quantitative classical realizability introduced by dal lago and hofmann quantitative realizability is a technique used to define models for logics based on multiplicative linear logic a particularity is that functions are interpreted as bounded time computable functions it has been used to give new and uniform proofs of soundness of several type systems with respect to certain time complexity classes we propose a reformulation of their ideas in the setting of krivine s classical realizability the framework obtained generalizes dal lago and hofmann s realizability and reveals deep connections between quantitative realizability and a linear variant of cohen s forcing

completeness ofl1 spaces for measures with values in complex vector spaces we provide a simple and direct proof of the completeness of the space of c valued integrable functions with respect to a vector measure taking its values in a complex frechet space a similar result is established via a quite different argument for closed measures taking their values in more general sequentially complete locally convex hausdorff spaces over c

the role of polymorphism in the characterisation of complexity by soft types soft type assignment systems sta sta and sta b characterise by means of reduction of terms computation in complexity classes ptime np and pspace respectively all these systems are inspired by linear logic and include polymorphism similar to the one of system f we show that the presence of polymorphism gives the undecidability of typechecking and type inference we also show that reductions in decidable monomorphic versions of these systems also capture the same complexity classes in a way sufficient for the traditional complexity theory the translations we propose show in addition that the monomorphic systems to serve as a programming language require some metalanguage support since the program which operates on data has form and type which depend on the size of the input

an arithmetic for polynomial time computation we define a restriction lha of heyting arithmetic ha with the property that all extracted programs are feasible the restrictions consist in linearity and ramification requirements

the undecidability of second order multiplicative linear logic the multiplicative fragment of second order propositional linear logic is shown to be undecidable

algebra structure for lp of a vector measure abstract in this paper we present new results about the space l p for being a vector measure defined in the borel algebra of a compact abelian group g and satisfying certain property concerning translation of simple functions namely we show that l p is a translation invariant space which can be endowed with an algebra structure via usual convolution product we apply these results to the optimal domain of the fourier transform and convolution operators

linear logic and elementary time a subsystem of linear logic elementary linear logic is defined and shown to represent exactly elementary recursive functions its choicest part consists in reducing the deductive power of the exponential also known as the bang which in linear logic is in charge of controlling duplication in the cut elimination process

a new approach on curves of constant precession in this paper we investigate a curve whose spherical images the tangent indicatrix and binormal indicatrix are spherical slant helices and called it as a c slant helix this notation is a generalization of a slant helix and special case of k slant helix defined by ali 2012 we obtain the axis of the curve and theorem 3 5 via the alternative moving frame and the sabban frame respectively furthermore we give some new characterizations of the c slant helix and prove that a curve of c constant precession is a c slant helix

phase semantics for light linear logic light linear logic girard inform comput 14 1998 175 204 is a refinement of the propositions as types paradigm to polynomial time computation a semantic setting for the underlying logical system is introduced here in terms of fibred phase spaces strong completeness is established with a purely semantic proof of cut elimination as a consequence a number of mathematical examples of fibred phase spaces are presented that illustrate subtleties of light linear logic

phase semantic cut elimination and normalization proofs of first and higher order linear logic abstract we give a natural extension of girard s phase semantic completeness proof of the first order linear logic girard theoret comput sci 1987 to a phase semantic cut elimination proof then we extend this idea to a phase semantic cut elimination proof for higher order linear logic we also extend the phase semantics for provability to a phase semantics like framework for proofs by modifying the phase space of monoid domain to that of proof structures untyped proofs domain in a natural way the resulting phase semantic like framework for proofs provides various versions of proof normalization theorem

on k type pseudo null darboux helices in minkowski 3 space abstract in this paper we introduce k type pseudo null darboux helices in minkowski 3 space for k 1 2 3 we obtain the relationship between 1 type and 2 type pseudo null darboux helices and show that all 3 type pseudo null darboux helices are also 1 type and 2 type helices but the converse does not hold finally we give some examples

bounded linear logic a modular approach to polynomial time computability typing is a way of describing the interactive behavior of algorithms usual typing systems are mainly concerned with input output specifications e g given terms f a b and a a the computation of f a by normalization yields a result of type b however one can dream of more refined typings that would not only ensure ethereal termination but would for instance yield feasible resource bounds it seems that time complexity does not lend itself naturally to modular manipulation we seek something more primitive

