the green s function and large time behavior of solutions for the one dimensional boltzmann equation we study the pointwise behavior of the green s function of the boltzmann equation our results reveal the particle and fluid aspects of the equation the particle aspect is represented by singular waves these waves are carried by transport equations and dominate the short time behavior of the solution we devise a picard type iteration for constructing the increasingly regular particlelike waves the fluidlike waves reveal the dissipative behavior of the type of navier stokes equations as usually seen by the chapman enskog expansion these waves are constructed as part of the long wave expansion in the spectrum of the fourier mode for the space variable the fluidlike waves represent the long time behavior of the solution as an application we obtain the pointwise description of the large time behavior of the convergence to the global maxwellian when the initial perturbation is not necessarily smooth in our analysis of the exchanges of the microscopic velocity decay and space decay we make essential use of the hard sphere models c 2004 wiley periodicals inc contents

comparing models of the intensional typed l calculus abstract in this paper we compare powerset models of the typed lambda calculus although the interpretation of the type constructor is in general not uniquely determined for intensional lambda calculi we can make a canonical choice in this particular case there exists a minimal interpretation of with respect to a certain class of interpretations which yields models with a maximal theory in that class

on the construction of stable models of untyped λ calculus abstract stable models of untyped calculus have been introduced by berry proceedings of icalp lecture notes in computer science vol 62 springer berlin 1978 pp 72 89 as a refinement of the continuous framework defined by scott unpublished manuscript 1969 53pp in this paper we show that even in the stable case we have a fair amount of freedom during the construction of models we introduce a uniform method to construct stable models that allows to obtain non standard models satisfying certain equational restraints we apply this method in two particular examples first we construct a simple stable model that distinguishes the terms y and second we construct a family of 2 0 stable models with pairwise distinct theories we show that the latter models are sensible using some results of david proceedings of tlca 1999 lecture notes in computer science vol 1581 springer berlin 1999 and kerth j symbolic logic 63 1998 1529 1548

an algebraic generalization of frege structures binding algebras abstract this paper introduces a kind of second order algebras called binding algebras which is an algebraic generalization of aczel s 1980 frege structure it investigates the universal property of binding algebras by following birkhoff s 1935 method an equational logic eba is introduced we obtain its admissible completeness the relationship between completeness and the admissible completeness is very close and will be discussed

retractions on dl domains as a model for type type abstract we introduce an extension with the type type assumption for a higher order functional language in the framework of p the extension of the type free lambda calculus introduced in r amadio and g longo 1986b in ifip conference formal description of programming concepts ebberup dk august 1986 then we find using the stable functions and berry s di domains a model for p this settles the consistency problem for p up to now open and gives very simple mathematical models for higher order calculi with type type such models are based on the relevant property that the range of a stable retraction on a di domain is still a di domain finally we give the categorical description of the models of p based on moggi asperti s notion of internalization in a asperti 1988 models of higher order calculi internal categories carnegie mellon report in preparation

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

order incompleteness and finite lambda reduction models many familiar models of the untyped lambda calculus are constructed by order theoretic methods this paper provides some basic new facts about ordered models of the lambda calculus we show that in any partially ordered model that is complete for the theory of or conversion the partial order is trivial on term denotations equivalently the open and closed term algebras of the untyped lambda calculus cannot be non trivially partially ordered our second result is a syntactical characterization in terms of so called generalized mal cev operators of those lambda theories which cannot be induced by any non trivially partially ordered model we also consider a notion of finite models for the untyped lambda calculus or more precisely finite models of reduction we demonstrate how such models can be used as practical tools for giving finitary proofs of term inequalities

set theoretical and other elementary models of the l calculus abstract part i pp 351 373 of this paper is the previously unpublished 1972 memorandum plotkin 1972 with editorial changes and some minor corrections part ii pp 373 409 presents what happened next together with some further development of the material the first part begins with an elementary set theoretical model of the calculus functions are modelled in a similar way to that normally employed in set theory by their graphs difficulties are caused in this enterprise by the axiom of foundation next based on that model a model of the calculus is constructed by means of a natural deduction method finally a theorem is proved giving some general properties of those nontrivial models of the calculus which are continuous complete lattices the second part begins with a brief discussion of models of the calculus in set theories with anti foundation axioms next the model of the calculus of part i and also the closely related but different models of scott 1976 1980 and of engeler 1981 1988 are reviewed then general frameworks in which elementary constructions of models can be given are discussed following longo 1982 one can employ certain scott engeler algebras following coppo et al 1983 one can obtain filter models from their extended applicative type structures an extended discussion is given of various ways of constructing models of the calculus and the connections between them finally an extension of the theorem to complete partial orders is given the theme of the paper is the consideration of means of constructing models there is hardly any analysis of their properties there is no discussion of their application

accelerated gcro dr method for solving sequences of systems of linear equations by recycling some error approximations generated during a previous cycle of iterations we present a technique for improving the convergence of gcro dr the scheme is able to mitigate the occurrence of small skip angles occurring in gcro dr and thus considerably accelerates the convergence the alternative phenomenon of the residual vectors is examined numerically and the effectiveness of the new method is illustrated by several examples from practical applications

universal retractions on di domains we prove that every di domain which contains an infinite set of pairwise incompatible elements in particular every reflexive di domain has an infinite set of universal retractions which moreover forms a di domain when ordered by the stable order isomorphic to n where is the divisibility relation

the discrete objects in the effective topos the original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two non trivial small i e internal full subcategories which are closed under all small limits in the topos and hence in particular are internally complete the interest in such subcategories arises from the fact that they provide a very natural notion of model for many of the strongly polymorphic type theories the nature of these models is that types are interpreted as objects of the topos and hence as sets in an intuitionistic sense they in fact provide set theoretic models in the sense of reynolds 16 except that the set theory involved is non classical at the end of the paper we indicate briefly how one only of the two full subcategories gives rise to a model of the theory of constructions this sheds some light on how far we can travel with a simple set theoretic picture of data types however our main concern is to present a clean mathematical characterization of the complete subcategories the basic idea due to peter freyd remains very simple it is that the collection of objects orthogonal to any given object is automatically closed under all existing limits and to look at the objects orthogonal to the subobject classifier in the effective topos however one of our original motivations for writing this paper was that we did not quite believe freyd s identification of this category during the course of the paper we shall show that the category of families of objects in orthogonal to q is the category of families of subquotients of n the chain of reasoning now proceeds by saying that this second category is equivalent to the externalisation of an internal category and that the internal category is therefore small complete somewhat to our surprise it was this second stage trivial for an internal category in sfe that we found difficulty in making precise and it is in the discussion of notions of equivalence and completeness for internal categories and fibrations that much of the content of the paper now lies we should warn the reader that even so our treatment is not exhaustive we discuss only two notions weak and strong of equivalence for internal or fibred categories and only two notions of completeness our definitions are welladapted for our present purposes corresponding more or less to the distinction between being told for any instance of a given set of parameters that there is an object with certain properties and being given a function that produces one let c be a small full subcategory of a topos see 3 for details which is closed under arbitrary products then c cannot contain q the object of truth values in fact c cannot contain any object into which q embeds for if a is the disjoint union of the sets in c then q ec and hence q y a which contradicts one of the constructive versions of cantor s theorem this argument runs parallel

maximality and totality of stable functions in the category of stable bifinite domains this paper studies maximality and totality of stable functions in the category of stable bifinite domains we present three main results as follows o 1 every maximum preserving function is a maximal element in the stable function spaces 2 a maximal stable function f d e is maximum preserving if d is maximum separable and e is completely separable and 3 a stable bifinite domain d is maximum separable if and only if for any locally distributive stable bifinite domain e each maximal stable function f d e is maximum preserving

free molecular flow with boundary effect we study the dissipative effect of the boundary condition in the kinetic theory we focus our study on the simplest situation of the free molecular flow with diffuse reflection boundary condition and constant boundary temperature t the geometry is also chosen to be the simplest ones a bounded symmetric domain in mathbb r d an interval for d 1 a disk for d 2 and a ball for d 3 it is shown that the solution converges to the global maxwellian with the given boundary temperature t we obtain the optimal convergence rates of t 1 d the stochastic formulation of shih hsien yu is refined and generalized for our analysis

the genericity theorem and parametricity in the polymorphic l calculus abstract this paper focuses on how terms of the polymorphic calculus which may take types as inputs depend on types these terms are generally understood in all models to have an essentially constant meaning on input types we show the proof theory of polymorphic calculus suggests a clear syntactic description of this phenomenon namely under a reasonable condition we show that if two polymorphic functions agree on a single type then they agree on all types equivalently types are generic inputs

uniformity and the taylor expansion of ordinary lambda terms we define the complete taylor expansion of an ordinary lambda term as an infinite linear combination with rational coefficients of terms of a resource calculus similar to boudol s lambda calculus with multiplicities or with resources in our resource calculus all applications are multi linear in the algebraic sense i e commute with linear combinations of the function or the argument we study the collective behaviour of the beta reducts of the terms occurring in the taylor expansion of any ordinary lambda term using in a surprisingly crucial way a uniformity property that they enjoy as a corollary we obtain the main part of a proof that this taylor expansion commutes with bohm tree computation syntactically

relative definability of boolean functions via hypergraphs the aim of this work is to show how hypergraphs can be used as a systematic tool in the classification of continuous boolean functions according to their degree of parallelism intuitively f is less parallel than g if it can be defined by a sequential program using g as its only free variable it turns out that the poset induced by this preorder is as for the degrees of recursion a sup semilattice although hypergraphs have already been used in bucciarelli theoret comput sci to appear as a tool for studying degrees of parallelism no general result relating the former to the latter has been proved in that work we show that the sup semilattice of degrees has a categorical counterpart we define a category of hypergraphs such that every object represents a monotone boolean function finite coproducts in this category correspond to lubs of degrees unlike degrees of recursion where every set has a recursive upper bound monotone boolean functions may have no sequential upper bound however the ones which do have a sequential upper bound can be nicely characterised in terms of hypergraphs these subsequential functions play a major role in the proof of our main result namely that f is less parallel than g if there exists a morphism between their associated hypergraphs

