kolmogorov sinai entropy and lyapunov spectra of a hard sphere gas abstract the mixing behavior of a hard sphere gas has its origin in the exponential growth of small perturbations in phase space this instability is characterized by the so called lyapunov exponents in this work we compute full spectra of lyapunov exponents for the hard sphere gas for a wide range of densities and particle numbers by using a recently developed algorithm in the dilute gas regime the maximum lyapunov exponent is found to obey the krylov relation ln a formula exactly derived for the low density lorentz gas by dorfman and van beijeren we study the system size dependence and the effect of the fluid solid phase transition on the spectra in the second part of this work we describe and test a direct simulation monte carlo method dsmc for the computation of lyapunov spectra and present results for dilute hard sphere gases excellent agreement is obtained with the results of the deterministic simulations this suggests that the lyapunov instability of a hard sphere gas may be analyzed within the framework of kinetic theory

lyapunov instabilities of lennard jones fluids recent work on many particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive lyapunov exponents les called hydrodynamic lyapunov modes until now however these modes have been found only for hard core systems here we report results on lyapunov spectra and lyapunov vectors lvs for lennard jones fluids by considering the fourier transform of the coordinate fluctuation density u ensuremath alpha x t it is found that the lvs with ensuremath lambda ensuremath approx 0 are highly dominated by a few components with low wave numbers these numerical results provide strong evidence that hydrodynamic lyapunov modes do exist in soft potential systems although the collective lyapunov modes are more vague than in hard core systems in studying the density and temperature dependence of these modes it is found that when the value of the lyapunov exponent ensuremath lambda ensuremath alpha is plotted as function of the dominant wave number k mathit max of the corresponding lv all data from simulations with different densities and temperatures collapse onto a single curve this shows that the dispersion relation ensuremath lambda ensuremath alpha vs k mathit max for hydrodynamical lyapunov modes appears to be universal for the low density cases studied here despite the wavelike character of the lvs no steplike structure exists in the lyapunov spectrum of the systems studied here in contrast to the hard core case further numerical simulations show that the finite time les fluctuate strongly we have also investigated localization features of lvs and propose a length scale to characterize the hamiltonian spatiotemporal chaotic states

a review of geometric transformations for nonrigid body registration this paper provides a comprehensive and quantitative review of spatial transformations models for nonrigid image registration it explains the theoretical foundation of the models and classifies them according to this basis this results in two categories physically based models described by partial differential equations of continuum mechanics e g linear elasticity and fluid flow and basis function expansions derived from interpolation and approximation theory e g radial basis functions b splines and wavelets recent work on constraining the transformation so that it preserves the topology or is diffeomorphic is also described the final section reviews some recent evaluation studies the paper concludes by explaining under what conditions a particular transformation model is appropriate

dynamic localization of lyapunov vectors in hamiltonian lattices the convergence of the lyapunov vector toward its asymptotic shape is investigated in two different one dimensional hamiltonian lattices the so called fermi pasta ulam and phi 4 chains in both cases we find an anomalous behavior i e a clear difference from the previously conjectured analogy with the kardar parisi zhang equation the origin of the discrepancy is eventually traced back to the existence of nontrivial long range correlations both in space and time as a consequence of this anomaly we find that in a hamiltonian lattice the largest lyapunov exponent is affected by stronger finite size corrections than standard space time chaos

comparison between covariant and orthogonal lyapunov vectors two sets of vectors covariant lyapunov vectors clvs and orthogonal lyapunov vectors olvs are currently used to characterize the linear stability of chaotic systems a comparison is made to show their similarity and difference especially with respect to the influence on hydrodynamic lyapunov modes hlms our numerical simulations show that in both hamiltonian and dissipative systems hlms formerly detected via olvs survive if clvs are used instead moreover the previous classification of two universality classes works for clvs as well i e the dispersion relation is linear for hamiltonian systems and quadratic for dissipative systems respectively the significance of hlms changes in different ways for hamiltonian and dissipative systems with the replacement of olvs with clvs for general dissipative systems with nonhyperbolic dynamics the long wavelength structure in lyapunov vectors corresponding to near zero lyapunov exponents is strongly reduced if clvs are used instead whereas for highly hyperbolic dissipative systems the significance of hlms is nearly identical for clvs and olvs in contrast the hlm significance of hamiltonian systems is always comparable for clvs and olvs irrespective of hyperbolicity we also find that in hamiltonian systems different symmetry relations between conjugate pairs are observed for clvs and olvs especially clvs in a conjugate pair are statistically indistinguishable in consequence of the microreversibility of hamiltonian systems transformation properties of lyapunov exponents clvs and hyperbolicity under changes of coordinate are discussed in appendices

