bounding the expectation of a saddle function with application to stochastic programming we previously obtained tight upper and lower bounds to the expectation of a saddle function of multivariate random variables using first and cross moments of the random variables without independence assumptions these bounds are applicable when domains of the random vectors are compact sets in the euclidean space in this paper we extend the results to the case of unbounded domains similar in spirit to the extensions by birge and wets in the pure convex case the relationship of these bounds to a certain generalized moment problem is also investigated finally for solving stochastic linear programs utilizing the above bounding procedures a computationally more appealing order cone decomposition scheme is proposed which behaves quadratically in the number of random variables moreover the resulting upper and lower approximations are amenable to efficient solution techniques

zeros of graph counting polynomials given a finite graph e we define a family of subgraphs f by restricting the number of edges of f with endpoint at any vertex of e defining we can in many cases give precise information on the location of zeros of zeros all real negative all imaginary etc extensions of these results to weighted and infinite graphs are given

the approximation of separable stochastic programs abstract stochastic programming sp problems with separable cost functions are considered special attention being paid to simple and linear quadratic recourse using barycentric approximation schemes and their features we derive convergent sub sequences of approximate solutions and of associated error bounds for the given problem based on structural properties that until now have not been exploited a further contribution of the present paper is seen in the convergence statements yielding argmin convergence assuming the sets of global minimizers of the approximate functions are nonempty argmin convergence is obtained by restricting the conditions of epi convergence to the sets of minimizers that correspond to the approximate functions

the zeros of derivatives of entire functions and the polya wiman conjecture an entire function f x which assumes only real values on the real axis is said to be a real entire function thus if f is a real entire function then its maclaurin coefficients are all real and consequently the zeros of f are symmetrically located with respect to the real axis one of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by polya p4 in his celebrated survey article entitled on the zeros of the derivatives of a function and its analytic character in this work p4 p 181 polya states the real axis seems to exert an influence on the complex zeros of f n z it seems to attract these zeros when the order is less than 2 and it seems to repel them when the order is greater than 2 indeed the first confirmation of this principle was made by alander a2 see also p6lya p2 who in 1930 showed that the polya wiman conjecture is true if the order of f x is less than 2 3 in wil and wi2 wiman established the validity of this conjecture for functions f x of order at most 1 while in 1937 polya p3 proved it for functions of order less than 4 3 a general survey of these results and related conjectures may be found in p2 and p4 for more recent surveys of this and related areas of research see boas b2 and prather pr it seems that no progress has been made on the p6lya wiman conjecture since the publication of polya s 4 3 theorem in 1937 our proof of the conjecture is self contained and does not rely on the results cited above in section 2 we will 1 introduce some definitions and notations 2 state some of the properties of functions in the laguerre p6lya class and 3 prove the polya wiman conjecture theorem 3 in the proof of theorem 3 we

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

polynomials with the half plane property and matroid theory a polynomial f is said to have the half plane property if there is an open half plane h subset of c whose boundary contains the origin such that f is non zero whenever all the variables are in h

on the convergence of algorithms with implications for stochastic and nondifferentiable optimization studies of the convergence of algorithms often revolve around the existence of a function with respect to which monotonic descent is required in this paper we show that under relatively lenient conditions stage dependent descent not necessarily monotonic is sufficient to guarantee convergence this development also provides the impetus to examine optimization algorithms we show that one of the important avenues in the study of convergence namely the theory of epi convergence imposes stronger conditions than are necessary to establish the convergence of an optimization algorithm working from a relaxation of epi convergence we introduce the notion of compatibility and prove several results that permit relaxations of conditions imposed by previous approaches to algorithmic convergence finally to illustrate the usefulness of the concepts we combine stage dependent descent with results derivable from compatibility to provide a basis for the convergence of a general algorithmic statement that might be used for stochastic and nondifferentiable optimization

matrix positivity preservers in fixed dimension i a classical theorem of i j schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size obtaining similar characterizations in fixed dimension is intricate in this note we provide a solution to this problem in the polynomial case as consequences we derive tight linear matrix inequalities for hadamard powers of positive semidefinite matrices and a sharp asymptotic bound for the matrix cube problem involving hadamard powers

