a priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1d for a next nearest neighbour pair interaction model in a periodic domain a priori and a posteriori analyses of the quasinonlocal quasicontinuum method qnl qc are presented the results are valid for large deformations and essentially guarantee a one to one correspondence between atomistic solutions and qnl qc solutions the analysis is based on consistency error estimates in negative norms novel a priori and a posteriori stability estimates and a quantitative inverse function theorem

on the inverse problem of statistical physics from irreversible semigroups to chaotic dynamics abstract we show that all measure preserving stationary markov processes arise as projections of kolmogorov dynamical systems which have positive entropy production and are prototypes of chaos this result not only contributes to the clarification of the relation of dynamics with stochastic processes but also shows the physical significance of the misra prigogine courbage theory of irreversibility in the more general context of the inverse problem of statistical physics because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

on the discrete time version of the brussels formalism we develop a discrete time version of the so called brussels formalism in nonequilibrium statistical mechanics for continuous endomorphisms of a banach space we show that if the evolution operator u and the projector p are such that pu is a compact operator and the spectral radius of i p u i p is strictly less than the spectral radius of u then the formalism holds and the evolution operator is quasicompact

jordan blocks in a one dimensional markov map abstract a simple one dimensional chaotic map whose spectral decomposition of the frobenius perron operator in the space of polynomials has jordan block structure is given and analyzed the jordan block structure implies that some correlation functions have exponential decay modified by coefficients which are polynomials in t such a correlation is given for this system and a numerical simulation which agrees with the theoretical prediction is presented

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

multiresolution molecular mechanics convergence and error structure analysis abstract we present a convergence and error structure analysis of the newly proposed energy based concurrent atomistic continuum method multiresolution molecular statics mms yang et al 2013 the ability to perform this study lies in the property that mms is fully consistent with full atomistics this work investigates the effects of 1 energy sampling schemes 2 linearity of interatomic interaction and 3 smoothness of the solution on the convergence characteristics and error structure of mms for this purpose the nonlocal harmonic and lennard jones lj potentials together with quadratic deformation uniformly distributed and random force loadings are employed to test the convergence for a one dimensional tensile loading model and a two dimensional cantilever beam bending model the key finding is that under certain conditions mms shows monotonic convergence and convergence rates that are similar to those in finite element method fem also the condition under which either discretization or sampling error dominates is determined for each specific sampling scheme based on the convergence and error structure analysis suggestions on how to effectively choose an mms model are given

on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

master equation for a quantum system driven by a strong periodic field in the quasienergy representation abstract the evolution of a quantum system weakly coupled with a thermal reservoir and influenced by an external periodic field is formulated in the quasienergy representation where the strong external periodic field disappears and the original model is transformed to a time dependent system weakly coupled with thermal reservoir based on this time dependent subdynamics we derive a master equation of the reduced density operator for the driven quantum system this equation is valid in the weak coupling limit our method can be useful for obtaining the master equation of any system coupled with a thermal reservoir and driven by strong external periodic fields

relativistic gamow vectors the friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes such eigenvectors are called gamow vectors and they acquire meaning in extensions of the conventional hilbert space of quantum theory to the so called rigged hilbert space in this paper gamow vectors are constructed for a solvable model of an unstable relativistic field as a result we obtain a time asymmetric relativistic extension of the fock space this extension leads to two distinct poincare semigroups the time reversal transformation maps one semigroup to the other as a result the usual pct invariance should be extended we show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity

probabilistic approach to homoclinic chaos three dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues i and giving rise to homoclinic chaos when the shil nikov condition is satisfied are studied the 2d poincare map and its 1d contractions capturing the essential features of the flow are given at homoclinicity these 1d maps are found to be piecewise linear this property allows one to reduce the frobenius perron equation to a master equation whose solution is analytically known the probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived

spectral decomposition of chebyshev maps abstract we construct a spectral decomposition of the frobenius perron operator for the chebyshev polynomials of the first kind we defined a suitable dual pair or rigged hilbert space which provides mathematical meaning to the spectral decomposition the spectra of the even chebyshev maps do not contain the odd powers of 1 m and the odd eigenfunctions are in the null space of the frobenius perron operator moreover the odd chebyshev maps have degenerate spectra without jordan blocks the eigenvalues in the decompositions are the resonances of power spectrum and have magnitudes less than one as in the case of the family of tent maps