dynamics of strongly localized lyapunov vectors in many hard disk systems the dynamics of the localized region of the lyapunov vector for the largest lyapunov exponent is discussed in quasi one dimensional hard disk systems at low density we introduce a hopping rate to quantitatively describe the movement of the localized region of the lyapunov vector and show that it is a decreasing function of the hopping distance implying a spatial correlation of the localized regions this behavior is explained quantitatively by a brick accumulation model derived from hard disk dynamics in the low density limit in which hopping of the localized lyapunov vector is represented as the movement of the highest brick position we also give an analytical expression for the hopping rate which is obtained as a sum of probability distributions for brick height configurations between two separated highest brick sites the results of these simple models are in good agreement with the simulation results for hard disk systems

fast image upsampling via the displacement field in this paper we present a fast image upsampling method within a two scale framework to ensure the sharp construction of upsampled image for both large scale edges and small scale structures in our approach the low frequency image is recovered via a novel sharpness preserving interpolation technique based on a well constructed displacement field which is estimated by a cross resolution sharpness preserving model within this model the distances of pixels on edges are preserved which enables the recovery of sharp edges in the high resolution result likewise local high frequency structures are reconstructed via a sharpness preserving reconstruction algorithm extensive experiments show that our method outperforms current state of the art approaches based on quantitative and qualitative evaluations as well as perceptual evaluation by a user study moreover our approach is very fast so as to be practical for real applications

dynamical behavior of hydrodynamic lyapunov modes in coupled map lattices in our previous study of hydrodynamic lyapunov modes hlms in coupled map lattices we found that there are two classes of systems with different ensuremath lambda text ensuremath k dispersion relations for coupled circle maps we found the quadratic dispersion relations ensuremath lambda ensuremath sim k 2 and ensuremath lambda ensuremath sim k for coupled standard maps here we carry out further numerical experiments to investigate the dynamic lyapunov vector lv structure factor which can provide additional information on the lyapunov vector dynamics the dynamic lv structure factor of coupled circle maps is found to have a single peak at ensuremath omega 0 and can be well approximated by a single lorentzian curve this implies that the hydrodynamic lyapunov modes in coupled circle maps are nonpropagating and show only diffusive motion in contrast the dynamic lv structure factor of coupled standard maps possesses two visible sharp peaks located symmetrically at ifmmode pm else textpm fi ensuremath omega u the spectrum can be well approximated by the superposition of three lorentzian curves centered at ensuremath omega 0 and ifmmode pm else textpm fi ensuremath omega u respectively in addition the ensuremath omega text ensuremath k dispersion relation takes the form ensuremath omega u c u k for k ensuremath rightarrow 2 ensuremath pi l these facts suggest that the hydrodynamic lyapunov modes in coupled standard maps are propagating the hlms in the two classes of systems are shown to have different dynamical behavior besides their difference in spatial structure moreover our simulations demonstrate that adding damping to coupled standard maps turns the propagating modes into diffusive ones alongside a change of the ensuremath lambda text ensuremath k dispersion relation from ensuremath lambda ensuremath sim k to ensuremath lambda ensuremath sim k 2 in cases of weak damping there is a crossover in the dynamic lv structure factors i e the spectra with smaller k are akin to those of coupled circle maps while the spectra with larger k are similar to those of coupled standard maps

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

single image super resolution using global regression based on multiple local linear mappings super resolution sr has become more vital because of its capability to generate high quality ultra high definition uhd high resolution hr images from low resolution lr input images conventional sr methods entail high computational complexity which makes them difficult to be implemented for up scaling of full high definition input images into uhd resolution images nevertheless our previous super interpolation si method showed a good compromise between peak signal to noise ratio psnr performances and computational complexity however since si only utilizes simple linear mappings it may fail to precisely reconstruct hr patches with complex texture in this paper we present a novel sr method which inherits the large to small patch conversion scheme from si but uses global regression based on local linear mappings glm thus our new sr method is called glm si in glm si each lr input patch is divided into 25 overlapped subpatches next based on the local properties of these subpatches 25 different local linear mappings are applied to the current lr input patch to generate 25 hr patch candidates which are then regressed into one final hr patch using a global regressor the local linear mappings are learned cluster wise in our off line training phase the main contribution of this paper is as follows previously linear mapping based conventional sr methods including si only used one simple yet coarse linear mapping to each patch to reconstruct its hr version on the contrary for each lr input patch our glm si is the first to apply a combination of multiple local linear mappings where each local linear mapping is found according to local properties of the current lr patch therefore it can better approximate nonlinear lr to hr mappings for hr patches with complex texture experiment results show that the proposed glm si method outperforms most of the state of the art methods and shows comparable psnr performance with much lower computational complexity when compared with a super resolution method based on convolutional neural nets srcnn15 compared with the previous si method that is limited with a scale factor of 2 glm si shows superior performance with average 0 79 db higher in psnr and can be used for scale factors of 3 or higher

localized behavior in the lyapunov vectors for quasi one dimensional many hard disk systems we introduce a definition of a localization width whose logarithm is given by the entropy of the distribution of particle component amplitudes in the lyapunov vector different types of localization widths are observed for example a minimum localization width where the components of only two particles are dominant we can distinguish a delocalization associated with a random distribution of particle contributions a delocalization associated with a uniform distribution and a delocalization associated with a wavelike structure in the lyapunov vector using the localization width we show that in quasi one dimensional systems of many hard disks there are two kinds of dependence of the localization width on the lyapunov exponent index for the larger exponents one is exponential and the other is linear differences due to these kinds of localizations also appear in the shapes of the localized peaks of the lyapunov vectors the lyapunov spectra and the angle between the spatial and momentum parts of the lyapunov vectors we show that the krylov relation for the largest lyapunov exponent ensuremath lambda ensuremath sim ensuremath ensuremath rho mathrm ln ensuremath rho as a function of the density ensuremath rho is satisfied apart from a factor in the same density region as the linear dependence of the localization widths is observed it is also shown that there are asymmetries in the spatial and momentum parts of the lyapunov vectors as well as in their x and y components

goldstone modes in lyapunov spectra of hard sphere systems in the study of chaotic behavior lyapunov exponents play an important part in this paper we demonstrate how the lyapunov exponents close to zero of a system of many hard spheres can be described as goldstone modes by using a boltzmann type of approach at low densities the correct form is found for the wave number dependence of the exponents as well as for the corresponding eigenvectors in tangent space the predicted values for the lyapunov exponents belonging to the transverse mode are within a few percent of the values found in recent simulations the propagation velocity for the longitudinal mode is within 1 but the value for the lyapunov exponent belonging to the longitudinal mode deviates from the simulations by 30 for higher densities the predicted values deviate more from the values calculated in the simulations these deviations may be due to contributions from ring collisions and similar terms which even at low densities can contribute to the leading order

perturbed phase space dynamics of hard disk fluids abstract the lyapunov spectrum describes the exponential growth or decay of infinitesimal phase space perturbations the perturbation associated with the maximum lyapunov exponent is strongly localized in space and only a small fraction of all particles contributes to the perturbation growth at any instant of time this fraction converges to zero in the thermodynamic large particle number limit for hard disk and hard sphere systems the perturbations belonging to the smallest of the non vanishing exponents are coherently spread out and form orthogonal periodic structures in space the lyapunov modes there are two types of mode polarizations transverse and longitudinal the transverse modes do not propagate but the longitudinal modes do with a speed about one third of the sound speed we characterize the symmetry and the degeneracy of the modes in the thermodynamic limit the lyapunov spectrum has a diverging slope near the intersection with the abscissa no positive lower bound exists for the positive exponents the mode amplitude scales with the inverse square root of the particle number as expected from the normalization of the perturbation vectors

image interpolation by adaptive 2 d autoregressive modeling and soft decision estimation the challenge of image interpolation is to preserve spatial details we propose a soft decision interpolation technique that estimates missing pixels in groups rather than one at a time the new technique learns and adapts to varying scene structures using a 2 d piecewise autoregressive model the model parameters are estimated in a moving window in the input low resolution image the pixel structure dictated by the learnt model is enforced by the soft decision estimation process onto a block of pixels including both observed and estimated the result is equivalent to that of a high order adaptive nonseparable 2 d interpolation filter this new image interpolation approach preserves spatial coherence of interpolated images better than the existing methods and it produces the best results so far over a wide range of scenes in both psnr measure and subjective visual quality edges and textures are well preserved and common interpolation artifacts blurring ringing jaggies zippering etc are greatly reduced

lyapunov modes in hard disk systems we consider simulations of a two dimensional gas of hard disks in a rectangular container and study the lyapunov spectrum near the vanishing lyapunov exponents to this spectrum are associated eigen directions called lyapunov modes we carefully analyze these modes and show how they are naturally associated with vector fields over the container we also show that the lyapunov exponents and the coupled dynamics of the modes where it exists follow linear laws whose coefficients only depend on the density of the gas but not on aspect ratio and very little on the boundary conditions

high resolution local structure constrained image upsampling with the development of ultra high resolution display devices the visual perception of fine texture details is becoming more and more important a method of high quality image upsampling with a low cost is greatly needed in this paper we propose a fast and efficient image upsampling method that makes use of high resolution local structure constraints the average local difference is used to divide a bicubic interpolated image into a sharp edge area and a texture area and these two areas are reconstructed separately with specific constraints for reconstruction of the sharp edge area a high resolution gradient map is estimated as an extra constraint for the recovery of sharp and natural edges for the reconstruction of the texture area a high resolution local texture structure map is estimated as an extra constraint to recover fine texture details these two reconstructed areas are then combined to obtain the final high resolution image the experimental results demonstrated that the proposed method recovered finer pixel level texture details and obtained top level objective performance with a low time cost compared with state of the art methods

origin of the hydrodynamic lyapunov modes recent studies of the lyapunov spectrum of the hard sphere fluid reveal that there are hydrodynamic lyapunov exponents corresponding to collective perturbations in phase space we show that these collective perturbations are due to the conservation of certain quantities during collisions these new conservation laws generate new hydrodynamic fields just as the conservation of mass momentum and energy generate the density velocity and temperature fields we then construct a detailed theory of the new hydrodynamic fields using a kinetic theory approach this theory predicts several properties of the modes but not all of them this suggests that the underlying idea is correct but a detailed theory must be elaborated in another way the hydrodynamic exponents are not related in a simple way to the transport coefficients

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

proof of lyapunov exponent pairing for systems at constant kinetic energy we present a proof that a system consisting of any finite number of particles that move under the action of a scalar potential at constant kinetic energy exhibits conjugate pairing of lyapunov exponents that is the lyapunov exponents come in pairs which sum to the same constant this result generalizes previous results because it is independent of the size of the system s1063 651x 96 51206 7

markov random field model based edge directed image interpolation this paper presents an edge directed image interpolation algorithm in the proposed algorithm the edge directions are implicitly estimated with a statistical based approach in opposite to explicit edge directions the local edge directions are indicated by length 16 weighting vectors implicitly the weighting vectors are used to formulate geometric regularity gr constraint smoothness along edges and sharpness across edges and the gr constraint is imposed on the interpolated image through the markov random field mrf model furthermore under the maximum a posteriori mrf framework the desired interpolated image corresponds to the minimal energy state of a 2 d random field given the low resolution image simulated annealing methods are used to search for the minimal energy state from the state space to lower the computational complexity of mrf a single pass implementation is designed which performs nearly as well as the iterative optimization simulation results show that the proposed mrf model based edge directed interpolation method produces edges with strong geometric regularity compared to traditional methods and other edge directed interpolation methods the proposed method improves the subjective quality of the interpolated edges while maintaining a high psnr level

a review of the hydrodynamic lyapunov modes of hard disk systems a review of the results obtained for hard disk fluids confined to a quasi one dimensional qod system is presented one of the main achievements in recent years has been determining the hydrodynamic lyapunov modes hlms as covariant stable and unstable manifolds defined by k vector analogues of the zero exponent subspace of conserved quantities the tangent space dynamics allows an interpretation of the hlms as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper the time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions each pair is seen to interact in a subspace almost completely separated from other vectors the modes undergo a rotation until they are collinear with the stable or unstable manifold the angle distributions between these covariant stable and unstable hlms are also determined and tangencies are not observed for a generic chaotic trajectory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

the qp carleson measure problem let be a nonnegative borel measure on the open unit disk this note shows how to decide that the mobius invariant space covering and is boundedly resp compactly embedded in the quadratic tent type space interestingly the embedding result can be used to determine the boundedness resp compactness of the volterra type and multiplication operators on

nearly separable behavior of fermi pasta ulam chains through the stochasticity threshold for the periodic fermi pasta ulam chain with quartic potential we prove the relation p k 2 t 1 k 2 q k 2 i e the proportionality already at early timest between averaged kinetic and harmonic energies of each mode the factor depends on the parameters of the model but not on the mode and the number of degrees of freedom it grows with the anharmonic strength from the value 0 of the harmonic limit virial theorem up to few units for the system much above the threshold in the stochastic regime the time necessary to reduce the fluctuations ink to a fixed percentage remains at least one order of magnitude smaller than the time necessary to reach a similar level of equipartition the persistence of such a behavior even above the stochasticity threshold clarifies a number of previous numerical results on the relaxation to equilibrium e g the existence of several time scales and the relevance of the harmonic frequency spectrum the difficulties in the numerical simulation of the thermodynamic limit are also discussed

