finite difference schemes for solving system equations of gas dynamic in a class of discontinuous functions in this paper the difference scheme for solving nonlinear system of equations of gas dynamic problems in a class of discontinuous functions is investigated firstly for some simple cases the nature of solution of the differential equations describing one dimensional constant pressure and isentropic flow of compressible fluids are considered it has been proved that the solution of this system equations has discontinuous points whose positions are unknown beforehand in order to obtain the solution of the main problem in a class of discontinuous functions the auxiliary problem is suggested the degree of differentiability of the solution of the auxiliary problem is higher than the degree of differentiability of solution of the main problem furthermore the suggested auxiliary problem provides to write out effective and higher order numerical algorithms the solutions obtained from these algorithms represent the physical nature of the problem with a high accuracy some properties of numerical solution are also investigated additionally some numerical experiments are carried out by means of using the auxiliary problem

spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

an automata theoretic decision procedure for the propositional mu calculus the propositional mu calculus is a propositional logic of programs which incorporates a least fixpoint operator and subsumes the propositional dynamic logic of fischer and ladner the infinite looping construct of streett and the game logic of parikh we give an elementary time decision procedure using a reduction to the emptiness problem for automata on infinite trees a small model theorem is obtained as a corollary

a high order space momentum discontinuous galerkin method for the boltzmann equation in this paper we present a discontinuous galerkin method for the boltzmann equation the distribution function f is approximated by a shifted maxwellian times a polynomial in space and momentum while the test functions are chosen as polynomials the first property leads to consistency with the euler limit while the second property ensures conservation of mass momentum and energy the focus of the paper is on efficient algorithms for the boltzmann collision operator we transform between nodal hierarchical and polar polynomial bases to reduce the inner integral operator to diagonal form

recursive programs and denotational semantics in absolute logics of programs abstract it has been proposed by meyer kfouri and others to extend the so called nonstandard logics of programs approach nlp henceforth to recursive programs and denotational semantics in some natural way the key step in doing nlp is selecting a semantics for programs which in the present paper is denotational semantics and then forming its kpu absolute version absoluteness is a set theoretic property of logics if a logic l is not absolute then some formulas of l do not speak about the models of l but about something else if a logic has this strange property then it is impossible to find a complete inference system for it and this because of purely administrative reasons which might not correspond to any of the inherent properties of the original situation we are modeling in devising this logic etc this administrative incompleteness is only one of the undesirable side effects of nonabsoluteness in nlp the purpose of constructing and studying absolute versions of logics is aimed at among other things the removal of these undesirable properties the absolute version of denotational semantics will turn out to be based on 2 sorted first order logic or more concretely on the usual nonstandard or weak second order logic some of the most relevant features of this logic are the ability to compare realistic existing program verification methods create new more powerful and still realistic ones completeness etc to mention only a very few

propositional dynamic logic of flowcharts following a suggestion of pratt we consider propositional dynamic logic in which programs are nondeterministic finite automata over atomic programs and tests i e flowcharts rather than regular expressions while the resulting version of pdl call it apdl is clearly equivalent in expressive power to pdl it is also in the worst case exponentially more succinct in particular deciding its validity problem by reducing it to that of pdl leads to a double exponential time procedure although pdl itself is decidable in exponential time we present an elementary combined proof of the completeness of a simple axiom system for apdl and decidability of the validity problem in exponential time the results are thus stronger than those for pdl since pdl can be encoded in apdl with no additional cost and the proofs simpler since induction on the structure of programs is virtually eliminated our axiom system for apdl relates to the pdl system just as floyd s proof method for partial correctness relates to hoare s

towards an ultra efficient kinetic scheme part i basics on the bgk equation in this paper we present a new ultra efficient numerical method for solving kinetic equations in this preliminary work we present the scheme in the case of the bgk relaxation operator the scheme being based on a splitting technique between transport and collision can be easily extended to other collisional operators as the boltzmann collision integral or to other kinetic equations such as the vlasov equation the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time the main difference between the method proposed and semi lagrangian methods is that here we do not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost of the method and it permits for the first time to the author s knowledge to compute solutions of full six dimensional kinetic equations on a single processor laptop machine numerical examples up to the full three dimensional case are presented which validate the method and assess its efficiency in 1d 2d and 3d

games for the m calculus abstract given a formula of the propositional calculus we construct a tableau of the formula and define an infinite game of two players of which one wants to show that the formula is satisfiable and the other seeks the opposite the strategy for the first player can be further transformed into a model of the formula while the strategy for the second forms what we call a refutation of the formula using martin s determinacy theorem we prove that any formula has either a model or a refutation this completeness result is a starting point for the completeness theorem for the calculus to be presented elsewhere however we argue that refutations have some advantages of their own they are generated by a natural system of sound logical rules and can be presented as regular trees of the size exponential in the size of a refuted formula this last aspect completes the small model theorem for the calculus established by emerson and jutla 1988 thus on a more practical side refutations can be used as small objects testifying incorrectness of a program specification expressed by a formula we illustrate this point by an example

a multiscale fast semi lagrangian method for rarefied gas dynamics in this paper we consider the extension of the method developed in dimarco and loubere 2013 22 23 with the aim of facing the numerical resolution of multi scale problems arising in rarefied gas dynamics the scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere this is the case of many realistic applications some regions of the computational domain require a microscopic description given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type our aim is to show how the kinetic scheme developed in the pre cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi scale realistic problems several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme

discrete velocity models and relaxation schemes for traffic flows we present simple discrete velocity models for traffic flows the novel feature in the corresponding relaxation system is the presence of nonnegative velocities only we show that in the small relaxation limit the discrete models reduce to the lighthill whitham richards equation in addition we propose second order schemes combined with implicit explicit time integrators as proper discretization of the relaxation type system numerical tests are carried out on various situations in traffic flow the results show that the proposed models are able to describe correctly the formation of backward waves induced by traffic jam

a convergence proof for bird s direct simulation monte carlo method for the boltzmann equation bird s direct simulation monte carlo method for the boltzmann equation is considered the limit as the number of particles tends to infinity of the random empirical measures associated with the bird algorithm is shown to be a deterministic measure valued function satisfying an equation close in a certain sense to the boltzmann equation a markov jump process is introduced which is related to bird s collision simulation procedure via a random time transformation convergence is established for the markov process and the random time transformation these results together with some general properties concerning the convergence of random measures make it possible to characterize the limiting behavior of the bird algorithm

results on the propositional μ calculus abstract in this paper we define and study a propositional calculus l which consists essentially of propositional modal logic with a least fixpoint operator l is syntactically simpler yet strictly more expressive than propositional dynamic logic pdl for a restricted version we give an exponential time decision procedure small model property and complete deductive system theory subsuming the corresponding results for pdl

on the two dimensional differential transform method in this paper two dimensional differential transform method of solution of the initial value problem for partial differential equations pdes have been studied new theorems have been added and some linear and nonlinear pdes solved by using this method the method can be easily applied to linear or nonlinear problems and is capable of reducing the size of computational work in this work additionally analytical form solutions of two diffusion problems have been obtained and the solutions are compared very well with those obtained by decomposition method

an approximation of the analytic solution of the shock wave equation in this article we discuss the analytic solution of the fully developed shock waves the adomian decomposition method is used to solve the shock wave equation which describes the flow of gases unlike the various numerical techniques which are usually valid for short period of time the solution of the presented equation is analytic for 0 t also the results presented here indicate that the method is reliable accurate and converges very rapidly

spectral method for a kinetic swarming model in this paper we present the first numerical method for a kinetic description of the vicsek swarming model the kinetic model poses a unique challenge as there is a distribution dependent collision invariant to satisfy when computing the interaction term we use a spectral representation linked with a discrete constrained optimization to compute these interactions to test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the vicsek model we observe that the kinetic model captures key features such as vortex formation and traveling waves

a logic to reason about likelihood abstract we present a logic ll which uses a modal operator l to help capture the notion of being likely despite the fact that likelihood is not assigned quantitative values through probabilities ll captures many of the properties of likelihood in an intuitively appealing way we give a possible worlds style semantics to ll and using standard techniques of modal logic we give a complete axiomatization for ll and show that satisfiability of ll formulas can be decided in exponential time we discuss how the logic might be used in areas such as medical diagnosis where decision making in the face of uncertainties is crucial we conclude by using ll to give a formal proof of correctness of some aspects of a protocol for exchanging secrets

numerical solutions of the space and time fractional coupled burgers equations by generalized differential transform method abstract in this paper by introducing the fractional derivative in the sense of caputo the generalized two dimensional differential transform method dtm is directly applied to solve the coupled burgers equations with space and time fractional derivatives the presented method is a numerical method based on the generalized taylor series formula which constructs an analytical solution in the form of a polynomial several illustrative examples are given to demonstrate the effectiveness of the generalized two dimensional dtm for the equations

simulating alternating tree automata by nondeterministic automata new results and new proofs of the theorems of rabin mcnaughton and safra abstract we give a proof that alternating tree automata can be simulated by nondeterministic tree automata which yields new complexity results and a unified proof of the theorems of rabin mcnaughton and safra we also give a simple axiomatic framework for uniformizing strategies

process logic with regular formulas the paper proposes an extension rpl of the process logic pl of harel kozen and parikh 1980 the pl formula operators f and suf are replaced by the operators chop and slice corresponding to kleene s regular operators and thus enabling formulas to express regular sets of paths the main result is that in expressive power pl rpl the hard part being in showing that pl rpl it is argued that this version of pl comes closer to the desired goal of a natural and powerful yet decidable logic for reasoning about the ongoing behavior of programs

looping vs repeating in dynamic logic two extensions of propositional dynamic logic for dealing with infinite computations lpdl and rpdl are compared in expressive power the first is obtained by adding the assertion loop for any program meaning contains an infinite computation and the second by adding repeat meaning can be repeated indefinitely while repeat can be used to encode loop and hence lpdl rpdl it is shown here that the converse fails thus lpdl

positivity preserving discontinuous galerkin schemes for linear vlasov boltzmann transport equations we develop a high order positivity preserving discontinuous galerkin dg scheme for linear vlasov boltzmann transport equations vlasov bte under the action of quadratically confined electrostatic potentials the solutions of such btes are positive probability distribution functions and it is very challenging to have a mass conservative high order accurate scheme that preserves positivity of the numerical solutions in high dimensions our work extends the maximum principle satisfying scheme for scalar conservation laws in a recent work by x zhang and c w shu to include the linear boltzmann collision term the dg schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy a discussion of the standard semi discrete dg schemes for the bte are included as a foundation for the stability and error estimates for this new scheme numerical results of the relaxation models are provided to validate the method

approximate analytical solutions of fractional gas dynamic equations in this article differential transform method dtm has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non homogeneous gas dynamic equations shock wave equation and shallow water equations with fractional order time derivatives the true beauty of the article is manifested in its emphatic application of caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations pdes the method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations numerical results for different particular cases of the equations are depicted through graphs

three logics for branching bisimulation three temporal logics are introduced that induce on labeled transition systems the same identifications as branching bisimulation a behavioral equivalence that aims at ignoring invisible transitions while preserving the branching structure of systems the first logic is an extension of hennessy milner logic with an until operator the second one is another extension of hennessy milner logic which exploits the power of backward modalities the third logic is ctl without the next time operator a relevant side effect of the last characterization is that it sets a bridge between the state and action based approaches to the semantics of concurrent systems

the propositional dynamic logic of deterministic well structured programs abstract we consider a restricted propositional dynamic logic strict deterministic propositional dynamic logic sdpdl which is appropriate for reasoning about deterministic well structured programs in contrast to pdl for which the validity problem is known to be complete in deterministic exponential time the validity problem for sdpdl is shown to be polynomial space complete we also show that sdpdl is less expressive than pdl and give a complete axiomatization for it the results rely on structure theorems for models of satisfiable sdpdl formulas and the proofs give insight into the effects of nondeterminism on intractability and expressiveness in program logics

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

on the power of bounded concurrency i finite automata we investigate the descriptive succinctness of three fundamental notions for modeling concurrency nondeterminism and pure parallelism the two facets of alternation and bounded cooperative concurrency whereby a system configuration consists of a bounded number of cooperating states our results are couched in the general framework of finite state automata but hold for appropriate versions of most concurrent models of computation such as petri nets statecharts or finite state versions of concurrent programming languages we exhibit exhaustive sets of upper and lower bounds on the relative succinctness of these features over s and s establishing that 1 each of the three features represents an exponential saving in succinctness of the representation in a manner that is independent of the other two and additive with respect to them 2 of the three bounded concurrency is the strongest representing a similar exponential saving even when substituted for each of the others for example we prove exponential upper and lower bounds on the simulation of deterministic concurrent automata by afas and triple exponential bounds on the simulation of alternating concurrent automata by dfas

a unifying theorem for algebraic semantics and dynamic logics abstract a unified single proof is given which implies theorems in such diverse fields as continuous algebras of algebraic semantics dynamic algebras of logics of programs and program verification methods for total correctness the proof concerns ultraproducts and diagonalization

a direct method for the boltzmann equation based on a pseudo spectral velocity space discretization a deterministic method is proposed for solving the boltzmann equation the method employs a galerkin discretization of the velocity space and adopts as trial and test functions the collocation basis functions based on weights and roots of a gauss hermite quadrature this is defined by means of half and or full range hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space the resulting semi discrete boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches the spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of maxwell molecules as an application the two dimensional cavity flow of a gas composed of hard sphere molecules is studied for different knudsen and mach numbers although computationally demanding the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows

generalized taylor s formula abstract in this paper a new generalized taylor s formula of the kind f x j 0 n a j x a j r n x where a j r x a 0 1 is established such an expression is precisely the classical taylor s formula in case of 1 in addition detailed expressions for r n x and a j that involving caputo fractional derivatives are well constructed some applications including approximation of functions and solutions of fractional differential equations are also given

a guide to completeness and complexity for modal logics of knowledge and belief abstract we review and re examine possible worlds semantics for propositional logics of knowledge and belief with three particular points of emphasis 1 we show how general techniques for finding decision procedures and complete axiomatizations apply to models for knowledge and belief 2 we show how sensitive the difficulty of the decision procedure is to such issues as the choice of modal operators and the axiom system and 3 we discuss how notions of common knowledge and distributed knowledge among a group of agents fit into the possible worlds framework as far as complexity is concerned we show among other results that while the problem of deciding satisfiability of an s5 formula with one agent is np complete the problem for many agents is pspace complete adding a distributed knowledge operator does not change the complexity but once a common knowledge operator is added to the language the problem becomes complete for exponential time

programs and program verifications in a general setting abstract definitions of programs and that of runs are given in a general setting which coincide with the standard programs and continuous traces in case of peano models using these notations we prove the completeness of the floyd hoare derivability in the second part of this paper we investigate the number of different halting configurations of a run in general a run may have a lot of them but in peano models they satisfy just the same formulas finally a sample run is given with two halting configurations distinguishable by a formula

balanced finite volume weno and central weno schemes for the shallow water and the open channel flow equations the goal of this work is to extend finite volume weno and central weno schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function in particular we apply proposed schemes to the shallow water and the open channel flow equations where the source term depends on the channel geometry for obtaining stable numerical schemes that are free of spurious oscillations it becomes crucial to use the decomposed source term evaluation which maintains the balancing between the flux gradient and the source term in addition the open channel flow equations contain spatially variable flux function the appropriate definitions of the terms that arise in the source term decomposition in combination with the roe approximate riemann solver that includes the spatial derivative of the flux function lead to the finite volume weno scheme that satisfies the exact conservation property the property of preserving the quiescent flow exactly when the central weno schemes are applied additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the weno reconstruction procedure the proposed central weno schemes also preserve the quiescent flow but only in prismatic channels in various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes as well as with some other first and second order numerical schemes

self similarity and power like tails in nonconservative kinetic models in this paper we discuss the large time behavior of solution of a simple kinetic model of boltzmann maxwell type such that the temperature is time decreasing and or time increasing we show that under the combined effects of the nonlinearity and of the time monotonicity of the temperature the kinetic model has non trivial quasi stationary states with power law tails in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution the same idea is applied to investigate the large time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and or decreasing of the mean wealth in this last case the large time behavior of the solution shows a pareto power law tail numerical results confirm the previous analysis

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

subject oriented spatial logic abstract we present a modal logic for subject oriented representation and reasoning about a two dimensional space which we call sosl the space is represented with the polar coordinate system where the subject occupies the central point and modal operators are interpreted by relations defined relatively to the position and orientation of the subject namely outwards inwards clockwise counter clockwise and the transitive closures of the first two such logic enables to express operators for the intrinsic relations in front behind to the left and to the right of the subject for the relative relations behind an object between the subject and an object to the left of an object and to the right of an object and hybrid or distance operators we prove that the satisfiability problem in sosl is pspace complete the same complexity holds over the classes of finite or infinite models however for models of fixed size the problem becomes np complete

on the undecidability of logics with converse nominals recursion and counting the evolution of description logics dls and propositional dynamic logics produced a hierarchy of decidable logics with multiple maximal elements it would be desirable to combine different maximal logics into one super logic but then inference may turn out to be undecidable then it is important to characterize the decidability threshold for these logics in this perspective an interesting open question pointed out by sattler and vardi proc ijcar 01 in lecture notes in artif intel vol 2083 springer 2001 pp 76 91 is whether inference in a hybrid calculus with restricted forms of graded modalities is decidable and which complexity class it belongs to in this paper we improve a previous result proc ijcai 03 morgan kaufmann 2003 pp 331 336 and prove that this calculus and the corresponding dl accio fa are undecidable we show also that nested fixpoints are not necessary for undecidability

temporal logics need their clocks abstract we investigate effective inference systems for first order temporal logics from the point of view of completeness and soundness among others the role of clocks in these issues will be somewhat clarified by our results some open problems from the literature of temporal logic will be solved

a fast algorithm for the energy space boson boltzmann collision operator abstract this paper introduces a fast algorithm for the energy space boson boltzmann collision operator compared to the direct o n3 calculation and the previous o n2 logn method markowich and pareschi 2005 the new algorithm runs in complexity o n log n which is optimal up to a logarithmic factor n is the number of grid points in energy space the basic idea is to partition the 3 d summation domain recursively into elementary shapes so that the summation within each shape becomes a special double convolution that can be computed efficiently by the fast fourier transform numerical examples are presented to illustrate the efficiency and accuracy of the proposed algorithm

complete axiomatizations for reasoning about knowledge and time sound and complete axiomatizations are provided for a number of different logics involving modalities for knowledge and time these logics arise from different choices for various parameters regarding the interaction of knowledge with time and regarding the language used all the logics considered involve the discrete time linear temporal logic operators next and until and an operator for the knowledge of each of a number of agents both the single agent and multiple agent cases are studied in some instances of the latter there is also an operator for the common knowledge of the group of all agents four different semantic properties of agents are considered whether they i have a unique initial state ii operate synchronously iii have perfect recall and iv learn the property of no learning is essentially dual to perfect recall not all settings of these parameters lead to recursively axiomatizable logics but sound and complete axiomatizations are presented for all the ones that do

galerkin petrov approach for the boltzmann equation abstract in this work we propose a new galerkin petrov method for the numerical solution of the classical spatially homogeneous boltzmann equation this method is based on an approximation of the distribution function by associated laguerre polynomials and spherical harmonics and test in a variational manner with globally defined three dimensional polynomials a numerical realisation of the algorithm is presented the algorithmic developments are illustrated with the help of several numerical tests

upper maxwellian bounds for the spatially homogeneous boltzmann equation for the spatially homogeneous boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above uniformly in time by a maxwellian distribution provided the initial data have a maxwellian upper bound the main technique is based on a comparison principle that uses a certain dissipative property of the linear boltzmann equation implications of the technique to propagation of upper maxwellian bounds in the spatially inhomogeneous case are discussed

hierarchical boltzmann simulations and model error estimation abstract a hierarchical simulation approach for boltzmann s equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics but a subsequent refinement allows to successively improve the result to the complete boltzmann result we use hermite discretization or moment equations for the steady linearized boltzmann equation for a proof of concept of such a framework all representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous galerkin formulation we demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains the hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations we present various model errors for a flow through a curved channel with obstacles

propositional dynamic logic with local assignment we propose an extension of propositional dynamic logic which allows a new kind of program terms local assignments to propositional variables they are very close to the known array assignments in dynamic logic beacuse they allow us to change the truth values of predicate variables in this logic many notions like equivalence of programs looping and finitely branching are expressible on a propositional level in fact we show that the resulting logic is equivalent in expressive power to first order logic augmented by a device to express transitive closures in other words it is modulo extra predicate symbols equivalent to first order dynamic logic not suprisingly therefore the validity problem for this extension is 11 complete

generalized differential transform method application to differential equations of fractional order abstract in this paper we propose a new generalization of the one dimensional differential transform method that will extend the application of the method to differential equations of fractional order the new generalization is based on generalized taylor s formula and caputo fractional derivative several illustrative examples are given to demonstrate the effectiveness of the obtained results the new generalization introduces a promising tool for many linear and nonlinear models containing fractional derivatives

