algebraic signal processing theory cooley tukey type algorithms for dcts and dsts in this paper we systematically derive a large class of fast general radix algorithms for various types of real discrete fourier transforms real dfts including the discrete hartley transform dht based on the algebraic signal processing theory this means that instead of manipulating the transform definition we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method the same method yields the well known cooley tukey fast fourier transform fft as well as general radix discrete cosine and sine transform algorithms the algebraic approach makes the derivation concise unifies and classifies many existing algorithms yields new variants enables structural optimization and naturally produces a human readable structural algorithm representation based on the kronecker product formalism we show for the first time that the general radix cooley tukey and the lesser known bruun algorithms are instances of the same generic algorithm further we show that this generic algorithm can be instantiated for all four types of the real dft and the dht

fluctuations convergence times correlation functions and power laws from many body lyapunov spectra for soft and hard disks and spheres the dynamical instability of many body systems is best characterized through the time dependent local lyapunov spectrum l j its associated comoving eigenvectors d j and the global time averaged spectrum l j we study the fluctuations of the local spectra as well as the convergence ratesand correlation functions associated with the d vectors as functions of j and system size n all the number dependences can be described by simple power laws the various powers depend on the thermodynamic state and force law as well as system dimensionality

dct algorithms for composite sequence lengths this paper presents fast algorithms for the type ii and iii discrete cosine transforms of composite sequence length in particular a radix q algorithm where q is an odd integer is derived for uniform or mixed radix decomposition of the discrete cosine transform by combining the radix q and radix 2 algorithms a general decomposition method for any composite length is developed reduction of computational complexity can be achieved for many sequence lengths compared with that needed by the well known radix 2 algorithm furthermore both the proposed and chan and siu s 1993 mixed radix algorithms achieve the same computational complexity for n 3 2 sup p and 9 2 sup p however our algorithm uses a simpler decomposition approach and provides a wider range of choices of sequence lengths

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

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

fast algorithm for computing discrete cosine transform an efficient method for computing the discrete cosine transform dct is proposed based on direct decomposition of the dct the recursive properties of the dct for an even length input sequence is derived which is a generalization of the radix 2 dct algorithm based on the recursive property a new dct algorithm for an even length sequence is obtained the proposed algorithm is very structural and requires fewer computations when compared with others the regular structure of the proposed algorithm is suitable for fast parallel algorithm and vlsi implementation

analysis of dyadic approximation error for hybrid video codecs with integer transforms in this paper we present an analysis of the dyadic approximation error introduced by the integerization of transform coding in h 264 avc like codecs we derive the analytical formulations for dyadic approximation error and nonorthogonality error we further classify the dyadic approximation error into a system error and a nonflat error and proposed two models for them we found that the nonflat error has a substantial impact on video quality if the number of shifting bits at decoder side dq bits is small we also give a theoretical justification on why scaling factors at encoder side are better to be adapted to the rescaling factors at decoder side in h 264 avc like codecs

on the validity of the conjugate pairing rule for lyapunov exponents for hamiltonian systems subject to an external potential which in the presence of a thermostat will reach a nonequilibrium stationary state dettmann and morriss proved a strong conjugate pairing rule scpr for pairs of lyapunov exponents in the case of isokinetic ik stationary states which have a given kinetic energy this scpr holds for all initial phases of the system all times t and all numbers of particles n this proof was generalized by wojtkowski and liverani to include hard interparticle potentials a geometrical reformulation of those results is presented the present paper proves numerically using periodic orbits for the lorentz gas that scpr cannot hold for isoenergetic ie stationary states which have a given total internal energy in that case strong evidence is obtained for cpr to hold for large n and t where it can be conjectured that the larger n the smaller t will be this suffices for statistical mechanics

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 algorithms for type iii dct of composite sequence lengths this correspondence presents an odd factor algorithm for the type iii discrete cosine transform dct for uniform or mixed radix decomposition by jointly using the old factor and the existing radix 2 algorithms a general decomposition method for arbitrarily composite sequence length is developed a reduction of computational complexity can be achieved compared with that needed by other reported algorithms for m 2 sup m the decomposition approach has a simple computational structure and supports a wider range of choices for different sequence lengths