rigged hilbert spaces associated with misra prigogine courbage theory of irreversibility abstract it is proved that in the misra prigogine courbage theory of irreversibility using the internal time superoperator fixing its associated non unitary transformation amounts to rigging the corresponding hilbert liouville space more precisely it is demonstrated that any determines three canonical riggings of the liouville space l first one with a hilbert space with a norm greater than the relative one from l a second one with a hilbertian space which is a kothe space if is compact and is a nuclear space if has certain nuclear properties and finally a third one with a smaller hilbertian space with a still stronger topology which is nuclear if n is hilbert schmidt for some positive integer n in contrast any rigging of this type originated in a dynamical system having an internal time superoperator defines a in a canonical way

unitary quasi affine transforms of contractions abstract the unitary quasi affine transforms of contractions are the cornerstone of the exact theory of irreversibility introduced by misra prigogine and courbage this work shows that the class of contractions with unitary quasi affine transforms is just c 1 and that every unitary quasi affine transform of a contraction is unitarily equivalent to the residual part of its sz nagy foias dilation some connections with the grossmann theory of nested hilbert spaces and generalized eigenvalues are established these results are applied to the study of the baker map

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

from irreversible markov semigroups to chaotic dynamics abstract we answer qualitatively the inverse coarse graining problem of statistical physics namely which microscopic dynamics give rise to a given physically observed markov semigroup as a result of exact coarse graining we prove the fact that all markov chains arise as projections of dynamical systems in larger spaces and show that in particular the irreversible markov chains arise as projections of chaotic systems of kolmogorov type this result generalizes our previous results on the misra prigogine courbage semigroups because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

from probabilistic descriptions to deterministic dynamics abstract we answer the question of dilating a markov semigroup to a deterministic evolution in a larger space we show that certain markov processes may arise as projections of kolmogorov dynamical systems this result shows in particular the physical significance of the misra prigogine courbage theory of irreversibility because we want positivity preserving transformations in both case our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of the natural extension of exact systems

ensemble dynamics of intermittency and power law decay a spectral decomposition of the frobenius perron operator is constructed for one dimensional maps with intermittent chaos using the method of coherent states a technique using the spectral density function is applied to the the well known cusp map which generates weak type ii intermittency higher order corrections are obtained to the leading 1 t long time behavior of the x x autocorrelation

periodic orbit expansions for classical smooth flows the authors derive a generalized selberg type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow this relation is a classical analogue of the quantum trace formulae and selberg type zeta functions

numerical solution of a virtual internal bond model for material fracture a virtual internal bond vib model is proposed recently in mechanical engineering literatures for simulating dynamic fracture the model is a nonlinear wave equation of mixed type hyperbolic or elliptic there is instability in the elliptic region and usual numerical methods might not work we examine the artificial viscosity method for the model and apply central type schemes directly to the corresponding viscous system to ensure appropriate numerical viscous term for such a mixed type problem we provide a formal justification of indicating convergence of the scheme despite the difficulty of the type change the exact solution of a riemann problem is used to demonstrate the numerical method for one dimensional case we then generalize the method to a two dimensional material with a triangular or hexagonal lattice structure computational results for a two dimensional example are given

resonances of dynamical systems and fredholm riesz operators on rigged hilbert spaces abstract resonances of dynamical systems are defined as the singularities of the analytically continued resolvent of the restriction of the frobenius perron operator to suitable test function spaces a sufficient condition for resonances to arise from a meromorphic continuation to the entire plane is that the frobenius perron operator is a fredholm riesz operator on a rigged hilbert space after a discussion of spectral theory in locally convex topological vector spaces we illustrate the approach for a simple chaotic system namely the renyi map

non unitary transformation of conservative to dissipative evolutions the authors briefly review in a general context the theory of non unitary transformations between conservative dynamics with internal time and dissipative evolutions developed previously in the context of statistical mechanics and apply the results to show the existence of a similarity transformation between the wave equation and the dissipative telegraphist equation

on the rate of convergence to equilibrium in one dimensional systems we determine the essential spectral radius of the perron frobenius operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one dimensional systems to the greatest eigenvalues of generalized perron frobenius operators of the transformations operators which yield singular invariant measures