on strong normalization and type inference in the intersection type discipline we introduce a new unification procedure for the type inference problem in the intersection type discipline it is well known that type inference in this case should succeed exactly for the strongly normalizing expressions we give a proof for the strong normalization result in the intersection type discipline which we obtain by putting together some well known results and proof techniques our proof uses a variant of klop s extended l calculus for which it is shown that strong normalization is equivalent to weak normalization this is proved here by means of a finiteness of developments theorem obtained following de vrijer s combinatory technique the main property of this extended calculus is uniformity i e weak and strong normalizabilities coincide the strong normalizability result is therefore a consequence of the fact first established by coppo and dezani for the l calculus that typability implies weak normalizability we then show that the unification process which is the basis for type inference exactly corresponds to reduction in the extended l calculus finally we show that our notion of unification allows us to compute a principal typing for any typable l expression

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

completeness of type assignment systems with intersection union and type quantifiers this paper develops type assignment systems with intersection and union types and type quantifiers we show that the known system for these types is not semantically complete however the following two hold for a certain class of typing statements called stable statements which include all statements without type quantifier 1 the validity of stable statements for kripke models is equivalent to that for standard models 2 if we add two axioms expressing the distributive laws of intersection over union and existential type quantifier then the resulting system is complete for kripke models as a consequence the known system with the axioms for distributive laws is complete for standard models if we restrict statements to stable ones all the results are obtained in a systematic way with sequent style formulations of type assignment and the cut elimination property for them

non representable hyperbolic matroids the generalized lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices hyperbolic polynomials give rise to a class of hyperbolic matroids which properly contains the class of matroids representable over the complex numbers this connection was used by the first author to construct counterexamples to algebraic stronger versions of the generalized lax conjecture by considering a non representable hyperbolic matroid the va mos matroid and a generalization of it are to this day the only known instances of non representable hyperbolic matroids we prove that the non pappus and non desargues matroids are non representable hyperbolic matroids by exploiting a connection due to jordan between euclidean jordan algebras and projective geometries we further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non representable finally we explore consequences to algebraic versions of the generalized lax conjecture

approximation to optimization problems an elementary review during the last two decades the concept of epi convergence was introduced and then was used in various investigations in optimization and related areas the aim of this review is to show in an elementary way how closely the arguments in the epi convergence approach are related to those of the classical theory of convergence of functions

complete restrictions of the intersection type discipline abstract in this paper the intersection type discipline as defined in barendregt 1983 is studied we will present two different and independent complete restrictions of the intersection type discipline the first restricted system the strict type assignment system is presented in section 2 its major feature is the absence of the derivation rule and it is based on a set of strict types we will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in barendregt we will show that the strict type assignment system is the nucleus of the full system i e for every derivation in the intersection type discipline there is a derivation in which is used only at the very end finally we will prove that strict type assignment is complete for inference semantics the second restricted system is presented in section 3 its major feature is the absence of the type we will show that this system gives rise to a filter i model and that type assignment without is complete for the i calculus finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable

filter models for conjunctive disjunctive l calculi abstract the distinction between the conjunctive nature of non determinism as opposed to the disjunctive character of parallelism constitutes the motivation and the starting point of the present work calculus is extended with both a non deterministic choice and a parallel operator a notion of reduction is introduced extending reduction of the classical calculus we study type assignment systems for this calculus together with a denotational semantics which is initially defined constructing a set semimodel via simple types we enrich the type system with intersection and union types dually reflecting the disjunctive and conjunctive behaviour of the operators and we build a filter model the theory of this model is compared both with a morris style operational semantics and with a semantics based on a notion of capabilities

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

the lee yang and polya schur programs iii zero preservers on bargmann fock spaces we characterize linear operators preserving zero restrictions on entire functions in weighted bargmann fock spaces this extends the characterization of linear operators on polynomials preserving stability due to borcea and the author to the realm of entire functions and translates into an optimal albeit formal lee yang theorem

