w symmetries of ito stochastic differential equations we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones

the arithmetic of the digital computer a new approach a new approach to the arithmetic of the digital computer is surveyed the methodology for defining and implementing floating point arithmetic is described shortcomings of elementary floating point arithmetic are revealed through sample problems the development of automatic computation with emphasis on the user control of errors is reviewed the limitations of conventional rule of thumb procedures for error control in scientific computation are demonstrated by means of examples computer arithmetic is extended so that the arithmetic operations in the linear spaces and their interval correspondents which are most commonly used in computation can be performed with maximum accuracy on digital computers a new fundamental computer operation the scalar product is introduced to develop this advanced computer arithmetic a process of automatic error control called validation which delivers high accuracy with guarantees for scientific computations is described validation of computations for a large class of nu

random lie point symmetries of stochastic differential equations we study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random lie point symmetries of stochastic differential equations both in ito and in stratonovich forms we also discuss relations with previous results in the literature

a probabilistic approach to semi classical approximations abstract we study in this paper the semi classical expansion of the schrodinger equation using a probabilistic approach based on the wiener measure using almost analytic extensions we exhibit a probabilistic ansatz for the wave function we show that this ansatz approximates very well the wave function in the semi classical regime and gives the semi classical expansion under mild hypothesis on the potential at infinity and no analyticity conditions in this paper the study takes place before the caustics

a symbol class for some schrodinger equations on r n on considere l equation de schrodinger dans r n id t dt h t et on interprete certains resultats de zelditch d une facon plus geometrique on suit les singularites de t a travers l espace x en introduisant une notion d ensemble front d onde qui est metaplectiquement covariant

symmetries in the stochastic calculus of variations given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space time transformations and a notion of prolonged transformations which extend the existing analogs in classical calculus of variations we prove that a family of prolonged transformations leaves the action integral invariant if and only if it leaves invariant the heat equation associated to it as well as the structure of the extremals we then prove a stochastic version of noether theorem to each family of transformations leaving the action invariant or symmetries we can associate a function which gives a martingale when taken along a process minimizing the action under endpoint constraints

the feynman integral for time dependent anharmonic oscillators we review some basic notions and results of white noise analysis that are used in the construction of the feynman integrand as a generalized white noise functional we show that the feynman integrand for the time dependent harmonic oscillator in an external potential is a hida distribution

malliavin calculus and euclidean quantum mechanics i functional calculus abstract we give a rigorous version of the functional calculus developed by r feynman in relation to his path integral formulation of quantum mechanics our approach is euclidean but distinct from the one founded on the feynman kac formula it uses two basic ingredients a new probabilistic interpretation of the classical heat equation introduced recently in the framework of euclidean quantum mechanics and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

probability and quantum symmetries ii the theorem of nœther in quantum mechanics for the largest class of physical systems having a classical analog a new rigorous but not probabilistic lagrangian version of nonrelativistic quantum mechanics is given in terms of a notion of regularized action function as a consequence of the study of the symmetries of this action an associated n ther theorem is obtained all the quantum symmetries resulting from the canonical quantization procedure follow in this way as well as a number of symmetries which are new even for the case of the simplest systems the method is based on the study of a corresponding lie algebra and an analytical continuation in the time parameter of the probabilistic construction given in paper i of this work generically the associated quantum first integrals are time dependent and the probabilistic model provides a natural interpretation of the new symmetries various examples illustrate the physical relevance of our results

efficient algorithms for fuzzy qualitative temporal reasoning fuzzy qualitative temporal relations have been proposed to reason about events whose temporal boundaries are ill defined although the corresponding reasoning tasks are in the same complexity class as their crisp counterparts in practice the scalability of fuzzy temporal reasoners may be insufficient for applications that require a high expressivity and deal with a large number of events on the other hand transitivity rules can be used to make sound but incomplete inferences in polynomial time utilizing a variant of allen s path consistency algorithm the aim of this paper is to investigate how this polynomial time algorithm can be improved without altering its time complexity to this end we establish a characterization of 2 consistency of fuzzy temporal relations and provide transitivity rules that are significantly stronger than those resulting from straightforwardly generalizing transitivity rules for crisp temporal relations we furthermore provide experimental evidence for the effectiveness of our improved algorithm