locating resonances for axiom a dynamical systems for a class of differentiable dynamical systems called axiom a systems it has been shown by pollicott and the author that correlation functions have fourier transforms which are meromorphic in a strip the poles or resonances are however not easy to locate this note reviews the results which are known and discusses a simple model where the position of resonances can be estimated the effect of noise is also discussed

kinetic time irreversible evolution of the unstable π meson abstract in liouville formalism the expression for density matrix determining the time evolution of unstable meson in the framework of unified formulation of quantum and kinetic dynamics is defined the eigenvalues problem is investigated in the framework of prigogine s principles of description of nonequilibrium processes at microscopic level the problem was solved on the basis of complex spectral representation it was shown that the approach leads to pauli master equation for the weakly interacting system

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

resonances and the extension of dynamics to rigged hilbert space abstract we propose a unified operator theoretic formulation of resonances and resonance states in rigged hilbert space our approach shows how resonances are the basis for the extension of dynamics for unstable systems and in a synthesis and a generalization of the work of the brussels austin groups directed by prigogine with the work of bohm and gadella on gamow vectors and with the extension theory developed by the st petersburg group directed by pavlov we illustrate the approach for prototypes of resonances namely resonances of the power spectrum scattering resonances and resonances associated with the unstable bloch states

quantum systems with diagonal singularity their logic and intrinsic irreversibility abstract the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables and the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity

time operators and shift representation of dynamical systems abstract we introduce and characterise time operators for unilateral shifts and exact endomorphisms the associated shift representation of evolution is related to the spectral representation by a generalized fourier transform we illustrate the results for a simple exact system namely the renyi map

space frequency quantization for wavelet image coding a new class of image coding algorithms coupling standard scalar quantization of frequency coefficients with tree structured quantization related to spatial structures has attracted wide attention because its good performance appears to confirm the promised efficiencies of hierarchical representation this paper addresses the problem of how spatial quantization modes and standard scalar quantization can be applied in a jointly optimal fashion in an image coder we consider zerotree quantization zeroing out tree structured sets of wavelet coefficients and the simplest form of scalar quantization a single common uniform scalar quantizer applied to all nonzeroed coefficients and we formalize the problem of optimizing their joint application we develop an image coding algorithm for solving the resulting optimization problem despite the basic form of the two quantizers considered the resulting algorithm demonstrates coding performance that is competitive often outperforming the very best coding algorithms in the literature

the extension of classical dynamics for unstable hamiltonian systems abstract classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the liouville equation it is shown that for the class of large nonintegrable poincare systems lps the two descriptions are not equivalent practically all dynamical systems studied in statistical mechanics belong to this class the basic step is the extension of the liouville operator l h outside the hilbert space to functions singular in their fourier transformation this function space plays an important role in statistical mechanics as functions of the hamiltonian and therefore equilibrium distribution functions belong to this class physically these functions correspond to situations characterized by persistent interactions as they are realized in macroscopic physics persistent interactions are introduced in contrast to transient interactions studied in quantum mechanics by the s matrix approach asymptotically free in and out states the eigenvalue problem for the liouville operator l h is solved in this generalized function space for lps we obtain a complex irreducible spectral representation complex means that the eigenvalues are complex numbers whose imaginary parts refer to the various irreversible processes such as relaxation times diffusion irreducible means that these representations cannot be implemented by trajectory theory as the result the dynamical group of evolution splits into two semigroups moreover the laws of classical dynamics take a new form as they have to be formulated on the statistical level they express possibilities and no more certitudes two examples of typical classical systems i e interacting particles and anharmonic lattices are studied

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

the differential zeta function for axiom a attractors closed orbit of least period x t an important problem is to construct the meromorphic extensions for these zeta functions a detailed knowledge of the domain of s gives information on the long term behaviour of the flow for example this is a standard approach to deriving asymptotic formulae for closed orbits cf 9 for the case of geodesic flows on surfaces of constant negative curvature the zeta function is known to have a meromorphic extension to the entire complex plane unfortunately an obstruction to a general theory along these lines is the existence of examples of axiom a flows for which d s does not have a meromorphic extension to c in a recent article william parry proposed a modified definition of a zeta function for axiom a attractors he defined the differential zeta function by ns s h i e s t l where the least period x x of a closed orbit is replaced by the expansion xu t along the unstable manifold around the orbit 8 parry also defined the useful notion of a synchronized flow i e one for which the measure of maximal entropy and the srb measure coincide the following remark is taken from page 273 of his article when the flow is synchronized the zeta functions coincide this is the case for example with a geodesic flow on a surface of constant negative curvature one wonders whether synchronization has something to do with the very desirable features of the zeta function of such a geodesic flow in this article we shall give a solution to parry s question which shows that in many cases du s has a meromorphic extension to the entire complex plane our main result can be stated as follows

gamow functionals on operator algebras we obtain the precise form of two gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially as a linear positive and continuous functional on an algebra containing observables these functionals do not admit normalization and with an appropriate choice of the algebra are time reversal of each other

order and disorder in the approach to equilibrium of a classical gas we study the approach to equilibrium of a classical gas the initial condition corresponds to a maxwell velocity distribution but to a nonequilibrium binary correlation we consider two cases in the first there are initially no spatial correlations while in the second initial correlations correspond to long range spatial order we show that the gas leaves the maxwell velocity distribution function in the process of building up equilibrium correlations the spatial correlations in the equilibrium state are seen to emerge from a self organization process in the gas non markovian effects play an essential role in this process by coupling the velocity distribution and the binary correlations for the case of initial long range correlations we obtain anti boltzmann behavior in the evolution of the velocity distribution as the boltzmann entropy decreases from the nonequilibrium to the equilibrium state for this case we also have nontrivial behavior on a short time scale due to the non markovian effects the approach used here is based on the theory of subdynamics as developed in previous publications the results obtained show the interplay between irreversible processes leading to disorder and to order in a classical gas

resonances in chaotic dynamics we present a discussion and some numerical results on the actual possibility of making accessible by numerical techniques the complex singularities of the power spectrum resonances for a chaotic signal henon s transformation is investigated in detail showing that the position of the leading resonance in the complex frequency plane determines the kind of mixing rate in the time evolution

poincare s theorem and subdynamics for driven systems abstract large dynamical systems with explicit time dependence will be studied for these systems which we call driven systems there exist resonances between internal frequencies and or between internal and external frequencies as in the time independent case the usual canonical or unitary perturbation theory leads to divergences due to the resonances as a result there exist no trajectories analytic in both the coupling constant and the initial data this is a generalization of poincare s theorem on non integrability and extends the notion of large poincare systems lps i e systems with a continuous spectrum and a continuous set of resonances here the resonances involve external frequencies along the line of the subdynamics theory developed by prigogine and his co workers we study lps within the liouville space formalism we construct projection operators which decompose the equations of motion and are analytic in the coupling constant our approach recovers coveney s theory of time dependent subdynamics but is based on recursion formulas which significantly simplify the construction of the projection operators these projectors are non hermitian and provide a description with broken time symmetry as an application we study the modification of the well known three stages of the decay process from a time dependent perturbation

goal oriented adaptive mesh refinement for the quasicontinuum approximation of a frenkel kontorova model abstract the quasicontinuum approximation e b tadmor m ortiz r phillips quasicontinuum analysis of defects in solids philos mag a 73 6 1996 1529 1563 is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation in regions of the crystal with a highly nonuniform deformation such as around defects every atom must be a representative atom to obtain sufficient accuracy but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy we present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized frenkel kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy

convergence analysis of a quasi continuum approximation for a two dimensional material without defects in many applications materials are modeled by a large number of particles or atoms where any particle can interact with any other the computational cost is very high since the number of atoms is huge recently much attention has been paid to a so called quasi continuum qc method which is a mixed atomistic continuum model the qc method uses an adaptive finite element framework to effectively integrate the majority of the atomistic degrees of freedom in regions where there is no serious defect however numerical analysis of this method is still in its infancy in this paper we will conduct a convergence analysis of the qc method in the case when there is no defect we will also remark on the case when the defect region is small the difference between our analysis and conventional analysis is that our exact atomistic solution is not a solution of a continuous partial differential equation but a discrete lattice scale solution which is not approximately related to any conventional partial differential equation

a priori error analysis of two force based atomistic continuum models of a periodic chain the force based quasicontinuum qcf approximation is a non conservative atomistic continuum hybrid model for the simulation of defects in crystals we present an a priori error analysis of the qcf method applied to a one dimensional periodic chain that is valid for an arbitrary interaction range large deformations and takes coarse graining into account our main tool in this analysis is a new concept of atomistic stress moreover we formulate a new atomistic continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method we show that the new method has several theoretical advantages over the original qcf method

theory based benchmarking of the blended force based quasicontinuum method we formulate an atomistic to continuum coupling method based on blending atomistic and continuum forces our precise choice of blending mechanism is informed by theoretical predictions we present a range of numerical experiments studying the accuracy of the scheme focusing in particular on its stability these experiments confirm and extend the theoretical predictions and demonstrate a superior accuracy of b qcf over energy based blending schemes

theoretical and numerical analysis for the quasi continuum approximation of a material particle model in many applications materials are modeled by a large number of particles or atoms where any one of particles interacts with all others near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community in this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full interaction problem so as to answer the question mathematically for a one dimensional model a few numerical methods have been designed in the engineering literature for the simplified model recently much attention has been paid to a finite element like quasicontinuum qc method which utilizes a mixed atomistic continuum approximation model no numerical analysis has been done yet in the paper we shall estimate the error of the qc method for this one dimensional model possible ill posedness of the method and its modification are discussed as well

an optimal order error analysis of the one dimensional quasicontinuum approximation we derive a model problem for quasicontinuum approximations that allows a simple yet insightful analysis of the optimal order convergence rate in the continuum limit for both the energy based quasicontinuum approximation and the quasi nonlocal quasicontinuum approximation for simplicity the analysis is restricted to the case of second neighbor interactions and is linearized about a uniformly stretched reference lattice the optimal order error estimates for the quasi nonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture the analysis is based on an explicit treatment of the coupling error at the atomistic to continuum interface combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation

resonances for axiom bf a flows etant donne un flot d axiome a sur m et des fonctions lisses b c m r on montre que la fonction de correlation en temps bc pour un etat de gibbs a une transformee de fourier bc mesomorphe dans une bande on etudie les residus des poles de bc

one dimensional discrete stark hamiltonian and resonance scattering by an impurity a one dimensional discrete stark hamiltonian with a continuous electric field is constructed by extension theory methods in the absence of an impurity the model is proved to be exactly solvable the spectrum is shown to be simple and continuous filling the real axis the eigenfunctions the resolvent and the spectral measure are constructed explicitly for this unperturbed system the resonance spectrum is shown to be empty the model considering an impurity in a single node is also constructed using the operator extension theory methods the spectral analysis is performed and the dispersion equation for the resolvent singularities is obtained the resonance spectrum is shown to contain an infinite discrete set of resonances one to one correspondence of the constructed hamiltonian to some lee friedrichs model is established

non hermitian hamiltonians in decoherence and equilibrium theory there are many formalisms to describe quantum decoherence however many of them give a non general and ad hoc definition of pointer basis or moving preferred basis and this fact is a problem for the decoherence program in this paper we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis these ideas are implemented in a well known open system model the obtained decoherence and the relaxation times are defined and compared with those of the literature for the lee friedrichs model this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

gamow vectors for degenerate scattering resonances in this paper we construct gamow vectors for resonances given by poles of the analytic continuation of the s matrix of any finite order we study their modes of decay or growth we obtain jordan block structures for the extended hamiltonians and evolution operators on the subspaces spanned by these gamow vectors we perform this study within the context of the rigged hilbert space extension of quantum theory we construct an explicit friedrichs model with a double pole resonance to illustrate the general formulation

an analysis of node based cluster summation rules in the quasicontinuum method we investigate two examples of node based cluster summation rules that have been proposed for the quasicontinuum qc method a force based approach and an energy based approach which is a generalization of the nonlocal qc method we show that even for the case of nearest neighbor interaction in a one dimensional periodic chain both of these approaches create large errors that cannot be removed by increasing the cluster size when used with graded and more generally nonsmooth meshes we offer some suggestions for how the accuracy of cluster summation rules may be improved