the boltzmann equation with large amplitude initial data in bounded domains abstract the paper is devoted to constructing the global solutions around global maxwellians to the initial boundary value problem on the boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries we allow a class of non negative initial data which have arbitrary large amplitude and even contain vacuum the result shows that the oscillation of solutions away from global maxwellians becomes small after some positive time provided that they are initially close to each other in l 2 this yields the disappearance of any initial vacuum and the exponential convergence of large amplitude solutions to equilibrium in large time the isothermal diffuse reflection boundary condition plays a vital role in the analysis the most key ingredients in our strategy of the proof include i l x v 2 l x l v 1 l x v estimates along a bootstrap argument ii pointwise estimates on the upper bound of the gain term by the product of l norm and l 2 norm iii an iterative procedure on the nonlinear term

the extended calculus of constructions ecc with inductive types abstract luo s extended calculus of constructions ecc is a higher order functional calculus based on coquand s and huet s calculus of constructions but has in addition strong sums and a predicative cumulative type hierarchy in this paper i introduce inductive types on the predicative type levels of ecc i also show how the set model for ecc can be extended to a model for this augmented calculus

a stable programming language it is well known that stable models as di domains qualitative domains and coherence spaces are not fully abstract for the language pcf this fact is related to the existence of stable parallel functions and of stable functions that are not monotone with respect to the extensional order which cannot be defined by programs of pcf in this paper a paradigmatic programming language named st pcf is proposed which extends the language pcf with two additional operators the operational description of the extended language is presented in an effective way although the evaluation of one of the new operators cannot be formalized in a pcf like rewrite system since st pcf can define all finite cliques of coherence spaces the above gap with stable models is filled consequently stable models are fully abstract for the extended language

type theories normal forms and d lambda models abstract a non standard inverse limit model d is constructed which has a non hilbert post complete theory moreover in d a simple semantic characterization of normalizable terms is given these results are proved using the properties of a generalized type assignment system which yields a filter model barendregt coppo and dezani ciancaglini 1983 j symbolic logic 48 931 940 coppo dezani ciancaglini honsell and longo 1983 pp 241 262 logic colloquium 82 north holland amsterdam isomorphic to d the type assignment system is also proved complete with respect to an interpretation of types in the term model of equality based only on normalization properties as an application a class of maximal monoids of normalizable terms is characterized

i categories as a framework for solving domain equations abstract an abstract notion of category of information systems or i category is introduced as a generalisation of scott s well known category of information systems as in the theory of partial orders i categories can be complete or algebraic and it is shown that algebraic i categories can be obtained from a certain completion of countable i categories the proposed axioms for a complete i category introduce a global partial order on the morphisms of the category making them a cpo an initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved thus giving canonical solution of domain equations an effective version of these results for algebraic i categories is also provided some basic examples of i categories representing the categories of sets boolean algebras scott domains and continuous scott domains are constructed

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

categories of embeddings abstract we present a categorical generalisation of the notion of domains which is closed under suitable exponentiation the goal was originally to generalise girard s model of polymorphism to f if we specialise this notion in the poset case we get new cartesian closed categories of domains

deflated and restarted symmetric lanczos methods for eigenvalues and linear equations with multiple right hand sides a deflated restarted lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors the restarting limits the storage so that finding eigenvectors is practical meanwhile the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting some reorthogonalization is necessary to control roundoff error and several approaches are discussed the eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides experiments are given with large matrices from quantum chromodynamics that have many right hand sides

categorical models of polymorphism abstract we present and discuss the relations between two classes of categorical models of the second order or polymorphic lambda calculus namely those based on internal categories internal models and those based on indexed categories external models we start in part i with a detailed introduction to internal categories and their relations to indexed categories the presentation is by means of equations between arrows in an ambient category with finite limits in part ii we recall the definition of the two classes of models and we present the externalization process that given an internal model yields an external model we show how one can go back in a straightforward way and that by making a full round trip from an internal model to an internal model via an external one or vice versa one does obtain equivalent models part iii discusses three major examples of models provable retractions inside a per model per inside set pl categories inside their grothendieck completions the appendix contains an account of internal adjunctions and internal cccs

a full continuous model of polymorphism we introduce a model of the second order lambda calculus such a model is a scott domain whose elements are themselves scott domains and in it polymorphic maps are interpreted by generic continous maps