a new version of bishop frame and an application to spherical images abstract in this work we introduce a new version of bishop frame using a common vector field as binormal vector field of a regular curve and call this frame as type 2 bishop frame thereafter by translating type 2 bishop frame vectors to the center of unit sphere of three dimensional euclidean space we introduce new spherical images and call them as type 2 bishop spherical images frenet serret apparatus of these new spherical images are obtained in terms of base curve s type 2 bishop invariants additionally we express some interesting relations and illustrate two examples of our main results

unit speed stationary points of the acceleration we consider the variational problem associated to the l2 norm of the angular acceleration for variations with constant length this is a classical problem related to recent models for the spinning relativistic particles both massive and massless and to bernoulli s elastica problem its interest arises also in computer graphics and in trajectory planning problems for rigid body motion explicit integration of the euler lagrange equations is considerably difficult in general we show how one can use a geometric integration procedure to achieve this goal for unit speed critical points in real space forms from the euler lagrange equations we obtain the geometric integration critical curves are a family of ordinary helices in a real space form of dimension 3 at most moreover ordinary helices in real space forms can be characterized in turn as geodesics in certain constant mean curvature flat cylinders this will allow us to obtain the analytical integration of the euler lagrange equations we also d

non null lie quadratics in e3 interpolation problems in the space so 3 of rotations of euclidean 3 space e3 are reviewed in secs i s2 as background and motivation to a study of curves in e3 called lie quadratics except for a special class called null lie quadratics have resisted analysis until now the rest of the present paper is devoted to new results showing non null lie quadratics have rich analytical geometrical and asymptotic structures rates of growth are studied using differential equations and inequalities lie quadratics are proved to be extendible over the whole of r and existence of axes is proved under fairly general conditions examples show sharpness of many results

on the p calculus and linear logic abstract we detail abramsky s proofs as processes paradigm for interpreting classical linear logic cll girard 1987 into a synchronous version of the calculus recently proposed by milner 1992 1993 the translation is given at the abstract level of proof structures we give a detailed treatment of information flow in proof nets and show how to mirror various evaluation strategies for proof normalization we also give soundness and completeness results for the process calculus translations of various fragments of cll the paper also gives a self contained introduction to some of the deeper proof theory of cll and its process interpretation

new curry howard terms for full linear logic in this paper we 1 provide a natural deduction system for full first order linear logic 2 introduce curry howard style terms for this version of linear logic 3 prove strong normalization for the system and 4 prove that given a proof of v x3y a y and any individual term t we can compute a term u such that a t u is provable

linear types and non size increasing polynomial time computation we propose a linear type system with recursion operators for inductive datatypes which ensures that all definable functions are polynomial time computable the system improves upon previous such systems in that recursive definitions can be arbitrarily nested in particular no predicativity or modality restrictions are made

an arithmetic for non size increasing polynomial time computation an arithmetical system is presented with the property that from every proof a realizing term can be extracted that is definable in a certain affine linear typed variant of godel s t and therefore defines a non size increasing polynomial time computable function

light types for polynomial time computation in lambda calculus we present a polymorphic type system for lambda calculus ensuring that well typed programs can be executed in polynomial time dual light affine logic dlal dlal has a simple type language with a linear and an intuitionistic type arrow and one modality it corresponds to a fragment of light affine logic lal we show that contrarily to lal dlal ensures good properties on lambda terms and not only on proof nets subject reduction is satisfied and a well typed term admits a polynomial bound on the length of any of its beta reduction sequences we also give a translation of lal into dlal and deduce from it that all polynomial time functions can be represented in dlal

a semantic measure of the execution time in linear logic we give a semantic account of the execution time i e the number of cut elimination steps leading to the normal form of an untyped mell net we first prove that 1 a net is head normalizable i e normalizable at depth 0 if and only if its interpretation in the multiset based relational semantics is not empty and 2 a net is normalizable if and only if its exhaustive interpretation a suitable restriction of its interpretation is not empty we then give a semantic measure of execution time we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link from the interpretations of the two cut free nets these results are inspired by similar ones obtained by the first author for the untyped lambda calculus

rearrangement invariant spaces of functions on lca groups in this paper the rearrangement invariant hulls and kernels of the classes f f lp and f f l1loc on a noncompact nondiscrete locally compact abelian group are characterized the spacel1loc s of tempered distributions serves as the extended domain of the fourier transform instead of the classical domainl1 l2 we also introduce and study a new function spacep0which arises as the rearrangement invariant hull of some of the classes under consideration

contributions to differential geometry of isotropic curves in the complex space abstract this work deals with classical differential geometry of isotropic curves in the complex space c 4 first we study spherical isotropic curves and pseudo helices besides in this section we introduce some special isotropic helices type 1 type 2 and type 3 isotropic slant helices and express some characterizations of them in terms of e cartan equations thereafter we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order finally we investigate position vector of an arbitrary curve with respect to e cartan frame by a system of complex differential equations whose solution gives components of the position vector solutions of the mentioned system and vector differential equation have not yet been found therefore in terms of special cases we present some special characterizations

cubic polynomials on lie groups reduction of the hamiltonian system this paper analyzes the optimal control problem of cubic polynomials on compact lie groups from a hamiltonian point of view and its symmetries the dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the lie group and its lie algebra using these control geometric tools the relation between the hamiltonian approach developed here and the known variational one is analyzed after making explicit the left trivialized system we use the technique of marsden weinstein reduction to remove the symmetries of the hamiltonian system in view of the reduced dynamics we are able to guarantee by means of the lie cartan theorem the existence of a considerable number of independent integrals of motion in involution

characterizing polynomial and exponential complexity classes in elementary lambda calculus abstract in this paper an implicit characterization of the complexity classes k exp and k fexp for k 0 is given by a type assignment system for a stratified calculus where types for programs are witnesses of the corresponding complexity class types are formulae of elementary linear logic ell and the hierarchy of complexity classes k exp is characterized by a hierarchy of types

paths based criteria and application to linear logic subsystems characterizing polynomial time abstract several variants of linear logic have been proposed to characterize complexity classes in the proofs as programs correspondence light linear logic lll ensures a polynomial bound on reduction time and characterizes in this way polynomial time p t i m e in this paper we study the complexity of linear logic proof nets and propose three semantic criteria based on context semantics stratification dependence control and nesting stratification alone entails an elementary time bound the three criteria entail together a polynomial time bound these criteria can be used to prove the complexity soundness of several existing variants of linear logic we define a decidable syntactic subsystem of linear logic s d n l l we prove that the proof nets of sdnll satisfy the three criteria which implies that s d n l l is sound for p t i m e several previous subsystems of linear logic characterizing polynomial time lll m l 4 maximal system of ms are embedded in s d n l l proving its p t i m e completeness

parabolic curves in lie groups to interpolate a sequence of points in euclidean space parabolic splines can be used these are curves which are piecewise quadratic to interpolate between points in a semi riemannian manifold we could look for curves such that the second covariant derivative of the velocity is zero we call such curves jupp and kent quadratics or jk quadratics because they are a special case of the cubic curves advocated by jupp and kent when the manifold is a lie group with bi invariant metric we can relate jk quadratics to null lie quadratics which arise from another interpolation problem we solve jk quadratics in the lie groups so 3 and so 1 2 and in the sphere and hyperbolic plane by relating them to the differential equation for a quantum harmonic oscillator

fixpoint logics relational machines and computational complexity we establish a general connection between fixpoint logic and complexity on one side we have fixpoint logic parameterized by the choices of 1st order operators inflationary or noninflationary and iteration constructs deterministic nondeterministic or alternating on the other side we have the complexity classes between p and exptime our parameterized fixpoint logics capture the complexity classes p np pspace and exptime but equally is achieved only over ordered structures there is however an inherent mismatch between complexity and logic while computational devices work on encodings of problems logic is applied directly to the underlying mathematical structures to overcome this mismatch we use a theory of relational complexity which bridges the gap between standard complexity and fixpoint logic on one hand we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complexity classes on the other hand the expressive power of fixpoint logic can be precisely characterized in terms of relational complexity classes this tight three way relationship among fixpoint logics relational complexity and standard complexity yields in a uniform way logical analogs to all containments among the complexity classes p np pspace and exptime the logical formulation shows that some of the most tantalizing questions in complexity theory boil down to a single question the relative power of inflationary vs noninflationary 1st order operators