subdynamics and nonintegrable systems work on the application of poincare s theorem to large classical or quantum systems with a continuous spectrum is continued in situations where it is applicable poincare s theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant in contrast the theory of subdynamics as developed by the brussels group permits the construction of a unique set of projectors for t 0 giving up the requirement of hermiticity which is replaced by star hermiticity the theory of subdynamics is presented in a new self contained way starting from the commutation relation lh lh where lh is the liouvillian this presentation is far more direct and avoids some of the lengthy discussions associated with previous presentations based mainly on the resolvent of the liouvillian subdynamics appears to be of interest from many points of view it generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed hamiltonian in contrast degrees of freedom are lost when going to the spectral representation e g in the friedrichs model subdynamics permits us to solve the initial value problem associated with the liouville equation retaining the non markovian contributions which appear in the standard presentation finally it introduces a classification of large dynamical systems classical or quantum into integrable and nonintegrable ones it is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems such as the interaction of matter with light the applications of this technique to these problems will be worked out in subsequent papers

analysis of energy based blended quasi continuum approximations the development of patch test consistent quasi continuum energies for multidimensional crystalline solids modeled by many body potentials remains a challenge the original quasi continuum energy qce r miller and e tadmor model simul mater sci eng 17 2009 053001 has been implemented for many body potentials in two and three space dimensions but it is not patch test consistent we propose that by blending the atomistic and corresponding cauchy born continuum models of qce in an interfacial region with thickness of a small number k of blended atoms a general blended quasi continuum energy bqce can be developed with the potential to significantly improve the accuracy of qce near lattice instabilities such as dislocation formation and motion in this paper we give an error analysis of the blended quasi continuum energy bqce for a periodic one dimensional chain of atoms with next nearest neighbor interactions our analysis includes the optimization of the blending function for an improved convergence rate we show that the ell 2 strain error for the nonblended qce energy which has low order o varepsilon 1 2 where varepsilon is the atomistic length scale m dobson and m luskin siam j numer anal 47 2009 pp 2455 2475 p ming and j z yang multiscale model simul 7 2009 pp 1838 1875 can be reduced by a factor of k 3 2 for an optimized blending function where k is the number of atoms in the blending region the qce energy has been further shown to suffer from a o 1 error in the critical strain at which the lattice loses stability m dobson m luskin and c ortner j mech phys solids 58 2010 pp 1741 1757 we prove that the error in the critical strain of bqce can be reduced by a factor of k 2 for an optimized blending function thus demonstrating that the bqce energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth

variational formulation of softening phenomena in fracture mechanics the one dimensional case we show that discrete models of atoms subject to nearest neighbour non linear interactions approximate continua allowing for softening and fracture a detailed study of local minima and stationary points is carried out scale effects are discussed

quadrature rule type approximations to the quasicontinuum method for long range interatomic interactions a quadrature rule type method is presented to approximate the quasicontinuum method for atomistic mechanics for both the short range and long range interaction cases the complexity of this method depends on the number of representative particles but not on the total number of particles simple analysis and numerical experiments are provided to illustrate the accuracy and performance of the method it is shown that for the same accuracy the quadrature rule type method is much less costly than the quasicontinuum method

resonances of the cusp family we study a family of chaotic maps with limit cases the tent map and the cusp map the cusp family we discuss the spectral properties of the corresponding frobenius perron operator in different function spaces including spaces of analytical functions and study numerically the eigenvalues and eigenfunctions

a posteriori error estimation and adaptive algorithm for atomistic continuum coupling in two dimensions atomistic continuum coupling methods aim to achieve optimal balance between accuracy and efficiency adaptivity is the key for the efficient implementation of such methods in this paper we carry

iterative methods for the force based quasicontinuum approximation analysis of a 1d model problem force based atomistic continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model for this reason and due to their algorithmic simplicity force based coupling methods have become a popular class of atomistic continuum hybrid models as well as other types of multiphysics models however the recently discovered unusual stability properties of the linearized force based quasicontinuum qcf approximation especially its indefiniteness present a challenge to the development of efficient and reliable iterative methods we present analytic and computational results for the generalized minimal residual gmres solution of the linearized qcf equilibrium equations we show that the gmres method accurately reproduces the stability of the force based approximation and conclude that an appropriately preconditioned gmres method results in a reliable and efficient solution method

one dimensional gibbs states and axiom a diffeomorphisms on etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle pour de tels systemes les transformees de fourier des fonctions de correlation de paires sont meromorphes dans une bande et les residus des poles peuvent s exprimer en termes de distributions de gibbs on applique ces resultats aux diffeomorphismes de l axiome a de smale