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

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

mixed radix discrete cosine transform presents two new fast discrete cosine transform computation algorithms a radix 3 and a radix 6 algorithm these two new algorithms are superior to the conventional radix 3 algorithm as they i require less computational complexity in terms of the number of multiplications per point ii provide a wider choice of the sequence length for which the dct can be realized and iii support the prime factor decomposed computation algorithm to realize the 2 sup m 3 sup n point dct furthermore a mixed radix algorithm is also proposed such that an optimal performance can be achieved by applying the proposed radix 3 and radix 6 and the well developed radix 2 decomposition techniques in a proper sequence

lyapunov instability of rough hard disk fluids the dynamical instability of rough hard disk fluids in two dimensions is characterized through the lyapunov spectrum and the kolmogorov sinai entropy h ks for a wide range of densities and moments of inertia i for small i the spectrum separates into translation dominated and rotation dominated parts with increasing i the rotation dominated part is gradually filled in at the expense of translation until such a separation becomes meaningless at any density the rate of phase space mixing given by h ks becomes less and less effective the more the rotation affects the dynamics however the degree of dynamical chaos measured by the maximum lyapunov exponent is only enhanced by the rotational degrees of freedom for high density gases but is diminished for lower densities surprisingly no traces of lyapunov modes were found in the spectrum for larger moments of inertia the spatial localization of the perturbation vector associated with the maximum exponent however persists for any i

front propagation techniques to calculate the largest lyapunov exponent of dilute hard disk gases a kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point measured by the largest lyapunov exponent for a dilute system of hard disks both in equilibrium and in a uniform shear flow we derive a generalized boltzmann equation for an extended one particle distribution that includes deviations from the reference phase space point the equation is valid for very low densities n and requires an unusual expansion in powers of 1 ln n it reproduces and extends results from the earlier more heuristic clock model and may be interpreted as describing a front propagating into an unstable state the asymptotic speed of propagation of the front is proportional to the largest lyapunov exponent of the system its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand for the equilibrium case an explicit expression for the largest lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest lyapunov exponent

on a class of elliptic problems in r symmetry and uniqueness results in the plane r 2 we classify all solutions for an elliptic problem of liouville type involving a radial weight function as a consequence we clarify the origin of the non radially symmetric solutions for the given problem as established by chanillo and kiessling for a more general class of liouville type problems we show that rather than radial symmetry the solutions always inherit the invariance of the problem under inversion with respect to suitable circles this symmetry result is derived with the help of a shrinking sphere method

a new algorithm to compute the dct and its inverse a novel algorithm to convert the discrete cosine transform dct to skew circular convolutions is presented the motivation for developing such an algorithm is the fact that vlsi implementation of distributed arithmetic is very efficient for computing convolutions it is also shown that the inverse dct idct can be computed using the same building blocks which are used for computing the dct a dct idct processor can be designed to compute either the dct or the idct depending on a 1 b control signal

singular limit solutions for a 2 dimensional semilinear elliptic system of liouville type in some general case adding singular sources part ii abstract we consider the existence of singular limit solutions for a nonlinear elliptic system of liouville type with singular sources terms given by dirac masses and dirichlet boundary conditions we use the nonlinear domain decomposition method and the pohozaev identity

input and output index mappings for a prime factor decomposed computation of discrete cosine transform a formal direct derivation of the prime factor decomposed computation algorithm is presented the derivation is direct in the sense that it is based on the real cosine function without resort to the discrete fourier transform expressions or the complex functions based on the equations obtained from the derivation input and output index mappings are introduced in the form of tables this tabulation enables any prime factor decomposable discrete cosine transform dct to be implemented in a straight forward manner the use of the index mapping tables is demonstrated for the 12 point dct

a new algorithm to compute the discrete cosine transform a new algorithm is introduced for the 2m point discrete cosine transform this algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms and it makes the system simpler

infinitely many solutions for a class of sublinear schrodinger equations with indefinite potentials in this paper we are concerned with qualitative properties of entire solutions to a schrodinger equation with sublinear nonlinearity and sign changing potentials our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions

fast algorithm for the 3 d dct ii recently many applications for three dimensional 3 d image and video compression have been proposed using 3 d discrete cosine transforms 3 d dcts among different types of dcts the type ii dct dct ii is the most used in order to use the 3 d dcts in practical applications fast 3 d algorithms are essential therefore in this paper the 3 d vector radix decimation in frequency 3 d vr dif algorithm that calculates the 3 d dct ii directly is introduced the mathematical analysis and the implementation of the developed algorithm are presented showing that this algorithm possesses a regular structure can be implemented in place for efficient use of memory and is faster than the conventional row column frame rcf approach furthermore an application of 3 d video compression based 3 d dct ii is implemented using the 3 d new algorithm this has led to a substantial speed improvement for 3 d dct ii based compression systems and proved the validity of the developed algorithm