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

an approach to time fractional gas dynamics equation in the present article a quadratic b spline finite element galerkin method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation while the caputo form is used for the time fractional derivative appearing in the equation the l1 discretization formula is applied to the equation in time a numerical example is given and the obtained results show the accuracy and efficiency of the method therefore the present method can be used as an efficient alternative one to find out the numerical solutions of other both linear and nonlinear fractional differential equations available in the literature

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

product measure steady states of generalized zero range processes we establish necessary and sufficient conditions for the existence of factorizable steady states of the generalized zero range process on a periodic or infinite lattice this process allows transitions from a site i to a site i q involving a bounded number of multiple particles with rates depending on the content of the site i the direction q of movement and the number of particles moving we also show the sufficiency of a similar condition for the continuous time mass transport process where the mass at each site and the amount transferred in each transition are continuous variables we conjecture that this is also a necessary condition

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

differential transformation technique for steady nonlinear heat conduction problems in this paper the differential transformation method is applied to solve steady nonlinear heat conduction problems the proposed approach provides a new iterative procedure to find the spectrum of the analytic solutions therefore approximate solutions can be obtained in the form of taylor s series expansion throughout the range of interest

weak second order characterizations of various program verification systems abstract we show the equivalence of leivant s characterization of floyd hoare logic in weak second order logic leivant 1985 with both csirmaz s 1980 and sain s 1985 characterizations of floyd hoare logic in nonstandard logics of programs our method allows us to spell out the precise role of the comprehension axiom in weak second order logic we then prove similar results for other program verification systems suggested by burstall and pnueli and identify exactly the comprehension axioms corresponding to those systems

towards an ultra efficient kinetic scheme part ii in a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the boltzmann type dimarco and loubere 2013 17 the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport part by following the characteristics backward in time on the contrary to classical semi lagrangian methods one does not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost and to perform efficient numerical simulations of kinetic equations up to the six dimensional case without parallelization however the main drawback of the method developed was the loss of spatial accuracy close to the fluid limit in the present work we modify the scheme in such a way that it is able to preserve the high order spatial accuracy for extremely rarefied and fluid regimes in particular in the fluid limit the method automatically degenerates into a high order method for the compressible euler equations numerical examples are presented which validate the method show the higher accuracy with respect to the previous approach and measure its efficiency with respect to well known schemes direct simulation monte carlo finite volume muscl weno

a rice style theorem for parallel automata we present a general result similar to rice s theorem concerning the complexity of detecting properties on finite automata enriched by bounded cooperative concurrency such as statecharts and abstract parallel automata which we denote by cfas concurrent finite automata on one extreme the complexity of detecting non trivial properties that preserve equivalence of machines i e properties of the accepted language on finite automata can be as little as o 1 on the other extreme rice s theorem states that all such properties on turing machines are undecidable we state that all the non trivial properties of the regular or w regular languages are pspace hard on cfas with e moves and on cfas without e moves accepting infinite words we also extend this result to cfas without e moves accepting finite words that satisfy a condition that holds for many properties

high order numerical methods for the space non homogeneous boltzmann equation in this paper we present accurate methods for the numerical solution of the boltzmann equation of rarefied gas the methods are based on a time splitting technique the transport is solved by a third order accurate in space positive and flux conservative pfc method the collision step is treated by a fourier approximation of the collision integral which guarantees spectral accuracy in velocity coupled with several high order integrators in time strang splitting is used to achieve second order accuracy in space and time several numerical tests illustrate the properties of the methods

an elementary proof of the completeness of pdl abstract we give an elementary proof of the completeness of the segerberg axions for propositional dynamic logic

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

concurrent program schemes and their logics abstract we define and investigate several classes of concurrent program schemes including goto schemes and two versions of structured schemes based on extensions of the regular expressions to trees the schemes are studied on the first order boolean variable and propositional levels we also define and study the dynamic logics based on these classes of schemes including issues of decidability and axiomatization

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

a simple dynamic logic abstract a system of dynamic predicate logic is presented it allows nonstandard modelling of time which makes recursive axiomatization possible and is essentially one sorted the same elements code data and time which makes the system reasonably simple and transparent the system is subjected to usual mathematical considerations in particular it is shown to be equivalent to the three sorted system due to andreka nemeti and sain 1982 a conservation result for the latter system is obtained as a by product

slip boundary conditions for the compressible navier stokes equations the slip boundary conditions for the compressible navier stokes equations are derived systematically from the boltzmann equation on the basis of the chapman enskog solution of the boltzmann equation and the analysis of the knudsen layer adjacent to the boundary the resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard sphere molecules as well as the bhatnagar gross krook model these formulas which can be applied to specific problems immediately help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis

discrete velocity models without nonphysical invariants models for mixtures of discrete velocity gases have been recently introduced by the authors and have produced unexpected results particularly with regard to the possible existence of nonphysical collision invariants here we discuss a method to construct models without spurious invariants the method can be extended to very general models including polyatomic gases chemical reactions etc

on the boltzmann equation for diffusively excited granular media we study the boltzmann equation for a space homogeneous gas of inelastic hard spheres with a diffusive term representing a random background forcing under the assumption that the initial datum is a nonnegative l 2 n function with bounded mass and kinetic energy second moment we prove the existence of a solution to this model which instantaneously becomes smooth and rapidly decaying under a weak additional assumption of bounded third moment the solution is shown to be unique we also establish the existence but not uniqueness of a stationary solution in addition we show that the high velocity tails of both the stationary and time dependent particle distribution functions are overpopulated with respect to the maxwellian distribution as conjectured by previous authors and we prove pointwise lower estimates for the solutions

two dimensional differential transform for partial differential equations the differential transform is a numerical method for solving differential equations in this paper we present the definition and operation of the two dimensional differential transform a distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations partial differential equation of parabolic hyperbolic elliptic and nonlinear types can be solved by the differential transform we demonstrate that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation

hoare s logic for nondeterministic regular programs a nonstandard approach abstract this paper studies hoare s logic for nondeterministic regular programs with unbounded nondeterminism from the point of view of nonstandard dynamic logic we define a so called continuous semantics which allows certain infinitely long computations and compare it with usual semantics proving among other things the equivalence of both over reasonable data types we also establish a completeness theorem for hoare s calculus relative to continuous semantics thereby generalizing a previous result of csirmaz the proof makes use of a normal form for regular programs which is perhaps interesting in its own right

determinism and looping in combinatory pdl abstract in this paper some propositional modal logics of programs are considered based on the system cpdl combinatory pdl an extension of pdl with proper names for states these proper names are atomic formulae which are satisfied at exactly one state in each model among other things e g decidability and finite model property results a version of streett s conjecture that his axioms do axiomatize the infinite repeating construct repeat is established with respect to cpdl

deterministic solution of the spatially homogeneous boltzmann equation using discontinuous galerkin discretizations in the velocity space abstract we present a new deterministic approach for the solution of the spatially homogeneous boltzmann kinetic equation based on nodal discontinuous galerkin dg discretizations in the velocity space in the new approach the collision operator has the form of a bilinear operator with a pre computed kernel its evaluation requires o n 5 operations at every point of the phase space where n is the number of degrees of freedom in one velocity dimension the method is generalized to any molecular potential results of numerical simulations are presented for the problem of spatially homogeneous relaxation for the hard spheres potential comparison with the method of direct simulation monte carlo showed excellent agreement

propositional dynamic logic of looping and converse is elementarily decidable propositional dynamic logic is a formal system for reasoning about the before after behavior of regular program schemes an extension of propositional dynamic logic which includes both an infinite looping construct and a converse or backtracking construct is considered and it is proved that the satisfiability problem for this logic is elementarily decidable in order to establish this result deterministic two way automata on infinite trees are defined and it is shown how they can be simulated by nondeterministic one way automata the satisfiability problem for propositional dynamic logic of looping and converse is then reduced to the emptiness problem for these two way automata

foundations of logic programming in hybrid logics with user defined sharing abstract the present contribution advances an abstract notion of hybrid logic by supplementing the definition of institution with an additional structure to extract frames the foundation of logic programming is set in the general framework proposed by defining the basic concepts such as horn clause query and solution and proving fundamental results such as the existence of initial model of horn clauses and herbrand s theorem the abstract results are then applied to hybrid logics with user defined sharing where the possible worlds share a common domain and the variables used for quantification are interpreted uniformly across the worlds

a complete logic for reasoning about programs via nonstandard model theory ii in part i of the present paper w z defined the first order dynamic language dld of type d in definition 13 3 defined a decidable proof concept i prn for dld and in theorem 2 we proved that kn pm is a strongly complete inferenr e system for dld that is for every theory th and formula qj of first order dynamic language we have th q iff th i cc by j ynamic logic of type d we understand dld bn pm here we investigate further properties of our dynamic logic its expressive power how it can be used for various purposes how it can be adapted to various situations then we investigate floyd s method using the framework of dl a complete characterization of the amount of information implicitly contained in floyd s method will be found but several questions remain open in this line the proof method i is proved to be strictly stronger than floyd s method in section 6 different semantics programming are compared in section 7 within the framework of dl comkrisons with several approaches related ill several ways are given in sections 7 9

on the self similar asymptotics for generalized nonlinear kinetic maxwell models maxwell models for nonlinear kinetic equations have many applications in physics dynamics of granular gases economics etc in the present manuscript we consider such models from a very general point of view including those with arbitrary polynomial non linearities and in any dimension space it is shown that the whole class of generalized maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual lipschitz conditions this property allows to describe in detail a behavior of solutions to the corresponding initial value problem in particular we prove in the most general case an existence of self similar solutions and study the convergence in the sense of probability measures of dynamically scaled solutions to the cauchy problem to those self similar solutions as time goes to infinity a new application of multi linear models to economics and social dynamics is discussed

on strictly arithmetical completeness in logics of programs we introduce and discuss a notion of strictly arithmetical completeness related to relative completeness of cook 1978 and arithmetical completeness of harel 1978 we present a powerful technique of obtaining strictly arithmetical axiomatizations of logics of programs given a model theoretic semantics of a logic and a set of formulae defining in a metalanguage its nonclassical connectives we automatically derive strictly arithmetically complete and sound proof systems for this logic as examples of application of the technique we obtain new axiomatizations of algorithmic logic concurrent dynamic logic and temporal logic

a bi fidelity method for the multiscale boltzmann equation with random parameters abstract in this paper we study the multiscale boltzmann equation with multi dimensional random parameters by a bi fidelity stochastic collocation sc method developed in 52 70 71 by choosing the compressible euler system as the low fidelity model we adapt the bi fidelity sc method to combine computational efficiency of the low fidelity model with high accuracy of the high fidelity boltzmann model with only a small number of high fidelity asymptotic preserving solver runs for the boltzmann equation the bi fidelity approximation can capture well the macroscopic quantities of the solution to the boltzmann equation in the random space a priori estimate on the accuracy between the high and bi fidelity solutions together with a convergence analysis is established finally we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method

analytical study of fractional bratu type equation arising in electro spun organic nanofibers elaboration abstract in this article we present a hybrid analytical scheme namely homotopy perturbation transform method to achieve the solutions of the non linear bratu type problem with fractional order derivatives the caputo type fractional derivatives are considered in the present article he s polynomial is used to tackle the nonlinearity which arise in our considered problem using the initial conditions the numerical solutions of the problem are computed the numerical procedure reveals that only a few iterations are needed for better approximation of the solutions which illustrates the effectiveness and reliability of the method effects of fractional order derivatives on the solutions for various particular cases are depicted through graphs

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

a modal logic for cyclic repeating abstract a simplified fragment of pdl denoted by mlcr a modal logic for cyclic repeating is introduced in which a cyclic repeating of a program can be expressed a completeness theorem for mlcr with respect to the intended standard semantics is given it is proved that mlcr does not possess the finite model property with respect to its standard models but nevertheless it possesses this property with respect to some nonstandard models and consequently is decidable

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem