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

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

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

partial information state based optimization of partially observable markov decision processes and the separation principle we propose a partial information state based approach to the optimization of the long run average performance in a partially observable markov decision process pomdp in this approach the information history is summarized at least partially by a or a few statistic s not necessary sufficient called a partial information state and actions depend on the partial information state rather than system states we first propose the single policy based comparison principle under which we derive an hjb type of optimality equation and policy iteration for the optimal policy in the partial information state based policy space we then introduce the q sufficient statistics and show that if the partial information state is q sufficient then the optimal policy in the partial information state based policy space is optimal in the space of all feasible information state based policies we show that with some further conditions the well known separation principle holds the results are obtained by applying the direct comparison based approach initially developed for discrete event dynamic systems

supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

decay rate of the green function in a random potential on the bethe lattice and a criterion for localization it is shown that in the tight binding anderson model on the bethe lattice the exponential decay rate of the green function can be obtained for arbitrary energies and arbitrary disorder analytical results in the case of the lorentzian distribution of site energies are presented as an application of these results a criterion for the localized region on the corresponding regular lattice on the level of the bethe peierls approximation is proposed our criterion yields exact results in the one dimensional limit and yields correct band edges for the hypercubic lattice in the vanishing limit of disorder the mobility edge trajectory obtained by our criterion is given by an elliptic curve in the case of the lorentzian distribution and its shape is found to be in qualitatively good agreement with that obtained by the finite size scaling method in the three dimensional system

preservation of a c spectrum for random decaying perturbations of square summable high order variation we consider random self adjoint jacobi matrices of the form j u n an u n 1 bn u n an 1 u n 1 on l2 n where an 0 and bn r are sequences of random variables on a probability space dp such that there exists q n such that for any l n 2l al al q and 2l 1 bl bl q are independent random variables of zero mean satisfying n 1 n2 dp let jp be the deterministic periodic of period q jacobi matrix whose coefficients are the mean values of the corresponding entries in j we prove that for a e the a c spectrum of the operator j equals to and fills the spectrum of jp if moreover n 1 n4 dp then for a e the spectrum of j is purely absolutely continuous on the interior of the bands that make up the spectrum of jp

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

generalization of the analytical inversion method for the solution of the point kinetics equations a method based on the analytical inversion of polynomials of the point kinetics matrix is applied to the solution of the reactor kinetics equations this method permits a fast inversion of polynomials by going temporarily to the complex plane several cases using various options of the method are presented for comparison the method developed was found to be very fast and accurate and has the ability to reproduce all the features of transients including prompt jump the analysis of the assumption of constant parameters reactivity and source during a time step are included it is concluded that the method provides a fast and accurate computational technique for the point kinetics equations with step reactivity

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

on a conjecture of alley and alder for fluids and lorentz models we discuss a conjecture of alley and alder predicting a relation between the four point and the two point velocity autocorrelation functions for fluids and lorentz models at sufficiently long times if the conjecture is correct a modified burnett coefficient can be defined which has a finite value contrary to the ordinary burnett coefficient which is divergent the conjecture is tested for four classes of models with different methods for three dimensional fluids mode coupling theory yields a negative result the conjecture is confirmed for thed dimensional deterministic lorentz gas d 2 and for a class ofd dimensional stochastic lorentz models d 1 by low density kinetic theory as well as by rigorous results available for one dimension for yet another class of one dimensional stochastic lorentz models which are exactly solvable in one dimension the result is negative again all four classes of models show long time tails in the velocity autocorrelation function and have a finite diffusion coefficient

on the essential spectrum of schrodinger operators with spherically symmetric potentials on determine le spectre essentiel des operateurs de schrodinger avec un potentiel a symetrie spherique v x q x

optimal average value convergence in nonhomogeneous markov decision processes abstract we address the undiscounted nonhomogeneous markov decision process with average reward criterion and prove two structural results first we establish equivalence of this problem to a discounted markov decision process by means of an ergodic coefficient embedded in the original problem second we prove for the original problem that the optimal finite horizon average values converge to the infinite horizon optimal average value under an ergodic condition