characterizing complexity classes by higher type primitive recursive definitions abstract higher type primitive recursive definitions also known as godel s system t defining first order functions i e functions of type ind x x ind ind ind for individuals note that the higher types are used as detour to define first order functions can be classificed into an infinite syntactic hierarchy a definition belongs to the n th stage of this hirarchy is of rank n iff n is an upper bound on the levels of the types occurring in it we interpret these definitions over finite structures and show rank 1 definitions characterize logspace in the sense of gurevich 1983 in fact this result is from his paper rank 2 definitions characterize ptime rank 3 pspace and rank 4 exptime dtime 2 poly

solid extensions of the cesaro operator on the hardy space h2 d abstract we introduce and study the largest banach space of analytic functions on the unit disc which is solid for the coefficient wise order and to which the classical cesaro operator c h 2 h 2 can be continuously extended while still maintaining its values in h 2 properties of this banach space h c e s 2 as well as a characterization of individual analytic functions which belong to h c e s 2 are presented in addition both the multiplier space of h c e s 2 and the spectrum of c h c e s 2 h c e s 2 are determined

rearrangements of functions on a locally compact abelian group and integrability of the fourier transform abstract we find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact abelian group these classes consist of all functions for which the fourier transform belongs to a given lorentz space on the dual group different special cases of the problems considered in this paper have been originally studied by hardy littlewood hewitt ross cereteli and the author

associated curves of a frenet curve and their applications abstract we introduce the notion of the principal binormal direction curve and principal binormal donor curve of a frenet curve in e 3 and give the relationship of curvature and torsion of its mates as application we characterize general helices and slant helices in terms of their associated curves and give a canonical method to construct them also we give a new characterization of bertrand curve

salem sets in local fields the fourier restriction phenomenon and the hausdorff young inequality abstract we establish the existence of salem sets in the ring of integers of any local field and study the fourier restriction phenomenon on such sets optimal extension of the hausdorff young inequality initially attained for the torus by g mockenhaupt and w ricker is also established in the local field setting

invariant higher order variational problems we investigate higher order geometric k splines for template matching on lie groups this is motivated by the need to apply diffeomorphic template matching to a series of images e g in longitudinal studies of computational anatomy our approach formulates euler poincar theory in higher order tangent spaces on lie groups in particular we develop the euler poincar formalism for higher order variational problems that are invariant under lie group transformations the theory is then applied to higher order template matching and the corresponding curves on the lie group of transformations are shown to satisfy higher order euler poincar equations the example of so 3 for template matching on the sphere is presented explicitly various cotangent bundle momentum maps emerge naturally that help organize the formulas we also present hamiltonian and hamilton ostrogradsky lie poisson formulations of the higher order euler poincar theory for applications on the hamiltonian side

null w slant helices in e13 abstract in this paper we give the necessary and sufficient conditions for null curves in e 1 3 to be w slant helix in terms of their curvature functions mainly throughout this paper relationships between the null w slant helices and their pseudo spherical images are obtained furthermore some illustrative examples of the null w slant helices and their pseudo spherical indicatrices in e 1 3 are demonstrated

optimal rearrangement invariant range for hardy type operators we characterize in the context of rearrangement invariant spaces the optimal range space for a class of monotone operators related to the hardy operator the connection between optimal range and optimal domain for these operators is carefully analyzed

linear logic by levels and bounded time complexity we give a new characterization of elementary and deterministic polynomial time computation in linear logic through the proofs as programs correspondence girard s seminal results concerning elementary and light linear logic achieve this characterization by enforcing a stratification principle on proofs using the notion of depth in proof nets here we propose a more general form of stratification based on inducing levels in proof nets by means of indices which allows us to extend girard s systems while keeping the same complexity properties in particular it turns out that girard s systems can be recovered by forcing depth and level to coincide a consequence of the higher flexibility of levels with respect to depth is the absence of boxes for handling the paragraph modality we use this fact to propose a variant of our polytime system in which the paragraph modality is only allowed on atoms and which may thus serve as a basis for developing lambda calculus type assignment systems with more efficient typing algorithms than existing ones

slant helices in three dimensional lie groups in this paper we define slant helices in three dimensional lie groups with a bi invariant metric and obtain a characterization of slant helices moreover we give some relations between slant helices and their involutes spherical images

a uniform semantic proof for cut elimination and completeness of various first and higher order logics we present a natural generalization of girard s first order phase semantics of linear logic theoret comput sci 50 1987 to intuitionistic and higher order phase semantics then we show that this semantic framework allows us to derive a uniform semantic proof of the first order and higher order cut elimination theorem as well as a first order and higher order phase semantic completeness theorem for various different logical systems at the same time our semantic proof works for various different logical systems uniformly in a strong sense without any change of the argument of proof it works for both first order and higher order versions and for linear substructural and standard logics uniformly and for both their intuitionistic and classical versions uniformly

compactness properties of sobolev imbeddings for rearrangement invariant norms compactness properties of sobolev imbeddings are studied within the context of rearrangement invariant norms attention is focused on the extremal situation namely when the imbedding is considered as defined on its optimal sobolev domain with the range space fixed the techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator of just one variable

extendability of kirchhoff elastic rods in complete riemannian manifolds the kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods and is defined to be a solution of the euler lagrange equations associated to the energy with the effect of bending and twisting we consider the initial value problem for the euler lagrange equations in a riemannian manifold in a previous paper the author proved the existence and uniqueness of global solutions of the initial value problem in the case where the ambient space is a space form in the present paper we extend this result to the case where the ambient space is a general complete riemannian manifold this implies that an arbitrary kirchhoff elastic rod of finite length in a complete riemannian manifold extends to that of infinite length

light logics and optimal reduction completeness and complexity typing of lambda terms in elementary and light affine logic eal and lal respectively has been studied for two different reasons on the one hand the evaluation of typed terms using lal eal respectively proof nets admits a guaranteed polynomial elementary respectively bound on the other hand these terms can also be evaluated by optimal reduction using the abstract version of lamping s algorithm the first reduction is global while the second one is local and asynchronous we prove that for lal eal respectively typed terms lamping s abstract algorithm also admits a polynomial elementary respectively bound we also give a proof of its soundness and completeness for eal and lal with type fixpoints by using a simple geometry of interaction model context semantics

a formal framework for specifying sequent calculus proof systems intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems in this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequences of the specified sequent calculus proof systems in particular derivability of an inference rule from a set of inference rules can be decided by bounded linear logic programming search on the specified rules we also present two simple and decidable conditions that guarantee that the cut rule and non atomic initial rules can be eliminated

differentials of complex interpolation processes for kothe function spaces we continue the study of centralizers on kothe function spaces and the commutator estimates they generate see 29 our main result is that if x is a super reflexive kothe function space then for every real centralizer on x there is a complex interpolation scale of kothe function spaces through x inducing as a derivative up to equivalence and a scalar multiple thus in a loose sense all real centralizers can be identified with derivatives of complex interpolation processes we apply our ideas in an appendix to show for example that there is a twisted sum of two hilbert spaces which fails to be a umd space

strong normalization property for second order linear logic the paper contains the first complete proof of strong normalization sn for full second order linear logic ll girard s original proof uses a standardization theorem which is not proven we introduce sliced pure structures sps a very general version of girard s proof nets and we apply to sps gandy s method to infer sn from weak normalization wn we prove a standardization theorem for sps if wn without erasing steps holds for an sps then it enjoys sn a key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness namely acyclicity slice by slice we conclude by showing how standardization for sps allows to prove sn of ll using as usual girard s reducibility candidates

additives of linear logic and normalization part i a restricted church rosser property we define a generalized cut elimination procedure for proof nets of fall linear logic without constants for which we prove a restricted church rosser property

derivation of slant helices on a given surface through new approaches abstract in this paper we provide some methods for generating slant helices lying on a parametric or an implicit surface in euclidean 3 space these methods are based on the initial value problems of first order ordinary differential equations although analytical solutions of the derived systems are not possible even for simple surfaces we provide some illustrative examples in which the slant helices have been obtained numerically the produced slant helices possess the given fixed direction constant angle and the curvature function

on the quaternionic b2 slant helices in the semi euclidean space e24 abstract in this paper we studied quaternionic slant helices which we called semi real quaternionic b2 slant helix in four dimensional semi euclidean space e 2 4 also we define the harmonic curvature functions for semi real quaternionic curves and we give a vector field d which we call darboux quaternion for semi real quaternionic b2 slant helix and then we obtained some characterizations for semi real quaternionic b2 slant helix in the terms of the harmonic curvatures and the darboux vector field d

an abstract approach to stratification in linear logic we study the notion of stratification as used in subsystems of linear logic with low complexity bounds on the cut elimination procedure the so called light subsystems from an abstract point of view introducing a logical system in which stratification is handled by a separate modality this modality which is a generalization of the paragraph modality of girard s light linear logic arises from a general categorical construction applicable to all models of linear logic we thus learn that stratification may be formulated independently of exponential modalities when it is forced to be connected to exponential modalities it yields interesting complexity properties in particular from our analysis stem three alternative reformulations of baillot and mazza s linear logic by levels one geometric one interactive and one semantic

null cubics and lie quadratics riemannian cubics are curves in a riemannian manifold m satisfying a variational condition they arise in computer graphics and in trajectory planning problems for rigid body motion where m is the group so 3 of rotations of euclidean three space e3 riemannian cubics on a lie group correspond to lie quadratics in the lie algebra there are only a few cases where closed form expressions are available for lie quadratics the present article is a qualitative analysis of null quadratics in so 3 focusing on long term dynamics and internal symmetries conclusions are drawn for asymptotics and symmetries of null cubics in so 3

a type assignment for λ calculus complete both for fptime and strong normalization one of the aims of implicit computational complexity is the design of programming languages with bounded computational complexity one of the most promising approaches to this aim is based on the use of lambda calculus as paradigmatic programming language and the design of type assignment systems for lambda terms where types guarantee both the functional correctness and the complexity bound along this line we propose a system of stratified types inspired by intersection types where intersection is a non associative operator this system is correct and complete for polynomial time when a uniform coding scheme is considered moreover all the strongly normalizing terms are typed in it thus increasing the typing power with respect to the previous proposals based on light logics

bounded combinatory logic and lower complexity we introduce a stratified version of combinatory logic1 in which there are two classes of terms called player and opponent such that the class of player terms is strictly contained in the class of opponent terms we show that the system characterizes polynomial time in a similar way to soft linear logic with the addition of explicit lazy products to the player terms and various notions of distributivity we obtain a characterization of polynomial space this paper is an expanded version of the abstract presented at dice 2013

on the expressivity of elementary linear logic elementary linear logic is a simple variant of linear logic due to girard and which characterizes in the proofs as programs approach the class of elementary functions that is to say functions computable in time bounded by a tower of exponentials of fixed height other systems like light and soft linear logics have then been defined to characterize in a similar way the more interesting complexity class of polynomial time functions but at the price of either a more complicated syntax or of more sophisticated encodings these logical systems can serve as the basis of type systems for calculus ensuring polynomial time complexity bounds on well typed terms this paper aims at reviving interest in elementary linear logic by showing that despite its simplicity it can capture smaller complexity classes than that of elementary functions for that we carry a detailed analysis of its normalization procedure and study the complexity of functions represented by a given type we then show that by considering a slight variant of this system with type fixpoints and free weakening elementary affine logic with fixpoints we can characterize the complexity of functions of type w k 2 b where w and b are respectively types for binary words and booleans the key point is a sharper study of the normalization bounds we characterize in this way the class p of polynomial time predicates and more generally the hierarchy of classes k exp for k 0 where k exp is the union of dtime 2 k n i for i 1

riemannian cubics in quadratic matrix lie groups abstract quadratic matrix lie groups are subgroups of the general linear group that satisfy a quadratic matrix identity the main purpose of this paper is to consider riemannian cubics in quadratic matrix lie groups with left invariant metrics results for riemannian cubics in quadratic matrix lie groups extend those in so n since the group so n with bi invariant metric is a very special case by examining riemannian cubics in so 2 1 and so 3 1 we find that the so called null lie quadratics in so p q p 0 q 0 and even more generally for any quadratic matrix lie group can be given in closed forms in terms of lie quadratics in so p and so q further we present some quantitative analyses of non null lie quadratics in so p q