r adic one dimensional maps and the euler summation formula for the r adic one dimensional maps the author explicitly constructs the decaying eigenstates and adjoint eigenstates associated with the ruelle resonances it is shown that the eigenfunctions of the corresponding frobenius perron operator are the well known bernoulli polynomials the adjoint eigendistributions are obtained as derivatives of the dirac distributions at the end points of the unit interval the resulting expansion of the initial density in terms of the decaying eigenstates is given by the euler summation formula

intrinsic irreversibility and the validity of the kinetic description of chaotic systems irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities the generalized representation is valid for one dimensional systems when the initial probability density satisfies certain physical conditions of smoothness the formalism is first applied to the one dimensional multi bernoulli map which is a simple map displaying deterministic diffusion the two dimensional invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered the generalized representation is constructed using a resolvent formalism the eigenstates of the diffusive systems are seen to be of a fractal nature

partial algebras of matrices and operators the set m of infinite matrices and the set q of squarable matrices are considered as partial algebras the connection between q and two partial algebras of closed operators is studied conditions for a matrix representation in von neumann s sense of a family of closed operators are given

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

generalized spectral decomposition of the β adic baker s transformation and intrinsic irreversibility abstract we construct a generalized spectral decomposition of the frobenius perron operator of the adic baker s transformation using a general iterative operator method applicable in principle for any mixing dynamical system the eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one we explicitly define appropriate generalized function spaces which provide mathematical meaning to the formally obtained spectral decomposition the unitary frobenius perron evolution of densities when extended to the generalized function spaces splits into two semigroups one decaying in the future and the other in the past this split which reflects the asymptotic evolution of the forward and backward k partitions shows the instrinsic irreversibility of the baker s transformation

rigged hilbert spaces for chaotic dynamical systems we consider the problem of rigging for the koopman operators of the renyi and the baker maps we show that the rigged hilbert space for the renyi maps has some of the properties of a strict inductive limit and give a detailed description of the rigged hilbert space for the baker maps

scalable wavelet video coding using aliasing reduced hierarchical motion compensation we describe a spatially scalable video coding framework in which motion correspondences between successive video frames are exploited in the wavelet transform domain the basic motivation for our coder is that motion fields are typically smooth and therefore can be efficiently captured through a multiresolutional framework a wavelet decomposition is applied to each video frame and the coefficients at each level are predicted from the coarser level through backward motion compensation to remove the aliasing effects caused by downsampling in the transform a special interpolation filter is designed with the weighted aliasing energy as part of the optimization goal and motion estimation is carried out with low pass filtering and interpolation in the estimation loop further to achieve robust motion estimation against quantization noise we propose a novel backward forward hybrid motion compensation scheme and a tree structured dynamic programming algorithm to optimize the backward forward mode choices a novel adaptive quantization scheme is applied to code the motion predicted residue wavelet coefficients experimental results reveal 0 3 2 db increase in coded psnr at low bit rates over the state of the art h 263 standard with all enhancement modes enabled and similar improvements over mpeg 2 at high bit rates with a considerable improvement in subjective reconstruction quality while simultaneously supporting a scalable representation

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

nonlocality of the misra prigogine courbage semigroup we show that the markov semigroups constructed by misra prigogine and courbage through nonunitary similarity transformations of kolmogorov systems are not implementable by local point transformations i e they are not the frobenius perron semigroups associated with noninvertible point transformations in contrast with the semigroups obtained by coarse graining projections our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by goldstein misra and courbage and also of the previous illustration by misra and prigogine for the baker transformation and completes the characterization of the misra prigogine courbage semigroups

alternative formulation of classical and quantum dynamics for non integrable systems we continue our work on large poincare systems lps characterized by a continuous spectrum and a continuous set of resonances poincare s theorem prevents the existence of solutions of the eigenvalue problem associated with the hamiltonian as well as the liouville operator which would be analytic in the coupling constant in simple cases such as the friedrichs model studied in a recent paper poincare s divergences can be avoided using a natural time ordering of the dynamical states in the hilbert space however in general the time ordering has to be performed in the liouville space of density matrices

partial algebras and volterra convolution of distribution kernels the set of distribution kernels is considered as a partial algebra when the multiplication of two kernels is defined as their volterra convolution the partial algebra of distribution kernels corresponding to a certain partial algebra of closable operators in l2 is studied

lie admissible perturbation methods for open quantum systems abstract we consider open quantum systems described by a hamiltonian of the type h0 v where is a small parameter for such systems we develop perturbative methods of solution of the corresponding liouville von neumann and schrodinger equations by introducing dissipation operators which connect conservative to dissipative systems in the case of the density matrix the corresponding operator is nothing but the non unitary transformation of misra prigogine and courbage our perturbative approach possesses a lie admissible structure since the dissipation operators obey time evolution equations whose brackets are the product of a lie admissible algebra explicit solutions for such operators are found in the form of series expansions in the matrix formulation of the above results is also given

on the long time behaviour of ensembles in a model of deterministic diffusion we describe a large model system based on the baker transformation where deterministic diffusion occurs the model is similar to one recently considered by gaspard and by hasegawa and driebe we point out the close relationship between this system and a simple random walk and analyse the evolution with time of ensembles in the system using the resolvent based version of the subdynamics formalism developed by prigogine and his collaborators we obtain an exact and rigorous description of the long time behaviour of ensembles including the irreversible approach to equilibrium for the case where the system has finite size we also consider the thermodynamic limit where the size of the system becomes infinite and derive a description of the long time behaviour in this case where correlations decay non exponentially with time

general formalism for the spectral decomposition of the hamiltonian in a quantum fractal network we present a general formulation for the spectral decomposition of the hamiltonian operator of a quantum fractal network qfn the qfn can be constructed by placing artificial neurons on each site of the fractal lattice an artificial neuron may consist of a cell of a quantum cellular automata or a quantum dot which confines a single electron the coulomb interaction or the spin spin interaction between neurons can be used to transmit signals and perform logic operations the local external field may be as input signal to influence output of the system we obtain explicitly the recursive formulae of the eigenvalues and eigenvectors between sublattices the intertwining relation between the collision operator and the hamiltonian operator by combining subdynamics and a reduced lattice approach furthermore the perturbation method to obtain the spectral decomposition for the time dependent hamiltonian is also discussed finally as an example we calculate the eigenvalues and eigenvectors of the hamiltonian operator for a sierpinski gasket based on our formulation analysis of the recursive formula for the spectrum of the sierpinski gasket reveals how its spectral structure changes in boundary conditions

meromorphic extensions of generalised zeta functions in this paper we give a full description of the spectrum of the ruelle perron frobenius operator acting on the banach space of holder continuous functions on a subshift of finite type theorem 1 these results are then used to extend the meromorphic domain of generalised zeta functions theorem 2 the most important application of these results is to the domain of the smale zeta function for axiom a flows theorem 3 in the course of this paper we settle questions raised by ruelle and sunada

intrinsic irreversibility of quantum systems with diagonal singularity the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity we illustrate the general ideas for the friedrichs model

the resonance spectrum of the cusp map in the space of analytic functions we prove that the frobenius perron operator u of the cusp map f 1 1 1 1 f x 1 2 x which is an approximation of the poincare section of the lorenz attractor has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1 we also prove that for any q 0 1 the spectrum of u in the hardy space in the disk z c z q 1 q is the union of the segment 0 1 and some finite or countably infinite set of isolated eigenvalues of finite multiplicity

quantum theory of non integrable systems in 1889 h poincare introduced a basic distinction between integrable and non integrable dynamical systems this distinction refers to the role of resonances which may lead to divergences a specially important class of non integrable systems are large poincare systems lps which have a continuous spectrum and present continuous sets of resonances as has been shown earlier lps play an essential role both in classical and quantum mechanics essentially all nontrivial problems of field theory as well as kinetic theory belong this class we consider here the well known friedrichs model in which an unstable discrete level is coupled to a continuum poincare s theorem prevents the existence of solutions of the eigenvalue problem associated to the hamiltonian which would be analytic in the coupling constant

a priori and a posteriori w 1 infty error analysis of a qc method for complex lattices in this paper we derive regularity results for equilibria of multilattices under an external force and prove a priori and a posteriori error estimates for a multiscale numerical method for computing such equilibria the estimates are derived in a w 1 infty norm in one space dimension one of the features of our analysis is that we establish an equivalent way of formulating the coarse grained problem which greatly simplifies derivation of the error bounds both a priori and a posteriori we illustrate our error estimates with numerical experiments