prime algebraicity a prime algebraic lattice can be characterised as isomorphic to the downwards closed subsets ordered by inclusion of its complete primes it is easily seen that the downwards closed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion the converse also holds any completely distributive algebraic lattice is isomorphic to such a set of downwards closed subsets of a partial order the partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes consequently prime algebraic lattices are precisely the completely distributive algebraic lattices the result extends to scott domains several consequences are explored briefly the representation of berry s di domains by event structures a simplified form of information systems for completely distributive scott domains and a simple domain theory for concurrency

the boltzmann equation with weakly inhomogeneous data in bounded domain this paper is concerned with the boltzmann equation with specular reflection boundary condition we construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum the result extends the case of cauchy problem considered by arkeryd esposito pulvirenti 4 to the specular reflection boundary value problem in bounded domain

improving the accuracy of gmres with deflated restarting in general restarting the generalized minimum residual method gmres slows down the convergence speed there exist a number of methods that improve the convergence of restarted gmres one of them is gmres with deflated restarting siam j sci comput 24 2002 pp 20 37 introduced by morgan this method retains a number of harmonic ritz vectors at each restart to mitigate convergence slowdown we investigate the accuracy of this method and propose a slight modification of it that is mathematically equivalent we show that the new implementation has better numerical properties especially if high accuracy of the solution is required numerical results are presented that confirm our theoretical results

program development in constructive type theory abstract we present the program development concept in a logical framework including constructive type theory and then show how to use such theories to derive programs from proofs of formal specifications we are interested in two important facts that are the mechanization of the proof construction and the possibility to express in the theory significant concepts for programming like inductively defined types and general recursion we give here a survey on some results and problems appearing in logical frameworks devoted to the programming with proofs approach

stable domains are generalized topological spaces abstract we present a generalization of the notion of topological structure on a set such that di or l domains can be naturally given such topologies in a way making the continuous functions exactly the stable functions as ordinary topological spaces give rise to locales these generalized spaces lead to a generalized form of locale that we define and explore

on maximal stable functions abstract this note studies maximal elements in di domains especially maximal stable functions first a sufficient condition for a stable function to be maximal is provided the condition requires that the function be total in the sense that it preserves maximal elements examples are then given to demonstrate that being total is not necessary for a stable function to be maximal secondly a characterization is given for di domains for which maximal also implies total finally topological properties of maximal elements are investigated di domains for which maximal is the same as total are those having a certain hausdorff property

information systems for continuous posets abstract the method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive this develops the results of raney on completely distributive lattices and of hoofman on continuous scott domains and also generalizes smyth s r structures various constructions on continuous posets have neat descriptions in terms of these continuous information systems here we describe hoffmann lawson duality which could not be done easily with r structures and vietoris power locales we also use the method to give a partial answer to a question of johnstone s in the context of continuous posets vietoris algebras are the same as localic semilattices

discrimination by parallel observers the algorithm abstract the main result of the paper is a constructive proof of the following equivalence two pure terms are observationally equivalent in the lazy concurrent calculus iff they have the same levy longo trees an algorithm which allows to build a context discriminating any two pure terms with different levy longo trees is described it follows that contextual equivalence coincides with behavioural equivalence bisimulation as considered by sangiorgi another consequence is that the discriminating power of concurrent lambda contexts is the same as that of boudol laneve s contexts with multiplicities

a relative pcf definability result for strongly stable functions and some corollaries we prove that in the hierarchy of simple types based on the type of natural numbers any finite strongly stable function is equal to the application of the semantics of a pcf definable functional to some strongly stable generally not pcf definable functionals of type two applying a logical relation technique we derive from this result that the strongly stable model of pcf is the extensional collapse of its sequential algorithms model

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

thermal transpiration for the linearized boltzmann equation the phenomena of thermal transpiration due to the boundary temperature gradient is studied on the level of the linearized boltzmann equation for the hard sphere model we construct such a flow for a highly rarefied gas between two plates and also in a circular pipe it is shown that the flow velocity parallel to the plates is proportional to the boundary temperature gradient for a highly rarefied gas that is for a sufficiently large knudsen number the flow velocity between two plates is of the order of log and the flow velocity in a pipe is of finite order our analysis is based on certain pointwise estimates of the solutions of the linearized boltzmann equation 2006 wiley periodicals inc

quantitative domains and infinitary algebras abstract we describe a class of models of the lambda calculus that generalize and simplify the quantitative domains of girard 1988 their salient feature is that these domains do not have enough points in the sense that a map of domains is not entirely determined by its value on the points i e the morphisms from the terminal object of the category we construct a universal domain whose structure is particularly simple and describe some of the features of the untyped lambda calculus associated to it finally we show how linear logic can be interpreted in these domains and construct another category of domains that supports linear logic

internal models of system f for decompilation this paper considers girard s internal coding of each term of system f by some term of a code type this coding is the type erasing coding definable already in the simply typed lambda calculus using only abstraction on term variables it is shown that there does not exist any decompiler for system f in system f where the decompiler maps a term of system f to its code an internal model of f is given by interpreting each type of f by some type equipped with maps between the type and the code type this paper gives a decompiler normalizer for this internal model in f where the decompiler normalizer maps any term of the internal model to the code of its normal form it is also shown that for any model of f the composition of this internal model and the model produces another model of f whose equational theory is below untyped beta eta equality

computability on di domains in this paper we investigate a subset of the class of scott computable stable functions called berry computable furthermore we introduce a corresponding notion of an effectively given di domain we obtain the result that the category with these domains as objects and berry computable functions as morphisms has all the important attributes of recursive domains and scott computable continuous functions e g cartesian closedness moreover we investigate the relationship between stable functions and sequential algorithms we show that any berry computable function can be computed in a sequential way by a relative algorithm additionally if the codomain of the considered stable function is an atomic di domain we obtain the result that the reverse holds as well

on the block gmres method with deflated restarting abstract in this paper we consider the block gmres method with deflated restarting for solving nonsymmetric linear systems with multiple right hand sides we modify slightly the restarted block gmres method with deflation of eigenvalues proposed by morgan to obtain a new one it is shown that the modified method is mathematically equivalent to the morgan s original method however the error analysis shows the modified version minimizes the numerical errors during a restart and therefore is better suited if the linear systems have to be solved with high precision numerical experiments report the effectiveness of the modified method

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

a k denotational semantics for map theory in zfc si abstract map theory or mt for short has been designed as an integrated foundation for mathematics logic and computer science by this we mean that most primitives and tools are designed from the beginning to bear the three intended meanings logical computational and set theoretic mt was originally introduced in 17 it is based on calculus instead of logic and sets and it fulfills church s original aim of introducing calculus in particular it embodies all of zfc set theory including classical propositional and classical first order predicate calculus mt also embodies the unrestricted untyped lambda calculus including unrestricted abstraction and unrestricted use of the fixed point operator mt is an equational theory we present here a semantic proof of the consistency of map theory within zfc si where si asserts the existence of an inaccessible cardinal the proof is in the spirit of denotational semantics and relies on mathematical tools which reflect faithfully and in a transparent way the intuitions behind map theory this gives a consistency proof but also for the first time gives a clear presentation of the semantics of map theory in a traditional framework from the metamathematical point of view the strength of mt lies somewhere between zfc and zfc si the lower bound is proved in 17 by means of a syntactical translation of zfc including classical propositional and predicate calculus into map theory and the upper bound by building an exceedingly complex model of map theory within zfc si the present paper confirms the upper bound by providing much simpler models the canonical models of the paper which are in fact the paradigm of a large class of quite natural models of mt that all these models interpret a model of zfc is a consequence of the syntactic translation f which is a difficult theorem of 17 we can however give here a direct proof of a stronger result namely that they interpret some vgs where is an inaccessible cardinal finally we return to the canonical models and show that they are adequate for the notion of computation which underlies mt

intersection types a la church in this paper we present a fully typed calculus based on the intersection type system discipline which is a counterpart a la church of the type assignment system as invented by coppo and dezani the relationship between and the intersection type assignment system is the standard isomorphism between typed and type assignment system and so the typed language inherits from the untyped system all the good properties like subject reduction and strong normalization moreover both type checking and type reconstruction are decidable

the scott model of linear logic is the extensional collapse of its relational model we show that the extensional collapse of the relational model of linear logic is the model of prime algebraic complete lattices a natural extension to linear logic of the well known scott semantics of the lambda calculus

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

completeness of intersection and union type assignment systems for call by value λ models we study a version of intersection and union type assignment system union elimination rule of which is allowed only when subject of its major premise is a value of call by value e calculus the system is shown to be sound and complete under some abstract notion of membership relation defined over simple semantics for call by value models and to be invariant under call by value conversion of subjects we prove it by constructing a filter call by value lgr model copyright 2002 elsevier science b v

on functors expressible in the polymorphic typed lambda calculus abstract given a model of the polymorphic typed lambda calculus based upon a cartesian closed category k there will be functors from k to k whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions we show that if t is such a functor then there is a weak initial t algebra and if in addition k possesses equalizers of all subsets of its morphism sets then there is an initial t algebra these results are used to establish the impossibility of certain models including those in which types denote sets and s s denotes the set of all functions from s to s

initial boundary value problem for one dimensional wave solutions of the boltzmann equation tai ping liu and his collaborator shih hsien yu solve the initial boundary value problem for the boltzmann equation because of the inclusion of microscopic velocity in the kinetic theory the boltzmann equation can model richer and physically more realistic boundary effects more so than the fluid equations such as the navier stokes this motivates our study the reflection of fluid waves off the boundary is indicated in the figure for the case only one of the euler wave velocities is positive