program equivalence in linear contexts program equivalence in linear contexts where programs are used or executed exactly once is an important issue in programming languages however existing techniques like those based on bisimulations and logical relations only target at contextual equivalence in the usual non linear functional languages and fail in capturing non trivial equivalent programs in linear contexts particularly when non determinism is present we propose the notion of linear contextual equivalence to formally characterize such program equivalence as well as a novel and general approach to studying it in higher order languages based on labeled transition systems specifically designed for functional languages we show that linear contextual equivalence indeed coincides with trace equivalence we illustrate our technique in both deterministic a linear version of pcf and non deterministic linear pcf in moggi s framework functional languages

phase semantics and decidability of elementary affine logic light elementary and soft linear logics are formal systems derived from linear logic enjoying remarkable normalization properties in this paper we prove decidability of elementary affine logic eal the result is obtained by semantical means first defining a class of phase models for eal and then proving soundness and strong completeness following okada s technique phase models for light affine logic and soft linear logic are also defined and shown complete

some associated curves of frenet non lightlike curves in e13 abstract in this paper we introduce the notion of the principal binormal directional curve and the principal binormal donor curve of the frenet curve in the minkowski space e 1 3 furthermore we study some associated curves of a given frenet curve to classify the general helices and the slant helices also we introduce the natural representation of general helices and slant helices from the natural representation of plane curves and general helices respectively as an example we give the natural representation of the closed slant helices constructed by circles

jupp and kent s cubics in lie groups we study a generalization of the cubic polynomial to riemannian manifolds and other affine connection spaces modifying the differential equation x 4 0 by replacing higher derivatives with covariant derivatives in matrix groups with two particular choices of a left invariant connection we can convert the equation into a system of first order linear differential equations we give asymptotics in a generic case in gl n and in the n sphere

finding interpolating curves minimizing l infty acceleration in the euclidean space via optimal control theory we study the problem of finding an interpolating curve passing through prescribed points in the euclidean space the interpolating curve minimizes the pointwise maximum length i e l infty norm of its acceleration we reformulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory we characterize solutions associated with singular and nonsingular controls some of the results we obtain are new even for the scalar interpolating function case we reduce the infinite dimensional interpolation problem to an ensuing finite dimensional one and derive closed form expressions for interpolating curves consequently we devise efficient numerical techniques and illustrate them with examples

type inference for light affine logic via constraints on words light affine logic lal is a system due to girard and asperti capturing the complexity class p in a proof theoretical approach based on linear logic lal provides a typing for lambda calculus which guarantees that a well typed program is executable in polynomial time on any input we prove that the lal type inference problem for lambda calculus is decidable for propositional lal to establish this result we reformulate the type assignment system into an equivalent one which makes use of subtyping and is more flexible we then use a reduction to a satisfiability problem for a system of inequations on words over a binary alphabet for which we provide a decision procedure

on an interpretation of safe recursion in light affine logic we introduce a subalgebra bc of bellantoni and cook s safe recursion function algebra bc functions of the subalgebra have safe arguments that are non contractible i e non duplicable we propose a definition of safe and normal variables in light affine logic lal and show that bc is the largest subalgebra that is interpretable in lal relative to that definition though bc itself is not pf complete there are extensions of it by additional schemes for defining functions with safe arguments that are and are still interpretable in lal and so preserve pf closure we focus on one such which is bc augmented by a definition by cases construct and a restricted form of definition by recursion scheme over safe arguments as a corollary we obtain a new proof of the pf completeness of lal

computational interpretations of linear logic abstract we study girard s linear logic from the point of view of giving a concrete computational interpretation of the logic based on the curry howard isomorphism in the case of intuitionistic linear logic this leads to a refinement of the lambda calculus giving finer control over order of evaluation and storage allocation while maintaining the logical content of programs as proofs and computation as cut elimination in the classical case it leads to a concurrent process paradigm with an operational semantics in the style of berry and boudol s chemical abstract machine this opens up a promising new approach to the parallel implementation of functional programming languages and offers the prospect of typed concurrent programming in which correctness is guaranteed by the typing

