recurrence for discrete time unitary evolutions we consider quantum dynamical systems specified by a unitary operator u and an initial state vector phi in each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state we call the system recurrent if this eventually happens with probability one we show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of u with respect to phi we also show that in the recurrent case the expected first return time is an integer or infinite for which we give a topological interpretation a key role in our theory is played by the first arrival amplitudes which turn out to be the complex conjugated taylor coefficients of the schur function of the spectral measure on the one hand this provides a direct dynamical interpretation of these coefficients on the other hand it links our definition of first return times to a large body of mathematical literature

stochastic dynamics of the scattering amplitude generating k distributed noise we derive the stochastic dynamics of the complex valued amplitude resulting from coherent scattering from a random population of scatterers when this becomes asymptotically large considerations of a random walk model introduced by jakeman are used to derive stochastic differential equations for the amplitude and corresponding intensity and phase stochastic processes an analysis of the correlation structure in the fluctuations is provided and interpreted geometrically in terms of the gauge invariant properties of the field and the markov property a fokker planck description for the evolution of the probability density is given and the equilibrium and detailed balance conditions shown to hold expressions for the intensity autocorrelation function and power spectral density are provided in closed form the practical implications of the stochastic theory are discussed

stochastic differential equation theory applied to wireless channels modeling wireless channels is essential to wireless communication systems an autoregressive ar process of order one for wireless channel has long been assumed but without a rigorous mathematical physical basis in this paper we derive a first order stochastic ar model for a flat stationary wireless channel which comes from stochastic differential equation sde theory concerning the nature of multipath fading channels the resulting ar model describes more of the origin of multipath fading channels than previous ar models and it can efficiently model and generate rayleigh distributed stationary fading channels the markovian property of the ar model is inherited through the sde approach

from quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

on the relationship between continuous and discrete time quantum walk quantum walk is one of the main tools for quantum algorithms defined by analogy to classical random walk a quantum walk is a time homogeneous quantum process on a graph both random and quantum walks can be defined either in continuous or discrete time but whereas a continuous time random walk can be obtained as the limit of a sequence of discrete time random walks the two types of quantum walk appear fundamentally different owing to the need for extra degrees of freedom in the discrete time case

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

some comments on the use of markov models in communications in their recent article feng field and haykin rediscovered an approach to the channel modelling based on stochastic differential equations sde in the following we would like to bring readers attention to a body of literature which deals with similar subjects using sde approach

entropy of some automorphisms of the ii 1 factor of the free group in infinite number of generators letl f be theii 1 factor defined by the free groupf in infinite number of generators it is shown that for a class of automorphisms ofl f arising from bijections of the set of generators off on itself and including the free shift the entropy is zero

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

orthogonal polynomials associated to almost periodic schro dinger operators a trend towards random orthogonal polynomials abstract we introduce a special class of schrodinger type h operators in l2 as h n 0 r n 1 n 1 r n n 1 rn being a nonnegative real number h satisfies the renormalization equation hd d h2 with real 2 d is the decimation operator defined by d n 0 n 2n a consequence of the renormalization equation is that the rn fulfil the recursion relation r0 0 r2nr2n 1 rn r2n r2n 1 from the above relations it can be shown that the rn are quasi periodic functions of their index n the components of the eigenfunctions of h corresponding to the eigenvalue x are the orthonormalized polynomials pn x satisfying r n 1 p n 1 x r n p n 1 x x p n x the spectrum of h is the support of the measure associated to the polynomials in the present case it is a compact perfect set of lebesque measure zero cantor set it is therefore purely singular continuous we are led to study classes of orthogonal polynomials whose three terms recursive relations are quasi periodic functions of their index we will present several results conjectures and open questions which may have relevant physical applications we study the randomness of the eigenfunctions and we discuss their algorithmic complexity

adaptive training for correlated fading channels with feedback we consider data transmission through a time selective correlated first order markov rayleigh fading channel subject to an average power constraint the channel is estimated at the receiver with a pilot signal and the estimate is fed back to the transmitter the estimate is used for coherent demodulation and to adapt the data and pilot powers we derive the hamilton jacobi bellman hjb equation for the optimal policy in a continuous time limit where the channel state evolves as an ornstein uhlenbeck diffusion process and is estimated by a kalman filter at the receiver finding an explicit solution to the hjb equation as well as proving that a twice differentiable solution exists appears to be quite challenging however assuming that such a solution does exist we explicitly determine the optimal pilot and data power control policies the optimal pilot policy switches between zero and the maximum peak constrained value bang bang control and approximates the optimal discrete time policy at low signal to noise ratios snrs equivalently large bandwidths the switching boundary is defined in terms of the system state estimated channel mean and associated error variance and can be explicitly computed under the optimal policy the transmitter conserves power by decreasing the training power when the channel is faded thereby increasing the data rate numerical results show a significant increase in achievable rate due to the adaptive training scheme with feedback relative to constant nonadaptive training which does not require feedback the gain is more pronounced at relatively low snrs and with fast fading results are further verified through monte carlo simulations