an edge guided image interpolation algorithm via directional filtering and data fusion preserving edge structures is a challenge to image interpolation algorithms that reconstruct a high resolution image from a low resolution counterpart we propose a new edge guided nonlinear interpolation technique through directional filtering and data fusion for a pixel to be interpolated two observation sets are defined in two orthogonal directions and each set produces an estimate of the pixel value these directional estimates modeled as different noisy measurements of the missing pixel are fused by the linear minimum mean square error estimation lmmse technique into a more robust estimate using the statistics of the two observation sets we also present a simplified version of the lmmse based interpolation algorithm to reduce computational cost without sacrificing much the interpolation performance experiments show that the new interpolation techniques can preserve edge sharpness and reduce ringing artifacts

on the concept of stationary lyapunov basis we propose the concept of stationary lyapunov basis the basis of tangent vectors e i x defined at every point x of the attractor of the dynamical system and show that one can reformulate some algorithms for calculation of lyapunov exponents i so that eachi can be treated as the average of a function si x this enables one to use measure averaging in theoretical arguments thus proposing the rigorous basis for a number of ideas for calculation of lyapunov exponents from time series we also study how the lyapunov vectors in benettin s algorithm converge to the stationary basis and show that this convergence rate determines continuity of the field of stationary lyapunov vectors 1998 elsevier science b v

the maximum number of highly localized lyapunov vectors at low density abstract the localization spectra which describes the magnitude of the localization of the lyapunov vectors in many particle systems exhibit a characteristic bending behavior at low density it is shown that this behavior is due to a restriction on the maximum number of the most localized lyapunov vectors determined by the system configuration and mutual orthogonality for a quasi one dimensional system using a randomly distributed brick model this leads to a predicted bending point at n c 0 432 n for an n particle system numerical evidence is presented that confirms this predicted bending point as a function of the number of particles n

lucas kanade 20 years on a unifying framework since the lucas kanade algorithm was proposed in 1981 image alignment has become one of the most widely used techniques in computer vision applications range from optical flow and tracking to layered motion mosaic construction and face coding numerous algorithms have been proposed and a wide variety of extensions have been made to the original formulation we present an overview of image alignment describing most of the algorithms and their extensions in a consistent framework we concentrate on the inverse compositional algorithm an efficient algorithm that we recently proposed we examine which of the extensions to lucas kanade can be used with the inverse compositional algorithm without any significant loss of efficiency and which cannot in this paper part 1 in a series of papers we cover the quantity approximated the warp update rule and the gradient descent approximation in future papers we will cover the choice of the error function how to allow linear appearance variation and how to impose priors on the parameters

image super resolution via sparse representation this paper presents a new approach to single image superresolution based upon sparse signal representation research on image statistics suggests that image patches can be well represented as a sparse linear combination of elements from an appropriately chosen over complete dictionary inspired by this observation we seek a sparse representation for each patch of the low resolution input and then use the coefficients of this representation to generate the high resolution output theoretical results from compressed sensing suggest that under mild conditions the sparse representation can be correctly recovered from the downsampled signals by jointly training two dictionaries for the low and high resolution image patches we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries therefore the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch the learned dictionary pair is a more compact representation of the patch pairs compared to previous approaches which simply sample a large amount of image patch pairs reducing the computational cost substantially the effectiveness of such a sparsity prior is demonstrated for both general image super resolution sr and the special case of face hallucination in both cases our algorithm generates high resolution images that are competitive or even superior in quality to images produced by other similar sr methods in addition the local sparse modeling of our approach is naturally robust to noise and therefore the proposed algorithm can handle sr with noisy inputs in a more unified framework

structure of characteristic lyapunov vectors in spatiotemporal chaos we study lyapunov vectors lvs corresponding to the largest lyapunov exponents in systems with spatiotemporal chaos we focus on characteristic lvs and compare the results with backward lvs obtained via successive gram schmidt orthonormalizations systems of a very different nature such as coupled map lattices and the continuous time lorenz 96 model exhibit the same features in quantitative and qualitative terms additionally we propose a minimal stochastic model that reproduces the results for chaotic systems our work supports the claims about universality of our earlier results i g szendro phys rev e 76 025202 r 2007 for a specific coupled map lattice

spatiotemporal structure of lyapunov vectors in chaotic coupled map lattices the spatiotemporal dynamics of lyapunov vectors lvs in spatially extended chaotic systems is studied by means of coupled map lattices we determine intrinsic length scales and spatiotemporal correlations of lvs corresponding to the leading unstable directions by translating the problem to the language of scale invariant growing surfaces we find that the so called characteristic lvs exhibit spatial localization strong clustering around given spatiotemporal loci and remarkable dynamic scaling properties of the corresponding surfaces in contrast the commonly used backward lvs obtained through gram schmidt orthogonalization spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction