approximate theories a those in which elastic deformations are of a different order of magnitude than plastic deformations and b those in which rotations are of a different order of magnitude than strains

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

diffusion in a one dimensional lattice with random asymmetric transition rates the authors study one dimensional continuous time random walks for which the pairs wn wn 1 of nearest neighbour transition rates are assumed to be independent equally distributed random variables the long time asymptotic behaviour of the mean displacement x t is determined exactly for a specific model system in which diodes u 0 and two way bonds lambda v v occur with probabilities p and 1 p respectively for lambda 1 p we find that x t approximately tnu f beta 1ln t where nu ln 1 p ln lambda and beta ln lambda and where f is a periodic function with period 1 the mean displacement thus not only increases slower than linearly in time but exhibits superimposed non decaying oscillations

even visiting random walks exact and asymptotic results in one dimension we reconsider the problem of even visiting random walks in one dimension this problem is mapped onto a non hermitian anderson model with binary disorder we develop very efficient numerical tools to enumerate and characterize even visiting walks the number of closed walks is obtained as an exact integer up to 1828 steps i e some 10535 walks on the analytical side the concepts and techniques of one dimensional disordered systems allow one to obtain explicit asymptotic estimates for the number of closed walks of 4k steps up to an absolute prefactor of the order of unity which is determined numerically all the cumulants of the maximum height reached by such walks are shown to grow as k1 3 with exactly known prefactors these results illustrate the tight relationship between even visiting walks trapping models and the lifshitz tails of disordered electron or phonon spectra

fokker planck equation of distributions of financial returns and power laws abstract our purpose is to relate the fokker planck formalism proposed by friedrich et al phys rev lett 84 2000 5224 for the distribution of stock market returns to the empirically well established power law distribution with an exponent in the range 3 5 we show how to use friedrich et al s formalism to predict that the distribution of returns is indeed asymptotically a power law with an exponent that can be determined from the kramers moyal coefficients determined by friedrich et al however with their values determined for the u s dollar german mark exchange rates the exponent predicted from their theory is found to be around 12 in disagreement with the often quoted value between 3 and 5 this could be explained by the fact that the large asymptotic value of 12 does not apply to real data that lie still far from the stationary state of the fokker planck description another possibility is that power laws are inadequate the mechanism for the power law is based on the presence of multiplicative noise across time scales which is different from the multiplicative noise at fixed time scales implicit in the arch models developed in the finance literature

linear stochastic dynamics with nonlinear fractal properties abstract stochastic processes with multiplicative noise have been studied independently in several different contexts over the past decades we focus on the regime found for a generic set of control parameters in which stochastic processes with multiplicative noise produce intermittency of a special kind characterized by a power law probability density distribution we present a review of applications highlight the common physical mechanism and summarize the main known results the distribution and statistical properties of the duration of intermittent bursts are also characterized in detail

log periodic critical amplitudes a perturbative approach log periodic amplitudes appear in the critical behavior of a large class of systems in particular when a discrete scale invariance is present here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial in this case the log periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated

the kalman levy filter abstract the kalman filter combines forecasts and new observations to obtain an estimation which is optimal in the sense of a minimum average quadratic error the kalman filter has two main restrictions i the dynamical system is assumed linear and ii forecasting errors and observational noises are projected onto gaussian distributions here we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and levy laws the main tool needed to solve this kalman levy filter is the tail covariance matrix which generalizes the covariance matrix in the case where it is mathematically ill defined i e for power law tail exponents 2 we present the general solution and discuss its properties on pedagogical examples the standard kalman gaussian filter is recovered for the case 2 the optimal kalman levy filter is found to deviate substantially from the standard kalman gaussian filter as deviates from 2 as decreases the filter favors more strongly the better one of the forecast and the observation based on the tail covariance matrix because a small exponent implies large errors with significant probabilities in terms of implementation the price to pay associated with the presence of heavy tail noise distributions is that the standard linear formalism valid for the gaussian case is transformed into a nonlinear matrix equation for the kalman levy filter direct numerical experiments in the univariate case confirms our theoretical predictions

viscosity for a periodic two disk fluid an existence proof we express the momentum current stress tensor for a periodic fluid with two hard disks per unit cell in terms of a single particle billiard we establish a central limit theorem for the time integrated stress tensor and thereby prove the existence of a strictly positive shear and bulk viscosity

