eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

towards a matrix theory for unbounded operator matrices sur l espace produit r l 2 r on considere l operateur 7b a a 0 ou d d dx est defini sur d a w 1 r 0 est la mesure de dirac en 0 c l 2 r est un operateur de r dans l 2 r et 7b a a un domaine d 7b a r w 1 r on introduit une fonction caracteristique non commutative donnant une facon efficace pour calculer le spectre de 7b a on obtient des resultats sur la generation de semi groupes par des matrices operateur

semi groupes sous markoviens engendres par des operateurs matriciels we characterize generators of sub markovian semigroups onlp by a version of kato s inequality this will be used to show under precise assumptions that the semigroup generated by a matrix operatora aij 1 i j n on lp n is sub markovian if and only if the semigroup generated by the sum of each rowai1 ain 1 i n is sub markovian the corresponding result on c0 x n characterizes dissipative operator matrices

probabilistic visual learning for object representation we present an unsupervised technique for visual learning which is based on density estimation in high dimensional spaces using an eigenspace decomposition two types of density estimates are derived for modeling the training data a multivariate gaussian for unimodal distributions and a mixture of gaussians model for multimodal distributions those probability densities are then used to formulate a maximum likelihood estimation framework for visual search and target detection for automatic object recognition and coding our learning technique is applied to the probabilistic visual modeling detection recognition and coding of human faces and nonrigid objects such as hands

towards a dynamical temperature of finite hamiltonian systems abstract with the aid of numerical simulations of the fermi pasta ulam fpu system we compare the different definitions of dynamical temperature for hamiltonian systems we have shown that each definition gives different values of temperature for a system with a small number of degrees of freedom dof only for systems with a sufficiently large number of dof do all the definitions of dynamical temperature approach the same value

possible physical meaning of the tsallis entropy parameter since the proposal of the tsallis generalized entropy the general explanation of the role played by the parameter q that defines which specific entropy to pick among a whole family remained somewhat partial and tentative although some particular examples were taken into account specifically the purpose of the present paper is to present a rigorous formal derivation of a mathematical expression giving the parameter q in terms of the momentum fluctuations of a stochastic process thus furnishing at least one of its possible physical origins

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

on linear volterra equations in banach spaces abstract in this paper we present the basic theory for a class of volterra differential integral equations of convolution type in banach spaces we show that existence of resolvent operator for such an equation is equivalent to its wellposedness and we obtain a hille yosida theorem unfortunately this result is not easy to apply and therefore it is important to have peturbation theorems available we present a result of this type which also contains an existence theorem and show by means of several examples that it cannot be improved

the spectrum of unbounded operator matrices with non diagonal domain abstract systems of linear evolution equations involving non diagonal boundary conditions yield operator matrices having non diagonal domains in products of banach spaces we show how to compute the spectrum of these matrices by using virtual matrix elements and characteristic operator functions

on the spectrum of evolution operators generated by hyperbolic systems abstract we prove that the asymptotic behavior of semigroups generated by mixed initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation the proof is based on the following reduction principle we construct a reduced system which is much simpler than the given one such that the difference of the semigroups is a compact operator the construction of the reduced system is presented in the non autonomous case because it has implifications in the study of systems which are time periodic

multivariable extremum seeking for pde dynamic systems in this article we address the design and analysis of multivariable newton based extremum seeking es for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations pdes multi input systems with distinct diffusion coefficients in each individual input channel are dealt with the phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation based averaging based estimate of the hessian s inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the diffusion pdes in addition the generalization of the results for different classes of parabolic reaction advection diffusion pdes wave equations and or first order hyperbolic transport dominated pdes is also discussed as for ordinary differential equations case the proposed newton approach removes the dependence of the algorithm s convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable unlike the gradient algorithm a numerical example illustrates the performance of our multivariable es for compensating pde based systems

fluctuation of energy in the generalized thermostatistics abstract we calculate the fluctuation of a system s energy in tsallis statistics following the finite heat bath canonical ensemble approach we obtain this fluctuation as the second derivative of the logarithm of the partition function plus an additional term we also find an explicit expression for the relative fluctuation as related to the number of degrees of freedom of the bath and the composite system

existence and stability for some partial neutral functional differential equations with infinite delay abstract in this paper we study a class of partial neutral functional differential equations with infinite delay we suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the hille yosida theorem we give some sufficient conditions ensuring the existence uniqueness and regularity of solutions a principle of linearized stability is also established in the autonomous case to illustrate our abstract results we conclude this work by an example

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

l2 regularity for parabolic partial integrodifferential equations with delay in the highest order derivatives abstract a partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay by means of abstract regularity results global existence and uniqueness of a strict solution are obtained moreover a characterization of the infinitesimal generator of the solution operator is given

possible canonical distributions for finite systems with nonadditive energy region des pays de la loire of france 2007 6088 science research fund 07bs105 huaqiao university people s republic of china

central limit behavior of deterministic dynamical systems we investigate the probability density of rescaled sums of iterates of deterministic dynamical systems a problem relevant for many complex physical systems consisting of dependent random variables a central limit theorem clt is valid only if the dynamical system under consideration is sufficiently mixing for the fully developed logistic map and a cubic map we analytically calculate the leading order corrections to the clt if only a finite number of iterates is added and rescaled and find excellent agreement with numerical experiments at the critical point of period doubling accumulation a clt is not valid anymore due to strong temporal correlations between the iterates nevertheless we provide numerical evidence that in this case the probability density converges to a q gaussian thus leading to a power law generalization of the clt the above behavior is universal and independent of the order of the maximum of the map considered i e relevant for large classes of critical dynamical systems

the closure of the generalised eigenspace of a class of infinitesimal generators in this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type this class includes the solution map for functional differential equations and certain partial differential equations in particular we present necessary and sufficient conditions for one to oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator

analysis of a viscoelastic spring mass model in this paper we consider a linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring mass damper ode our purpose is to compare analytically these two approaches to the same physical system we take a functional analysis point of view based on semigroup theory spectral perturbation analysis and dominant eigenvalues

complete second order linear differential operator equations in hilbert space and applications in hydrodynamics we study the cauchy problem for a complete second order linear differential operator equation in a hilbert space h of the form d 2 u dt 2 f ik du dt bu f u 0 u 0 u 1 0 u 1 problems of this kind arise e g in hydrodynamics where the coefficients f k and b are unbounded selfadjoint operators it is assumed that f is the dominating operator in the cauchy problem above i e d f d b d f d k we also suppose that f and b are bounded from below but the operator coefficients are not assumed to commute the main results concern the existence of strong solutions to the stated cauchy problem and applications of these results to the cauchy problem associated with small motions of some hydrodynamical systems

quasicanonical gibbs distribution and tsallis non extensive statistics abstract we derive and study quasicanonical gibbs distribution function which is characterized by the thermostat with finite number of particles quasithermostat we show that this naturally leads to tsallis non extensive statistics and thermodynamics with tsallis parameter q is found to be related to the number of particles in the quasithermostat we show that the chi square distribution of fluctuating temperature used recently by beck can be partially understood in terms of normal random momenta of particles in the quasithermostat also we discuss on the importance of the time scale hierarchy and fluctuating probability distribution functions in understanding of tsallis distribution within the framework of kinetics of dilute gas and weakly inhomogeneous systems

numerical calculation of the energy relative fluctuation for a system in contact with a finite heat bath abstract we have recently shown that the dispersion and the energy square relative fluctuations of a finite system in contact with a finite heat bath depends on the number of degrees of freedom of the system and of the bath in this paper we present numerical simulations which support this result

dissipative dynamics in a finite chaotic environment relationship between damping rate and lyapunov exponent we consider the energy flow between a classical one dimensional harmonic oscillator and a set of n two dimensional chaotic oscillators which represents the finite environment using linear response theory we obtain an analytical effective equation for the system harmonic oscillator which includes a frequency dependent dissipation a shift and memory effects the damping rate is expressed in terms of the environment mean lyapunov exponent a good agreement is shown by comparing theoretical and numerical results even for environments with mixed regular and chaotic motion resonance between system and environment frequencies is shown to be more efficient to generate dissipation than larger mean lyapunov exponents or a larger number of bath chaotic oscillators

eventual norm continuity for neutral semigroups on banach spaces in this paper we prove under suitable hypotheses eventually norm continuity and compactness of the solution semigroups of certain neutral differential equations in banach spaces our approach is based on a general perturbation theorem obtained from closed loop systems of infinite dimensional control systems with unbounded control and observation operators in fact the solution semigroup can be viewed as the semigroup of an appropriate closed loop system in product state spaces

on a maximal lp lq approach to the compressible viscous fluid flow with slip boundary condition abstract in this paper we prove a local in time unique existence theorem for the compressible viscous fluids in the general domain with slip boundary condition for the purpose we use the contraction mapping principle based on the maximal l p l q regularity by means of the weis operator valued fourier multiplier theorem for the corresponding time dependent problem to obtain the maximal l p l q regularity we prove the sectorial r boundedness of the solution operator to the generalized stokes equations

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

statistical mechanics of quartic oscillators we study statistical mechanics of quartic oscillator with two degrees of freedom which is known to be chaotic almost everywhere except in a few regions of the parameter range we obtain exact expressions for temperature entropy and distribution functions temperature is also obtained numerically by time averaging the kinetic energy and using equipartition theorem and agrees with our expressions when the system is almost chaotic we further generalize our model to quartic oscillators with n degrees of freedom and exact expressions for thermodynamic quantities are obtained as n standard statistical mechanics results are recovered we also discuss pressure density and equation of state of this system

wellposedness and wave propagation for a class of integrodifferential equations in banach space in recent years there has been an increasing interest in the mathematical treatment of problems which originate in the theory of viscoelastic materials in such problems the constitutive relations for these materials incorporate the history of the strain or the strain rate and in the linear theory lead to linear volterra integrodifferential equations with infinite memory in a banach space x the purpose of this paper is to examine the problems of wellposedness and wave propagation for one such equation in particular we are concerned with equations of the form x t 4 x r j f t s x s ds al

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

asymptotic stability of the solution of the m mb 1 queueing model by using positive c 0 semigroup theory we study the asymptotic stability of the solution of a bulk queueing model m m b 1

a mixed discrete continuous fragmentation model abstract motivated by the occurrence of shattering mass loss observed in purely continuous fragmentation models this work concerns the development and the mathematical analysis of a new class of hybrid discrete continuous fragmentation models once established the model which takes the form of an integro differential equation coupled with a system of ordinary differential equations is subjected to a rigorous mathematical analysis using the theory and methods of operator semigroups and their generators most notably by applying the theory relating to the kato voigt perturbation theorem honest substochastic semigroups and operator matrices the existence of a unique differentiable solution to the model is established this solution is also shown to preserve nonnegativity and conserve mass

on nonautonomous functional differential equations abstract in this paper we prove a theorem on boundary perturbation of nonautonomous cauchy problems and then apply this result to show the existence and uniqueness of classical solutions of the nonautonomous banach space valued functional differential equation formula

generalized entropies from first principles a derivation of power law canonical distributions from first principle statistical mechanics including the exponential distribution as a particular case is presented it is shown that these distributions arise naturally and that the heat capacity of the heat bath is the condition that determines its type as a consequence a physical interpretation for the parameter q of the generalized entropy is given

necessary conditions for partial differential equations with delay to generate c0 semigroups on etudie l existence des solutions ou le caractere bien pose d equations aux derivees partielles a retard du type du t dt f u t ou f est un operateur d un espace de fonction 7b f r 0 y dans l espace de banach y dans lequel l equation evolue

hopf bifurcation for semilinear dissipative hyperbolic systems abstract we consider boundary value problems for semilinear hyperbolic systems of the type t u j a j x x u j b j x u 0 x 0 1 j 1 n with smooth coefficient functions a j and b j such that b j x 0 0 for all x 0 1 r and j 1 n we state conditions for hopf bifurcation i e for existence local uniqueness up to phase shifts smoothness and smooth dependence on of time periodic solutions bifurcating from the zero stationary solution furthermore we derive a formula which determines the bifurcation direction the proof is done by means of a liapunov schmidt reduction procedure for this purpose fredholm properties of the linearized system and implicit function theorem techniques are used

thermodynamical relations for systems in contact with finite heat baths abstract we obtain the relations between macroscopic thermodynamical properties and microscopic ensemble averages of a system in contact with a heat bath which has a finite number of degrees of freedom our approach is based on the geometry of the accessible phase space of the system by using a suitable definition of temperature for such kind of system energy equipartition theorem it is shown that the relations obtained here have the same dependence on the system s partition function z 1 as in boltzmann gibbs statistics

existence of asymptotically almost automorphic solutions to some abstract partial neutral integro differential equations abstract the paper establishes the existence and uniqueness of asymptotically almost automorphic mild solution to an abstract partial neutral integro differential equation with unbounded delay an example is given to illustrate our results

existence and uniqueness of pseudo almost periodic solutions to some classes of partial evolution equations abstract some sufficient conditions for the existence and uniqueness of pseudo almost periodic mild solutions to some classes of partial evolution equations are given we then make use of our abstract results to discuss the existence of pseudo almost periodic solutions to some partial differential equations

existence of solutions for neutral integro differential equations with state dependent delay in this paper we study the existence and regularity of solutions for a neutral functional integro differential equation with state dependent delay in banach space the mild solutions are investigated by sadovskii fixed point theorem under compactness condition for the resolvent operator and the theory of fractional power and a norm are also used in the discussion since the nonlinear terms of the system involve spacial derivatives the strict solutions are obtained under holder continuity condition hence this work extends and develops the existed results on this topic in the end an example is provided to illustrate the application of the obtained results

modeling the manifolds of images of handwritten digits this paper describes two new methods for modeling the manifolds of digitized images of handwritten digits the models allow a priori information about the structure of the manifolds to be combined with empirical data accurate modeling of the manifolds allows digits to be discriminated using the relative probability densities under the alternative models one of the methods is grounded in principal components analysis the other in factor analysis both methods are based on locally linear low dimensional approximations to the underlying data manifold links with other methods that model the manifold are discussed

spectral enclosures for a class of block operator matrices abstract we prove new spectral enclosures for the non real spectrum of a class of 2 2 block operator matrices with self adjoint operators a and d on the diagonal and operators b and b as off diagonal entries one of our main results resembles gershgorin s circle theorem the enclosures are applied to j frame operators

tsallis power laws and finite baths with negative heat capacity it is often stated that heat baths with finite degrees of freedom i e finite baths are sources of tsallis distributions for classical hamiltonian systems by using well known fundamental statistical mechanics expressions we rigorously show that tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails however the correspondence between tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium we comment on the implications of the finite bath for the recent attempts towards a q generalized central limit theorem

local existence for retarded volterra integrodifferential equations with hille yosida operators abstract this paper is devoted to finding some existence and uniqueness theorems for classical solutions of the integrodifferential equations with infinite delay we give some sufficient conditions ensuring the existence and uniqueness of solutions we assume the linear part is not necessary to be densely defined and satisfies a hille yosida condition so that it is the generator of a nondegenerated locally lipschitz continuous integrated semigroup by using matrix operators and fixed point theorems we obtain new results for the retarded integrodifferential equations

local existence and stability for some partial functional differential equations with infinite delay we study the existence regularity and stability of solutions for some partial functional dierential equations with infinite delay we assume that the linear part is not necessarily densely defined and satisfies the hille yosida condition on a banach space x the nonlinear term takes its values in space larger than x namely the extrapolated favard class of the extrapolated semigroup corresponding to the linear part our approach is based on the theory of the extrapolation spaces

eigenvalue asymptotics in isothermal forced elongation in this paper we consider the formal linearization of the governing equations of forced elongation in the isothermal regime we give an asymptotic characterization of the spectrum of the associated strongly continuous semigroup to this end we develop rigorous asymptotic techniques to determine the asymptotic behavior of the eigenvalues by solving a characteristic integral equation in a generalized sector of the complex plane the spectral mapping theorem relating the spectrum of the semigroup to the spectrum of its generator proves essential numerical evidence will be given to demonstrate the accuracy of our asymptotic predictions

stabilizability of systems of one dimensional wave equations by one internal or boundary control force we study the internal and boundary stabilizability of a system of wave equations by one control force we prove that the classical internal damping applied to only one of the equations never gives exponential stability if the wave speeds are different and if the wave speeds are the same we give explicit necessary and sufficient conditions for the stability to occur we also study the simultaneous boundary stabilization of the same system

dynamical stability of non constant equilibria for the compressible navier stokes equations in eulerian coordinates in this paper we establish global existence and uniqueness of strong solutions to the non isothermal compressible navier stokes equations in bounded domains the initial data have to be near equilibria that may be non constant due to considering large external forces we are able to show exponential stability of equilibria in the phase space and above all to study the problem in eulerian coordinates the latter seems to be a novelty since in works by other authors global strong lp solutions have been investigated only in lagrangian coordinates eulerian coordinates are even declared as impossible to deal with the proof is based on a careful derivation and study of the associated linear problem

face recognition using the second order mixture of eigenfaces method the well known eigenface method uses an eigenface set obtained from principal component analysis however the single eigenface set is not enough to represent the complicated face images with large variations of poses and or illuminations to overcome this weakness we propose a second order mixture of eigenfaces method that combines the second order eigenface method iso mpg m5750 noordwijkerhout march 2000 and the mixture of eigenfaces method a k a gaussian mixture model proceedings ijcnn2001 2001 in this method we use a couple of mixtures of multiple eigenface sets one is a mixture of multiple approximate eigenface sets for face images and another is a mixture of multiple residual eigenface sets for residual face images each mixture of multiple eigenface sets has been obtained from expectation maximization learning consecutively based on two mixture of multiple eigenface sets each face image is represented by a couple of feature vectors obtained by projecting the face image onto a selected approximate eigenface set and then by projecting the residual face image onto a selected residual eigenface set recognition is performed by the distance in the feature space between the input image and the template image stored in the face database simulation results show that the proposed second order mixture of eigenfaces method is best for face images with illumination variations and the mixture of eigenfaces method is best for the face images with pose variations in terms of average of the normalized modified retrieval rank and false identification rate

polynomial operator matrices as semigroup generators the 2 2 case on considere des systemes d equations d evolution non lineaires ecrits sous la forme u t au t ou u est une fonction a valeurs dans un espace produit e n et a a ij n m est une matrice dont les a ij sont des operateurs lineaires sur e on etudie le cas ou les a ij sont des polynomes p ij a par rapport a un seul operateur a sur e et on se restreint au cas des matrices 2 2

on the free boundary problem for the oldroyd b model in the maximal lp lq regularity class abstract in the present work we prove the local well posedness of non newtonian compressible viscous barotropic fluid flow of oldroyd b type with free surface in a bounded domain of n dimensional euclidean space n 2 the key step is to prove the maximal l p l q regularity theorem for the linearized equation with the help of the r bounded solution operators for the corresponding resolvent problem and weis s operator valued fourier multiplier theorem

global existence for semilinear parabolic systems on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme u t a t u f t x u du d 2m 1 u dans t 0 b t u 0 sur t 0 u t 0 u 0 sur ou est un domaine de r n

approximate controllability of semilinear neutral evolution systems with delay this paper considers the approximate controllability for a class of neutral control systems governed by semilinear neutral delayed equations sufficient conditions for approximate controllability are established by constructing fundamental solutions and using resolvent condition banach contraction mapping principle and techniques on fractional power operators the obtained results improve some existing analogous ones on this topic to some extent an example is presented finally to illustrate the applications of the obtained results

brief paper arbitrary decay rate for two connected strings with joint anti damping by boundary output feedback in this paper we are concerned with the boundary stabilization of two connected strings with middle joint anti damping for which all eigenvalues of the control free system are located on the right complex plane we first design an explicit state feedback controller to achieve exponential stability for the closed loop system consequently we design the output feedback by using infinite dimensional observer the backstepping approach is adopted in investigation it is shown that by using one boundary stabilizer only the output feedback can make the closed loop system exponentially stable with arbitrary decay rate

large time behavior of the semigroup on lp spaces associated with the linearized compressible navier stokes equation in a cylindrical domain abstract large time behavior of solutions to the linearized compressible navier stokes equation around the motionless state in a cylindrical domain is investigated the l p decay estimates of the associated semigroup are established for all 1 p it is also shown that the time asymptotic leading part of the semigroup is given by a one dimensional heat semigroup

matrices of operators and regularized semigroups where p is a matr ix of polynomials in n variables and ia1 ia is a family of c o m m u t i n g generators of bounded linear strongly cont inuous groups of operators intuitively one wants to take the exponent ia l of the matr ix of opera to r s p a since the spec t rum of ak is real for all k this means we want the complex matr ix pi j x to be contained in a left half plane in some sense for all x in r the strongest sense is to have the numerical range of the complex matr ix conta ined in a half plane r e z 0 xer a sense that is easier to calculate or at least estimate is to have the spectrum of p d x conta ined in a left half plane for all x r the system 1 1 is then said to be petrowsky correct see 9 the intuit ion we just described is not quite true in general wi thout some