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

event based optimization of markov systems recent research indicates that markov decision processes mdps and perturbation analysis pa based optimization can be derived easily from two fundamental performance sensitivity formulas with this sensitivity point of view an event based optimization approach including event based sensitivity analysis and event based policy iteration was proposed via an example by x r cao discrete event dyn syst theory appl vol 15 pp 169 197 2005 this approach utilizes the special feature of a system and illustrates how the potentials can be aggregated using the special feature the approach applies to many practical problems that do not fit well the standard mdp formulation this note provides a mathematical formulation and proves the main results for this approach

optimization of average rewards of time nonhomogeneous markov chains we study the optimization of average rewards of discrete time nonhomogeneous markov chains in which the state spaces transition probabilities and reward functions depend on time the analysis encounters a few major difficulties 1 notions crucial to homogeneous markov chains such as ergodicity stationarity periodicity and connectivity no longer apply 2 the average reward criterion is under selective i e it does not depend on the decisions in any finite period and thus dynamic programming is not amenable and 3 because of the under selectivity an optimal average reward policy may not be the best in any finite period

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

an improved class of generalized runge kutta methods for stiff problems part ii the separated system case in order to study a generalization of the explicit p stage runge kutta methods we introduce a new family of p stage formulas for the numerical integration of some special systems of odes that provides better order and stability results with the same number of stages in our recent paper entitled an improved class of generalized runge kutta methods for stiff problems part i the scalar case we studied new schemes for the numerical integration of scalar autonomous odes in this second part we will show that it is possible to generalize our grk methods so that they can be applied to some non autonomous scalar odes and systems obtaining linearly implicit a stable and l stable methods these methods do not require jacobian evaluations in their implementation some numerical examples are discussed in order to show the good performance of the new schemes

singularities in spectra of disordered systems abstract this paper starts with a review of work on singularities in the spectral density of random harmonic chains detailed descriptions of singular behavior near the maximal frequency near special frequencies and near island frequencies are discussed next the site disordered anderson model is considered here the discussion is limited to the singularity in the density of states near the low energy band edge for systems with arbitrary disorder in dimensions d 1 2 3 in a specific example it is shown that renormalization improves the prediction for the band tail quantitatively

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

a generalized thouless formula as a criterion for anderson localization in two and three dimensional systems abstract a generalized thouless formula is proposed as an approximate criterion for anderson localization in two and three dimensional systems the criterion is exact for ordered systems and in the mean field limit of infinite dimensionality for random systems it is expected that possible corrections are small ample numerical evidence is provided through an exact diagonalization of several large size hamiltonians the predictions of the thouless formula whose evaluation only needs the eigenvalues are critically checked against the inverse participation ratio ipr of the corresponding eigenvectors included are the lloyd model for sufficiently low disorder mobility edges cannot be excluded and surprising new results on the binary alloy problem the criterion predicts four as the upper and two as the lower critical dimension

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

a study of rosenbrock type methods of high order this paper deals with the solution of nonlinear stiff ordinary differential equations the methods derived here are of rosenbrock type this has the advantage that they area stable or stiffly stable and nevertheless do not require the solution of nonlinear systems of equations we derive methods of orders 5 and 6 which require one evaluation of the jacobian and onelu decomposition per step we have written programs for these methods which use richardson extrapolation for the step size control and give numerical results

the laplacian on a random one dimensional lattice the authors study the spectrum of minus the laplacian on a random one dimensional lattice it extends monotonically throughout the positive real axis with a continuum limit behaviour for small values the authors find as expected localisation effects throughout the whole range they use a variety of techniques which are shown to be consistent and in agreement with a monte carlo simulation they include small and large disorder expansions direct integral equations for probability distributions as well as the replica method the latter is also used to investigate green functions