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

a characterization of f complete type assignments the aim of this paper is to investigate the soundness and completeness of the intersection type discipline for terms of the untyped calculus with respect to the f semantics f soundness and f completeness as pointed out by scott if d is the domain of a model there is a subset f of d whose elements are the canonical representatives of functions the f semantics of types takes into account that theintuitive meaning of is the type of functions with domain and range and interprets as a subset of f the type theories which induce f complete type assignments are characterized it follows that a type assignment is f complete iff equal terms get equal types and whenever m has a type n where is a type variable and is the universal type the term z1 zn mz1 zn has type here we assume that z1 z n do not occur free in m

the ising partition function zeros and deterministic approximation we study the problem of approximating the partition function of the ferromagnetic ising model with both pairwise as well as higher order interactions equivalently in graphs as well as hypergraphs our approach is based on the classical lee yang theory of phase transitions along with a new lee yang theorem for the ising model with higher order interactions and on an extension of ideas developed recently by barvinok and patel and regts that can be seen as an algorithmic realization of the lee yang theory our first result is a deterministic polynomial time approximation scheme an fptas for the partition function in bounded degree graphs that is valid over the entire range of parameters beta the interaction and lambda the external field except for the case left lambda right 1 the zero field case a polynomial time randomized approximation scheme fpras for all graphs and all beta lambda based on markov chain monte carlo simulation has long been known unlike most other deterministic approximation algorithms for problems in statistical physics and counting our algorithm does not rely on the decay of correlations property but as pointed out above on lee yang theory this approach extends to the more general setting of the ising model on hypergraphs of bounded degree and edge size where no previous algorithms even randomized were known for a wide range of parameters in order to achieve this latter extension we establish a tight version of the lee yang theorem for the ising model on hypergraphs improving a classical result of suzuki and fisher

principal type scheme and unification for intersection type discipline abstract the intersection type discipline for the calculus itd is an extension of the classical functionality theory of curry in the itd a term satisfying a given property has a principal type scheme in an extended meaning i e there is a type scheme deducible for it from which all and only the type schemes deducible for it are reachable by means of suitable operations the problem of finding the principal type scheme for a term if it exists is semidecidable in the paper a procedure is shown building the principal type scheme of a term through the construction of the most general unifier for intersection type schemes

embedding a second order type system into an intersection type system this paper presents the relationship between a second order type assignment system t and an intersection type assignment system t first we define a translation tr from intersection types to second order types then we define a system t obtained from t by restricting the use of the intersection type introduction rule and show that t and t are equivalent in the following senses a if a term m has a type in t then m has the type tr in t and conversely b if m has a type t in t then m has a type in t such that tr is equivalent to t these two theorems mean that t is embedded into t

partition function zeros at first order phase transitions a general analysis we present a general rigorous theory of partition function zeros for lattice spin models depending on one complex parameter first we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper 5 under these assumptions we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions up to an error which we prove is generically exponentially small in the linear size of the system for asymptotically large systems the zeros concentrate on phase boundaries which are simple curves ending in multiple points for models with an ising like plus minus symmetry we also establish a local version of the lee yang circle theorem this result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle while in the complement of this region the zeros concentrate on less symmetric curves

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

computability in higher types pω and the completeness of type assignment p the powerset of the natural numbers may be turned into an applicative structure by myhill and shepherdson then for a b p set a b d p a a da b any effectively given domain in the sense of scott 1982 can be embedded into p by a continuous and computable retraction notation x c ax for some ax p which is also an effectively given domain we first prove that if x c ax and y c ay then also ax ay is an effectively given domain and cont x y c ax ay i e the continuous functions can be embedded into ax ay let now p p be the collection of single valued sets i e p is isomorphic to the effectively given domain of the partial functions on and let t be the function type symbols with 1 t then for p1 p p p p extends the classical recursive operators at higher types by ershov s model of the kleene kreisel countable functionals can effectively be embedded by some g s into the type structure p t in p thus the recursive functionals correspond to the r e sets in the due types for example f has type iff g f is an r e set in p p p t clearly yields a model for formal type assingnment to terms of calculus i e for any assignment b of types to variables and any t one has b m m b p where b var p according to b we prove that also the reverse implication holds for typable terms thus a completeness theorem for type checking is established over a model defined by an independent recursion theoretical motivation

intersection types for λ trees we introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating terms bhm trees lvy longo trees etc then we prove in an almost uniform way that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms more precisely for each family of trees two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system copyright 2002 elsevier science b v

intersection types for combinatory logic abstract two different translations of the usual formulation of intersection types for calculus into combinatory logic are proposed in the first one the rule is unchanged while in the second one the rule is replaced by three new rules and four axiom schemes which seem to be simpler than rule itself

