existence and uniqueness of solutions for the couette problem we study existence and uniqueness results for the one dimensional boltzmann equation with inflow and diffusive boundary conditions our focus partly encompasses some of the properties of the boltzmann collision gain term which play a significant role in existence and uniqueness results a series of estimates are proven on the collision term which is shown to produce a suitable function space in which the contraction mapping arguments are available

on diffuse reflection at the boundary for the boltzmann equation and related equations the paper considers diffuse reflection at the boundary with nonconstant boundary temperature and unbounded velocities the solutions obtained are proved to conserve mass at the boundary after a preliminary study of the collisionless case the main results obtained are existence for the boltzmann equation in a diperna lions framework with the above boundary conditions in a bounded measure sense and existence together with uniqueness for the bgk equation with maxwellian diffusion on the boundary in anl framework

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

generalized quantum onsager coefficients from a symmetrized campbell hausdorff expansion the analogue of the classical onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite dimensional hilbert space the dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of gorini kossakowski sudarshan type the basic spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics to this end an appropriate lie series is constructed from a particular symmetrization procedure applied to the ordinary campbell hausdorff expansion in this way only hermitian contributions by higher order commutators are generated which allow an identification with so called generalized onsager coefficients the explicit derivations concentrate on second third and fourth order coefficients whereas complete detailed expressions are worked out for second and third order in a suitable coherence vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the lie algebra of su n as an illustration an application to generalized bloch equations for two level systems is studied where the onsager like expansion can be compared with exact results for entropy production we find that convergence is good even for rather large deviations between initial and final state if the calculation includes second and third order coefficients only the formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes

the cphd filter with target spawning in its classical form the cardinalized probability hypothesis density cphd filter does not model the appearance of new targets through spawning yet there are applications for which spawning models more appropriately account for newborn objects when compared to spontaneous birth models in this paper we propose a principled derivation of the cphd filter prediction step including spontaneous birth and spawning a gaussian mixture implementation of the cphd filter with spawning is then presented illustrated with three applicable spawning models on a simulated scenario involving two parent targets spawning a total of five objects

simulation of boundary value problems for the boltzmann equation the paper presents numerical results on the simulation of boundary value problems for the boltzmann equation in one and two dimensions in the one dimensional case we use prescribed fluxes at the left and diffusive conditions on the right end of a slab to study the resulting steady state solution moreover we compute the numerical density function in velocity space and compare the result with the chapman enskog distribution obtained in the limit for continuous media the aim of the two dimensional simulations is to investigate the possibility of a symmetry break in the numerical solution

quantum theory of dissipative processes the markov approximation revisited adopting the standard mathematical framework for describing reduced dynamics we derive two formal identities for the density operator of an open quantum system each of these is equivalent to the old nakajima zwanzig equation the first identity is local in time it contains the inverse of the dynamical map which govern the evolution of the density operator we indicate a time interval on which this inverse exists the second identity constitutes a suitable starting point for going beyond the markov approximation in a controlled way on the basis of the bloch equations we argue once more that in studying quantum dissipation one has to pay attention to the von neumann conditions in the nakajima zwanzig equation we make the first born approximation the ensuing master equation possesses the correct weak coupling limit while proving this rather obvious but at the same time important statement we elucidate the mathematical methods which underlie the weak coupling limit moving to a two dimensional hilbert space we find that both for short and for long times our approximate master equation respects the von neumann conditions assuming exponential decay for correlation functions we propose a physical limit in which the solutions for the density operator become markovian in character we confirm the well known statement that as seen from a macroscopic standpoint the system starts from an effective initial condition the approach to equilibrium is exponential the accessory relaxation constants can differ from the usual bloch parameters and by more than 50

lyapunov functionals and l 1 stability for discrete velocity boltzmann equations we devise lyapunov functionals and prove uniform l 1 stability for one dimensional semilinear hyperbolic systems with quadratic nonlinear source terms these systems encompass a class of discrete velocity models for the boltzmann equation the lyapunov functional is equivalent to the l 1 distance between two weak solutions and non increasing in time they result from computations of two point interactions in the phase space for certain models with only transversal collisional terms there exist generalizations for three and multi point interactions

minimax or feared value l 1 l control abstract this is the third paper after bernhard expected value feared value and partial information optimal control in g j olsder ed new trends in dynamic games and applications annals of the international society of dynamic games vol 3 birkhauser boston usa 1995 pp 95 100 a separation theorem for expected value and feared value discrete time control cocv vol 1 pp 191 206 smai www emath fr cocv 1996 where we attempt to develop and exploit the parallel between stochastic control and min max control induced by the use of the max plus algebra using the concept of feared value as the parallel to expected value the present paper builds on the main formula of bernhard 1996 the results of which are a subset of those given here its new contribution is twofold on the one hand we clarify the role of the integral part of the cost in that parallel this leads to a more extensive theory of the so called imperfect information minimax l control problem than apparently available in the literature including a certainty equivalence theorem on the other hand we extend the parallel to the continuous time case as much as we can in that direction the present paper slightly extends the classical framework of the variational inequality of stopping time games and gives a formal treatment of the partial information case altogether this might be the first new results obtained with the tool of mathematical fear

