random magnetic fields on line graphs we study the spectral and transport properties of schrodinger operators on line graphs with random magnetic fields we show that it has a pure point spectrum with exponentially decaying eigenfunctions on spectral edges whereas there appears an eigenvalue with infinite multiplicity due to the structure of line graphs we compute the electrical conductivity which is zero on spectral edges but is nonzero and finite on the isolated eigenvalue mentioned above some related problems are also discussed

quasiperiodicity and phase locking in stochastic circle maps a spectral approach abstract while there are clear definitions of what it means for a deterministic dynamical system to be periodic quasiperiodic or chaotic it is unclear how to define such notions for a noisy system in this article we study markov chains on the circle which is a natural stochastic analog of deterministic dynamical systems the main tool is spectral analysis of the transition operator of the markov chain we analyze path wise dynamic properties of the markov chain such as stochastic periodicity or phase locking and stochastic quasiperiodicity and show how these properties are read off of the geometry of the spectrum of the transition operator

ferromagnetic ground states for the hubbard model on line graphs the author discusses some of the properties of the hubbard model on a line graph with n vertices it is shown that the model has ferromagnetic ground states if the interaction is repulsive u 0 and if the number of electrons n satisfies 2n or n or m m is a natural number that depends on the line graph for example the kagome lattice is a line graph it has m 5n 3 1

a rate distortion theorem for arbitrary discrete sources a rate distortion theorem for arbitrary not necessarily stationary or ergodic discrete time finite alphabet sources is given this result which provides the expression of the minimum spl epsiv achievable fixed length coding rate subject to a fidelity criterion extends a recent data compression theorem by steinberg and verdu see ibid vol 42 p 63 86 jan 1996

noise induced effects on period doubling bifurcation for integrate and fire oscillators this study provides a method for calculating first order approximations of the lyapunov exponents of systems with discontinuity in the presence of noise in order to characterize the stability of motions in those systems this paper in particular illustrates the results of the ways in which noise influences period doubling bifurcation in the parameter space of an integrate and fire model with a periodically modulated reset level for evaluating a stochastic version of period doubling bifurcation the first passage time problem of the ornstein uhlenbeck process is used to define a markov operator governing the transition of a reset level phase density the results on the stochastic bifurcation are thus compared with numerical computations of angles and moduli of eigenvalues of the markov operator that describes the firing properties of the model

the spectrum of infinite regular line graphs let g be an infinite d regular graph and l g its line graph we consider discrete laplacians on g and l g and show the exact relation between the spectrum of ag and that of al g our method is also applicable to di d2 semiregular graphs subdivision graphs and para line graphs

fluxes laplacians and kasteleyn s theorem the genesis of this paper was an attempt to understand a problem in condensed matter physics related to questions about electron correlations superconductivity and electron magnetic field interactions the basic idea which was proposed a few years ago is that a magnetic field can lower the energy of electrons when the electron density is not small certain very specific and very interesting mathematical conjectures about eigenvalues of the laplacian were made and the present paper contains a proof of some of them furthermore those conjectures lead to additional natural conjectures about determinants of laplacians which we both present and prove here it is not clear whether these determinantal theorems have physical applications but they might conceivably in the context of quantum field theory some but not all of the results given here were announced earlier in le

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

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

zariski closure of subgroups of the symplectic group and lyapunov exponents of the schrodinger operator on the strip we consider the schrodinger equation with a random potential y n 1 q n ei y n y n 1 0 whereqn is a sequence of independent identically distributed random symmetricm m matrices with real valued elements yn m n e is the real parameter andi is the identity matrix we show that if the smallest jordan algebra of matrices containing the support of the distribution of matricesqn coincides with jordan algebra of all real valued symmetric matrices then for all but maybe a finite number of values ofe all the lyapunov exponents of our schrodinger equation are different and thus the spectrum of the corresponding schrodinger operator is pure point

minimal attractors and bifurcations of random dynamical systems we consider attractors for certain types of random dynamical systems these are skew product systems whose base transformations preserve an ergodic invariant measure we discuss definitions of invariant sets attractors and invariant measures for deterministic and random dynamical systems under assumptions that include for example iterated function systems but that exclude stochastic differential equations we demonstrate how random attractors can be seen as examples of milnor attractors for a skew product system we discuss the minimality of these attractors and invariant measures supported by them as a further connection between random dynamical systems and deterministic dynamical systems we show how dynamical or d bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations and we relate the issue of branching at such d bifurcations to branching at blowout bifurcations

graph invertibility and median eigenvalues abstract let g w be a weighted graph with a weight function w e g r 0 a weighted graph g w is invertible to a new weighted graph if its adjacency matrix is invertible graph inverses have combinatorial interests and can be applied to bound median eigenvalues of graphs such as have physical meanings in this paper we characterize the inverse of a weighted graph based on its sachs subgraphs that are spanning subgraphs with only k 2 or cycles or loops as components the characterization can be used to find the inverse of a weighted graph based on its structures instead of its adjacency matrix if a graph has its spectra split about the origin i e half of eigenvalues are positive and half of them are negative then its median eigenvalues can be bounded by estimating the largest and smallest eigenvalues of its inverse we characterize graphs with a unique sachs subgraph and prove that these graphs has their spectra split about the origin if they have a perfect matching as applications we show that the median eigenvalues of stellated graphs of trees and corona graphs belong to different halves of the interval 1 1

simulation of random processes and rate distortion theory we study the randomness necessary for the simulation of a random process with given distributions on terms of the finite precision resolvability of the process finite precision resolvability is defined as the minimal random bit rate required by the simulator as a function of the accuracy with which the distributions are replicated the accuracy is quantified by means of various measures variational distance divergence orstein 1973 prohorov 1956 and related measures of distance between the distributions of random process in the case of ornstein prohorov and other distances of the kantorovich vasershtein type we show that the finite precision resolvability is equal to the rate distortion function with a fidelity criterion derived from the accuracy measure this connection leads to new results on nonstationary rate distortion theory in the case of variational distance the resolvability of stationary ergodic processes is shown to equal entropy rate regardless of the allowed accuracy in the case of normalized divergence explicit expressions for finite precision resolvability are obtained in many cases of interest and connections with data compression with minimum probability of block error 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

stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise a first passage time approach noise effects on phase lockings in a system consisting of a piecewise linear van der pol relaxation oscillator driven by a periodic input are studied the problem of finding the period of the oscillator is reduced to the first passage time problem of the ornstein uhlenbeck process with time varying boundary the probability density functions of the first passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator

spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally forced van der pol oscillator with additive noise noise effects on the phase lockings and bifurcations in the sinusoidally forced van der pol relaxation oscillator are investigated deterministic noise free one dimensional poincare mapping is extended to the iteration of the operator defined by a stochastic kernel function stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator in particular a new method which uses spectra eigenvalues and eigenfunctions of the operator to analyze stochastic bifurcations intensively is proposed

mode locking and arnold tongues in integrate and fire neural oscillators an analysis of mode locked solutions that may arise in periodically forced integrate and fire if neural oscillators is introduced based upon a firing map formulation of the dynamics a q p mode locked solution is identified with a spike train in which p firing events occur in a period q ensuremath delta where ensuremath delta is the forcing period a linear stability analysis of the map of firing times around such solutions allows the determination of the arnold tongue structure for regions in parameter space where stable solutions exist the analysis is verified against direct numerical simulations for both a sinusoidally forced if system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current this approach is extended to the case of two synaptically coupled if oscillators showing that mode locked states can exist for some self consistently determined common period of repetitive firing numerical simulations show that such solutions have a bursting structure where regions of spiking activity are interspersed with quiescent periods before repeating the influence of the synaptic current upon the arnold tongue structure is explored in the regime of weak coupling

characterization of stochastic bifurcations in a simple biological oscillator this study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first passage time problem of the ornstein uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density stochastic phase locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase a firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

eigenbasis of the evolution operator of 2 tessellable quantum walks abstract staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well known discrete time quantum walk models in this work we address the class of 2 tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator by interpreting the evolution operator as a quantum markov chain on an underlying multigraph we define the concept of quantum detailed balance which helps to obtain the eigenbasis a subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum markov chain to obtain the remaining eigenvectors we have to use the quantum detailed balance conditions if the quantum markov chain has a quantum detailed balance there is an eigenvector for each fundamental cycle of the underlying multigraph if the quantum markov chain does not have a quantum detailed balance we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors we exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice the line graph of the hexagonal lattice which has symmetry properties that help in the calculation process

a quantum walk induced by hoffman graphs and its periodicity abstract in order to define the discrete time quantum walks we divide the vertex set of a graph by two partitions and give a time evolution operator by a product of two local unitary operators on each partition in this paper we make such partitions from a hoffman graph which gives generalization of line graphs and study a staggered walk we first show that the square of the time evolution operator of the grover walk is unitarily equivalent to that of the staggered walk on the generalized line graph induced from a hoffman graph furthermore we focus on the periodicity of quantum walks and show that the staggered walk on the generalized line graph is periodic whenever the grover walk on the original graph is periodic by using the unitary equivalence this work is the first trial to develop the relation between quantum walks and hoffman graphs

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

applications of automata and graphs labeling operators in hilbert space ii we introduced a family of infinite graphs directly associated with a class of von neumann automaton model ag these are finite state models used in symbolic dynamics stimuli models and in control theory in the context of groupoid von neumann algebras and an associated fractal group we prove a classification theorem for representations of automata

fractafolds based on the sierpinski gasket and their spectra we introduce the notion of fractafold which is to a fractal what a manifold is to a euclidean half space we specialize to the case when the fractal is the sierpinski gasket sg we show that each such compact fractafold can be given by a cellular construction based on a finite cell graph g which is 3 regular in the case that the fractafold has no boundary we show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph using the spectral decimation method of fukushima and shima this enables us to obtain isospectral pairs of nonhomeomorphic fractafolds we also show that although sg is topologically rigid there are fractafolds based on sg that are not topologically rigid