the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

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

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

hamiltonian systems near relative equilibria abstract we give explicit differential equations for the dynamics of hamiltonian systems near relative equilibria these split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit the form of the differential equations that is inherited from the symplectic structure and symmetry properties of the hamiltonian system is analysed and the effects of time reversing symmetries are included the results will be applicable to the stability and bifurcation theories of relative equilibria of hamiltonian systems

a model for nonlinear axisymmetric droplet vibrations abstract nonlinear inviscid droplet vibrations can be described by luke s variational principle the model equations for axisymmetric motion are derived from this principle with the rayleigh ritz method using four surface and four bulk modes the equations fulfill the energy mass and axial momentum conservation exactly poincare sections are shown for various energies several bifurcations are obtained a singularity of the type of a weak collapse which is associated to the splitting of the drop is also found for higher energies irregular motion associated to the collapse singularity occurs

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

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

on the minimum probability of error of classification with incomplete patterns abstract some expected features of sample patterns in a classification system may be missing immeasurable or obscured when a sample pattern with such an incomplete set of features is presented to a classifier how reliable is the decision thus made by the classifier what is the effect of those absent features to the accuracy and correctness of the classification how to evaluate or estimate the probability of error of such a decision compared to that made on the use of more features or a supposedly complete set of features we derive the solutions of above problems from a point of view of bayes classifier the traditional decision theory is extended to the classifications with incomplete feature sets it shows that in case of classifying incomplete patterns the bayes classifier constructed on the marginal distributions of presented features also minimizes the probability of error we then use bhattacharyya bound to obtain an analytic form for estimating the error probabilities of classifications made on the partially available features with respect to that using more or a complete set of features directions are given to decision making on problems of in which situation the effect of missing or obscured features can be neglected or must be seriously considered experimental results have shown consistence with the theoretical analysis

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

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

using information systems to solve recursive domain equations abstract this paper aims to make the following main contribution to show how to use the concrete nature of scott s information systems to advantage in solving recursive domain equations the method is based on the substructure relation between information systems this essentially makes a complete partial order cpo of information systems standard domain constructions like function space can be made continous on this cpo so the solution of recursive domain equations reduces to the more familiar construction of forming the least fixed point of a continuous function

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

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

automatic pattern recognition a study of the probability of error a test sequence is used to select the best rule from a class of discrimination rules defined in terms of the training sequence the vapnik chervonenkis and related inequalities are used to obtain distribution free bounds on the difference between the probability of error of the selected rule and the probability of error of the best rule in the given class the bounds are used to prove the consistency and asymptotic optimality for several popular classes including linear discriminators nearest neighbor rules kernel based rules histogram rules binary tree classifiers and fourier series classifiers in particular the method can be used to choose the smoothing parameter in kernel based rules to choose k in the k nearest neighbor rule and to choose between parametric and nonparametric rules

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

nonlinear stability for saddle solutions of ideal flows and symmetry breaking in this paper we generalize arnol d s method for nonlinear stability of the two dimensional incompressible euler equation s stationary solutions the method is extended so as to apply for saddle solutions which include many cases of practical interest in particular the case of solutions which do not share the symmetry of the domain the extension is made possible by our technique of supporting functionals and the use of additional flow invariants

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

kernel classification rules from missing data nonparametric kernel classification rules derived from incomplete missing data are studied a number of techniques of handling missing observation in the training set are taken into account in particular the straightforward approach of designing a classifier only from available data deleting missing values is considered the class of imputation techniques is also taken into consideration in the latter case one estimates missing values and then calculates classification rules from such a completed training set consistency and speed of convergence of proposed classification rules are established results of simulation studies are presented

nonlinear stability of a rotating planar liquid drop incompressible inviscid free boundary fluid flows are hamiltonian systems i e the evolution of the flow is determined by a poisson bracket and a hamiltonian function the poisson structure for such flows generalizes a previous structure of zakharov for irrotational free boundary flow and a structure of arnold for flow of an incompressible fluid in a region with fixed boundary the poisson bracket is determined by reduction from canonical variables in the lagrangian material description

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

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

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