on complex lee and yang polynomials we introduce a family of polynomialsp n z 1 z n with complex coefficients having the same property as the real lee and yang polynomials the zeros ofp n z z all lie on the unit circle we deduce from this construction a class of inner functions in several complex variables

completeness of type assignment in continuous lambda models abstract the completeness of curry s rules for assigning type schemes to terms of the pure lambda calculus has been proved by hindley 1983 and barendregt et al 1983 using models of syntactic nature a first result of this paper is a completeness proof with respect to the model p as asked by scott 1976 moreover an extension of curry s system in which type schemes can be assigned to the fixed point combinator is introduced together with a notion of type semantics for which it is proved sound and complete answering a question of scott 1980 also in this case completeness is proved with respect to the model p all results also hold for the alternative notions of type semantics proposed by hindley 1983 and scott 1976 1980

curry s type rules are complete with respect to the f semantics too abstract a formal system for assigning type schemes to untyped terms due in essence to h b curry was given in hindley the completeness theorem for typing terms theoret comput sci 22 1983 1 17 this issue the system was proved complete with respect to two different semantics and conjectured to be complete with respect to a third the f semantics this paper proves that conjecture the type schemes have exponentiation as their only connective

bounds for two stage stochastic programs with fixed recourse this paper develops upper and lower bounds on two stage stochastic linear programs using limited moment information the case considered is when both the right hand side as well as the objective coefficients of the second stage problem are random random variables are allowed to have arbitrary multivariate probability distributions with bounded support first upper and lower bounds are obtained using first and cross moments from which we develop bounds using only first moments the bounds are shown to solve the respective general moment problems

multivariate polya schur classification problems in the weyl algebra a multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts we classify all linear partial differential operators in the weyl algebra a n that preserve stability an important tool that we develop in the process is the higher dimensional generalization of polya schur s notion of multiplier sequence we characterize all multivariate multiplier sequences as well as those of finite order next we establish a multivariate extension of the cauchy poincare interlacing theorem and prove a natural analog of the lax conjecture for real stable polynomials in two variables using the latter we describe all operators in a 1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in a n preserves stability if and only if its fischer fock adjoint does these are powerful multivariate extensions of the classical hermite poulain jensen theorem polya s curve theorem and schur malo szegocomposition theorems examples and applications to strict stability preservers are also discussed

lee yang theorems and the complexity of computing averages we study the complexity of computing average quantities related to spin systems such as the mean magnetization and susceptibility in the ferromagnetic ising model and the average dimer count or average size of a matching in the monomer dimer model by establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function we show that these averages are p hard to compute and hence under standard assumptions computationally intractable in the case of the ising model our approach requires us to prove an extension of the famous lee yang theorem from the 1950s

model checking higher order programs we propose a novel verification method for higher order functional programs based on higher order model checking or more precisely model checking of higher order recursion schemes recursion schemes for short the most distinguishing feature of our verification method for higher order programs is that it is sound complete and automatic for the simply typed calculus with recursion and finite base types and for various program verification problems such as reachability flow analysis and resource usage verification we first show that a variety of program verification problems can be reduced to model checking problems for recursion schemes by transforming a program into a recursion scheme that generates a tree representing all the interesting possible event sequences of the program we then develop a new type based model checking algorithm for recursion schemes and implement a prototype recursion scheme model checker to our knowledge this is the first implementation of a recursion scheme model checker experiments show that our model checker is reasonably fast despite the worst case time complexity of recursion scheme model checking being hyperexponential in general altogether the results provide a new promising approach to verification of higher order functional programs

type inference with recursive types syntax and semantics abstract in this paper we study type inference systems for calculus with a recursion operator over types the main syntactical properties notably the existence of principal type schemes are proved to hold when recursive types are viewed as finite notations for infinite regular type expressions representing their infinite unfoldings exploiting the approximation structure of a model for the untyped language of terms types are interpreted as limits of sequences of their approximations we show that the interpretation is essentially unique and that two types have equal interpretation if and only if their infinite unfoldings are identical finally a completeness theorem is proved to hold w r t the specific model we consider for a natural infinitary extension of the type inference system

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

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

characterization theorems for a filter lambda model the filter model is a model of the calculus based on a system of type assignment which extends the basic functionality theory of curry invented in order to give a completeness proof for scott s semantics of curry s type assignment in this paper the local structure of this model i e the syntactical characterization of the equality and inclusion relation is given

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

principal type schemes for an extended type theory we consider the extension of curry s basic functionality theory presented by barendregt et al to appear and we define for any term x a principal type scheme p t s we prove that all and the only type schemes deducible for x can be obtained from the p t s of x by suitable operations

principality and type inference for intersection types using expansion variables principality of typings is the property that for each typable term there is a typing from which all other typings are obtained via some set of operations type inference is the problem of finding a typing for a given term if possible we define an intersection type system which has principal typings and types exactly the strongly normalizable terms more interestingly every finite rank restriction of this system using leivant s first notion of rank has principal typings and also has decidable type inference this is in contrast to system f where the finite rank restriction for every finite rank at 3 and above has neither principal typings nor decidable type inference furthermore the notion of principal typings for our system involves only one operation substitution rather than several operations not all substitution based as in earlier presentations of principality for intersection types without rank restrictions in our system the earlier notion of expansion is integrated in the form of expansion variables which are subject to substitution as are ordinary variables a unification based type inference algorithm is presented using a new form of unification unification

polymorphic type inference and containment abstract type expressions may be used to describe the functional behavior of untyped lambda terms we present a general semantics of polymorphic type expressions over models of untyped lambda calculus and give complete rules for inferring types for terms some simplified typing theories are studied in more detail and containments between types are investigated

lagrange multipliers and optimality lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first order optimality conditions formally as a system of equations modern applications with their emphasis on numerical methods and more complicated side conditions than equations have demanded deeper understanding of the concept and how it fits into a larger theoretical picture a major line of research has been the nonsmooth geometry of one sided tangent and normal vectors to the set of points satisfying the given constraints another has been the game theoretic role of multiplier vectors as solutions to a dual problem interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black and white constraints to be replaced by penalty expressions this paper traces su

lambda calculi for strict parallel functions we introduce two calculi and show that they are expressive for two canonical domains of parallel functions the first calculus is an enrichment of the lazy call by name calculus with call by value abstractions and parallel composition while in the second the usual call by name abstractions are disallowed the corresponding domains are respectively abramsky s domain d d d a lifted function space and d d d a lifted domain of strict functions these domains are lattices and we show that the parallelism is adequately represented by the join operator while call by value abstractions correspond to strict functions the proofs of the results rely on a completeness theorem for the logical presentation of the semantics

outer preserving linear operators abstract a natural question about linear operators on the hilbert hardy space is answered motivated by work in geophysical imaging namely which bounded linear operators on the hardy space preserve the set of all shifted outer functions a complete characterization is determined which allows an explicit construction of all such operators every operator that preserves the set of shifted outer functions is necessarily a product composition operator consisting of composition with a shifted outer function followed by multiplication with a possibly different shifted outer function such operators represent important physical processes including the propagation of seismic wave energy through the earth applications to seismic imaging are briefly discussed

polya schur master theorems for circular domains and their boundaries we characterize all linear operators on finite or infinite dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region omega subset of c for arbitrary

principal types and unification for simple intersection type systems in this paper we introduce a decidable restriction of the intersection type discipline the restriction is based on a uniform principle rather than on a notion of rank for this system we give a type checking algorithm that we prove to be sound and complete an extended language of types with labelled intersections and intersection schemes is needed to describe the principal type of terms which turns out to be unique module a set of equations between labels the unification algorithm far labelled types that extends first order unification could be used as a basis to define inference algorithms for other more powerful systems with intersection

partition function zeros at first order phase transitions pirogov sinai theory this paper is a continuation of our previous analysis 2 of partition functions zeros in models with first order phase transitions and periodic boundary conditions here it is shown that the assumptions under which the results of ref 2 were established are satisfied by a large class of lattice models these models are characterized by two basic properties the existence of only a finite number of ground states and the availability of an appropriate contour representation this setting includes for instance the ising potts and blume capel models at low temperatures the combined results of ref 2 and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class

principal types of bck lambda terms abstract bck terms are the terms in which each variable occurs at most once the principal type of a term is the most general type of the term in this paper we prove that if two bck terms in normal form have the same principal type then they are identical this solves the following problem y komori 1987 in more general form if two closed normal form bck terms are assigned to the same minimal bck formula are they identical a minimal bck formula is the most general formula among bck provable formulas with respect to substitutions for type variables to analyze type assignment the notion of connection is introduced a connection is a series of occurrences of a type in a type assignment figure connected occurrences of a type have the same meaning the occurrences of the type in distinct connection classes can be rewritten separately as a result we have more general type assignment by formulae as types correspondence the result implies the uniqueness of the normal proof figure for principal bck formulas the result is valid for bci logic or implicational fragment of linear logic as well

the heart of intersection type assignment normalisation proofs revisited this paper gives a new proof for the approximation theorem and the characterisation of normalisability using intersection types for a system with w and a relation that is contra variant over arrow types the technique applied is to define reduction on derivations and to show a strong normalisation result for this reduction from this result the characterisation of strong normalisation and the approximation result will follow easily the latter in its turn will lead to the characterisation of head normalisability

intersection types for explicit substitutions we present a new system of intersection types for a composition free calculus of explicit substitutions with a rule for garbage collection and show that it characterizes those terms which are strongly normalizing this system extends previous work on the natural generalization of the classical intersection types system which characterized head normalization and weak normalization but was not complete for strong normalization an important role is played by the notion of available variable in a term which is a generalization of the classical notion of free variable

a category theoretic characterization of functional completeness abstract functional languages are based on the notion of application programs may be applied to data or programs by application one may define algebraic functions and a programming language is functionally complete when any algebraic function f x1 xn is representable i e there is a constant a such that f x1 xn a x1 xn combinatory logic is the simplest type free language which is functionally complete in a sound category theoretic framework the constant a may be considered as an abstract godel number for f when godel numberings are generalized to principal morphisms in suitable categories by this models of combinatory logic are categorically characterized and their relation is given to lambda calculus models within cartesian closed categories finally the partial recursive functionals in any finite higher type are shown to yield models of combinatory logic

normalization approximation and semantics for combinator systems this paper studies normalization of typeable terms and the relation between approximation semantics and filter models for combinator systems it presents notions of approximants for terms intersection type assignment and reduction on type derivations the last will be proved to be strongly normalizable with this result it is proved that every typeable term has an approximant with the same type and a characterization of the normalization behaviour of terms using their assignable types is given then the two semantics are defined and compared and it is shown that the approximants semantics is fully abstract but the filter semantics is not

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

intersection and union types type assignment systems with intersection and union types are introduced although the subject reduction property with respect to s reduction does not hold for a natural deduction like system we manage to overcome this problem in two different ways the first is to adopt a notion of parallel reduction which is a refinement of gross knuth reduction the second is to introduce type theories to refine the system among which is the theory called that induces an assignment system preserving s reduction this type assignment system further clarifies the relation with the intersection discipline through the decomposition first of a disjunctive type into a set of conjunctive types and second of a derivation in the new type assignment system into a set of derivations in the intersection type assignment system for this system we propose three semantics and prove soundness and completeness theorems

infinite l calculus and types recent work on infinitary versions of the lambda calculus has shown that the infinite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus working in the framework of the intersection type disciplines we devise a type assignment system such that two terms are equal in the infinite lambda calculus iff they can be assigned the same types in any basis a novel feature of the system is the presence of a type constant to denote the set of all terms of order zero and the possibility of applying a type to another type we prove a completeness and an approximation theorem for our system our results can be considered as a first step towards the goal of giving a denotational semantics for the lambda calculus which is suited for the study of the unsolvable terms however some noncontinuity phenomena of the infinite lambda calculus make a full realization of this idea namely the construction of a filter model a quite difficult task