position vector of a time like slant helix in minkowski 3 space in this paper position vector of a time like slant helix with respect to standard frame of minkowski space e31 is studied in terms of frenet equations first a vector differential equation of third order is constructed to determine position vector of an arbitrary time like slant helix in terms of solution we determine the parametric representation of the slant helices from the intrinsic equations thereafter we apply this method to find the representation of a time like salkowski and time like anti salkowski curves as examples of a slant helices by means of intrinsic equations moreover we present some new characterizations of slant helices and illustrate some examples of our main results

on constant slope spacelike surfaces in 3 dimensional minkowski space abstract a spacelike surface in the minkowski 3 space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface in this work we study such surfaces and classify all of them

introduction to clarithmetic ii the earlier paper introduction to clarithmetic i constructed an axiomatic system of arithmetic based on computability logic and proved its soundness and extensional completeness with respect to polynomial time computability the present paper elaborates three additional sound and complete systems in the same style and sense one for polynomial space computability one for elementary recursive time and or space computability and one for primitive recursive time and or space computability

godel s system t revisited the linear lambda calculus where variables are restricted to occur in terms exactly once has a very weak expressive power in particular all functions terminate in linear time in this paper we consider a simple extension with natural numbers and a restricted iterator only closed linear functions can be iterated we show properties of this linear version of godel s tusing a closed reduction strategy and study the class of functions that can be represented surprisingly this linear calculus offers a huge increase in expressive power over previous linear versions of t which are closed at construction rather than closed at reduction we show that a linear twith closed reduction is as powerful as t

realizability models for a linear dependent pcf abstract recently dal lago and gaboardi have proposed a type system named d l pcf as a framework for implicit computational complexity d l pcf is a non standard type system for pcf programs which is relatively complete with respect to quantitative properties thanks to the use of linear types inspired by bounded linear logic and dependent types a la dependent ml in this work we adapt the framework of quantitative realizability and obtain a model for d l pcf the quantitative realizability model aims at a better understanding of d l pcf type decorations and at giving an abstract semantic proof of intensional soundness

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

on slant helix and its spherical indicatrix in this paper we investigate spherical images the tangent indicatrix and binormal indicatrix of a slant helix we obtain that the spherical images are spherical helices moreover we show that a curve of constant precession is a slant helix for involute of a curve c we have obtain that c is a slant helix if and only if its involute is a general helix

realizability models for bll like languages we give a realizability model of girard scedrov scott s bounded linear logic bll this gives a new proof that all numerical functions representable in that system are polytime our analysis naturally justifies the design of the bll syntax and suggests further extensions

quantum implicit computational complexity we introduce a quantum lambda calculus inspired by lafont s soft linear logic and capturing the polynomial quantum complexity classes eqp bqp and zqp the calculus is based on the classical control and quantum data paradigm this is the first example of a formal system capturing quantum complexity classes in the spirit of implicit computational complexity it is machine free and no explicit bound e g polynomials appears in its syntax

generating plans in linear logic i actions as proofs abstract there is an increasing interest in the relation between logic and the changes involved in reasoning and specifically in plan generation up to now several attempts in this direction have been made either by embedding actions into a classical framework or by using nonstandard formalisms we think that these attempts though promising miss their objectives for a lack of a suitable logic and that the effort must be pursued in this paper we show how to obtain a strong and clean correspondence between proofs and sequences of actions by using only girard s linear logic eliminating from the classical logic the structural rules which are not adapted to our purpose a theorem is presented which expresses the new adequacy between proofs and actions

soft linear logic and polynomial time we present a subsystem of second order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms and vice versa

a type system for lock free processes advanced type systems for the calculus have recently been proposed to guarantee deadlock freedom in the sense that certain communications will eventually succeed unless the whole process diverges although such guarantees are useful for reasoning about the behavior of concurrent programs there still remains the weakness that the success of a communication is not completely guaranteed due to the possibility of divergence for example although a server process that has received a request message cannot discard the request it is allowed to infinitely delegate the request to other processes causing a livelock in this paper we present a type system which guarantees that certain communications will eventually succeed under fair scheduling regardless of whether processes diverge we also present a variant of the type system which guarantees that a communication will succeed within a given number of reduction steps