accurate sum and dot product algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time we show that the computed results are as accurate as if computed in twice or k fold working precision k ge 3 for twice the working precision our algorithms for summation and dot product are some 40 faster than the corresponding xblas routines while sharing similar error estimates our algorithms are widely applicable because they require only addition subtraction and multiplication of floating point numbers in the same working precision as the given data higher precision is unnecessary algorithms are straight loops without branch and no access to mantissa or exponent is necessary

point algebras for temporal reasoning algorithms and complexity we investigate the computational complexity of temporal reasoning in different time models such as totally ordered partially ordered and branching time our main result concerns the satisfiability problem for point algebras and point algebras extended with disjunctions for these problems we identify all tractable subclasses we also provide a number of additional results for instance we present a new time model suitable for reasoning about systems with a bounded number of unsynchronized clocks we investigate connections with spatial reasoning and we present improved algorithms for deciding satisfiability of the tractable point algebras

smoothness and non smoothness of the fundamental solution of time dependent schrodinger equations the fundamental solutione t s x y of time dependent schrodinger equationsi u t 1 2 u v t x u is studied it is shown that e t s x y is smooth and bounded fort s if the potential is sub quadratic in the sense thatv t x o x 2 at infinity in one dimension ifv t x v x is time independent and super quadratic in the sense thatv x c 1 x 2 e at infinity c 0 and e 0 thene t s x y is nowherec1

accurate evaluation of the k th derivative of a polynomial and its application this paper presents a compensated algorithm for the evaluation of the k th derivative of a polynomial in power basis the proposed algorithm makes it possible the direct evaluation without obtaining the k th derivative expression of the polynomial itself with a very accurate result to all but the most ill conditioned evaluation forward error analysis and running error analysis are performed by an approach based on the data dependency graph numerical experiments illustrate the accuracy and efficiency of the algorithm

the method of stationary phase for oscillatory integrals on hilbert spaces we develop the method of stationary phase for the normalized oscillatory integral on hilbert space giving borel summable expansions the developments that we obtain hold for more general situations than the ones of previous papers on the same subject

accurate floating point product and exponentiation several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision here we focus on a method to improve the accuracy of the product of floating point numbers we show that the computed result is as accurate as if computed in twice the working precision the algorithm is simple since it only requires addition subtraction and multiplication of floating point numbers in the same working precision as the given data such an algorithm can be useful for example to compute the determinant of a triangular matrix and to evaluate a polynomial when represented by the root product form it can also be used to compute the integer power of a floating point number

accurate summation dot product and polynomial evaluation in complex floating point arithmetic several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision here we focus on methods to improve the accuracy of summation dot product and polynomial evaluation such algorithms exist real floating point numbers in this paper we provide new algorithms which deal with complex floating point numbers we show that the computed results are as accurate as if computed in twice the working precision the algorithms are simple since they only require addition subtraction and multiplication of floating point numbers in the same working precision as the given data

reconstruction of singularities for solutions of schrodinger s equation we determine the behavior in time of singularities of solutions to some schrodinger equations onr n we assume the hamiltonians are of the formh 0 v where h 0 1 2 delta 1 2 sum limits k 1 n omega k 2 x k 2 and wherev is bounded and smooth with decaying derivatives when all k 0 the kernelk t x y of exp ith is smooth inx for every fixed t y when all 1 are equal but non zero the initial singularity reconstructs at times t frac m pi omega 1 and positionsx 1 m y just as ifv 0 k is otherwise regular in the general case the singular support is shown to be contained in the union of the hyperplanes x x js 1 l js y js when j t l j forj j 1 j r

fuzzifying allen s temporal interval relations when the time span of an event is imprecise it can be represented by a fuzzy set called a fuzzy time interval in this paper we propose a framework to represent compute and reason about temporal relationships between such events since our model is based on fuzzy orderings of time points it is not only suitable to express precise relationships between imprecise events ldquoroosevelt died before the beginning of the cold warrdquo but also imprecise relationships ldquoroosevelt died just before the beginning of the cold warrdquo we show that unlike previous models our model is a generalization that preserves many of the properties of the 13 relations allen introduced for crisp time intervals furthermore we show how our model can be used for efficient fuzzy temporal reasoning by means of a transitivity table finally we illustrate its use in the context of question answering systems