higher order quantum onsager coefficients from dynamical invariants functionals representing dynamical invariants under unitary quantum dynamics of open systems are used to derive onsager coefficients for entropy production in irreversible processes if the nonunitary time evolution is determined by quantum dynamical semigroups the procedure allows a derivation from first principles of the quantum analogue to the classical case

a discrete velocity stationary wigner equation this paper is concerned with the one dimensional stationary linear wigner equation a kinetic formulation of quantum mechanics specifically we analyze the well posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete velocity model we find that the problem is uniquely solvable if zero is not a discrete velocity otherwise one obtains a differential algebraic equation of index 2 and hence the inflow data make the system overdetermined

the boltzmann grad limit for a one dimensional boltzmann equation in a stationary state in this paper we consider a one dimensional model of interacting particles in a bounded interval with possibly not homogeneous diffusive boundary conditions we prove that when the number of particlesn goes to infinity and the interaction is suitably rescaled the boltzmann grad limit the one particle distribution function of the unique invariant measure for the particle system converges to the unique solution of the boltzmann equation of the model provided that the mean free path is sufficiently large

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

evolution matrix in a coherence vector formulation for quantum markovian master equations of n level systems quantum markovian master equation in n dimensions are systematically transformed into vector form by using the lie algebra of the special unitary group su n density operators are represented through coherence vectors and the infinitesimal kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable in general a complete classification of the spectrum of the latter and its jordan canonical form together with all types of associated stationary states is given in detail the results of this analysis are particularly suited for computations in practical applications

derivation of slip boundary conditions for the navier stokes system from the boltzmann equation rarefied gas flow behavior is usually described by the boltzmann equation the navier stokes system being valid when the gas is less rarefied slip boundary conditions for the navier stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level boltzmann equation these slip conditions are explicitly written in terms of asymptotic behavior of some linear half space problems the validity of this analysis is established in the simple case of the couette flow for which it is proved that the right boundary conditions are obtained

the stationary nonlinear boltzmann equation in a couette setting this paper studies the stationary nonlinear boltzmann equation for hard forces in a couette setting between two coaxial rotating cylinders with given indata of maxwellian type on the cylinders a priori estimates are obtained mainly in l2 leading to multiple isolated solutions together with a hydrodynamic limit control based on asymptotic expansions together with a rest term

regularization of quantum relative entropy in finite dimensions and application to entropy production the fundamental concept of relative entropy is extended to a functional that is regular valued also on arbitrary pairs of nonfaithful states of open quantum systems this regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution on this basis a generalized formula for entropy production is proposed the applicability of which is tested in models of irreversible processes the dynamics of the latter is determined by either markovian or non markovian master equations and involves all types of states

weighted l bounds and uniqueness for the boltzmann bgk model we prove rather general l bounds for hydrodynamical fields in terms of weighted l norms of the kinetic density we use these estimates to prove l bounds and uniqueness for the solution of the bgk equation which is a simple relaxation model introduced by bhatnagar gross krook to mimic boltzmann flows

global weak solutions of the boltzmann equation a new definition of the concept of weak solution of the nonlinear boltzmann equation is introduced it is proved that without any truncation on the collision kernel the boltzmann equation in the one dimensional case has a global weak solution in this sense global conservation of energy follows

existence and uniqueness for nonlinear boundary value problems in kinetic theory a boundary value problem for the stationary nonlinear boltzmann equation in a slab has been examined in a weightedl space it has been proved that the problem possesses a unique solution for boundary data small enough the proof is based on the implicit function theorem it has also been shown that for the linearized problem the fredholm alternative applies

chain differentials with an application to the mathematical fear operator abstract we introduce a concept of derivative in a topological vector space that yields the chain rule of differentiation for composition of functions akin to but simpler than the epiderivative of aubin frankowska we show that if a function has a g teaux differential with suitable continuity properties it is a chain differential as an example we show that the mathematical fear operator is chain differentiable with respect to the cost density distribution in the space of continuous functions endowed with the topology of pointwise convergence uniform on every compact subsets

unfolding a quantum master equation into a system of real valued equations computationally effective expansion over the basis of su n generators dynamics of an open n state quantum system is often modeled with a markovian master equation describing the evolution of the system density operator by using generators of su n group as a basis the density operator can be transformed into a real valued coherence vector a generator of the dissipative evolution so called lindbladian can be expanded over the same basis and recast in the form of a real matrix together these expansions result is a nonhomogeneous system of n 2 1 real valued linear ordinary differential equations now one can e g implement standard high performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace norm and hermiticity of the density operator however when performed in a straightforward way the expansion turns to be an operation of the time complexity o n 10 the complexity can be reduced when the number of dissipative operators is independent of n which is often the case for physically meaningful models here we present an algorithm to transform quantum master equation into a system of real valued differential equations and propagate it forward in time by using a specific scalable model we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with n 10 3 states on a single node of a computer cluster

regional variance for multi object filtering recent progress in multi object filtering has led to algorithms that compute the first order moment of multi object distributions based on sensor measurements the number of targets in arbitrarily selected regions can be estimated using the first order moment in this work we introduce explicit formulae for the computation of the second order statistic on the target number the proposed concept of regional variance quantifies the level of confidence on target number estimates in arbitrary regions and facilitates information based decisions we provide algorithms for its computation for the probability hypothesis density phd and the cardinalized probability hypothesis density cphd filters we demonstrate the behaviour of the regional statistics through simulation examples

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

nonlinear functionals of multi d discrete velocity boltzmann equations in this paper we study nonlinear functionals measuring potential interactions and l1 distance between two mild solutions for the multi dimensional discrete velocity boltzmann equations when the initial data are a small perturbation of a vacuum we employ bony s dispersion estimates to show that these functionals satisfy lyapunov type estimates which are useful for the study of time asymptotics and l1 stability of mild solutions

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

measure of irreversibility and entropy production in open quantum systems a new concept of a measure of irreversibility for quantum dynamics in open systems is introduced as a suitably regularized substitute for the common notion of entropy production which unfortunately yields infinite values for so many irreversible processes of physical relevance

measure valued solutions and the phenomenon of blow down in logarithmic diffusion abstract the existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory a prominent example being the fundamental solutions of the linear theory in nonlinear equations the existence of such solutions may find special obstacles that can be either essential or otherwise they may lead to more general concepts of solution we give a particular review of results in the field of nonlinear diffusion as a new contribution we study in detail the case of logarithmic diffusion associated with ricci flow in the plane where we can prove existence of measure valued solutions the surprising thing is that these solutions become classical after a finite time in that general setting the standard concept of weak solution is not adequate but we can solve the initial value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions we prove that the problem is well posed the phenomenon of blow down in finite time is precisely described initial point masses diffuse into the medium and eventually disappear after a finite time t i m i 4

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

a second order phd filter with mean and variance in target number the probability hypothesis density phd and cardinalized phd cphd filters are popular solutions to the multitarget tracking problem due to their low complexity and ability to estimate the number and states of targets in cluttered environments the phd filter propagates the first order moment i e mean of the number of targets while the cphd propagates the cardinality distribution in the number of targets albeit for a greater computational cost introducing the panjer point process this paper proposes a second order phd so phd filter propagating the second order moment i e variance of the number of targets alongside its mean the resulting algorithm is more versatile in the modeling choices than the phd filter and its computational cost is significantly lower compared to the cphd filter this paper compares the three filters in statistical simulations which demonstrate that the proposed filter reacts more quickly to changes in the number of targets i e target births and target deaths than the cphd filter in addition a new statistic for multiobject filters is introduced in order to study the correlation between the estimated number of targets in different regions of the state space and propose a quantitative analysis of the spooky effect for the three filters

initial boundary value problems for the boltzmann equation global existence of weak solutions for an open set of 3 bounded or not we consider initial boundary value problems for the boltzmann equation for general gas surface interaction laws and for hard potentials we prove a global existence result for weak solutions the proof uses the regularization of the collision operator and the renormalization method for the regularized problem by using weak compactness in l1 and averaged stability ofq f f we prove the existence of weak solutions of our problem

global weak solutions and uniform lp stability of the boltzmann enskog equation abstract we present the regularity theory of renormalized solutions and uniform l p stability estimates of the boltzmann enskog model we use a multi dimensional bony type functional to control the time phase space integral of a truncated collision operator we employ cercignani s arguments and a bony type functional to show that the renormalized solutions to the boltzmann enskog model are weak solutions in usual distributional sense for the uniform l p stability estimate we use lu s trick together with the holder inequality

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

continuity bounds on the quantum relative entropy ii the quantum relative entropy is frequently used as a distance measure between two quantum states and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory we have derived many such inequalities in previous work the present paper is a follow up on this and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned the result obtained here is more general than the corresponding one from our previous work as a corollary we obtain a sharp upper bound on the regularised relative entropy introduced by lendi farhadmotamed and van wonderen

a boundary value problem for the two dimensional broadwell model it is shown that a certain boundary value problem for the steady two dimensional broadwell model on a rectangle has a solution the boundary conditions specify the ingoing particle densities on each side of the rectangle