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

suffridge s convolution theorem for polynomials and entire functions having only real zeros abstract we present a q extension of polya s and schur s characterization of multiplier sequences and a q extension and a converse of newton s inequalities

a type assignment system for game semantics we present a type assignment system that provides a finitary interpretation of lambda terms in a game semantics model traditionally type assignment systems describe the semantic interpretation of terms in domain theoretic models quite surprisingly the type assignment system presented in this paper is very similar to the traditional ones the main difference being the omission of the subtyping rules

characterization of lee yang polynomials the lee yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle jzj d 1 these polynomials are obtained by taking z1 d d zn d z in certain multiaffine polynomials z1 zn which we call lee yang polynomials they do not vanish when jz1j jznj 1 we characterize the lee yang polynomials in nc1 variables in terms of polynomials in n variables those such that z1 zn 0 when jz1j jznj 1 including jzi j d 1 in a sense to be made precise later our current understanding of lee yang polynomials is based on the concept of asano contraction 1 we shall define an inner radius associated with a multiaffine polynomial and see that it behaves supermultiplicatively with respect to asano contraction proposition 2 using the properties of the inner radius we shall characterize the 2 lync1 nc 1 variables in terms of polynomials in n variables such that z1 zn 0 when jz1j jznj 1 theorem 3 this characterization will give us a good understanding of lync1 proposition 5 and allow us to exhibit elements of lync1 outside of the pair interaction class originally considered by lee and yang see in particular example 7 d the original

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

the completeness theorem for typing λ terms abstract rules for assigning type schemes to untyped terms are given three different semantics are described and the rules are proved complete with respect to two of these semantics the type schemes are built up from type variables not constants by the semantics are defined in arbitrary models of the untyped calculus such models do not come with a type structures as part of their definition the fact that two distinct semantics are completely captured by one set of rules says that the usual type language with as its only connective is not expressive enough to describe the differences between them i conjecture that the rules are also complete with respect to the third semantics

type inference abstract interpretation and strictness analysis abstract filter domains coppo et al 1984 can be seen as abstract domains for the interpretation of functional type free programming languages what is remarkable is the fact that in filter domains the interpretation of a term is given by the set of its types in the intersection type discipline with inclusion thus reducing the computation of an abstract interpretation to typechecking as a main example an abstract filter domain for strictness analysis of type free functional languages is presented the inclusion relation between types representing strictness properties has a complete recursive axiomatization type inference rules cannot be complete strictness being a 0 1 property but a complete extension of the type inference system is presented

multivariate stable polynomials theory and applications univariate polynomials with only real roots while special do occur often enough that their properties can lead to interesting conclusions in diverse areas due mainly to the recent work of two young mathematicians julius borcea and petter branden a very successful multivariate generalization of this method has been developed the first part of this paper surveys some of the main results of this theory of multivariate stable polynomials the most central of these results is the characterization of linear transformations preserving stability of polynomials the second part presents various applications of this theory in complex analysis matrix theory probability and statistical mechanics and combinatorics

semantic types and approximation for featherweight java we consider semantics for the class based object oriented calculus featherweight java without casts based upon approximation we also define an intersection type assignment system for this calculus and show that it satisfies subject reduction and expansion i e types are preserved under reduction and its converse we establish a link between type assignment and the approximation semantics by showing an approximation result which leads to a sufficient condition for the characterisation of head normalisation and termination we show the expressivity of both our calculus and our type system by defining an encoding of combinatory logic into our calculus and showing that this encoding preserves typeability we also show that our system characterises the normalising and strongly normalising terms for this encoding we thus demonstrate that the great analytic capabilities of intersection types can be applied to the context of class based object orientation

subtypes in fuzzy type theory abstract in this paper we introduce extension of the fuzzy type theory ftt by subtypes this demand arises in situations when for example it is useful to distinguish rational and real numbers where the set of rational numbers is a subset of the reals in the original ftt this causes a problem because we cannot assign rational numbers a special type the solution is to introduce special types for both of them and declare that the former is a subtype of the latter the resulting fuzzy type theory is denoted by sftt when constructing the frame for sftt we must realize that though all the considered functions are total if we take them as functions between subsets then they become partial

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

f semantics for type assignment systems abstract yokouchi h f semantics for type assignment systems theoretical computer science 129 1994 39 77 this paper develops f semantics for type systems that assign types to untyped terms curry s system is complete for f semantics but a polymorphic type assignment system is not so we introduce two additional rules fi and fe and show that a polymorphic type assignment system becomes complete for f semantics if we add these two rules furthermore we apply the same method to a polymorphic type assignment system without the equality rule and obtain a similar completeness result we also show that an intersection type assignment system is complete for f models if we add two rules defined on the basis of fi and fe