fine structure and complex exponents in power law distributions from random maps discrete scale invariance dsi has recently been documented in time to failure rupture earthquake processes and financial crashes in the fractal geometry of growth processes and in random systems the main signature of dsi is the presence of log periodic oscillations correcting the usual power laws corresponding to complex exponents log periodic structures are important because they reveal the presence of preferred scaling ratios of the underlying physical processes here we present new evidence of log periodicity overlaying the leading power law behavior of probability density distributions of affine random maps with parametric noise the log periodicity is due to intermittent amplifying multiplicative events we quantify precisely the progressive smoothing of the log periodic structures as the randomness increases and find a large robustness our results provide useful markers for the search of log periodicity in numerical and experimental data

physically nonlinear and related approximate theories of elasticity and their invariance properties the existing developments of physically nonlinear elasticity have several shortcomings with the aim of remedying these deficiencies a number of approximate theories of elasticity are discussed in the present paper and in particular a theory of physically nonlinear elasticity is systematically developed more specifically stress strain relations for anisotropic physically nonlinear materials are derived the method of casey naghdi 1 is then applied to obtain properly invariant results this method involves the use of auxiliary motions obtained by removing from any given motion the translation and rotation at any one particle called a pivot the auxiliary motions represent the original motions in the approximate theory the connection between the transformation of fields under a change of pivot and invariance requirements associated with the auxiliary motions is investigated

localization of random walks in one dimensional random environments we consider a random walk on the one dimensional semi lattice 0 1 2 we prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment the size of this neighbourhood is estimated the limit parameters of the walks are also determined

exact solutions for ising chains in a random field the authors present an exact solution for the one dimensional ising model in a random field the distribution of field strengths is site independent symmetric and has a three peak structure the free energy is obtained for all finite temperatures the low temperature behaviour is studied in detail they find an expansion for the free energy in integer powers of temperature the ground state energy and the zero point entropy are calculated explicitly the specific heat is linear for low temperatures in contradiction with mean field theory the origin of an additional exponential parameter is discussed

holonomy groups in general relativity the infinitesimal holonomy group structure of space time is discussed and related to the petrov type of the weyl tensor and the algebraic segre type of the energy momentum tensor the number of covariant derivatives of the curvature tensor required to determine the infinitesimal holonomy group is determined in each case and the complete classification scheme is tabulated some special cases of physical interest are investigated in more detail a geometrical approach is followed throughout

convergence to stable laws for a class of multidimensional stochastic recursions we consider a markov chain x n n 0 infty on mathbb r d defined by the stochastic recursion xn mn xn 1 qn where qn mn are i i d random variables taking values in the affine group a mathbb r d mathbb r d rtimes rm gl mathbb r d assume that mn takes values in the group of similarities of mathbb r d and the markov chain has a unique stationary measure which has unbounded support we denote by mn the expansion coefficient of mn and we assume mathbb e m alpha 1 for some positive we show that the partial sums s n sum k 0 n x k properly normalized converge to a normal law 2 or to an infinitely divisible law which is stable in a natural sense 2 or of the tails of and of stationary measure for an associated markov operator 2 we extend the results to the situation where mn is a random generalized similarity

an invariant infinitesimal theory of motions superposed on a given motion abstract the development is applicable to any material but special attention is given to elastic solids included as a special case is an infinitesimal theory of elasticity with the following properties 1 it is properly invariant under arbitrary not necessarily infinitesimal superposed rigid body motions 2 it reduces by specialization to the theory of rigid bodies undergoing finite motion and 3 it can be brought into correspondence with the classical linear elasticity through a suitable reinterpretation of the symbols in the constitutive equation of the latter author

curvature structure in general relativity this paper investigates the extent to which the curvature structure of space time determines the metric stucture it continues the work of earlier papers by prescribing the curvature structure and the curvature covariant derivatives up to certain orders it is shown that with the exception of the so called generalized pp waves the curvature and its first and second covariant derivatives essentially determine the metric up to coordinate transformations

collineations of a symmetric 2 covariant tensor ricci collineations the infinitesimal transformations that leave invariant a two covariant symmetric tensor are studied the interest of these symmetry transformations lays in the fact that this class of tensors includes the energy momentum and ricci tensors we find that in most cases the class of infinitesimal generators of these transformations is a finite dimensional lie algebra but in some cases exhibiting a higher degree of degeneracy this class is infinite dimensional and may fail to be a lie algebra as an application we study the ricci collineations of a type b warped spacetime

on the convergence of a class of random geometric series with application to random walks and percolation theory we consider a class of random geometric series with an underlying tree like structure that has a number of applications in statistical physics convergence criteria for these series are discussed and consistency with different criteria known to hold in the one dimensional limit is established a multiplicative two component model of percolation on a cayley tree is defined and analysed the order of the percolation transition and certain critical exponents are altered compared to conventional bond percolation

affine collineations in general relativity and their fixed point structure a general discussion of affine vector fields collineations in space time is given following a previous article by hall and da costa some general remarks are presented on the zeros of affine vector fields and then the general theory of such zeros is established some preliminary results regarding homothetic symmetries in two and three dimensional space times are also given the impossibility of reducing an affine algebra containing proper affine vector fields to homothetic or killing vector fields by means of a change of compatible metric is demonstrated

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

fluctuations of the lyapunov exponent in disordered systems the fluctuations of the maximum lyapunov exponent of a product of random matrices are studied analytically and numerically it is shown that they may be expressed as a sum of two terms one related to the order of matrices within the product the other to fluctuations of the number of matrices of a given type this result is then applied to the one dimensional random field ising model and the discrete schrodinger equation with a random potential

generalized lyapunov exponents in high dimensional chaotic dynamics and products of large random matrices we study the behavior of the generalized lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsd for products of random matrices without any particular structure the generalized lyapunov exponents become equal in this limit and the value of one of the generalized lyapunov exponents is obtained by simple arguments on the contrary for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized lyapunov exponents remains different ford indicating that high dimensionality cannot always destroy intermittency

curvature collineations in general relativity ii this paper is the second of a set of two papers on curvature collineations in general relativity the first paper presented the mathematical basis of curvature collineations and a possible approach to their study this paper continues from the first one by investigating in detail many of the cases where curvature collineations can occur in space time it is based on a classification of the curvature tensor which is discussed in the first paper and reviewed briefly here this together with the geometrical approach favored in this paper leads to a rather general discussion of the problem which it is hoped does not obscure those physical aspects of the situation that are important in einstein s theory

tail homogeneity of stationary measures for some multidimensional stochastic recursions we consider a stochastic recursion x n 1 m n 1 x n q n 1 n in mathbb n where q n m n are i i d random variables such that q n are translations m n are similarities of the euclidean space mathbb r d and x n in mathbb r d in the present paper we show that if the recursion has a unique stationary measure with unbounded support then the weak limit of properly dilated exists and defines a homogeneous tail measure the structure of is studied and the supports of and are compared in particular we obtain a product formula for

diffusion in a one dimensional random medium and hyperbolic brownian motion classical diffusion in a random medium involves an exponential functional of brownian motion this functional also appears in the study of brownian diffusion on a riemann surface of constant negative curvature we analyse in detail this relationship and study various distributions using stochastic calculus and functional integration

one dimensional random ising models a study of several one dimensional random ising chains and strips is presented the free energy is expressed in terms of a probability distribution the support of which is studied with use of a recurrence relation the shape of the support is shown to vary from a connected set to a cantor set depending on the type of randomness monte carlo calculations and analytical approximations are presented

power law behavior of lyapunov exponents in some conservative dynamical systems abstract three billiards whose border depends on a parameter e are considered for e 0 they are integrable while for e 0 they are known to be k systems the maximal lyapunov exponent 1 is numerically computed and a power law behavior 1 e is found with 1 2 for all billiards a related abstract problem is then considered precisely the case of an infinite product of conservative 2 2 random matrices which are perturbations of commuting parabolic ones analogous computations give here two power laws with 1 2 or 2 3 depending on the probability law used to construct random matrices all these phenomena seem to have an universal character a heuristic mean field theory is also proposed based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

low temperature thermodynamics of random field ising chains exact results the authors consider the random field ferromagnetic ising chain for a class of continuous symmetric probability distributions of the random magnetic fields of a diluted power times exponential type this class of distribution is exactly solvable in the sense that the disorder can be integrated explicitly at any temperature a detailed analysis of the low temperature thermodynamics is presented exact expressions are obtained for the ground state energy the zero temperature entropy and the amplitude of the specific heat which vanishes linearly at low temperature the diluted symmetric binary distribution where the magnetic fields can only assume the values or hb or zero can be viewed as a limiting case of the class of exactly solvable distributions the low temperature physics of this discrete model is investigated in detail including the exponential fall off of the specific heat

predicted and verified deviations from zipf s law in ecology of competing products zipf s power law distribution is a generic empirical statistical regularity found in many complex systems however rather than universality with a single power law exponent equal to 1 for zipf s law there are many reported deviations that remain unexplained a recently developed theory finds that the interplay between i one of the most universal ingredients namely stochastic proportional growth and ii birth and death processes leads to a generic power law distribution with an exponent that depends on the characteristics of each ingredient here we report the first complete empirical test of the theory and its application based on the empirical analysis of the dynamics of market shares in the product market we estimate directly the average growth rate of market shares and its standard deviation the birth rates and the death hazard rate of products we find that temporal variations and product differences of the observed power law exponents can be fully captured by the theory with no adjustable parameters our results can be generalized to many systems for which the statistical properties revealed by power law exponents are directly linked to the underlying generating mechanism

functional feynman kac equations for limit lognormal multifractals a novel technique of functional feynman kac equations is developed for the probability distribution of the limit lognormal multifractal process introduced by mandelbrot in statistical models and turbulence m rosenblatt and c van atta eds springer new york 1972 and constructed explicitly by bacry delour and muzy phys rev e 64 026103 2001 the distribution of the process is known to be determined by the complicated stochastic dependence structure of its increments sdsi it is shown that the sdsi has two separate layers of complexity that can be captured in a precise way by a pair of functional feynman kac equations for the laplace transform exact solutions are obtained as power series expansions in the intermittency parameter using a novel intermittency differentiation rule the expansion of the moments gives a new representation of the selberg integral

a theory of shells with small strain accompanied by moderate rotation this paper is concerned with a constrained theory of shells in the presence of small strain accompanied by moderate rotation the constrained theory accounts for the effect of transverse normal strain and includes of course the special case corresponding to the kirchhoff love theory of shells in which the effect of transverse normal strain is absent after precise estimates for local moderate rotation and relative displacement gradients in terms of infinitesimal strain have been effected a complete theory is formulated with the use of linear constitutive equations the nature of the complete theory is further examined when initially the shell like body is a plate and it is shown that our kinematical formulae strain displacement relations as well as the relevant differential equations of the theory in the absence of the effect of transverse normal strain systematically reduce to those used in the von karman plate equations also in the light of the present results an assessment of kinematical aspects of previously developed theories of shells undergoing small strain and moderate rotation is indicated

symmetries of bianchi i space times all diagonal proper bianchi i space times are determined which admit certain important symmetries it is shown that for homothetic motions conformal motions and kinematic self similarities the resulting space times are defined explicitly in terms of a set of parameters whereas affine collineations ricci collineations and curvature collineations if they are admitted they determine the metric modulo certain algebraic conditions in all cases the symmetry vectors are explicitly computed the physical and the geometrical consequences of the results are discussed and a new anisotropic fluid physically valid solution which admits a proper conformal killing vector is given

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

a borel like estimate for the rest of the heat kernel expansion it is shown that under some simple assumptions the rest of the truncated asymptotic expansion for the heat kernel has the form required for borel summability if n the number of terms in the truncation is sufficiently large rigorously speaking this result does not imply borel summability which requires a uniform bound in n but it could be useful as a starting point for further investigation on global estimates for the zeta function

on an imaginary exponential functional of brownian motion we investigate a random integral which provides a natural example of an imaginary exponential functional of brownian motion this functional shows up in the study of the binary annihilation process within the doi peliti formalism for reaction diffusion systems the main emphasis is put on the complementarity between the usual langevin approach and another approach based on the similarity with kesten variables and other one dimensional disordered systems even though neither of these routes leads to the full solution of the problem we have obtained a collection of results describing various regimes of interest