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

conditions for bounded solutions of non markovian quantum master equations for the most general quantum master equations also called nakajima zwanzig or non markovian equations we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions as required by the von neumann conditions such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states the derivation is based on an equivalent coherence vector representation in finite dimension n leading to coupled volterra integro differential equations of second kind and convolution type in an n2 1 dimensional real vector space as examples analytical and numerical model solutions are worked out for 2 level systems in order to test suitable trial functions for input quantities all this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics

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

boundary value problems for the steady boltzmann equation we discuss steady boundary value problems for the boltzmann equation with inflow and diffusive boundary conditions in one two and three dimensions with suitable truncations of the collision kernel general existence and uniqueness results are obtained if the domain is sufficiently small in one dimension the existence of solutions on general intervals is obtained by abstract fixed point theory

quantum skew divergence in this paper we study the quantum generalisation of the skew divergence which is a dissimilarity measure between distributions introduced by lee in the context of natural language processing we provide an in depth study of the quantum skew divergence including its relation to other state distinguishability measures finally we present a number of important applications new continuity inequalities for the quantum jensen shannon divergence and the holevo information and a new and short proof of bravyi s small incremental mixing conjecture

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

trend to equilibrium of weak solutions of the boltzmann equation in a slab with diffusive boundary conditions recently r illner and the author proved that under a physically realistic truncation on the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense here it is proved that when the maxwellians associated with the boundary conditions atx 0 andx 1 are the same maxwellianmw then the solution is uniformly bounded and tends tomw fort

global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions it is proved that under a physically realistic truncation of the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense this solution satisfies the boundary conditions almost everywhere and has at worst exponentially growing total energy

on nonlinear stationary half space problems in discrete kinetic theory we show that every steady discrete velocity model of the boltzmann equation on the real line i d dx fi ci f which satisfies anh theorem and for which all i 0 has solutions on the half line 0 which take prescribed non negativefi o if i 0 and approach a certain manifold of maxwellians asx such solutions give the density distribution in a knudsen boundary layer in the discrete velocity case

loeb solutions of the boltzmann equation existence problems for the boltzmann equation constitute a main area of research within the kinetic theory of gases and transport theory the present paper considers the spatially periodic case with l1 initial data the main result is that the loeb subsolutions obtained in a preceding paper are shown to be true solutions the proof relies on the observation that monotone entropy and finite energy imply loeb integrability of non standard approximate solutions and uses estimates from the proof of the h theorem two aspects of the continuity of the solutions are also considered

virtues and limitations of markovian master equations with a time dependent generator a markovian master equation with time dependent generator is constructed that respects basic constraints of quantum mechanics in particular the von neumann conditions for the case of a two level system bloch equations with time dependent parameters are obtained necessary conditions on the latter are formulated by employing a time local counterpart of the nakajima zwanzig equation we establish a relation with unitary dynamics we also discuss the relation with the weak coupling limit on the basis of a uniqueness theorem a standard form for the generator of time local master equations is proposed the jaynes cummings model with atomic damping is solved the solution explicitly demonstrates that reduced dynamics can be described by time local master equations only on a finite time interval this limitation is caused by divergencies in the generator a limit of maximum entropy is presented that corroborates the foregoing statements a second limiting case demonstrates that divergencies may even occur for small perturbations of the weak coupling regime

a constructive approach to steady nonlinear kinetic equations abstract we study steady boundary value problems of nonlinear kinetic theory using a continuation argument based on the variation of the knudsen number we derive a method for the construction of steady solutions of discrete velocity models in a slab this method is readily transformed into a numerical code in a preliminary numerical test case the numerical scheme turns out to yield solutions even for knudsen numbers small enough to calculate with high precision the asymptotic flow field adjacent to a kinetic boundary layer thus we are able to numerically simulate in a simplified situation the transition from a mesoscopic kinetic boundary layer to the macroscopic far field

fast generation of low discrepancy sequences abstract the paper presents a fast implementation of a constructive method to generate a special class of low discrepancy sequences which are based on van neumann kakutani transformations such sequences can be used in various simulation codes where it is necessary to generate a certain number of uniformly distributed random numbers on the unit interval from a theoretical point of view the uniformity of a sequence is measured in terms of the discrepancy which is a special distance between a finite set of points and the uniform distribution on the unit interval numerical results are given on the cost efficiency of different generators on different hardware architectures as well as on the corresponding uniformity of the sequences as an example for the efficient use of low discrepancy sequences in a complex simulation code results are presented for the simulation of a hypersonic rarefied gas flow

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

a boundary value problem for discrete velocity models on considere deux problemes aux valeurs limites non lineaires concernant un ecoulement stationnaire a une dimension entre des plaques paralleles