orthogonalization process by recurrence relations an orthogonalization process is proposed applicable to spaces which are realizations of abstract hilbert space it is simpler than the gram schmidt process a recurrence relation which orthogonalizes a physical space is proposed and it is shown that the generalized langevin equation is contained therein this process serves as a basis for understanding the nature of the dynamic many body formalism

a new approach to multipath correction of constant modulus signals an adaptive digital filtering algorithm that can compensate for both frequency selective multipath and interference on constant envelope modulated signals is presented the method exploits the fact that multipath reception and various interference sources generate incidental amplitude modulation on the received signal a class of so called constant modulus performance functions is developed which sense this am term but are insensitive to the angle modulation simple adaptive algorithms for finite impulse response fir digital filters are developed which employ a gradient search of the performance function one of the resulting algorithms is simulated for the example of an fm signal degraded by specular multipath propagation substantial improvements in noise power ratio npr are observed e g 25 db with moderately rapid convergence time these results are then extended to include tonal interference on a fm signal and intersymbol interference on a qpsk data signal

observation of symmetry breaking state selection and sensitivity in a noisy electronic system abstract measurements of the time evolution of the statistical density as a control parameter is swept through a bifurcation point in the presence of additive noise are presented we have used these densities to study branch selectivity which results from a small additive interaction a universal time scale for all such processes is established the results are in good agreement with a recent theory

estimates on the fundamental solution to heat flows with uniformly elliptic coefficients we obtain estimates from above and below on the fundamental solution to the cauchy initial value problem for parabolic equations having uniformly elliptic coefficients we impose minimal regularity requirements on the coefficients the estimates are expressed in terms of an energy functional associated with the coefficients

diffusion determines the manifold we prove under a weak smoothness condition that two riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the dirichlet type heat semigroups on the manifolds

dimensionality dependence of dynamical correlations exact results from a quantum many body system using one of the simplest model interacting quantum many body systems it is shown that the relaxation function behaves remarkably differently at low temperatures depending upon whether the lattice dimensionality of the system d t t c in quantum many body systems

geometric bounds on the growth rate of null controllability cost for the heat equation in small time given a control region omega on a compact riemannian manifold m we consider the heat equation with a source term g localized in omega it is known that any initial data in l 2 m can be steered to 0 in an arbitrarily small time t by applying a suitable control g in l 2 0 t times omega and as t tends to 0 the norm of g grows like exp c t times the norm of the data we investigate how c depends on the geometry of omega 72 words we prove c geq d 2 4 where d is the largest distance of a point in m from omega when m is a segment of length l controlled at one end we prove c leq alpha l 2 for some alpha 2 moreover this bound implies c leq alpha l omega 2 where l omega is the length of the longest generalized geodesic in m which does not intersect omega the em control transmutation method used in proving this last result is of a broader interest

deformation of the three term recursion relation and generation of new orthogonal polynomials we find solutions for a linear deformation of the three term recursion relation the orthogonal polynomials of the first and second kind associated with the deformed relation are obtained the new density weight function is written in terms of the original one and the deformation parameters

spin dynamics of the quantum xy chain and ladder in a random field abstract we investigate the hamiltonian dynamics of two low dimensional quantum spin systems in a random field at the infinite temperature limit the xy chain and the two leg xy ladder with interchain ising interactions we determine the longitudinal spin autocorrelation functions of the spin 1 2 xy chain and ladder in the presence of disordered fields by using the method of recurrence relations the first six basis vectors for the chain and the first four basis vectors for the ladder of the dynamic hilbert spaces of jz t as well as the corresponding recurrents and moments of the time dependent autocorrelation function are analytically computed for bimodal distributions of the fields we did find a remarkable result in the disordered models cases with a fraction of p sites under field bb and a fraction of 1 p sites under the field ba have the same longitudinal dynamics as those with p sites under field ba and 1 p sites under the field bb we also find that both the xy chain and the two leg xy ladder with ising interchain coupling in the presence of random fields are sensitive to the percentage of disorder but not to the intensity of the fields

