infinitary lambda calculus in a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems in this paper we perform the same task for the lambda calculus from the viewpoint of infinitary rewriting the bohm model of the lambda calculus can be seen as an infinitary term model in contrast to term rewriting there are several different possible notions of infinite term which give rise to different bohm like models which embody different notions of lazy or eager computation

adding algebraic rewriting to the untyped lambda calculus abstract we investigate the system obtained by adding an algebraic rewriting system r to an untyped lambda calculus in which terms are formed using the function symbols from r as constants on certain classes of terms called here stable we prove that the resulting calculus is confluent if r is confluent and is terminating if r is terminating the termination result has the corresponding theorems for several typed calculi as corollaries the proof of the confluence result suggests a general method for proving confluence of typed reduction plus rewriting we sketch the application to the polymorphic lambda calculus

some new results on easy lambda terms abstract given two closed terms a and b we consider the question whether the equation a b is consistent with the calculus in general the problem is undecidable however if a is a 0 term we can give good sufficient conditions for the consistency of a b this allows us to prove some counterintuitive results such as 1 there is a closed term x which can be consistently equated to every closed term with the exception of the identity 2 there is a closed term which can be consistently equated to every closed normal form but not to the curry fixed point operator y

modularity of simple termination of term rewriting systems with shared constructors abstract kurihara m and a ohuchi modularity of simple termination of term rewriting systems with shared constructors theoretical computer science 103 1992 273 282 a term rewriting system is simply terminating if there exists a simplification ordering showing its termination let r and r be term rewriting systems which share no defined symbol but may share constructors constructors are function symbols which do not occur at the leftmost position in left hand sides of rewrite rules the rest of the function symbols are defined symbols in this paper we prove that r u r is simply terminating if and only if both r and r are so 1 introduction a property p of term rewriting systems trss 4 6 is modular if r u r has the property p if and only if both r0 and r have that property starting with toyama 141 several authors studied modular aspects of trss under the disjointness condition which requires that

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

modularity in term rewriting revisited we revisit modularity in term rewriting which for the last 25 years has been a very active and fruitful research field starting with the pioneering works of yoshihito toyama on the modularity of confluence and the non modularity of termination he thus initiated an extremely productive line of research with many non trivial and deep results striking counterexamples and a substantial amount of systematic theoretical foundations methodological principles and novel proof techniques in this focused summary we will revisit the modularity analysis for ordinary term rewriting systems considering various confluence and termination properties and restricting ourselves mainly to the case of disjoint unions we will summarize known results on the non modularity of various confluence and termination properties and exhibit crucial ideas constructions and phenomena later on we will also briefly consider various extensions applications related questions and open problems as well as recent developments

limiting partial combinatory algebras from every partial combinatory algebra a we construct another partial combinatory algebra a lim a in a lim a every representable partial numerical function n is exactly of the form limt t n for some representable partial numerical function t n of a the partial combinatory algebra a lim a is a quotient of a by a partial equivalence relation and is equipped with a limit structure in the sense that each element of a lim a is the limit of a countable sequence of a elements in this paper we discuss the limit structures for a in terms of barendregt s range property if the range of a combinator is finite then it is a singleton moreover we repeat the construction a lim transfinite times to interpret infinitary calculi finally we attempt to interpret type free calculus by introducing another partial applicative structure which has an asynchronous application operator that allows a parallel limit operation

innocent game models of untyped λ calculus we present a new denotational model for the untyped l calculus using the techniques of game semantics the strategies used are innocent in the sense of hyland and ong inform and comput to appear and nickau hereditarily sequential functionals a game theoretic approach to sequentiality shaker verlag 1996 dissertation universitat gesamthochschule siegen shaker verlag 1996 but the traditional distinction between question and answer moves is removed we first construct models d and drec as global sections of a reflexive object in the categories a and arec of arenas and innocent and recursive innocent strategies respectively we show that these are sensible le algebras but are neither extensional nor universal we then introduce a new representation of innocent strategies in an economical form we show a strong connexion between the economical form of the denotation of a term in the game models and a variable free form of the nakajima tree of the term using this we show that the definable elements of drec are precisely what we call effectively almost everywhere copycat eac strategies the category aeac with these strategies as morphisms gives rise to a model deac which we show has the same expressive power as d i e the equational theory of deac is the maximal consistent sensible theory h we show that the model deac is sensible order extensional and universal i e every strategy is the denotation of some term to our knowledge this is the first syntax free model of the untyped calculus with the universality property copyright 2002 elsevier science b v

a finite equational axiomatization of the functional algebras for the lambda calculus abstract a lambda theory satisfies an equation between contexts where a context is a term with some holes in it if all the instances of the equation fall within the lambda theory in the main result of this paper it is shown that the equations between contexts valid in every lambda theory have an explicit finite equational axiomatization the variety of algebras determined by the above equational theory is characterized as the class of isomorphic images of functional lambda abstraction algebras these are algebras of functions and naturally arise as the coordinatizations of environment models or lambda models the natural combinatory models of the lambda calculus the main result of this paper is also applied to obtain a completeness theorem for the infinitary lambda calculus recently introduced by berarducci

on termination and confluence properties of disjoint and constructor sharing conditional rewrite systems abstract we investigate the modularity behavior of termination and confluence properties of join conditional term rewriting systems we give counterexamples showing that the properties weak termination weak innermost termination and strong innermost termination are not preserved in general under signature extensions then we develop sufficient conditions for the preservation of these properties under signature extensions and more generally for their modularity in the disjoint union case this leads to new criteria for modularity of termination and completeness generalizing known results for unconditional systems finally combining our analysis with recent related results on the preservation of semi completeness we show how to cover the non disjoint case of combined conditional rewrite systems with shared constructors too

highlights in infinitary rewriting and lambda calculus we present some highlights from the emerging theory of infinitary rewriting both for first order term rewriting systems and l calculus in the first section we introduce the framework of infinitary rewriting for first order rewrite systems so without bound variables we present a recent observation concerning the continuity of infinitary rewriting in the second section we present an excursion to the infinitary l calculus after the main definitions we mention a recent observation about infinite looping l terms that is terms that reduce in one step to themselves next we describe the fundamental trichotomy in the semantics of l calculus bohm trees levy longo trees and berarducci trees we conclude with a short description of a new refinement of bohm tree semantics called clocked semantics

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

roast rapid orthogonal approximate slepian transform in this paper we provide a rapid orthogonal approximate slepian transform roast for the discrete vector that one obtains when collecting a finite set of uniform samples from a baseband analog signal the roast offers an orthogonal projection which is an approximation to the orthogonal projection onto the leading discrete prolate spheroidal sequence dpss vectors also known as slepian basis vectors as such the roast is guaranteed to accurately and compactly represent not only oversampled bandlimited signals but also the leading dpss vectors themselves moreover the subspace angle between the roast subspace and the corresponding dpss subspace can be made arbitrarily small the complexity of computing the representation of a signal using the roast is comparable to the fast fourier tranform which is much less than the complexity of using the dpss basis vectors we also give non asymptotic results to guarantee that the proposed basis not only provides a very high degree of approximation accuracy in a mean squared error sense for bandlimited sample vectors but also that it can provide high quality approximations of all sampled sinusoids within the band of interest

applications of infinitary lambda calculus we present an introduction to infinitary lambda calculus highlighting its main properties subsequently we give three applications of infinitary lambda calculus the first addresses the non definability of surjective pairing which was shown by the first author not to be definable in lambda calculus we show how this result follows easily as an application of berry s sequentiality theorem which itself can be proved in the setting of infinitary lambda calculus the second pertains to the notion of relative recursiveness of number theoretic functions the third application concerns an explanation of counterexamples to confluence of lambda calculus extended with non left linear reduction rules adding non left linear reduction rules such as dxx x or the reduction rules for surjective pairing to the lambda calculus yields non confluence as proved by the second author we discuss how an extension to the infinitary lambda calculus where bohm trees can be directly manipulated as infinite terms yields a more simple and intuitive explanation of the correctness of these church rosser counterexamples

an improved surface radiation condition for high frequency acoustic scattering problems a new artificial boundary condition for two dimensional acoustic scattering problems is constructed in this paper it is designed from the principal symbol of the dirichlet to neumann map so that both the propagating and evanescent fields are properly modeled in the high frequency regime the new artificial boundary condition can be used as an on surface radiation condition for efficient approximation of the wave field the accuracy and efficiency of our method are demonstrated in numerical examples

three dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries we report on a generalization of the bayliss gunzburger turkel non reflecting boundary conditions to arbitrarily shaped convex artificial boundaries for elongated scatterers such as submarines we show that this generalization can improve significantly the computational efficiency of finite element methods applied to the solution of three dimensional acoustic scattering problems copyright 2001 john wiley sons ltd

on the modularity of termination of term rewriting systems it is well known that termination is not a modular property of term rewriting systems i e it is not preserved under disjoint union the objective of this paper is to provide a uniform framework for sufficient conditions which ensure the modularity of termination we will prove the following result whenever the disjoint union of two terminating term rewriting systems is nonterminating then one of the systems is not ce terminating i e it loses its termination property when extended with the rules cons x y x and cons x y y and the other is collapsing this result has already been achieved by gramlich 7 for finitely branching term rewriting systems a more sophisticated approach is necessary however to prove it in full generality most of the known sufficient criteria for the preservation of termination 24 15 13 7 follow as corollaries from our result and new criteria are derived this paper particularly settles the open question whether simple termination is modular in general we will moreover shed some light on modular properties of combined systems with shared constructors for instance it will be shown that the above result does not carry over to constructor sharing systems

theory and computation of spheroidal wavefunctions in this paper we report on a package written in the mathematica computer algebra system which has been developed to compute the spheroidal wavefunctions of meixner and schafke 1954 mathieusche funktionen und spharoidfunktionen and is available online physics uwa edu au falloon spheroidal spheroidal html this package represents a substantial contribution to the existing software since it computes the spheroidal wavefunctions to arbitrary precision for general complex parameters and argument z existing software can only handle integer and does not give arbitrary precision the package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter the spheroidal wavefunctions of flammer 1957 spheroidal wave functions are included as a special case of meixner s more general functions this paper presents a concise review of the general theory of spheroidal wavefunctions and a description of the formulae and algorithms used in their computation and gives high precision numerical examples

unique normal forms for lambda calculus with surjective pairing we consider the equational theory of calculus extended with constants 0 1 and axioms for surjective pairing 0 xy x 1 xy y 0x 1x x two reduction systems yielding the equality of are introduced the first is not confluent and for the second confluence is an open problem it is shown however that in both systems each term possessing a normal form has a unique normal form some additional properties and problems in the syntactical analysis of and the corresponding reduction systems are discussed

bohm theorem and bohm trees for the λμ calculus parigot s email protected parigot 1992 1 is now a standard reference about the computational content of classical logic as well as for the formal study of control operators in functional languages in addition to the fine grained curry howard correspondence between minimal classical deductions and simply typed email protected and to the ability to encode many usual control operators such as call cc in the email protected in its historical call by name presentation or in call by value versions the success of email protected m calculus comes from its simplicity its good meta theoretical properties both as a typed and an untyped calculus confluence strong normalization etc as well as the fact that it naturally extends church s l calculus though in 2001 david and py proved 2 that bohm s theorem which is a fundamental result of the untyped l calculus cannot be lifted to parigot s calculus in the present article we exhibit a natural extension to parigot s calculus email protected m calculus in which bohm s property also known as separation property can be stated and proved this is made possible by a careful and detailed analysis of david and py s proof of non separability and of the characteristics of the email protected which break the property we identify that the crucial point lies in the design of parigot s email protected with a two level syntax in addition we establish a standardization theorem for the extended calculus deduce a characterization of solvability describe email protected trees and connect the calculus with stream computing and delimited control

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

infinitary lambda calculus and discrimination of berarducci trees we propose an extension of lambda calculus for which the berarducci trees equality coincides with observational equivalence when we observe rootstable or rootactive behavior of terms in one direction the proof is an adaptation of the classical bohm out technique in the other direction the proof is based on confluence for strongly converging reductions in this extension

confluence of the coinductive λ calculus the coinductive calculus aco arises by a coinductive interpretation of the grammar of the standard calculus a and contains non well founded terms an appropriate notion of reduction is analyzed and proven to be confluent by means of a detailed analysis of the usual tait martin lof style development argument this yields bounds for the lengths of those joining reduction sequences that are guaranteed to exist by confluence these bounds also apply to the well founded calculus thus adding quantitative information to the classic result

infinitary combinatory reduction systems we define infinitary combinatory reduction systems icrss thus providing the first notion of infinitary higher order rewriting the systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary l calculus are special cases furthermore we generalise a number of known results from first order infinitary rewriting and infinitary l calculus to icrss in particular for fully extended left linear icrss we prove the well known compression property and for orthogonal icrss we prove that 1 if a set of redexes u has a complete development then all complete developments of u end in the same term and that 2 any tiling diagram involving strongly convergent reductions s and t can be completed iff at least one of s t and t s is strongly convergent we also prove an ancillary result of independent interest a set of redexes in an orthogonal icrs has a complete development iff the set has the so called finite jumps property

modularity in noncopying term rewriting it is known that termination and completeness are not modular properties of term rewriting systems the disjoint union of terminating complete term rewriting systems need not be terminating complete in this paper we introduce a class of noncopying term rewriting systems as a new term based formalism for a kind of graph rewriting systems and prove that this class enjoys the modularity of termination and of completeness actually our results are stronger in the sense that the disjointness condition is relaxed for allowing the systems to share constructors and some defined symbols while preserving the main results

on modularity in infinitary term rewriting we study modular properties in strongly convergent infinitary term rewriting in particular we show that confluence is not preserved across direct sum of a finite number of systems even when these are noncollapsing confluence modulo equality of hypercollapsing subterms is not preserved across direct sum of a finite number of systems normalization is not preserved across direct sum of an infinite number of left linear systems unique normalization with respect to reduction is not preserved across direct sum of a finite number of left linear systems together these facts constitute a radical departure from the situation in finitary term rewriting positive results are confluence is preserved under the direct sum of an infinite number of left linear systems iff at most one system contains a collapsing rule confluence is preserved under the direct sum of a finite number of non collapsing systems if only terms of finite rank are considered top termination is preserved under the direct sum of a finite number of left linear systems normalization is preserved under the direct sum of a finite number of left linear systems all of the negative results above hold in the swtting of weakly convergent rewriting as well as do the positive results concerning modularity of top termination and normalization for left lineat systmes

analytical and numerical investigation of the performance of the bgt2 condition for low frequency acoustic scattering problems a mathematical and numerical analysis is performed to assess the performance of the second order bayliss gunzburger turkel bgt2 condition when applied to solving low frequency acoustic scattering problems in the case of elongated scatterers this investigation suggests that bgt2 retains an acceptable level of accuracy for relatively low wavenumber a damping effect is incorporated to the bgt2 condition in order to extend the range of satisfactory performance this damping procedure consists in adding only a constant imaginary part to the wavenumber the numerical results indicate that the modified version of bgt2 extends the range of satisfactory performance by improving the level of accuracy by up to two orders of magnitude guidelines on the appropriate choice of the damping coefficient are provided

on the church rosser property for the direct sum of term rewriting systems the direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols it is shown that if two term rewriting systems both have the chruch rosser property then the direct sum of these systems also has this property

coinductive big step operational semantics using a call by value functional language as an example this article illustrates the use of coinductive definitions and proofs in big step operational semantics enabling it to describe diverging evaluations in addition to terminating evaluations we formalize the connections between the coinductive big step semantics and the standard small step semantics proving that both semantics are equivalent we then study the use of coinductive big step semantics in proofs of type soundness and proofs of semantic preservation for compilers a methodological originality of this paper is that all results have been proved using the coq proof assistant we explain the proof theoretic presentation of coinductive definitions and proofs offered by coq and show that it facilitates the discovery and the presentation of the results

on the algebraic models of lambda calculus the variety equational class of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way boolean algebras algebraize the classical propositional calculus the equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a first order algebraic description of lambda calculus which allows to keep the lambda notation and hence all the functional intuitions in this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety for example the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras this result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus to the infinitary lambda calculus another result of the paper is an algebraic proof of consistency of the infinitary lambda calculus finally some algebraic constructions by krivine are generalized to lambda abstraction algebras

transfinite reductions in orthogonal term rewriting systems strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems otrss for these we prove the transfinite parallel moves lemma and the compressing lemma strongness is necessary as shown by counterexamples normal forms which we allow to be infinite are unique in contrast to normal forms strongly converging fair reductions result in normal forms

adapting innocent game models for the bohm tree λ theory we present a game model of the untyped calculus with equational theory equal to the bohm tree theory b which is universal i e every element of the model is definable by some term this answers a question of di gianantonio franco and honsell we build on our earlier work which uses the methods of innocent game semantics to develop a universal model inducing the maximal consistent sensible theory h to our knowledge these are the first syntax independent universal models of the untyped calculus

from diagrammatic confluence to modularity this paper builds on a fundamental notion of rewriting theory that characterizes confluence of a binary rewriting relation klop s cofinal derivations cofinal derivations were used by van oostrom to obtain another characterization of confluence of a rewriting relation via the existence of decreasing diagrams for all local peaks in this paper we show that cofinal derivations can be used to give a new concise proof of toyama s celebrated modularity theorem and its recent extensions to rewriting modulo in the case of strongly coherent systems an assumption discussed in depth here this is done by generalizing cofinal derivations to cofinal streams allowing us in turn to generalize van oostrom s result to the modulo case

bubbles in modularity we provide a global technique called neatening for the study of modularity of left linear term rewriting systems objects called bubbles are identified as the responsibles of most of the problems occurring in modularity and the concept of well behaved from the modularity point of view reduction called neat reduction is introduced neatening consists of two steps the first is proving a property is modular when only neat reductions are considered the second is to neaten a generic reduction so to obtain a neat one thus showing that restricting to neat reductions is not limitative this general technique is used to provide a unique uniform method able to elegantly prove all the existing results on the modularity of every basic property of left linear term rewriting systems and also to provide new results on the modularity of termination

on interreduction of semi complete term rewriting systems for a complete i e confluent and terminating term rewriting system trs it is well known that simplification also called interreduction into an equivalent canonical i e complete and interreduced trs is easily possible this can be achieved by first normalizing all right hand sides of the trs and then deleting all rules with a reducible left hand side here we investigate the logical and operational preservation properties of the same simplification operations for semi complete i e confluent and weakly terminating trss surprisingly it turns out that for semi complete trss these simplifications are neither operationally harmless nor logically correct semi completeness may get lost and the induced equational theory need not be preserved we also provide sufficient criteria for the preservation of semi completeness and of the induced equational theory in particular we show that orthogonal trss enjoy all these preservation properties in the general case for a given semi complete trs an equivalent semi canonical i e semi complete and interreduced system need not even exist

perturbation analysis of an svd based linear prediction method for estimating the frequencies of multiple sinusoids a linear prediction approach is studied for estimating the frequencies of sinusoids in white noise it is shown that in the first step the continuity of the generalized inverse and the concept of angle between subspaces play an important role the continuity concept helps explain the need for a low rank approximation and the quality of the approximation is appraised by using the notion of angle between subspaces for the second step the sensitivity of the zeros of the predictor polynomial becomes an important consideration and is examined it is shown that increasing the order of the predictor polynomial and computing the minimum norm solution provides a mechanism to reduce parameter sensitivity

relative undecidability in term rewriting ii the confluence hierarchy for a hierarchy of properties of term rewriting systems related to confluence we prove relative undecidability i e for implications x y in the hierarchy the property x is undecidable for term rewriting systems satisfying y for some of the implications either x or x is semi decidable for others neither x nor x is semi decidable we prove most of these results for linear term rewrite systems

fast nearly ml estimation of the parameters of real or complex single tones or resolved multiple tones this paper presents new computationally efficient algorithms for estimating the parameters frequency amplitude and phase of one or more real tones sinusoids or complex tones cisoids in noise from a block of n uniformly spaced samples the first algorithm is an interpolator that uses the peak sample in the discrete fourier spectrum dfs of the data and its two neighbors we derive cramer rao bounds crbs for such interpolators and show that they are very close to the crb s for the maximum likelihood ml estimator the new algorithm almost reaches these bounds a second algorithm uses the five dfs samples centered on the peak to produce estimates even closer to ml enhancements are presented that maintain nearly ml performance for small values of n for multiple complex tones with frequency separations of at least 4 spl pi n rad sample unbiased estimates are obtained by incorporating the new single tone estimators into an iterative cyclic descent algorithm which is a computationally cheap nonlinear optimization single or multiple real tones are handled in the same way the new algorithms are immune to nonzero mean signals and provided n is large remain near optimal in colored and non gaussian noise

polymorphic rewriting conserves algebraic confluence we study combinations of many sorted algebraic term rewriting systems and polymorphic lambda term rewriting algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus as higher order constants we show that if a many sorted algebraic rewrite system r has the church rosser property is confluent then r s type s type rewriting of mixed terms has the church rosser property too reduction does not commute with algebraic reduction in general however using long normal forms we show that if r is canonical confluent and strongly normalizing then equational provability from r s type s type is still decidable

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

substitution in non wellfounded syntax with variable binding inspired from the recent developments in theories of non wellfounded syntax coinductively defined languages and of syntax with binding operators the structure of algebras of wellfounded and non wellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution this is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category the main technical tool is a novel concept of heterogeneous substitution systems

bayliss turkel like radiation conditions on surfaces of arbitrary shape this paper addresses the extension of the bayliss turkel second order radiation condition to an arbitrarily shaped surface the derivation is based mainly on the pseudo differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition the radiation condition then ranges among the most accurate of those of order two as a by product of the derivation almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared

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

abstract data type systems abstract this paper is concerned with the foundations of an extension of pure type systems by abstract data types hence the name of abstract data type systems adts generalize inductive types as they are defined in the calculus of constructions by providing definitions of functions by pattern matching on the one hand and relations among constructors of the inductive type on the other it also generalizes the first order framework of abstract data types by providing function types and higher order equations the first half of the paper describes the framework of adts while the second half investigates cases where adts are strongly normalizing this is shown to be the case for the polymorphic lambda calculus with possibly subtypes enriched by higher order algebraic rules obeying a strong generalization of primitive recursion of higher type that we call the general schema this covers in particular the case of inductive types whose constructors do not have functional arguments we conjecture that this result holds true for all calculi of the so called barendregt s cube on the other hand the definition of a schema for the higher order rules allowing for more general inductive types is left open

termination for direct sums of left linear complete term rewriting systems a term rewriting system is called complete if it is confluent and terminating we prove that completeness of trss is a modular property meaning that it stays preserved under direct sums provided the constituent trss are left linear here the direct sum r 0 r is the union of trss r 0 r 1 with disjoint signature the proof hinges crucially upon the non deterministic collapsing behavior of terms from the sum trs

modular termination of r consistent and left linear term rewriting systems a modular property of term rewriting systems is one that holds for the direct sum of two disjoint term rewriting systems iff it holds for every involved term rewriting system a term rewriting system is r consistent iff there is no term that can be rewritten to two different variables we show that the subclass of left linear and r consistent term rewriting systems has the modular termination property this subclass may also contain nonconfluent term rewriting systems since confluence implies r consistency this constitutes a generalisation of the theorem of toyama klop and barendregt on the modularity of termination for confluent and left linear term rewriting systems

on the confluence of lambda calculus with conditional rewriting the confluence of untyped l calculus with unconditional rewriting is now well un derstood in this paper we investigate the confluence of l calculus with conditional rewriting and provide general results in two directions first when conditional rules are algebraic this extends results of muller and dougherty for unconditional rewriting two cases are considered whether b reduction is allowed or not in the evaluation of conditions moreover dougherty s result is improved from the assumption of strongly normalizing b reduction to weakly normalizing b reduction we also provide examples showing that outside these conditions modularity of confluence is difficult to achieve second we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules

productivity of stream definitions we give an algorithm for deciding productivity of a large and natural class of recursive stream definitions a stream definition is called productive if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit whereas productivity is undecidable for stream definitions in general we show that it can be decided for pure stream definitions for every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements called pebbles in a finite pebbleflow net work and the production of a pebbleflow net associated with a pure stream definition that is the amount of pebbles the net is able to produce at its output port can be calculated by reducing nets to trivial nets

projected nonlinear least squares for exponential fitting the modern ability to collect vast quantities of data provides a challenge for parameter estimation when posed as a nonlinear least squares problem fitting a model to data the cost of each iterat