the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

nonlinear stability of boundary layer solution to the boltzmann equation with diffusive effect at the boundary in this paper the stability of boundary layer solutions to the boltzmann equation with diffusive effect at the boundary for hard sphere model is considered when the mach number of the far field is less than 1 the exponential decay in time is proven for linearized operator first at last based on this property nonlinear stability of the boundary layers is obtained by bootstrap argument

contraction of the matrix unit disk abstract a mobius transformation m that is j equivalent to a contraction decreases the pseudochordal distance between every pair of points in the unit matrix disk it may keep unchanged the distance between some pairs and strictly decrease the distance between others

energy method for boltzmann equation abstract a basic simple energy method for the boltzmann equation is presented here it is based on a new macro micro decomposition of the boltzmann equation as well as the h theorem this allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method as an illustration we apply the method for the study of the time asymptotic nonlinear stability of the global maxwellian states previous energy method starting with grad and finishing with ukai involves the spectral analysis and regularity of collision operator through sophisticated weighted norms

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

a half space problem for the boltzmann equation with specular reflection boundary condition there are many open problems on the stability of nonlinear wave patterns to the boltzmann equation even though the corresponding stability theory has been comparatively well established for the gas dynamical systems in this paper we study the nonlinear stability of a rarefaction wave profile to the boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut off the analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection and the decomposition on both the solution and the boltzmann equation introduced in 24 26 for energy method

nonlinear stability of rarefaction waves for compressible navier stokes equations it is shown that expansion waves for the compressible navier stokes equations are nonlinearly stable the expansion waves are constructed for the compressible euler equations based on the inviscid burgers equation our result shows that navier stokes equations and euler equations are time asymptotically equivalent on the level of expansion waves the result is proved using the energy method making essential use of the expansion of the underlining nonlinear waves and the specific form of the constitutive eqution for a polytropic gas

nonlinear stability of rarefaction waves for the boltzmann equation it is well known that the boltzmann equation is related to the euler and navier stokes equations in the field of gas dynamics the relation is either for small knudsen number or for dissipative waves in the time asymptotic sense in this paper we show that rarefaction waves for the boltzmann equation are time asymptotic stable and tend to the rarefaction waves for the euler and navier stokes equations our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro macro decomposition of the boltzmann equation the expansion nature of the rarefaction waves and the suitable microscopic version of the h theorem are essential elements of our analysis

m band compactly supported orthogonal symmetric interpolating scaling functions in many applications wavelets are usually expected to have the following properties compact support orthogonality linear phase regularity and interpolation to construct such wavelets it is crucial designing scaling functions with the above properties in two and three band cases except for the haar functions there exists no scaling function with the above five properties in m band case m spl ges 4 more free degrees available in design enable us to construct such scaling functions a novel approach to designing such scaling functions is proposed first we extend the two band dubuc 1986 filters to the m band case next the m band fir regular symmetric interpolating scaling filters are parameterized and then m band fir regular orthogonal symmetric interpolating scaling filters osisfs are designed via optimal selection of parameters finally two family of four band and five band osisfs and scaling functions are developed and their smoothness are estimated

model validation of multirate systems from time domain experimental data the model validation problem using time domain experimental data is studied for multirate linear fractional uncertain models in this note as a technical tool the caratheodory fejer cf interpolation problem with a nest operator constraint is first investigated this problem is itself of interest mathematically and has potential applications in addressing other problems in control signal processing and circuit theory a necessary and sufficient solvability condition for this interpolation problem is given the validation tests are then presented based on this condition and the lifting technique tractable convex optimization methods can be used to solve the validation problems

on a boundary layer problem for the nonlinear boltzmann equation this article deals with a boundary layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero the model considered here is the stationary nonlinear boltzmann equation in one dimension with a slightly perturbed reflection boundary condition we restrict our attention to the case of hard spheres collisions with grad s cutoff assumption existence uniqueness and asymptotic behavior are derived by means of energy estimates

multirate sampled data systems all h sub spl infin suboptimal controllers and the minimum entropy controller for a general multirate sampled data sd system the authors characterize explicitly the set of all causal stabilizing controllers that achieve a certain h sub spl infin norm bound moreover they give explicitly a particular controller that further minimizes an entropy function for the sd system the characterization lays the groundwork for synthesizing multirate control systems with multiple mixed control specifications

large time behaviors of solutions to an inflow problem in the half space for a one dimensional system of compressible viscous gas the inflow problem for a one dimensional compressible barotropic flow on the half line r 0 is investigated not only classical waves but also the new wave which is called the boundary layer solution arise large time behaviors of the solutions to be expected have been classified in terms of the boundary values by a matsumura inflow and outflow problems in the half space for a one dimensional isentropic model system of compressible viscous gas to appear in proceedings of ims conference on differential equations from mechanics hong kong 1999 in this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave

multiscale difference equation signal models i theory the paper studies multiscale difference equation models for l d and m d signals in this modeling technique the signal of interest is viewed as a solution to a multiscale difference equation msde the model completely characterizes the signal as well as a number of its higher derivatives it provides a recursive signal interpolation scheme as a function of scale it also leads naturally to multigrid signal filtering detection and estimation algorithms an msde model must be uniquely decodable i e it must correspond to a unique signal therefore one must guarantee that the modeling msde has a unique solution the authors investigate the existence and uniqueness of l sub 1 and l sub 2 solutions to multiscale difference equations using fourier domain techniques they derive conditions for the existence of l sub 1 solutions to an msde they provide conditions under which the l sub 1 solution is unique up to a multiplicative constant and has compact support they also derive sufficient but not necessary conditions for the existence of a unique l sub 2 solution to a subclass of msdes the results extend known facts about the solutions of two scale difference equations the paper concludes with several examples of msde signal models that highlight the modeling advantages of msdes over two scale difference equation models

multiscale representations of markov random fields recently a framework for multiscale stochastic modeling was introduced based on coarse to fine scale recursive dynamics defined on trees this model class has some attractive characteristics which lead to extremely efficient statistically optimal signal and image processing algorithms the authors show that this model class is also quite rich in particular they describe how 1 d markov processes and 2 d markov random fields mrfs can be represented within this framework the recursive structure of 1 d markov processes makes them simple to analyze and generally leads to computationally efficient algorithms for statistical inference on the other hand 2 d mrfs are well known to be very difficult to analyze due to their noncausal structure and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification in contrast their multiscale representations are based on scale recursive models and thus lead naturally to scale recursive algorithms which can be substantially more efficient computationally than those associated with mrf models in 1 d the multiscale representation is a generalization of the midpoint deflection construction of brownian motion the representation of 2 d mrfs is based on a further generalization to a midline deflection construction the exact representations of 2 d mrfs are used to motivate a class of multiscale approximate mrf models based on one dimensional wavelet transforms they demonstrate the use of these latter models in the context of texture representation and in particular they show how they can be used as approximations for or alternatives to well known mrf texture models

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

strictly contractive and positive completions for block matrices abstract this paper deals with the following two problems of completion of block matrices 1 let a ij i j q be given matrices find additional matrices a ij such that the block matrix a a ij n i 1 m j 1 has norm less than one 2 let b ij j i q be given matrices find additional b ij such that the completion b b ij n i j 1 is positive definite the analysis is based on a study of certain linear fractional maps this approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data the maximum entropy principle appears as a corollary of a general formula for the determinant of a completion special attention is paid to the toeplitz case

completion of hankel partial contractions of extremal type we find concrete necessary and sufficient conditions for the existence of contractive completions of hankel partial contractions of size 4 4 in the extremal case along the way we introduce a new approach that allows us to solve algorithmically the contractive completion problem for 4 4 hankel matrices as an application we obtain a concrete example of a partially contractive 4 4 hankel matrix which does not admit a contractive completion

unitary dilation approach to contractive matrix completion in this paper a new method to construct all contractive matrix completion to a block triangular matrix is given this new method uses only elementary matrix operations and has the advantages of better numerical reliability and ease in implementation on computers

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

a classification of well posed kinetic layer problems in the first part of this paper we study the half space boundary value problem for the boltzmann equation with an incoming distribution obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero we linearize it about a drifting maxwellian and prove that as conjectured by cercignani 4 the problem is well posed when the drift velocity u exceeds the sound speed c but that one respectively four five additional conditions must be imposed when 0 u c respectively c u 0 and u c in the second part we show that the well posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions

multirate periodic systems and constrained analytic function interpolation problems multirate periodic systems and some related constrained analytic function interpolation problems are studied in this paper after showing how to convert a general multirate periodic system to an equivalent linear time invariant lti system with a structural constraint we formulate some analytic function interpolation problems with such a constraint that can find various applications in the study of multirate and periodic systems both the solvability conditions and characterization of all solutions are presented to these constrained interpolation problems

global stability of the rarefaction wave of a one dimensional model system for compressible viscous gas this paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one dimensional barotropic model system for compressible viscous gas we assume that the initial data tend to constant states atx respectively and the riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave if the adiabatic constant satisfies 1 2 then the solution is proved to tend to the rarefaction wave ast under no smallness conditions of both the difference of asymptotic values atx and the initial data the proof is given by an elementaryl 2 energy method

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

the maximum distance problem and band sequences abstract we solve the following problem for 1 j k n and j k m let a jk be a given complex number with a kj jk we wish to find linearly independent vectors x 1 x n such that x k x j a jk for j k m and such that the distance from x k to the linear span of x 1 x k 1 is maximal for 2 k n we construct and characterize all such sequences of vectors our solution leads naturally to the class of m band sequences of vectors in an inner product space we study these sequences and characterize their equivalence classes under unitary transformations