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

proff search in type theoretic languages an introduction we introduce the main concepts and problems in the theory of proof search in type theoretic languages and survey some specific connected topics we do not claim to cover all of the theoretical and implementation issues in the study of proof search in type theoretic languages rather we present some key ideas and problems starting from well motivated points of departure such as a definition of a type theoretic language or the relationship between languages and proof objects the strong connections between different proof search methods in logics type theories and logical frameworks together with their impact on programming and implementation issues are central in this context

parallel and serial hypercoherences abstract it is known that the strongly stable functions which arise in the semantics of pcf can be realized by sequential algorithms which can be considered as deterministic strategies in games associated to pcf types studying the connection between strongly stable functions and sequential algorithms two dual classes of hypercoherences naturally arise the parallel and serial hypercoherences the objects belonging to the intersection of these two classes are in bijective correspondence with the so called serial parallel graphs that can essentially be considered as games we show how to associate to any hypercoherence a parallel hypercoherence together with a projection onto the given hypercoherence and present some properties of this construction intuitively it makes explicit the computational time of a hypercoherence

simple consequence relations abstract we provide a general investigation of logic in which the notion of a simple consequence relation is taken to be fundamental our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets we use our notion to characterize several known logics including linear logic and non monotonic logics and for a general semantics independent classification of standard onnectives via equations on consequence relations these include girard s multiplicatives and additives we next investigate the standard methods for uniformly representing consequence relations hilbert type natural deduction and gentzen type the advantages and disadvantages of using each system and what should be taken as good representations in each case especially from the implementation point of view are explained we end by briefly outlining with examples some methods for developing non uniform but still efficient representations of consequence relations

continuous information system abstract in this paper we generalise the notion of algebraic information system to continuous information system just as algebraic information systems are concrete representations of scott domains continuous information systems are concrete representations of continuous scott domains the class of continuous information systems can be generated from the subclass of qualitative information systems in the sense that the category of continuous information systems is equivalent to the karoubi envelope of the category of qualitative information systems because qualitative information systems correspond to the qualitative domains of girard theoret comput sci 45 159 192 1986 this implies that the category of continuous scott domains is equivalent to the karoubi envelope of the category of qualitative domains we show how constructions on qualitative information systems such as product and function space can be translated to constructions on continuous information systems among other things we prove that the category of qualitative information systems is weak cartesian closed finally we define two universal information systems

a typed calculus based on a fragment of linear logic abstract linear logic we concisely write ll has been introduced recently by jean yves girard in theoretical computer science 50 1987 born from the semantics of second order lambda calculus ll is more expressive than traditional logic both classical and intuitionistic due to the absence of structural rules and to a particular treatment of negation which is denoted by proofs in ll do not have a directional character the constructive meaning of a proof of a b is a function mapping all proofs of a into proofs of b in ll a b has a similar meaning but b a represents the same formula and has the same proofs so one of such proofs can map a proof of a into one of b as well as a proof of b into one of a in this paper we are interested in the multiplicative second order subsystem ll of linear logic we introduce a calculus called v calculus whose terms are canonical representations of proofs the aim of the calculus is to give a better comprehension of the computational aspects of the process of cut elimination we prove that the v calculus obeys strong normalization and the church rosser properties

information systems revisited the general continuous case in this paper a new notion of continuous information system is introduced it is shown that the information systems of this kind generate exactly the continuous domains the new information systems are of the same logic oriented style as the information systems first introduced by scott in 1982 they consist of a set of tokens a consistency predicate and an entailment relation satisfying a set of natural axioms it is shown that continuous information systems are closely related to abstract bases indeed both categories are equivalent since it is known that the categories of abstract bases and or continuous domains are equivalent it follows that the category of continuous information systems is also equivalent to that of continuous domains in applications mostly subclasses of continuous domains are considered for example the domains have to be pointed algebraic bounded complete or fs conditions are presented that when fulfilled by a continuous information system force the generated domain to belong to the required subclass in most cases the requirements are not only sufficient but also necessary

domain theoretic models of polymorphism abstract we give an illustration of a construction useful in producing and describing models of girard and reynolds polymorphic calculus the key unifying ideas are that of a grothendieck fibration and the category of continuous sections associated with it constructions used in indexed category theory the universal types of the calculus are interpreted as the category of continuous sections of the fibration as a major example a new model for the polymorphic calculus is presented in it a type is interpreted as a scott domain in fact understanding universal types of the polymorphic calculus as categories of continuous sections appears to be useful generally for example the technique also applies to the finitary projection model of bruce and longo and a recent model of girard indeed the work here was inspired by girard s and arose through trying to extend the construction of his model to scott domains it is hoped that by pin pointing a key construction this paper will help towards a deeper understanding of models for the polymorphic calculus and the relations between them

re visiting axioms of information systems based on the investigation of the axioms used in the theory of information system the notion of simplified continuous information system is introduced which is considered as a generalization of continuous information systems just like the continuous information system it is also a concrete representation of continuous domains and categorically equivalent to the category of continuous domains and scott continuous functions what is more all types of information systems studied in the literature so far turn out to be special cases of the new type by defining linear information systems and their entailment functions we show that the states of a scott style information system are exactly the fixpoints of its entailment function some axioms about an entailment relation can be characterized by the properties of the corresponding entailment function and in particular the simplified continuous information systems are solely determined by the idempotence of entailment functions

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

filter models with polymorphic types abstract using ideas and results from barendrecht 1983 and coppo 1984 on intersection types a comparable theory is developed for second order polymorphic types the set of filters constructed with polymorphic type forms with inclusion a continuous lattice which yields a model of what we call expansion i e the value of a term increases under reduction but not of conversion combining intersection with polymorphic types does give filter models but the two standard ways of interpreting terms do not coincide

the semantics of second order lambda calculus abstract in the second order polymorphic typed lambda calculus lambda abstraction over type variables leads to terms denoting polymorphic functions straightforward cardinality considerations show that a naive set theoretic interpretation of the calculus is impossible we give two definitions of semantic models for this language and prove them equivalent our syntactical environment model definition and a more algebraic combinatory model definition for the polymorphic calculus correspond to analogous model definitions for untyped lambda calculus soundness and completeness theorems are proved using the environment model definition we verify that some specific interpretations of the calculus proposed in the literature indeed yield models in our sense

a synthetic axiomatization of map theory this paper presents a subtantially simplified axiomatization of map theory and proves the consistency of this axiomatization in zfc under the assumption that there exists an inaccessible ordinal map theory axiomatizes lambda calculus plus hilbert s epsilon operator all theorems of zfc set theory including the axiom of foundation are provable in map theory and if one omits hilbert s epsilon operator from map theory then one is left with a computer programming language map theory fulfills church s original aim of introducing lambda calculus map theory is suited for reasoning about classical mathematics as well ascomputer programs furthermore map theory is suited for eliminating thebarrier between classical mathematics and computer science rather than just supporting the two fields side by side map theory axiomatizes a universe of maps some of which are wellfounded the class of wellfounded maps in map theory corresponds to the universe of sets in zfc the first version mt0 of map theory had axioms which populated the class of wellfounded maps much like the power set axiom et al populates the universe of zfc the new axiomatization mt of map theory is synthetic in the sense that the class of wellfounded maps is defined inside maptheory rather than being introduced through axioms in the paper we define the notion of kappa and kappasigma expansions and prove that if sigma is the smallest strongly inaccessible cardinal then canonical kappasigma expansions are models of mt which proves the consistency furthermore in the appendix we prove that canonical omega expansions are fully abstract models of the computational part of map theory

probabilistic coherence spaces as a model of higher order probabilistic computation we study a probabilistic version of coherence spaces and show that these objects provide a model of linear logic we build a model of the pure lambda calculus in this setting and show how to interpret a probabilistic version of the functional language pcf we give a probabilistic interpretation of the semantics of probabilistic pcf closed terms of ground type last we suggest a generalization of this approach using banach spaces

non isothermal boundary in the boltzmann theory and fourier law in the study of the heat transfer in the boltzmann theory the basic problem is to construct solutions to the following steady problem v cdot nabla x f frac 1 rm k rm n q f f qquad x v in omega times mathbf r 3 quad quad 0 1 f x v n x cdot v 0 f x v prime n x cdot v prime dv prime quad x in partial omega quad quad 0 2 where is a bounded domain in mathbf r d 1 leq d leq 3 kn is the knudsen number and mu theta frac 1 2 pi theta 2 x rm exp frac v 2 2 theta x is a maxwellian with non constant non isothermal wall temperature x based on new constructive coercivity estimates for both steady and dynamic cases for theta theta 0 leq delta ll 1 and any fixed value of kn we construct a unique non negative solution fs to 0 1 and 0 2 continuous away from the grazing set and exponentially asymptotically stable this solution is a genuine non equilibrium stationary solution differing from a local equilibrium maxwellian as an application of our results we establish the expansion f s mu theta 0 delta f 1 o delta 2 and we prove that if the fourier law holds the temperature contribution associated to f1 must be linear in the slab geometry

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

linear logic coherence and dinaturality a general coherence theorem for monoidal closed structures is obtained by modifying the logical approach to coherence questions due to lambek 1969 1990 by making use of linear logic linear logic introduced by girard has many advantages which are of use in studying coherence most notably its resource sensitive nature makes it ideal for studying monoidal closed structures the logical approach is also modified by using natural deduction rather than sequent calculus the natural deduction system in question is proof nets also introduced by girard proof nets have several important properties which are exploited to prove the coherence theorem in particular the cut elimination procedure is confluent and strongly normalizing the approach to coherence is to define a general structure the autonomous deductive system for defining many theories of monoidal closed categories an autonomous deductive system is a deductive system with several added features which are suggested by the properties of proof nets it is then possible to give a straightforward criterion for whether a given theory of monoidal closed categories specified by an autonomous deductive system is coherent finally a relationship is established between coherence and the composition problem for dinatural transformations thus the dinatural approach to modelling polymorphic types due to bainbridge et al 1990 can be extended to linear polymorphism

stability and computability in coherent domains abstract the relation between stability and sequentiality is investigated in the category of girard s coheent domains we introduce and discuss a notion of computability for stable functions based on the recursive enumerability of their traces in a way similar to the definition of computable functions in scott s effectively given domains we then relate this notion of computability to regular sets and relative algorithms oracle machines of the theory of relativized computability the notion of oracle machine is used to formalize the idea of a main sequential program which calls an unspecified external agent o a sort of subroutine call in particular we prove that a function f between two coherent domains x and y is stable and computable if and only if f may be computed by an oracle machine questioning in a positive way a simple class of oracles that supply informations about elements in x

glueing and orthogonality for models of linear logic we present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic in particular we treat the exponentials in detail we indicate simple applications of the methods and show that they cover familiar examples

adjunction of semifunctors categorical structures in nonextensional lambda calculus abstract some connections between calculus and category theory have been known among them it has been shown by lambek that cartesian closed categories ccc for short can be identified with extensional typed calculus cf lambek 1980 and lambek and scott 1986 in this paper we introduce the notion of adjunction of semifunctors for simplicity we refer to this as semiadjunction and by the aid of this notion we define the notion of semi cartesian closed category semi ccc for short some categorical or algebraic systems aimed to represent calculus will turn out to be special cases of semi ccc another intersting connection between ccc and calculus is scott s embedding of theory into a ccc cf scott 1980 this will be referred to as scott embedding we will show that any semiadjunction is embeddable in an adjunction of functors and scott embedding is a special case

weak algebraic information systems and a new equivalent category of dom of domains abstract in this paper concepts of generalized algebraic information system in short ga inf and weak algebraic information system in short wa inf are introduced relationships among several algebraicities of information systems are explored main results are 1 a dcpo d is an algebraic domain iff d is represented by an algebraic information system in short a inf and iff d is represented by a ga inf 2 every a inf is a ga inf and a wa inf 3 every c inf s d b induced by a domain with a basis b is a wa inf 4 the category wainf of wa infs and approximable mappings is equivalent to the category dom of domains and scott continuous functions 5 every wa inf s satisfying condition mt is a retract of w s the information structure induced by s in category wainf of wa infs and approximable mappings particularly c inf s d b induced by a domain d with a basis b is a retract of a inf w s d b in category wainf

a higher order calculus and theory abstraction abstract we present a higher order calculus ecc which naturally combines coquand huet s calculus of constructions and martin lof s type theory with universes ecc is very expressive both for structured abstract reasoning and for program specification and construction in particular the strong sum types together with the type universes provide a useful module mechanism for abstract description of mathematical theories and adequate formalization of abstract mathematics this allows comprehensive structuring of interactive development of specifications programs and proofs after a summary of the meta theoretic properties of the calculus an set realizability model of ecc is described to show how its essential properties can be captured set theoretically the model construction entails the logical consistency of the calculus and gives some hints on how to adequately formalize abstract mathematics theory abstraction in ecc is discussed as a pragmatic application

building continuous webbed models for system f we present here a large family of concrete models for girard and reynolds polymorphism system f in a noncategorical setting the family generalizes the construction of the model of barbanera and berardi tech report university of turin 1997 hence it contains complete models for f a complete model for system f preprint june 1998 and we conjecture that it contains models which are complete for f it also contains simpler models the simplest of them e2 being a second order variant of the engeler plotkin model e all the models here belong to the continuous semantics and have underlying prime algebraic domains all have the maximum number of polymorphic maps the class contains models which can be viewed as two intertwined compatible webbed models of untyped calculus in the sense of berline from computations to foundations the calculus and its webbed models revised version theoret comput sci 86 pp to appear but it is much larger than this finally many of its models might he read as two intertwined strict intersection type systems

behavioural inverse limit λ models we construct two inverse limit models which completely characterise sets of terms with similar computational behaviours the sets of normalising head normalising weak head normalising terms those corresponding to the persistent versions of these notions and the sets of closable closable normalising and closable head normalising terms more precisely for each of these sets of terms there is a corresponding element in at least one of the two models such that a term belongs to the set if and only if its interpretation in a suitable environment is greater than or equal to that element we use the finitary logical description of the models obtained by defining suitable intersection type assignment systems to prove this

a representation of l domains by information systems in this paper we propose the notion of l information systems which provides a concrete representation of l domains in particular we prove that the category of l information systems with approximable mappings as morphisms is equivalent to that of l domains with scott continuous functions as morphisms

a natural semantics of first order type dependency abstract we develop a natural model of first order type dependency by considering types as stable families of di domains indexed on a di domain or in an equivalent way as stable functors from a di domain to the category of di domains with rigid embedding projections the fundamental idea of the interpretation is to interpret any construction in the type system by an algebraic construction in this category we present here a model of the so called generalised algebraic fragment of martin lof s and we replace it in the d category general framework