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

state classification of time nonhomogeneous markov chains and average reward optimization of multi chains in a discrete time nonhomogeneous markov chain tnhmc the states spaces transition probabilities and reward functions at different times may be different in this paper with the confluencity previously introduced we show that the states of a tnhmc can be classified into the branching states and a number of classes of confluent states versus the transient and recurrent states in the time homogeneous case the optimization of average reward in tnhmc s consisting of a single confluent class uni chain have been addressed in a previous paper by the author in this paper we show that with confluencity and the state classification and under some bound conditions we can obtain the necessary and sufficient conditions for optimal policies of the average reward of tnhmcs consisting of multiple confluent classes multi chains just like in the uni chain tnhmc case the sufficient condition does not need to hold in any zero frequently visited time sequence this under selectivity makes the problem not amenable to dynamic programming a direct comparison based approach is used to prove the results the results enhance our understanding of state classification and performance optimization with the notion of confluencity

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

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

special energies and special frequencies special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators our investigation includes both this model and the random one dimensional schrodinger operator describing an alloy or its discrete analogue using the phase method we exactly determine a bilateral lifsic asymptotic of the integrated density of statesk e at special energiese s which is not only of the classical type exp c e e s 1 2 but also exp c e e s is a typical behaviour in addition other asymptotics occur e g e e s c which show thatk e need not bek

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

localization for off diagonal disorder and for continuous schrodinger operators we extend the proof of localization by delyon levy and souillard to accommodate the anderson model with off diagonal disorder and the continuous schrodinger equation with a random potential

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

optimality conditions for long run average rewards with underselectivity and nonsmooth features we study three existing issues associated with optimization of long run average rewards of time nonhomogeneous markov processes in continuous time with continuous state spaces 1 the underselectivity i e the optimal policies do not depend on their actions in any finite time period 2 its related issue the bias optimality i e policies that optimize both long run average and transient total rewards and 3 the effects of a nonsmooth point of a value function on performance optimization these issues require considerations of the performance in the entire period with an infinite horizon and therefore are not easily solvable by dynamic programming which works backwards in time and takes a local view at a particular time instant in this paper we take a different approach called the relative optimization theory which is based on a direct comparison of the performance measures of any two policies we derive tight necessary and sufficient optimality conditions that take the underselectivity into consideration we derive bias optimality conditions for both long run average and transient rewards and we show that the effect of a wide class of nonsmooth points called semismooth points of a value function on the long run average performance is zero and can be ignored

stability of generalized runge kutta methods for stiff kinetics coupled differential equations a stability and efficiency improved class of generalized runge kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and or nonlinear coupled differential equations the determination of the coefficients required by the method is precisely obtained from the so called equations of condition which in turn are derived by an approach based on butcher series since the equations of condition are fewer in number free parameters can be chosen for optimizing any desired feature of the process a further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced in addition the a stability properties of the method are investigated implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback

l 2 exponential lower bounds to solutions of the schrodinger equation we study decay properties of solutions of the schrodinger equation v e typical of our results is one which shows that ifv o x 1 2 at infinity or ifv is a homogeneousn body potential for example atomic or molecular then ife 0 and alpha sqrt e e alpha left x right psi notin l 2 left mathbb r n right we also construct examples to show that previous essential spectrum dependent upper bounds can be far from optimal if is not the ground state

two classes of internally s stable generalized runge kutta processes which remain consistent with an inaccurate jacobian generalized runge kutta processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a jacobian at every step however it is possible to derive processes which are internally s stable when an accurate jacobian is used but still remain consistent and highly stable if an approximate jacobian is used it is shown that these processes require at least as many function evaluations as an explicit runge kutta process of the same order and second and third order processes are developed a second class of generalized runge kutta is introduced which requires that the jacobian be evaluated accurately less than once every step a third order process of this class is developed and all three methods contain an error estimator similar to those of fehlberg or england

picard s iterative method for nonlinear advection reaction diffusion equations picard s iterative method for the solution of nonlinear advection reaction diffusion equations is formulated and its convergence proved the method is based on the introduction of a complete metric space and makes uses of a contractive mapping and banach s fixed point theory from picard s iterative method the variational iteration method is derived without making any use at all of lagrange multipliers and constrained variations some examples that illustrate the advantages and shortcomings of the iterative procedure presented here are shown

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

lifshitz singularities in the total and the wavenumber dependent spectral density of random harmonic chains abstract lifshitz has shown that the density of states of random systems becomes exponentially small near the band edge we perform a detailed computation in a one dimensional system and calculate the periodic prefactor of this essential singularity the analysis is extended to the wavenumber dependent spectral density the periodic amplitude is derived for the exactly soluble cases where the distribution of masses is exponential for binary distributions the leading behavior in the concentration of light masses is also derived

reexamination of the long range potts model a multicanonical approach we investigate the critical behavior of the one dimensional q state potts model with long range lr interaction 1 r d sigma using a multicanonical algorithm the recursion scheme initially proposed by berg is improved so as to make it suitable for a large class of lr models with unequally spaced energy levels the choice of an efficient predictor and a reliable convergence criterion is discussed we obtain transition temperatures in the first order regime which are in far better agreement with mean field predictions than in previous monte carlo studies by relying on the location of spinodal points and resorting to scaling arguments we determine the threshold value sigma c q separating the first and second order regimes to two digit precision within the range 3 leq q leq 9 we offer convincing numerical evidence supporting sigma c q 1 0 for all q by virtue of an unusual finite size effect namely finite size scaling predicts a continuous transition in the thermodynamic limit despite the first order nature of the transition at finite size a qualitative account in terms of correlation lengths is provided finally we find the crossover between the lr and short range regimes to occur inside a narrow window 1 0 sigma 1 2 thus lending strong support to sak s scenario

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

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

a new optimality criterion for nonhomogeneous markov decision processes we propose a new definition of optimality for nonhomogeneous markov decision processes called periodic forecast horizon pfh optimality using measures of discounting and ergodicity we establish conditions under which pfh optimal strategies exist and pfh optimality implies the more conventional notions of optimality and average optimality finally we use this definition of optimality as the basis for a direct development of forecast horizon results for discounted and undiscounted nonhomogeneous markov decision processes

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

the n th order bias optimality for multichain markov decision processes in this paper we propose a new approach to the theory of finite multichain markov decision processes mdps with different performance optimization criteria we first propose the concept of nth order bias then using the average reward and bias difference formulas derived in this paper we develop an optimization theory for finite mdps that covers a complete spectrum from average optimality bias optimality to all high order bias optimality in a unified way the approach is simple direct natural and intuitive it depends neither on laurent series expansion nor on discounted mdps we also propose one phase policy iteration algorithms for bias and high order bias optimal policies which are more efficient than the two phase algorithms in the literature furthermore we derive high order bias optimality equations this research is a part of our effort in developing sensitivity based learning and optimization theory

anderson localization for the 1 d discrete schrodinger operator with two frequency potential we prove the complete exponential localization of eigenfunctions for the 1 d discrete schrodinger operators with quasi periodic potentials having two basic frequencies it is shown also that for such operators there is no forbidden zones in the spectrum unlike the operators with one basic frequency

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

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

solution of the point kinetics equations in the presence of newtonian temperature feedback by pade approximations via the analytical inversion method a method based on the pade approximations is applied to the solution of the point kinetics equations with a time varying reactivity the technique consists of treating explicitly the roots of the inhour formula a significant improvement has been observed by treating explicitly the most dominant roots of the inhour equation which usually would make the pade approximation inaccurate also the analytical inversion method which permits a fast inversion of polynomials of the point kinetics matrix is applied to the pade approximations results are presented for several cases of pade approximations using various options of the method with different types of reactivity the formalism is applicable equally well to non linear problems where the reactivity depends on the neutron density through temperature feedback it was evident that the presented method is particularly good for cases in which the reactivity can be represented by a series of steps and performed quite well for more general cases

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

a probabilistic analysis of bias optimality in unichain markov decision processes focuses on bias optimality in unichain finite state and action space markov decision processes using relative value functions we present methods for evaluating optimal bias this leads to a probabilistic analysis which transforms the original reward problem into a minimum average cost problem the result is an explanation of how and why bias implicitly discounts future rewards

stochastic feedback control with one dimensional degenerate diffusions and nonsmooth value functions we study the stochastic control problems with degenerate semimartingales and nonsmooth value functions we first derive the optimality conditions for finite horizon problems with nonsmooth value functions and singular control of infinite horizon discounted problems then we show that the local time at the degenerate point is zero so if the value function has nonsmooth first order derivatives at the degenerate points of all policies the nonsmoothness can be ignored and the optimality equation for stochastic control including optimal stopping is simply the classical hamilton jacobi bellman equation at the smooth points furthermore we model the singular control by the reflecting force in the skorokhod problem and show that there are two classes of degenerate points in the first class singular control at a degenerate point can be treated in the same way as regular control and in the second class the density of the singular control force may be of the order sqrt dt 1 gamma 0 with gamma 0 for nondegenerate points we apply the direct comparison based approach to derive all the results in all the analysis viscosity solution is not needed

unbounded trace orbits of thue morse hamiltonian it is well known that an energy is in the spectrum of fibonacci hamiltonian if and only if the corresponding trace orbit is bounded however it is not known whether the same result holds for the thue morse hamiltonian in this paper we give a negative answer to this question more precisely we construct two subsets sigma ii and sigma iii of the spectrum of the thue morse hamiltonian both of which are dense and uncountable such that each energy in sigma ii cup sigma iii corresponds to an unbounded trace orbit exact estimates on the norm of the transfer matrices are also obtained for these energies for e in sigma ii cup sigma iii the norms of the transfer matrices behave like begin aligned e c 1 gamma sqrt n le vert t n e vert le e c 2 gamma sqrt n end aligned however two types of energies are quite different in the sense that each energy in sigma ii is associated with a two sided pseudo localized state while each energy in sigma iii is associated with a one sided pseudo localized state the difference is also reflected by the local dimensions of the spectral measure the local dimension is 0 for energies in sigma ii and is larger than 1 for energies in sigma iii as a comparison we mention another known countable dense subset sigma i each energy in sigma i corresponds to an eventually constant trace map and the associated eigenvector is an extended state in summary the thue morse hamiltonian exhibits mixed spectral nature

picard iteration algorithm combined with gauss seidel technique for initial value problems abstract in this paper we introduce and implement a relatively new improvement for the well known picard s method for studying linear and nonlinear systems of ordinary differential equations in this method the solution takes the form of a rapidly convergent series with easily computable components a mathematica computer algebra system mathematica 5 1 is used to introduce a code which generates the series different illustrative examples have been considered as well as different comparison criterion between the proposed method and modern reliable and conventional methods the deeply mathematical roots of the method fixed point theorem and the correspondence with the theory of algebraic systems makes it is very interesting theoretical considerations are being discussed and convergence of the method is addressed other extensions and applications for further work are mentioned

a new approach to the problem of disordered harmonic chains abstract in the problem of harmonic chains with random masses the exponential growth rate and the spectral distribution function are shown to be the real and imaginary part of dyson s characteristic function along its cut in the complex plane a new function satisfying a certain integral equation is introduced from which the characteristic function follows immediately the functions introduced by dyson schmidt and matsuda and ishii are recovered as special cases and it is shown that schmidt s distributions w n converge as n relations between dyson s and schmidt s functions are obtained finally the same method is applied to the problem of harmonic chains with random variables k n m n and k n 1 m n and to the tight binding electron model with random site energies

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

correlation function of random field ising chains is it lorentzian or not the two point connected correlation function or wavevector dependent susceptibility of the ferromagnetic ising chain in a random field is calculated exactly at any temperature for a two parameter family of diluted symmetric exponential distributions of the magnetic fields thermodynamic properties of this model have been derived in a previous work by the authors besides the correlation function itself the solution provides exact results for the usual susceptibility the correlation length and the edwards anderson parameter the low temperature regime is examined in full detail the authors obtain in closed form the limit value at zero temperature of various quantities and the first correction to this limit which behaves linearly with temperature the correlation length is discontinuous at zero temperature they also derive the scaling form of the correlation function for small p where the dilution p is the probability for a spin to be subjected to a non zero random field even in this limit the correlation function is more complicated than the simple lorentzian predicted e g by mean field theory

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

long time tails in lattice lorentz gases we consider a lorentz gas on a square lattice with a fraction c of scattering sites the collision laws are deterministic fixed mirror model or stochastic with transmission reflection and deflection probabilities and respectively if all mirrors are parallel the mirror model is exactly solvable for the general case a self consistent ring kinetic equation is used to calculate the longtime tails of the velocity correlation function 0 t and the tensor correlation q 0 q t withq x y both functions showt 2 tails as opposed to the continuous lorentz gas where the tails are respectivelyt 2 andt 3 inclusion of the self consistent ring collisions increases the low density coefficient of the tail in 0 t by 30 100 as compared to the simple ring collisions depending on the model parameters

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

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

an improved class of generalized runge kutta methods for stiff problems part i the scalar case a new family of p stage methods for the numerical integration of some scalar equations and systems of odes is proposed these methods can be seen as a generalization of the explicit p stage runge kutta ones while providing better order and stability results we will show in this first part that at the cost of losing linearity in the formulas it is possible to obtain explicit a stable and l stable methods for the numerical integration of scalar autonomous odes scalar autonomous odes are of very little interest in current applications however be begin studying this kind of problems because most of the work can be easily extended to a more general situation in fact we will show in a second part entitled the separated system case that it is possible to generalize our methods so that they can be applied to some non autonomous scalar odes and systems we will obtain linearly implicit l stable methods which do not require jacobian evaluations in both parts some numerical examples are discussed in order to show the good performance of the new schemes

denumerable state nonhomogeneous markov decision processes we consider denumerable state nonhomogeneous markov decision processes and extend results from both denumerable state homogeneous and finite state nonhomogeneous problems we show that under weak ergodicity accumulation points of finite horizon optima termed algorithmic optima are average cost optimal we also establish the existence of solution horizons finally an algorithm is presented to solve problems of this class for the case where there is a unique algorithmic optimum

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

schrodinger operators with an electric field and random or deterministic potentials we prove that the schrodinger operatorh d2 dx2 v x f x has purely absolutely continuous spectrum for arbitrary constant external fieldf for a large class of potentials this result applies to many periodic almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenf 0 the spectrum is almost surely pure point with exponentially decaying eigenfunctions

picard iteration and pade approximations for stiff fractional point kinetics equations a model of stiff point kinetics equations is one of the important models in the nuclear reactor dynamics this model describes the neutron density and the precursor concentrations of delayed neutrons into nuclear reactors in this work a fractional model of the stiff point kinetics equations is studied to describe the neutron density behavior by the fractional order picard iteration and pade approximations are presented to solve the stiff fractional point kinetics equations with multi group of delayed neutrons the validity of the fractional model is tested for different cases of step ramp and sinusoidal reactivities the numerical results of picard iteration and pade approximations are computed for various fractional order the results of pade11 approximation are in good agreement with the results of picard iteration than pade01 approximation in addition the numerical results confirm that the neutron density for a positive negative reactivity is increasing decreasing quicker with decreases the fractional order

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

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies