properties of the error linear complexity spectrum this paper studies the error linear complexity spectrum of binary sequences with period 2n a precise categorization of those sequences having two distinct critical points in their spectra as well as an enumeration of these sequences is given an upper bound on the maximum number of distinct critical points that the spectrum of a sequence can have is proved and a construction which yields a lower bound on this number is given in the process simpler proofs of some known results on the linear complexity and k error linear complexity of sequences with period 2n are provided

pushdown automata multiset automata and petri nets in this paper we consider various classes of infinite state automata generated by simple rewrite transition systems these classes are defined by two natural hierarchies one given by interpreting concatenation of symbols in the rewrite system as sequential composition and the other by interpreting concatenation as parallel composition in this way we provide natural definitions for commutative parallel context free automata multiset parallel or random access pushdown automata and petri nets we provide example automata which demonstrate the strictness of this hierarchy in particular we provide a proof of an earlier conjecture by moller that multiset automata form a proper subset of petri nets this result contrasts with the result of caucal for the analogous question in the sequential case where the hierarchy collapses

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

a note on binary arrays with a certain window property corresp for some rs 2 mn there exist periodic r times s binary arrays such that all m times n binary matrices will appear in it as two dimensional versions of de bruijn sequences these kinds of arrays have possible applications in cryptography and the design of mask configurations for spectrometers

sequence folding lattice tiling and multidimensional coding folding a sequence b into a multidimensional box is a well known method which is used as a multidimensional coding technique the operation of folding is generalized in a way that the sequence b can be folded into various shapes and not just a box the novel definition of folding is based on a lattice tiling for the given shape s and a direction in the d dimensional integer grid necessary and sufficient conditions that a lattice tiling for s combined with a direction define a folding of a sequence into s are derived the immediate and most impressive applications are some new lower bounds on the number of dots in two dimensional synchronization patterns asymptotically optimal such patterns were known only for rectangular shapes we show asymptotically optimal such patterns for a large family of hexagons this is also generalized for multidimensional synchronization patterns the best known patterns in terms of dots for circles and other polygons are also given the technique and its application for two dimensional synchronization patterns raises some interesting problems in discrete geometry we will also discuss these problems it is also shown how folding can be used to construct multidimensional error correcting codes finally by using the new definition of folding new types of multidimensional pseudo random arrays with various shapes are generated

computing the error linear complexity spectrum of a binary sequence of period 2 sup n binary sequences with high linear complexity are of interest in cryptography the linear complexity should remain high even when a small number of changes are made to the sequence the error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed we present an algorithm which computes the error linear complexity for binary sequences of period spl lscr 2 sup n using o spl lscr log spl lscr sup 2 bit operations the algorithm generalizes both the games chan 1983 and stamp martin 1993 algorithms which compute the linear complexity and the k error linear complexity of a binary sequence of period spl lscr 2 sup n respectively we also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of reed muller codes

coupling of the wang chang uhlenbeck equations with the multispecies euler system after having recalled the basic properties of the wang chang uhlenbeck equations we describe a class of relaxation schemes to solve the multispecies euler system closed with a non classical state equation system which is the fluid limit of these kinetic equations then we show how to couple the resolution of the wang chang uhlenbeck equations with the resolution of this euler system by using a particular relaxation scheme namely a kinetic scheme which allows to define a natural boundary condition at the kinetic fluid interface and by using a marshak condition to take into account the effect of the knudsen layer in the fluid domain through an asymptotic matching finally we show applications in the field of the atomic vapor laser isotopic separation avlis

simple and robust binary self location patterns a simple method to generate a 2 d binary grid pattern which allows for absolute and accurate self location in a finite planar region is proposed the pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly

nonlinear boundary layers of the boltzmann equation i existence we study the half space problem of the nonlinear boltzmann equation assigning the dirichlet data for outgoing particles at the boundary and a maxwellian as the far field we will show that the solvability of the problem changes with the mach number of the far maxwellian if 1 respectively we also show that the same is true for the linearized problem at the far maxwellian and the manifold is then a hyperplane the proof is essentially based on the macro micro or hydrodynamics kinetic decomposition of solutions combined with an artificial damping term and a spatially exponential decay weight

constructions for perfect maps and pseudorandom arrays a construction of perfect maps i e periodic r v binary arrays in which each n m binary matrix appears exactly once is given a similar construction leads to arrays in which only the zero n m matrix does not appear and to a construction in which only a few n m binary matrices do not appear a generalization to the nonbinary case is given the constructions involve an interesting problem in shift register theory the solution is given for almost all the case 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

half space problem for the discrete boltzmann equation condensing vapor flow in the presence of a non condensable gas we consider a non linear half space problem related to the condensation problem for the discrete boltzmann equation and extend some known results for a single component gas to the case when a non condensable gas is present the vapor is assumed to tend to an assigned maxwellian at infinity as the non condensable gas tends to zero at infinity we assume that the vapor is completely absorbed and that the non condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution the conditions on the given distribution needed for the existence of a unique solution of the problem are investigated we also find exact solvability conditions and solutions for a simplified six four velocity model as the given distribution is a maxwellian at rest and study a simplified twelve six velocity model

asymptotic analysis and coupling conditions for kinetic and hydrodynamic equations abstract this article is concerned with domain decomposition for kinetic and hydrodynamic equations coupling conditions at the interface between the equations are developed and investigated in particular for nonequilibrium situations at the interface new coupling conditions are developed by considering interface layers this leads to kinetic half space problems a fast procedure to solve these problems is given

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

stability of contact discontinuity for the boltzmann equation the boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics that is euler equations and navier stokes equations as for a basic wave pattern to euler equations we consider the nonlinear stability of contact discontinuities to the boltzmann equation even though the stability of the other two nonlinear waves i e shocks and rarefaction waves has been extensively studied there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves its derivative has no definite sign and decays slower than a rarefaction wave moreover it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism based on the new definition of contact waves to the boltzmann equation corresponding to the contact discontinuities for the euler equations we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate in our analysis an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates

a rigorous derivation of the coupling of a kinetic equation and burgers equation in this article we investigate the kinetic fluid coupling on a toy model which we obtain rigorously from a hydrodynamical limit the idea is that at the level of the full kinetic model the coupling is obvious we then investigate the coupling obtained when passing into the limit we show that especially in presence of a shock stuck on the interface the coupling involves a kinetic layer known as the milne problem due to this layer the limit process is quite delicate and some blow up techniques are needed to ensure its strong convergence

construction of de bruijn sequences of minimal complexity it is well known that the linear complexity of a de bruijn sequence s of length 2 n is bounded below by 2 n 1 n for n geq 3 it is shown that this lower bound is attainable for all n

existence of nonlinear boundary layer solution to the boltzmann equation with physical boundary conditions abstract in this paper the existence of boundary layer solutions to the boltzmann equation with two physical boundary conditions for hard sphere model is considered the boundary condition is first imposed on incoming particles of diffuse reflection type and the solution tends to a global maxwellian in the far field similar to the problem with dirichlet boundary condition studied in s ukai t yang s h yu nonlinear boundary layers of the boltzmann equation i existence comm math phys 236 3 2003 373 393 the existence of a solution is shown to depend on the mach number of the far field maxwellian and there is an implicit solvability conditions yielding the co dimensions of the boundary data at last the specular reflection boundary condition is considered and the similar conclusions are obtained

symmetry of the linearized boltzmann equation and its application a symmetric relation of macroscopic quantities between two different steady problems of the linearized boltzmann equation is derived a few applications to half space problems are presented first then for the gas in bounded or unbounded domains such that solid bodies or condensed phases are confined in a finite region general representations of the mass momentum and heat fluxes through the boundary possibly at a point on or on a part of it are derived from the symmetric relation linked to the separability of boundary data this result implies a reduction of the original problem to a single elemental problem in the same domain as far as the fluxes are concerned many applications are also presented

contact discontinuity with general perturbations for gas motions the contact discontinuity is one of the basic wave patterns in gas motions the stability of contact discontinuities with general perturbations for the navier stokes equations and the boltzmann equation is a long standing open problem general perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties in this paper we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the navier stokes equations and the boltzmann equation in a uniform way one of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of thus this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the l8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern

the nonlinear boundary layer to the boltzmann equation with mixed boundary conditions for hard potentials in this paper the existence of boundary layer solutions to the boltzmann equation for hard potential with mixed boundary condition i e a linear combination of dirichlet boundary condition and diffuse reflection boundary condition at the wall is considered the boundary condition is imposed on the incoming particles and the solution is supposed to approach to a global maxwellian in the far field as for the problem with dirichlet boundary condition chen et al 2004 5 the existence of a solution highly depends on the mach number of the far field maxwellian furthermore an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given

discrete velocity models for mixtures without nonphysical collision invariants an important aspect of constructing discrete velocity models dvms for the boltzmann equation is to obtain the right number of collision invariants it is a well known fact that dvms can also have extra collision invariants so called spurious collision invariants in plus to the physical ones a dvm with only physical collision invariants and so without spurious ones is called normal for binary mixtures also the concept of supernormal dvms was introduced meaning that in addition to the dvm being normal the restriction of the dvm to any single species also is normal here we introduce generalizations of this concept to dvms for multicomponent mixtures we also present some general algorithms for constructing such models and give some concrete examples of such constructions one of our main results is that for any given number of species and any given rational mass ratios we can construct a supernormal dvm the dvms are constructed in such a way that for half space problems as the milne and kramers problems but also nonlinear ones we obtain similar structures as for the classical discrete boltzmann equation for one species and therefore we can apply obtained results for the classical boltzmann equation

aperiodic and semi periodic perfect maps paterson see ibid vol 40 p 743 1993 has shown that the trivial necessary conditions are sufficient for the existence of a binary perfect map these periodic structures can be transformed very simply into corresponding aperiodic and semi periodic perfect maps however aperiodic and semi periodic perfect maps can exist for parameter sets for which the corresponding periodic perfect maps cannot in this paper it is shown by construction that binary aperiodic and semi periodic perfect maps exist for all possible parameter sets

measure solutions of the steady boltzmann equation in a slab it is shown that the steady boltzmann equation in a slab 0 a has solutionsx x such that the ingoing boundary measures 0 0 and 0 can be prescribed a priori the collision kernel is truncated such that particles with smallx component of the velocity have a reduced collision rate

quickest sequence phase detection a phase detection sequence is a length n cyclic sequence such that the location of any length k contiguous subsequence can be determined from a noisy observation of that subsequence in this paper we derive bounds on the minimal possible k in the limit of n to infty and describe some sequence constructions we further consider multiple phase detection sequences where the location of any length k contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences we study the optimal trade offs between the lengths of the sequences and describe some sequence constructions we compare these phase detection problems to their natural channel coding counterparts and show a strict separation between the fundamental limits in the multiple sequence case both adversarial and probabilistic noise models are addressed

kinetic boundary layers numerical simulations abstract we take interest here in nonequilibrium boundary layers for polytropic viscous flows a quick physical introduction is given then we propose an extension of an existing numerical scheme for the compressible navier stokes equations to take into account slip boundary conditions 2 dimensional numerical simulations are presented

pointwise convergence to knudsen layers of the boltzmann equation in this paper green s functions for the boltzmann equation around a global maxwellian are used to construct the non characteristic nonlinear knudsen layers as well as their time asymptotic stability furthermore the detailed pointwise structures nonlinear wave couplings and wave interactions with boundary are studied

the solutions for the boundary layer problem of boltzmann equation in a half space we study the half space boundary layer problem for boltzmann equation with cut off potentials in all the cases 3 1 while the boundary condition is imposed on the incoming particles of dirichlet type and the solution is assumed to approach to a global maxwellian at the far field the same as for cut off hard sphere model there is an implicit solvability condition on the boundary data which gives the co dimensions of the boundary data in terms of positive characteristic speeds

structured light using pseudorandom codes we solve the correspondence problem in active stereo vision using a novel pseudorandom coded structured light sl this coding scheme performs well in the presence of occlusion in settings where color coding is feasible 3d information can be obtained using a single image

entropy inequalities for evaporation condensation problem in rarefied gas dynamics the present paper is devoted mainly to the half space problem for stationary boltzmann type equations using only conservation laws and the boltzmann h theorem we derive an inequality for unknown constant fluxes of mass energy and momentum this inequality is expressed in terms of three parameters pressure p temperature t and the mach number m of the asymptotic maxwellian at infinity geometrically the inequality describes a physical domain with positive entropy production in the 3 d space of the parameters the domain appears to be qualitatively different for evaporation and condensation problems we show that for given m the curve p p m t t m of maximal entropy production practically coincides with the experimental evaporation curve obtained by sone et al on the basis of numerical solutions of bgk equation similar consideration for the condensation problem is also in qualitative agreement with known numerical results

half space problems for the boltzmann equation a survey this paper reviews recent mathematical results on the half space problem for the boltzmann equation the case of a phase transition is discussed in detail

pseudo wigner matrices we consider the problem of generating pseudo random matrices based on the similarity of their spectra to wigner s semicircular law we introduce the notion of an r independent pseudo wigner matrix ensemble and prove the closeness of the spectra of its matrices to the semicircular density in the kolmogorov distance we give an explicit construction of a family of n times n pseudo wigner ensembles using dual bch codes and show that the kolmogorov complexity of the obtained matrices is of the order of mathrm log n bits for a fixed designed kolmogorov distance precision we compare our construction with the quasi random graphs introduced by chung et al and demonstrate that the pseudo wigner matrices pass stronger randomness tests than the adjacency matrices of these graphs lifted by the mapping 0 rightarrow 1 and 1 rightarrow 1 do finally we provide numerical simulations verifying our theoretical results

from boltzmann s kinetic theory to euler s equations abstract the incompressible euler equations are obtained as a weak asymptotics of the boltzmann equation in the fast relaxation limit the knudsen number kn goes to zero when both the mach number ma defined as the ratio between the bulk velocity and the speed of sound and the inverse reynolds number kn ma which measures the viscosity of the fluid go to zero the entropy method used here consists in deriving some stability inequality which allows us to compare the sequence of solutions of the scaled boltzmann equation to its expected limit provided that it is sufficiently smooth it thus leads to some strong convergence result one of the main points to be understood is how to deal with the corrections to the weak limit i e the contributions converging weakly but not strongly to 0 such as the initial layer or the acoustic waves

a fast algorithm for determining the complexity of a binary sequence with period 2 n corresp the complexity of a periodic sequence s is defined to be the least number of stages in a linear feedback shift register that generates s

the solution of the nonlinear boltzmann equation a survey of analytic and computational methods abstract this paper provides a survey and a critical overview of the mathematical results analytic and computational on the solution of the nonlinear boltzmann equation the topics dealt with in this paper are the following mathematical formulation of initial and or boundary value problems existence theorems computational treatment of fluid dynamical problems

nonlinear stability of boundary layers of the boltzmann equation i the case m 1 this is a continuation of the paper 15 on nonlinear boundary layers of the boltzmann equation where the existence is established and shown to be strongly dependent on the mach number m of the maxwellian state at far field in this paper when m 1 we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers

symmetry of the linearized boltzmann equation ii entropy production and onsager casimir relation this is the second part of the study by the author on the symmetry of the linearized boltzmann equation the issue of the present part is the entropy production and the onsager casimir reciprocity relation in the steady non equilibrium systems after the discussions on the definition of the entropy entropy flow and entropy production in the non equilibrium gas systems the expression of the entropy production in the steady state is presented then for the systems weakly perturbed from a uniform equilibrium state the entropy production is shown to be expressed in terms of the solution of the linearized boltzmann equation the thermodynamic forces and fluxes and the kinetic coefficients are defined solely from the expression of the entropy production the conventional type onsager casimir relation is shown to hold for the entire range of the knudsen number in bounded and unbounded domain systems provided that the state of the gas in a far field is a local maxwellian satisfying the boltzmann equation for the latter as to the other unbounded domain systems a nonconventional reciprocity is shown to hold

a numerical method for computing asymptotic states and outgoing distributions for kinetic linear half space problems linear half space problems can be used to solve domain decomposition problems between boltzmann and aerodynamic equations a new fast numerical method computing the asymptotic states and outgoing distributions for a linearized bgk half space problem is presented relations with the so called variational methods are discussed in particular we stress the connection between these methods and chapman enskog type expansions

half space problems for a linearized discrete quantum kinetic equation we study typical half space problems of rarefied gas dynamics including the problems of milne and kramer for a general discrete model of a quantum kinetic equation for excitations in a bose gas in the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations these systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete boltzmann equation then a classification of well posed half space problems for the homogeneous as well as the inhomogeneous linearized discrete kinetic equation can be made the number of additional conditions that need to be imposed for well posedness is given by some characteristic numbers these characteristic numbers are calculated for discrete models axially symmetric with respect to the x axis when the characteristic numbers change is found in the discrete as well as the continuous case as an illustration explicit solutions are found for a small sized model

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

asymptotic theory for the time dependent behavior of a slightly rarefied gas over a smooth solid boundary a time evolution of a slightly rarefied monoatomic gas namely a gas for small knudsen numbers which is perturbed slowly and slightly from a reference uniform equilibrium state at rest is investigated on the basis of the linearized boltzmann equation by a systematic asymptotic analysis a set of fluid dynamic type equations and its boundary conditions that describe the gas behavior up to the second order of the knudsen number are derived the developed theory covers a general intermolecular potential and a gas surface interaction it is shown that i the compressibility of the gas manifests itself from the leading order in the energy equation and from the first order in the continuity equation ii although the momentum equation is the stokes equation it contains a double laplacian of the leading order flow velocity as a source term at the second order iii a double laplacian source term also appears in the energy equation at the second order iv the slip and jump conditions are the same as those in the time independent case up to the first order and the difference occurs at the second order in the jump conditions as the terms of the divergence of flow velocity and of the laplacian of temperature numerical values of all the slip and jump coefficients are obtained for a hard sphere gas by the use of a symmetric relation developed recently

diffusion approximations and domain decomposition method of linear transport equations in this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations we treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist the diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data we apply the half space solver in 20 to resolve the boundary layer equation and obtain the boundary data for the diffusion equation the algorithms are validated by numerical experiments and also by error analysis for the pure diffusive scaling case

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

nonlinear stability of boundary layers of the boltzmann equation for cutoff hard potentials many physical models have boundaries for the boltzmann equation the study on the boundary layer in the region of the width in the order of the kundsen number along the boundary is important both in mathematics and physics in this paper we consider the nonlinear stability of boundary layer solutions to the boltzmann equation for hard potentials with angular cut off the boundary condition is imposed on incoming particles of the dirichlet type and the solution tends to a global maxwellian in the far field for the existence of the boundary layer solutions it is proved by chen et al anal appl 2 337 363 2004 by introducing a weight function which is a function of both position and velocity to overcome the difficulty from the sublinear growth in the collision frequency unlike the hard sphere model even for stability in the case when the mach number of the far field is less than 1 exponential decay in time cannot be expected for the cutoff hard potentials instead an algebraic decay in time to

symmetric pseudo random matrices we consider the problem of generating symmetric pseudo random sign 1 matrices based on the similarity of their spectra to wigner s semicircular law using binary m sequences golomb sequences of lengths n 2 m 1 we give a simple explicit construction of circulant n times n sign matrices and show that their spectra converge to the semicircular law when n grows the kolmogorov complexity of the proposed matrices equals to that of golomb sequences and is at most 2 mathrm log 2 n bits

invariant manifolds for steady boltzmann flows and applications we develop a theory of invariant manifolds for the steady boltzmann equation and apply it to the study of boundary layers and nonlinear waves the steady boltzmann equation is an infinite dimensional differential equation so the standard center manifold theory for differential equations based on spectral information does not apply here instead we employ a time asymptotic approach using the pointwise information of green s function for the construction of the linear invariant manifolds at the resonance cases when the mach number at the far field is around one of the critical values of 1 0 or 1 the truly nonlinear theory arises in such a case there are wave patterns combining the fast decaying knudsen type and slow varying fluid like waves the key knudsen manifolds consisting of only knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical mach numbers around mach number 1 the fluidlike waves are compressive and expansive waves and around the mach number 0 they are linear thermal layers the quantitative analysis of the fluid like waves is done using the reduction of dimensions to the center manifolds two scale nonlinear dynamics based on those on the knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns there are striking bifurcations in the transition of evaporation to condensation and in the transition of the milne s problem with a subsonic far field to one with a supersonic far field the analysis of these wave patterns allows us to understand the sone diagram for the study of the complete condensation boundary value problem the monotonicity of the boltzmann shock profiles a problem that initially motivated the present study is shown as a consequence of the quantitative analysis of the nonlinear fluid like waves

the milne problem for high field kinetic equations half space problems of the linear boltzmann equation with a constant driving force are considered such problems model boundary layers between kinetic zones and fluid zones described by a high field limit of the boltzmann equation existence uniqueness and asymptotic behavior of solutions are studied for positive and negative driving forces in the positive case the force field accelerates the particles and we show that the solution of the half space problem is determined only by the inflow data in contrast for negative forces the behavior at infinity has to be prescribed in order to insure uniqueness due to the nonvanishing forces the problem does not possess any entropy the existence and uniqueness issues are dealt with by supersolution techniques while the asymptotic behavior is analyzed by semiexplicit integration of the equations along the characteristics in the case of relaxation time approximation a fast numerical method for computing the asymptotic state method is presented and tested