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

wrapped and unwrapped phase of radiation scattered by a discrete number of particles this paper investigates wrapped and unwrapped phase differences generated by a non gaussian scattering model the two dimensional random walk mean square values for these quantities are obtained for one and two scatterers as well as the large scatterer limit when the field constitutes a circular complex gaussian process numerical simulation is used to investigate the phase under more general fluctuation conditions and reveals that the wrapped phase difference correlation converges rapidly to that result predicted for a gaussian speckle field analytical results for the unwrapped phase indicate that this quantity transitions from a stationary process for one and two scatterers to a non stationary process in the large scatterer limit the nature of this transition is examined using numerical simulation for arbitrary scatterer number phase correlations are of consequence in various phase sensitive detection systems and this paper examines both gaussian and non gaussian fields

dual birth and death processes and orthogonal polynomials we discuss connections between birth and death processes and orthogonal polynomials we survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes birth and death processes with linear birth and death rates are studied in some detail we also mention some results concerning birth and death processes whose transition rates are rational functions

birth and death processes and quantum spin chains this paper underscores the intimate connection between the quantum walks generated by certain semi infinite spin chain hamiltonians and classical birth and death processes it is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the karlin mcgregor representation formula of the transition probability functions for classes of birth and death processes as an application we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1

dynamical entropy for automorphisms of exact c algebras abstract some estimations of dynamical entropies are given by applying the entropy ht for an automorphism of a c algebra a with respect to invariant state this ht coincides with the connes narnhofer thirring cnt entropy h when a is abelian let a be the reduced free product of ai i i i if ai i i are finite dimensional then h ht 0 for all aut a concerning the free product 1 2 of i aut ai i we have always that ht 1 2 1 2 max ht 1 1 ht 2 2 and the cnt entropy satisfies this formula for all automorphisms in the ergodic theory

linear interference cancellation in cdma based on iterative techniques for linear equation systems it has previously been shown that well known iterations for solving a set of linear equations correspond to linear interference cancellation structures here we suggest applying a block wise iteration that consists of an outer and an inner iteration the outer iteration used is the gauss seidel gs method while for the inner iteration we study direct matrix inversion the jacobi over relaxation iteration and the conjugate gradient iteration when a true inner iteration is used this approach allows for a timely derivation of the acceleration parameters required by advanced iterations the block iteration is based on a symbol level implementation which leads to the same detection delay profile for both parallel and serial structures at the expense of differences in the amount of serial processing required this is discussed in some detail and quantified for comparison the performance of the detectors is studied via computer simulations where it is found that the block approach can provide significantly faster convergence leading to improved detection delay over the simpler gs iteration the improvements are obtained at the expense of an increase in the required serial processing speed

matrix valued szegő polynomials and quantum random walks we consider quantum random walks qrw on the integers a subject that has been considered in the last few years in the framework of quantum computation we show how the theory of cmv matrices gives a natural tool to study these processes and to give results that are analogous to those that karlin and mcgregor developed to study classical birth and death processes using orthogonal polynomials on the real line in perfect analogy with the classical case the study of qrws on the set of nonnegative integers can be handled using scalar valued laurent polynomials and a scalar valued measure on the circle in the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions we show how our tools yield results in the well known case of the hadamard walk but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods more precisely we consider qrws on the set of nonnegative integers the analysis of these cases leads to phenomena that are absent in the case of qrws on the integers even if one restricts oneself to a constant coin this is illustrated here by studying recurrence properties of the walk but the same method can be used for other purposes the presentation here aims at being self contained but we refrain from trying to give an introduction to quantum random walks a subject well surveyed in the literature we quote for two excellent reviews see 1 9 see also the recent notes 20 2009 wiley periodicals inc

dynamical entropy of generalized quantum markov chains over infinite dimensional algebras we compute the dynamical entropy in the sense of connes narnhofer and thirring of generalized quantum markov chain over infinite dimensional algebras for the case in which the transition expectation is defined by a set of conditional density amplitudes we show that the dynamical entropy is equal to the mean entropy thus we extend the result of park lett math phys 32 63 74 1994 to non af type quantum markov chains we employ the main method developed in park lett math phys 32 63 74 1994 with necessary modifications

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

on first order markov modeling for the rayleigh fading channel previous models for the received signal amplitude of the flat fading channel that use first order finite state markov chains are examined the stochastic properties of a proposed first order model based on these models are examined the limitations of using an information theoretic metric which is sometimes used to justify a first order markov chain as a sufficient model for very slowly fading channels are discussed a simple method of qualitatively comparing autocorrelation functions is instead proposed the usefulness of the first order markov chain in representing the flat fading channel is examined by looking at two specific problems in wireless system applications that represent two disparate cases the first case involves analysis over a short duration of time relative to the inverse of the normalized doppler frequency while the second involves analysis over a long duration of time contrary to previous reports the results indicate that first order markov chains are not generally suitable for very slowly fading channels rather first order markov chains can be suitable for very slowly fading applications which require analysis over only a short duration of time

three routes to the exact asymptotics for the one dimensional quantum walk we demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one coin quantum walk on the infinite line via the jacobi polynomials that arise in the path integral representation we calculate the asymptotics using a method that is significantly easier to use than the darboux method it also provides a single integral representation for the wavefunction that works over the full range of positions n including throughout the transitional range where the behaviour changes from oscillatory to exponential previous analyses of this system have run into difficulties in the transitional range because the approximations on which they were based break down here the fact that there are two different kinds of approach to this problem path integral versus schrodinger wave mechanics is ultimately a manifestation of the equivalence between the path integral formulation of quantum mechanics and the original formulation developed in the 1920s we also discuss how and why our approach is related to the two methods that have already been used to analyse these systems

coupled dynamics of populations supported by discrete sites and their continuum limit the illumination of single population behaviour subject to the processes of birth death and immigration has provided a basis for the discussion of the non gaussian statistical and temporal correlation properties of scattered radiation as a first step towards the modelling of its spatial correlations we consider the populations supported by an infinite chain of discrete sites each subject to birth death and immigration and coupled by migration between adjacent sites to provide some motivation and illustrate the techniques we will use the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived a certain continuum limit is identified and its properties studied via asymptotic analysis this forms the basis of the multi particle model of a coupled population subject to single site birth death and immigration processes in addition to inter site migration a discrete rate equation is formulated and its generating function dynamics derived this facilitates derivation of the equations of motion for the first and second order cumulants thus generalizing the earlier results of bailey through the incorporation of immigration at each site we present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter site variances in the closed form the results for the first two moments at a single time are used to derive expressions for the asymptotic time delayed correlation functions which relates to glauber s analysis of an ising model the paper concludes with an analysis of the continuum limit of the birth death immigration migration process in terms of a path integral formalism the continuum rate equation and evolution equation for the generating function are developed from which the evolution equation of the mean occupancy is derived in this limit its solution is provided in closed form

quantum intermittency in almost periodic lattice systems derived from their spectral properties abstract hamiltonian tridiagonal matrices characterized by multi fractal spectral measures in the family of iterated function systems can be constructed by a recursive technique described here we prove that these hamiltonians are almost periodic they are suited to describe quantum lattice systems with nearest neighbours coupling as well as chains of linear classical oscillators and electrical transmission lines we investigate numerically and theoretically the time dynamics of the systems so constructed we derive a relation linking the long time power law behaviour of the moments of the position operator expressed by a scaling function of the moment order and spectral multi fractal dimensions dq via d1 we show cases in which this relation is exact and cases where it is only approximate unveiling the reasons for the discrepancies

dynamical models of weak scattering we extend a recent diffusion model in which the continuous time dynamics of the k scattering process have been proposed to include the effect of the presence of a coherent offset in the scattering amplitude the weak scattering amplitudes are characterized in terms of continuous time biased random walk models and the corresponding stochastic dynamics derived the stochastic differential geometry of the resultant amplitude fluctuations is derived in relation to that of pure k scattering asymptotic distributions of amplitude intensity and phase are provided and the condition for detailed balance shown to hold