dynamics of a transverse ising model with four spin interactions abstract we report the first study of the dynamics of a transverse ising model with four spin interactions we consider finite chains of increasing size in order to determine the time dependent correlation function as well as the longitudinal relaxation function of the infinite chain up to 8 exact 5 approximate moments of the infinite chain were also obtained the correlation function is oscillatory for large values of the transverse field and is monotonically decreasing for small values of the transverse field the system undergoes a crossover from a collective mode excitation regime to a central mode type of dynamics as the strenght of field is lowered the present model does not allow gaussian decay for the correlation function

analytic expressions for asymptotic forms of continued fraction coefficients in the presence of a spectral gap it is shown that the recursion method i e the continued fraction expansion of the diagonal element of the resolvent is equivalent to the exponential toda lattice in the sense that the limiting identities in the former method have the same forms as the conservation laws in the latter lattice the asymptotic continued fraction coefficients recursion coefficients are then related to a particular motion in the toda lattice as a result of this relation the analytic expressions for the simply oscillating asymptotic recursion coefficients in the two band single gap case are given explicitly in terms of the jacobian elliptic functions and in the fourier series forms

heisenberg langevin and current equations via the recurrence relations approach some years ago the heisenberg equation of motion was formally solved by the recurrence relations approach it is shown here that the langevin equation represents a structural property of the recurrence relations the langevin equation is useful for studying the time evolution of the current the resulting current current correlation function is compared with luttinger s phenomenological theory geometric interpretations are made for the conductivity and the dielectric function

intrinsic geometry and analysis of diffusion processes and l variational problems the aim of this paper is twofold first we obtain a better understanding of the intrinsic distance of diffusion processes precisely a for all n 1 the diffusion matrix a is weak upper semicontinuous on if and only if the intrinsic differential and the local intrinsic distance structures coincide b if n 1 or if n 2 and a is weak upper semicontinuous on the intrinsic distance and differential structures always coincide c if n 2 and a fails to be weak upper semicontinuous on the non coincidence of the intrinsic distance and differential structures depend on the geometry of the non weak upper semicontinuity set of a second for an arbitrary diffusion matrix a we show that the intrinsic distance completely determines the absolute minimizer of the corresponding l variational problem and then obtain the existence and uniqueness for given boundary data we also give an example of a diffusion matrix a for which there is an absolute minimizer that is not of class c1 when a is continuous we also obtain the linear approximation property of the absolute minimizer

a generalized langevin equation for dealing with nonadditive fluctuations a suitable extension of the mori memory function formalism to the non hermitian case allows a multiplicative process to be described by a langevin equation of non markoffian nature this generalized langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well known theoretical approach developed by kubo the stochastic liouville equation sle theory it is shown furthermore that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath the stochastic process under study can also be described by a fokker planck like equation which results in a gaussian equilibrium distribution for the variable of interest the main flaw of the sle theory that resulting in an uncorrect equilibrium distribution is therefore completely eliminated

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

ruelle resonances in kicked quantum spin chain abstract we study exponential decay of high temperature time correlation functions in a non integrable quantum spin chain problem namely ising spin 1 2 chain kicked with tilted homogeneous magnetic field for this purpose we define a master propagator over a suitable banach space of quantum observables quantum many body analogue of perron frobenius operator whose leading eigenvalue determines the asymptotic decay of correlations this is demonstrated with explicit calculation for which a fast algorithm for the construction of the master propagator is developed

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

wavelet analysis of time series for the duffing oscillator the detection of order within chaos we consider a wavelet analysis of various time series for the duffing oscillator for which there is a potential maximum and a harmonic forcing term and we focus on time series that return to the region of the potential maximum when the dynamics is chaotic and the time series is highly nonstationary there are many significant higher harmonics in a fourier expansion and the usual fourier analysis is problematic especially for short total times we show that the wavelet analysis is a robust tool that may be used to obtain qualitative information for highly nonstationary time series specifically that it may be used to detect a small amplitude harmonic forcing term even when the dynamics is chaotic and even for short total times

properties of noise correlation functions of langevin like equations abstract the stochastic properties of the noise in langevin type equations are studied it is demonstrated that the noise is non gaussian and that the three point noise correlation function has an exponential decay on the slow time scale explicit calculations for systems containing one slow linear variable are carried out using mode coupling techniques the results are easily extended to four and higher order correlation functions and to systems in which there is more than one slow linear variable in particular the noise in the hydrodynamic langevin equations has the properties described above

on the relaxation of fluctuations in the steady state of the stratonovich model a detailed investigation has been carried out of the relaxation of fluctuations in the steady state of the stratonovich model also known as the random growing rate model the autocorrelation time of this system driven by parametric white noise in the linear term has been measured for an electronic circuit model computed by digital simulation and calculated by use of a continued fraction expansion method the results are all consistent with each other and with those obtained from the matrix continued fraction method of jung and risken 1985 provided that explicit account is taken of the ways in which any macroscopic real physical system is bound to differ from the idealisation represented by the original form of the stratonovich model in particular it is necessary to recognise the existence of weak additive noise and a small additive constant the physical origins of some earlier seemingly discrepant calculations and experimental data are discussed

dynamics of the independent oscillator io model in the einstein and pseudo debye limits abstract the independent oscillator model is thought to be a simple model for brownian motion explored extensively by ford lewis and o connell this model does not specify the distribution of the oscillators frequencies by considering the distribution to be of two standard limits otherwise preserving the hermiticity of the model we obtain the memory function exactly by the method of recurrence relations in one limit the memory function is purely oscillatory while in the other limit it shows a decreasing amplitude with an oscillation relevance to transport theory is briefly pointed out

asymptotic solution for first and second order linear volterra integro differential equations with convolution kernels this paper addresses the problem of finding an asymptotic solution for first and second order integro differential equations containing an arbitrary kernel by evaluating the corresponding inverse laplace and fourier transforms the aim of the paper is to go beyond the tauberian theorem in the case of integral differential equations which are widely used by the scientific community the results are applied to the convolute form of the lindblad equation setting generic conditions on the kernel in such a way as to generate a positive definite density matrix and show that the structure of the eigenvalues of the correspondent liouvillian operator plays a crucial role in determining the positivity of the density matrix

solving the liouville equation for conservative systems continued fraction formalism and a simple application abstract there are very few formalisms available to solve the liouville equation for energy conserved systems the continued fraction formalism introduced through the works of zwanzig mori kubo lee grigolini and others is described here a detailed discussion is presented on ways to apply the continued fraction formalism to solve for relaxation functions and for the dynamical variable itself for a simple and exactly solvable quantum spin system

coloured thermal noise driven dynamical system upper bound of time derivative of information entropy in this paper we have studied the upper bound of the time derivative of information entropy for non markovian and thermodynamically closed system s using reduced model theory rmt the upper bound is calculated on the basis of the fokker planck equation and the schwartz inequality principle our calculation shows that the upper bound exhibits extremal nature in the variation of system parameters such as noise correlation time dissipation strength the present calculation also considers how the upper bound does change if we increase the number of auxiliary variables involved in the rmt

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

on the blind equalization of continuous phase modulated signals using the constant modulus criterion the use of the constant modulus cm criterion to achieve blind equalization of continuous phase modulated cpm signals is investigated in this paper solutions to the cm criterion are characterized it is shown that blind equalization using the cm criterion may be unsuccessful in the sense that the output of the equalizer does not necessarily coincides with a delayed and rotated version of the input signal an indeterminacy indeed remains and this residual indeterminacy is characterized as an application of our results a blind equalization criterion that allows us to eliminate the latter indeterminacy is proposed simulation results sustain theoretical claims and illustrate the attractive performance of the proposed criterion

fluctuation dissipation dispersion relation and quality factor for slow processes a generalization of the callen welton formula for systems with slowly varying parameters is given using the momentum method and the time multiscale technique it is shown that not only the dissipation but also the dispersive contributions determine the spectral function of the fluctuations in these systems the general formalism is illustrated for an oscillating electrical circuit and the influence of the dispersion contributions on the quality factor of the system is discussed

short time asymptotics in dirichlet spaces this work is involved with the short time asymptotics of the heat semigroup in a general setting a generalization of fang s version of varadhan s formula is proved a spectral gap or the possibility of obtaining one by an appropriate change of measure is required 2001 john wiley sons inc

relaxation in a duffing potential abstract the solution of the equation of motion for a particle in a duffing potential v x 1 x 2 2 2 x 4 4 1 2 0 for arbitrary anharmonicity strength is characterized by the presence of odd frequencies which implies that velocity and position autocorrelation functions of such an oscillator in a microcanonical ensemble are also characterized by odd frequencies it is however non trivial to determine whether such discrete frequencies also characterize the autocorrelation functions in a canonical ensemble as discussed recently by fronzoni et al j stat phys 41 1985 553 we recover and extend upon the results of fronzoni et al to show analytically via mori lee theory that essentially discrete i e well defined peaks with finite but small width temperature dependent frequencies characterize the autocorrelation functions in a canonical ensemble

a large deviation result for a class of dirichlet processes let 0 be a constant and let 1 q x be a bounded measurable function we consider the large deviation problem for the diffusion process with generator we are going to prove that its density function satisfies mathop lim limits t to 0 t log h y t x frac left x y right 2 2

universal formula for extreme first passage statistics of diffusion the timescales of many physical chemical and biological processes are determined by first passage times fpts of diffusion the overwhelming majority of fpt research studies the time it takes a single diffusive searcher to find a target however the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers this fastest fpt depends on extremely rare events and has a drastically faster timescale than the fpt of a given single searcher in this work we prove a simple explicit formula for every moment of the fastest fpt the formula is remarkably universal as it holds for d dimensional diffusion processes i with general space dependent diffusivities and force fields ii on riemannian manifolds iii in the presence of reflecting obstacles and iv with partially absorbing targets our results rigorously confirm generalize correct and unify various conjectures and heuristics about the fastest fpt

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

large deviations for the yang mills measure on a compact surface we prove the first mathematical result relating the yang mills measure on a compact surface and the yang mills energy we show that at the small volume limit the scaled yang mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the yang mills energy

approximate gaussian representation of evolution equations i single degree of freedom nonlinear equations we have developed a generalization of the method of statistical linearization to enable us to describe transient and other nonstationary phenomena obeying stochastic nonlinear differential equations this approximation technique provides an optimal gaussian representation with time dependent parameters the algorithm specifies a set of ordinary differential equations for the gaussian parameters in terms of the time dependent average nonlinearities we apply the general formalism developed herein for single degree of freedom dissipative systems to a particular example

classical relaxation in a double well abstract the canonical ensemble position and velocity autocorrelation functions x and v acf s and the power spectra are obtained for a classical particle at t barrier height in a symmetric double well dw potential we report the presence of well defined temperature dependent characteristics frequencies in the vacf and the xacf both above and below the harmonic frequency in this strongly anharmonic regime a striking feature of the dw relaxation is that the v x acf s are weakly temperature dependent at t

the analytic inversion of any finite symmetric tridiagonal matrix we use the theory of orthogonal polynomials to write down explicit expressions for the polynomials of the first and second kind associated with a given infinite symmetric tridagonal matrix h the green s function is the inverse of the infinite symmetric tridiagonal matrix h zi by calculating the inverse of the finite symmetric tridiagonal matrix we can find the analytical form of the inverse of the finite symmetric tridiagonal matrix

optimal kernel estimates for elliptic operators with second order discontinuous coefficients abstract we consider the family of second order elliptic operators l a 1 i j 1 n x i x j x 2 d i j c x x 2 b x 2 a 0 b c r which includes the schrodinger operator with inverse square potential and we prove optimal upper bounds for the heat kernel

derivation of the generalized langevin equation by a method of recurrence relations the generalized langevin equation was first derived by mori using the gram schmidt orthogonalization process this equation can also be derived by a method of recurrence relations for a physical space commonly used in statistical mechanics the recurrence relations are simple and they lead directly to the langevin equation the langevin equation is shown to be composed of one homogeneous and one inhomogeneous equation

dynamical correlations and the direct summation method of evaluating infinite continued fractions the mori lee formalism for solving the liouville or heisenberg equation of motion for hermitian systems demonstrates that the laplace transformed dynamical correlations in canonical ensembles can be written as in finite continued fractions we show that a model independent direct summation method allows accurate numerical evaluation of all known classes of these in finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many body and few body systems some studies on dynamical correlations in s 1 2 quantum spin chains are cited as applications of the method presented

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

lattice green function in uniform magnetic fields the green function of a square tight binding model in magnetic fields is expressed by means of continued fractions for rational fields the expression is reduced to a more useful one the green function expressing the out going wave is numerically computed at arbitrary sites for the first time it is found that the absolute value of the green function takes maximum values on the reciprocal lattice of the magnetic brillouin zone

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

a new method in the many body problem ii the earlier proposed method for exact determination of the energy spectra of hamiltonians h and the quantities where w is a finite dimensional perturbation is extended to treat a still larger class of hamiltonians in particular some two and three dimensional problems became amenable to this approach in addition the expressions for the density and local densities of states are deduced as an example an impurity problem is considered in a simple cubic lattice with short range and in a linear chain with long range interactions

oscillatory behavior of solutions of forced second order differential equations with alternating coefficients abstract in this paper we study the oscillatory behavior of solutions of the forced second order nonlinear ordinary functional differential equation x t g x t p t d dt sufficient conditions are established for all solutions of the equation to be oscillatory as well as bounded

aspects of non ergpdicity in hermitian systems abstract it is shown that some of the simplest interacting many body systems exhibiting non ergodicity necessarily satisfy certain identities involving their equilibrium multipoint static correlations these identities may be useful in the study of model non ergodic systems being analysed or simulated some examples of pertinent model non ergodic systems are given to illustrate the material discussed

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

relaxation in nonlinear systems nonconvergent infinite continued fractions and sensitive relaxation processes abstract the mori lee treatment of linear response theory demonstrates that the laplace transform of any relaxation function of a dynamical variable can be expressed as a continued fraction for certain simple nonlinear systems the continued fraction representation of the relaxation functions cannot be evaluated perturbatively it turns out that many body systems with even a single on site nonlinearity can dramatically alter the relaxation behavior in such systems we argue that nonperturbative continued fractions in the mori lee formalism are necessarily associated with systems that exhibit relaxation that is sensitive to the presence of nonlinearities

on the linearization of nonlinear langevin type stochastic differential equations we show that a logical extension of the piecewise optimal linearization procedure leads to the gaussian decoupling scheme where no iteration is required the scheme is equivalent to solving a few coupled equations the method is applied to models which represent a a single steady state b passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state and c a bivariate case of passage from an unstable state to a final stable state the results are shown to be in very good agreement with the monte carlo calculations carried out for these cases the method should be of much value in multidimensional cases

the duffing oscillator in the low friction limit theory and analog simulation by a joint use of theory and analog simulation the low friction regime of the duffing oscillator is explored in the weak temperature case it is shown that the low friction regime in turn must be divided in two well distinct subregimes in the former one characterized by the friction ranging from 2 0 0 is the harmonic frequency up to a lower bound t the method of statistical linearization applies and a continued fraction procedure cfp generated by the zwanzig mori projection techniques is shown to provide correct information for both the renormalized frequency of the oscillator and the corresponding line shape the latter subregime characterized by the friction ranging from 0 to t is fraught with the complete breakdown of the statistical linearization method the cfp is shown to provide an incorrect picture of the line shape while suggesting a novel mean field approximation which is then proven analytically via an alternative method of calculation this method consists of expressing the system in a form reminiscent of the model of kubo s stochastic oscillator by using this alternative approach we are in a position to account for the residual linewidth which is shown by the analog experiment to survive for 0

ill convergence of godard blind equalizers in data communication systems the existence of stable undesirable equilibria for the godard algorithm is demonstrated through a simple autoregressive ar channel model these undesirable equilibria correspond to local but nonglobal minima of the underlying mean cost function and thus permit the ill convergence of the godard algorithms which are stochastic gradient descent in nature simulation results confirm predicted misbehavior for channel input of constant modulus it is shown that attaining the global minimum of the mean cost necessarily implies correct equalization a criterion is also presented for allowing a decision at the equalizer as to whether a global or nonglobal minimum has been reached

ergodicity in simple and not so simple systems and kubo s condition abstract the ergodic hypothesis is approached from the side of time averaging to ascertain regions of its validity a necessary and sufficient condition is formulated in terms of certain infinite products and tested in models of nn coupled harmonic oscillators the validity of the hypothesis requires energy delocalizability indicated by the existence of coherent translation modes kubo s condition is re examined to see the origin of its limitations

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

generalized langevin equation and recurrence relations the generalized langevin equation gle is a reformulation of the heisenberg equation of motion and hence an exact equation it is the basis of the memory function approach a very widely used method for studying dynamics of classical and quantum fluids the gle was first derived by mori in a very formal way a much simpler and more physically motivated derivation was given by us some years later in this work we provide perhaps the simplest possible derivation of the gle the simplicity of the derivation helps to bring out the subtleties present in this important dynamical relationship

why irreversibility is not a sufficient condition for ergodicity khinchin s theorem of ergodicity is examined by means of linear response theory the resulting ergodic condition shows that contrary to the theorem irreversibility is not a sufficient condition for ergodicity by the recurrence relations method we prove that irreversibility is broader in scope than ergodicity showing why it can only be a necessary condition for ergodicity

momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses iii some limiting cases abstract the momentum autocorrelation function of a mass impurity in a classic diatomic chain is studied using the recurrence relations method general expressions for the contributions of branch cuts and resonant poles have been derived and illustrated in previous papers i and ii respectively in the present paper a series of limiting cases that any one of the three masses m 0 m 1 m 2 approaches to zero or infinity are analyzed it is found that the cases m 0 0 and 2 m 2 are closely related to each other and that the general expressions for the amplitudes are valid also in the limits 0 and the ergodicity in the case m 2 0 is studied and the ratio of two specific infinite products is obtained

analytic properties of the recursion method in the presence of band gaps the recursion method gives a good approximation of the density of states in the single band case where the recursion coefficients converge when band gaps exist it is known that they exhibit asymptotic undamped oscillations in order to clarify these asymptotic behaviours an analytic study of the recursion method applicable to the band gap case is presented the asymptotic properties of the system of orthogonal polynomials to which this method is closely related are investigated by use of the concept of the exterior mapping function various relations which connect the means of the recursion coefficients to the support of a spectrum are obtained the period of their asymptotic oscillations is written as a function of the support the numerical results by turchi ducastelle and treglia 1982 can be well interpreted by the present theory the asymptotic properties of the truncated basis are also examined

characterization of the undesirable global minima of the godard cost function case of noncircular symmetric signals the deconvolution of a filtered version of a zero mean normalized independent and identically distributed i i d signal s sub n sub n spl isin z having a strictly negative kurtosis spl gamma sub 2 e s sub n sup 4 2 e s sub n sup 2 sup 2 e s sub n sup 2 sup 2 is addressed this correspondence focuses on the global minimizers of the godard function a well known result states that these minimizers achieve deconvolution at least if the input signal shows the symmetry e s sup 2 0 when this constraint is relaxed s sub n sub n spl isin z is said to be noncircular symmetric it is shown that the minimizers achieve deconvolution if and only if 2 e s sub n sup 2 sup 2 spl gamma sub 2 s if this condition is not met the global minimizers are found to be finite impulse response filters with two taps

momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses ii illustrations abstract a classic diatomic chain with a mass impurity is studied using the recurrence relations method the dependence of frequency and amplitude upon two mass ratios is illustrated in detail by examining the figures we can learn a lot about the model such as the widths of the acoustic and optical branches as well as the gap between them the frequency and amplitude of the resonant modes etc the momentum autocorrelation functions of the impurity in different regions of the configuration space are numerically calculated and illustrated a theorem is obtained governing the upper and lower frequencies of the acoustic and optical branches the envelope of the integrand of the cut contribution is determined the convergence of numerical calculations of time integration for parameter w to its theoretical value is confirmed

new criteria for blind deconvolution of nonminimum phase systems channels a necessary and sufficient condition for blind deconvolution without observing the input of nonminimum phase linear time invariant systems channels is derived based on this condition several optimization criteria are proposed and their solution is shown to correspond to the desired response these criteria involve the computation only of second and fourth order moments implying a simple tap update procedure the proposed methods are universal in the sense that they do not impose any restrictions on the probability distribution of the unobserved input sequence it is shown that in several important cases e g when the additive noise is gaussian the proposed criteria are essentially unaffected

rough metrics on manifolds and quadratic estimates we study the persistence of quadratic estimates related to the kato square root problem across a change of metric on smooth manifolds by defining a class of rough riemannian like metrics that are permitted to be of low regularity and degenerate on sets of measure zero we also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi lipschitz as a consequence we demonstrate the invariance of the kato square root problem under lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric furthermore we show that a lower bound on the injectivity radius is not a necessary condition to solve the kato square root problem

momentum autocorrelation function of an impurity in a classical oscillator chain with alternating masses i general theory abstract a chain composed of classic harmonic oscillators with alternating masses and one mass impurity is studied using the recurrence relations method the dynamics of the momentum autocorrelation function of the impurity mass is contributed by three branch cuts corresponding to acoustic and optic modes and by two pairs of poles for resonant modes one pair of the poles are located beyond the optic branch and the other in the gaps between the branches criteria for the resonances are established general expressions for frequencies and amplitudes of the resonant modes are derived ergodic behavior of the momentum of the impurity is analyzed

spectral analysis of a nonlinear oscillator driven by random and periodic forces i linearized theory we have combined the techniques of statistical and harmonic linearization to develop a linearized approximation theory for the calculation of the second order statistics i e autocorrelation functions and spectral densities of nonlinear systems driven by both random and periodic forces for the special case of a duffing oscillator a damped anharmonic oscillator with a cubic nonlinearity driven by gaussian white noise and by a sinusoidal force explicit expressions for the renormalized linearized frequency the autocorrelation function and the spectral density of the oscillator displacement in terms of all the system parameters have been derived we have determined the region of the parameter space in which the applied periodic force has a significant influence on the second order statistics of the oscillator

transition behaviour in the asymptotic long time limit and theoretical approach to explore the two different long time limit behaviours decaying and non decaying to zero for t of a dynamical quantity we present a theoretical approach which analyses a continued fraction cf representation of the quantity on the basis of our analysis we show that the two asymptotic behaviours are characterized by a structure of cf coefficients and that there exists a certain condition that determines the transition between the two limits in addition a measure is introduced to predict the transition we apply our theory to various systems to confirm its validity the nondecay behaviour involves a kind of feedback process in a dynamical system the appearance of a dynamical similarity near the transition is discussed

the sinkhorn algorithm parabolic optimal transport and geometric monge ampere equations we show that the discrete sinkhorn algorithm as applied in the setting of optimal transport on a compact manifold converges to the solution of a fully non linear parabolic pde of monge ampere type in a large scale limit the latter evolution equation has previously appeared in different contexts e g on the torus it can be be identified with the ricci flow this leads to algorithmic approximations of the potential of the optimal transport map as well as the optimal transport distance with explicit bounds on the arithmetic complexity of the construction and the approximation errors as applications we obtain explicit schemes of nearly linear complexity at each iteration for optimal transport on the torus and the two sphere as well as the far field antenna problem connections to quasi monte carlo methods are exploited

testing boltzmann s ergodic hypothesis with electron gas models kubo proposed a physical approach to proving the validity of boltzmann s ergodic hypothesis it is to perform the time averages on dynamical functions thereby avoiding the difficulties of measure theory inherent in the classical approach to perform time averaging properly one must have a general solution for the heisenberg equation of motion such as by the recurrence relations method a time averaging carried over with a recurrence relations solution is found to yield an ergodic condition in the form of an infinite product it is linked to the energy transfer mechanisms hence to the ergodicity itself the electron gas models are fertile ground for testing ergodicity by this approach for several static domains we have evaluated the infinite product and drawn from them a general physical picture that underlies the ergodic hypothesis

fractional kinetics in kac zwanzig heat bath models we study a variant of the kac zwanzig model of a particle in a heat bath the heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data as n the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro differential equation a generalized langevin equation gle with power law memory kernel and driven by 1 f noise the limiting process exhibits fractional sub diffusive behaviour we further consider the approximation of non markovian processes by higher dimensional markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting gle in contrast we show the inadequacy of a so called fractional fokker planck equation in the present context all results are supported by direct numerical experiments

relaxation and phase space singularities in time series of human magnetoencephalograms as indicator of photosensitive epilepsy to analyze the crucial role of fluctuation and relaxation effects for the function of the human brain we studied some statistical quantifiers that support the information characteristics of neuromagnetic brain responses magnetoencephalogram meg the signals to a flickering stimulus of different color combinations have been obtained from a group of control subjects which is then contrasted with those of a patient suffering photosensitive epilepsy pse we found that the existence of the specific stratification of the phase clouds and the concomitant relaxation singularities of the corresponding nonequilibrium dynamics of the chaotic behavior of the signals in separate areas in a patient provide likely indicators for the zones which are responsible for the appearance of pse