finite dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions abstract we give an approach to the theory of oscillatory integrals in infinite dimensions starting from finite dimensional oscillatory integrals we also provide detailed asymptotic expansions by a method of stationary phase in infinite dimensions with convergence and estimates for the remainder provided by a strong control on finite dimensional approximations

accurate validated and fast evaluation of elementary symmetric functions and its application this paper is concerned with the fast accurate and validated evaluation of elementary symmetric functions in floating point arithmetic we present two new compensated algorithms with real and complex floating point inputs respectively by applying error free transformations to improve the accuracy of the so called summation algorithm that is used by example in the matlab s poly function we derive forward roundoff error bounds and running error bounds for our new algorithms the roundoff error bounds imply that the computed results are as accurate as if computed with twice the working precision and then rounded to the current working precision the running error analysis provides a shaper bound along with the result without increasing significantly the computational cost numerical experiments illustrate that our algorithms run much faster than the algorithms using the classic double double library while sharing similar error estimates such algorithms can be widely applicable for example to compute characteristic polynomials from eigenvalues or polynomial s coefficients from zeros some simple applications are presented to show that the proposed algorithms compute the coefficients of the characteristic polynomials of some real and complex matrices to high relative accuracy

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

feynman path integrals for polynomially growing potentials abstract a general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied a representation formula of the parseval type is proved as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals these results are applied to provide a rigorous feynman path integral representation for the solution of the time dependent schrodinger equation with a quartic anharmonic potential the borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved

the algebra ia fuz a framework for qualitative fuzzy temporal reasoning the aim of this work is to integrate the ideas of flexibility and uncertainty into allen s interval based temporal framework defining a new formalism called ia f u z which extends classical interval algebra ia in order to express qualitative fuzzy constraints between intervals we generalize the classical operations between ia relations to ia f u z relations as well as the concepts of minimality and local consistency referring to the framework of fuzzy constraint satisfaction problem we analyze the most interesting reasoning tasks in our framework which generalize the classical problems of checking consistency finding a solution and computing the minimal network in the context of ia in order to solve these tasks we devise two constraint propagation algorithms and a branch bound algorithm since these tasks are np difficult we address the problem of finding tractable sub algebras of ia f u z by extending to our fuzzy framework the classical pointizable sub algebras sa c and sa as well as the maximal tractable subalgebra h introduced by nebel in particular we prove that the fuzzy extension of the latter called h f u z shares with its classical counterpart a maximality property in that it is the unique maximal subalgebra of ia f u z which contains the fuzzy extensions of allen s atomic relations ions

accurate floating point summation part ii sign k fold faithful and rounding to nearest in part ii of this paper we first refine the analysis of error free vector transformations presented in part i based on that we present an algorithm for calculating the rounded to nearest result of s sum p i for a given vector of floating point numbers p i as well as algorithms for directed rounding a special algorithm for computing the sign of s is given also working for huge dimensions assume a floating point working precision with relative rounding error unit eps we define and investigate a k fold faithful rounding of a real number r basically the result is stored in a vector mathtt res nu of k nonoverlapping floating point numbers such that sum mathtt res nu approximates r with relative accuracy mathtt eps k and replacing mathtt res k by its floating point neighbors in sum mathtt res nu forms a lower and upper bound for r for a given vector of floating point numbers with exact sum s we present an algorithm for calculating a k fold faithful rounding of s using solely the working precision furthermore an algorithm for calculating a faithfully rounded result of the sum of a vector of huge dimension is presented our algorithms are fast in terms of measured computing time because they allow good instruction level parallelism they neither require special operations such as access to mantissa or exponent they contain no branch in the inner loop nor do they require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithms are proved to be optimal

the feynman integrand as a hida distribution it is shown that feynman integrals are weighted averages over brownian paths with suitable hida distributions as complex weights for simple cases the construction of these distributions is be performed explicitly for a much larger class of interactions their existence is shown

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities