homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

periodic solutions of a second order ordinary differential equation a necessary and sufficient condition for nonresonance the nonlinearity g in 1 1 is a continuous function from r to r and the forcing term h is taken in l o 27r nonresonance means that 1 1 admits at least one solution x for any given h integrating eq 1 1 over a period one immediately sees that a necessary condition for nonresonance is that the function g be unbounded from above and from below on r it will appear later cf 1 6 that this unboundedness can be looked at as a condition relating the behaviour at infinity of the nonlinearity g with respect to the first eigenvalue 1 0 of the associated linear problem xf ax in 0 27r x 0 x 27r x 0 x1 271 1 2

numerical methods for two classes of singularly oscillatory bessel transforms and their error analysis abstract this work introduces and analyzes both the complex integration method and the modified complex integration method for two classes of highly oscillatory bessel transforms with algebraic or algebraic logarithmic singularities we first transform the original integrals into the infinite integrals on 1 by the change of variable t x where 0 with the help of the integral representation of the bessel function the integral representation and the asymptotic series of the whittaker function we then use the cauchy s residue theorem and convert the above infinite integrals into the infinite integrals on 0 the integrands of the resulting infinite integrals are not oscillatory and decay exponentially fast fortunately they can be efficiently evaluated by using the gauss laguerre quadrature rule furthermore we conduct their error analysis in inverse powers of the frequency finally numerical experiments are provided to confirm our theoretical analysis we can see from theoretical analysis and numerical experiments that the accuracy of the quadrature increases both for the case of growing the number of nodes n and fixed and for the case of an increasing and fixed the number of nodes n the comparison of the proposed two methods and the filon type method presented in kang and ma 2017 is briefly discussed as well

recursive algorithm for arrays of generalized bessel functions numerical access to dirac volkov solutions in the relativistic and the nonrelativistic theoretical treatment of moderate and high power laser matter interaction the generalized bessel function occurs naturally when a schrodinger volkov and dirac volkov solution is expanded into plane waves for the evaluation of cross sections of quantum electrodynamic processes in a linearly polarized laser field it is often necessary to evaluate large arrays of generalized bessel functions of arbitrary index but with fixed arguments we show that the generalized bessel function can be evaluated in a numerically stable way by utilizing a recurrence relation and a normalization condition only without having to compute any initial value we demonstrate the utility of the method by illustrating the quantum classical correspondence of the dirac volkov solutions via numerical calculations

periodic solution to nonlinear differential equation at resonance abstract in this paper a class of nonlinear sturm liouville problems at resonance is discussed an extended form of a non variational version of a max min principle is applied to show that the equation possesses unique 2 periodic solution under a less restrictive condition this result extends what we have already known

uniform approximation to finite hilbert transform of oscillatory functions and its algorithm abstract for the finite hilbert transform of oscillatory functions q f c 1 1 f x e i x x c d t with a smooth function f and real 0 for c 1 1 in the sense of cauchy principal value or for c 1 of hadamard finite part we present an approximation method of clenshaw curtis type and its algorithm interpolating f by a polynomial p n of degree n and expanding in terms of the chebyshev polynomials with o n log n operations by the fft we obtain an approximation q p n c q f c we write q p n c as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree n 1 we efficiently evaluate the oscillatory integral with a combination of authors previous method and keller s method for f z analytic on the interval 1 1 in the complex plane z the error of q p n c is bounded uniformly with respect to c and numerical examples illustrate the performance of our method

periodic solutions for duffing equations with a p laplacian like operator in this paper sufficient conditions are established to guarantee the existence of at least one periodic solution to a second order duffing equation with a p laplacian like operator generalized polar coordinates are applied in our proof

dynamical systems and sequence transformations this paper discusses the connections between numerical methods for ordinary differential equations fixed point iterations and sequence transformations books on chaos and fractals are concerned on the one hand with continuous dynamical systems and on the other hand with the dynamics of iterations in discrete dynamical systems there exists a strong connection between numerical methods for the integration of ordinary differential equations odes and fixed point iterations fpis this connection has already been studied see 14 chapter 5 pp 197 343 or 25 chapter 6 pp 165 201 for an introduction and 30 for an extended review but the dynamics of the iterations generated by a numerical method for odes was mainly used for understanding the behaviour of the solution of the differential equation in this paper we are mostly interested in fpis we will see that fpis can be considered as coming from methods for the numerical integration of odes reciprocally numerical methods for odes can be used in the solution of fixed point problems so fpis can lead to new methods for the numerical integration of odes and conversely the link with sequence transformations used in numerical analysis to accelerate the convergence will also be discussed

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable on the lattice x s q 2 abstract we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable on a non uniform lattice x s q2s i e the q analogues of charlier meixner krawtchouk and hahn polynomials in terms of the coefficients and of the pearson equation satisfied by the weight function and the coefficients of the three term recurrence relation and of two structure relations obeyed by these polynomials

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

an analogue for szego polynomials of the clenshaw algorithm abstract linear combinations of polynomials that are orthogonal with respect to an inner product defined on part of the real axis are commonly evaluated by the clenshaw algorithm we present an analogous algorithm for the evaluation of a linear combination n j 0 j j of polynomials j that are orthogonal with respect to an inner product defined on part of the unit circle the j are known as szeg polynomials and find applications e g in signal processing we also discuss how to express n j 0 j j as a linear combination of monomials

halley maps for a trigonometric and rational function among the methods available for the characterization of complicated mathematical and physical phenomena computers with graphics are emerging as an important tool in this article i present computational and graphical results on halley s method for one parameter functions of the form 1 0 and sin 0 in order to gain insight as to where the method can be relied upon and where it behaves strangely the resulting plots reveal a visually striking and intricate class of patterns indicating behavior ranging from stable attractive and repulsive points to chaos iterative approximation methods such a halley s method occur frequently in science and engineering

a complete method for the computations of mathieu characteristic numbers of integer orders a complete method and expressions for computing mathieu characteristic numbers mcns of integer orders are given the paper reviews available algorithms and their shortcomings and then suitable methods are selected on the basis of simplicity and accuracy for small orders n 4 new approximate formulas for mcns are developed these have much larger range than available expressions a new normalization technique for the mcns is proposed with the aid of this normalization a new approximate recurrence relation for mcns is obtained using the same normalization general formulas are obtained for regional limits where the approximate and asymptotic expressions can be used to compute mcns with these expressions the algorithms for computing mcns can handle very high orders up to n 1000 and beyond

existence and uniqueness of periodic solution for a class of differential systems in this paper sufficient conditions are obtained for the existence of a unique periodic solution for a class of differential systems

uniform approximation to cauchy principal value integrals with logarithmic singularity abstract an approximation of clenshaw curtis type is given for cauchy principal value integrals of logarithmically singular functions i f c 1 1 f x log x c x c d x c 1 1 with a given function f using a polynomial p n of degree n interpolating f at the chebyshev nodes we obtain an approximation i p n c i f c we expand p n in terms of chebyshev polynomials with o n log n computations by using the fast fourier transform our method is efficient for smooth functions f for which p n converges to f fast as n grows and so simple to implement this is achieved by exploiting three term inhomogeneous recurrence relations in three stages to evaluate i p n c for f z analytic on the interval 1 1 in the complex plane z the error of the approximation i p n c is shown to be bounded uniformly using numerical examples we demonstrate the performance of the present method

solutions of linear recurrence equations solutions are given to general homogeneous and non homogeneous recurrence equations defined on the set of integers solutions to homogeneous recurrence equations are given as i an infinite series of terms involving the partial ordinary bell polynomial ii an infinite series of terms involving the complete ordinary bell polynomial iii a weighted finite sum of terms involving powers solutions to non homogeneous recurrence equations are given as i a finite series of terms involving the complete ordinary bell polynomial ii a weighted infinite sum of terms involving powers computational issues of these solutions are also discussed

current trends in asymptotics some problems and some solutions abstract i discuss five topics of current interest in asymptotic analysis the use of probabilistic methods to estimate the growth of combinatorial sequences asymptotic methods in the theory of random walks the asymptotic estimation of solutions of difference equations asymptotic expansions in generalized scales and the computation by asymptotic methods of distributions whose moments are known i give a number of examples and mention some outstanding unsolved problems

on the computation of fourier transforms of singular functions abstract in this paper we consider a method for the computation of finite fourier transforms of functions having endpoint singularities of algebraic or logarithmic type this method may be considered as a modification of clenshaw curtis quadrature the nonoscillatory and nonsingular part of the integrand is replaced by a truncated chebyshev series approximation and the integral is then evaluated using recurrence relations the numerical stability of these recurrence relations is investigated

convergence of piessens method for numerical inversion of the laplace transform on the real line a method proposed by piessens j inst math appl 10 1972 pp 185 192 for numerical inversion of the laplace transform on the real line is shown to converge exponentially fast for transforms f s s gamma g s with gamma 0 and g s analytic at infty the method is based on an orthogonal polynomial expansion for the transform f s convergence is obtained for a broad class of orthogonal polynomials with the jacobi polynomial expansions of the original formulation as a special case implications for practical use are discussed

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

reduced cancellation in the evaluation of entire functions and applications to the error function a general concept for the reduction of cancellation problems in the evaluation of taylor sections of certain entire functions is proposed the resulting method is applied to and tested in the case of the complementary error function

the abc of hyper recursions each member of the family of gauss hypergeometric functions fn 2f1 a e1n b e2n c e3n z where a b c and z do not depend on n and ej 0 1 not all ej equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions the set of 26 cases for different ej values can be reduced to a set of 5 basic forms of difference equations in this paper the coefficients an bn and cn of these basic forms are given in addition domains in the complex z plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified the determination of such a pair asks for a detailed study of the asymptotic properties of the gauss functions fn for large values of n and of other gauss functions outside this group this will be done in a later paper

difference equations in statistical mechanics i cluster statistics models a review and some new results are presented for several cluster statistics models solutions of which can be reduced to difference equations mathematical techniques suitable for solving these equations are surveyed

uniform approximation to fractional derivatives of functions of algebraic singularity fractional derivative d qf x 0 1 with g x being a well behaved function we propose a quadrature method for uniformly approximating d q x ag x the present method consists of interpolating g x at abscissae in 0 1 by a finite sum of chebyshev polynomials it is shown that the use of the lower endpoint x 0 as an abscissa is essential for the uniform approximation namely to bound the approximation errors independently of x 0 1 numerical examples demonstrate the performance of the present method

continuation theorems for boundary value problems abstract this paper gives constructive proofs of some important continuation theorems for boundary value problems via the homotopy method this leads to a global convergent method of finding solutions

efficient calculation and asymptotic expansions of many different oscillatory infinite integrals abstract this paper introduces and analyzes quadrature rules and asymptotic expansions of a few highly oscillatory infinite integrals we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter which clarify the large behavior of these integrals then based on the resulting asymptotic expansions two different interpolatory quadrature rules are given one is the so called filon type methods based on standard hermite interpolation of the non oscillatory and non singular part of the integrands at equidistant nodes the other is the filon clenshaw curtis type method fcc by using special hermite interpolation at n 1 clenshaw curtis points and the fast computation of modified moments the interpolation coefficients needed in the fcc method can be computed by a numerically stable algorithm in o nlog n operations based on fast fourier transform fft the required modified moments can be accurately and efficiently calculated by some recurrence relation formulae moreover for these quadrature rules their error analyses in inverse powers of the frequency are provided the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

computation of jacobi functions of the second kind for use in nearside farside scattering theory abstract the nearside farside decomposition of a partial wave series is currently being used to understand the angular scattering of atom diatom collision systems in this theory it is necessary to compute jacobi functions of the second kind on the cut these functions are denoted by q n cos where n may be large positive integers the q n cos can be computed from a three term linear recurrence relation provided the initial values corresponding to n 0 and 1 are known we derive explicit formulas for q 0 cos q 1 cos in terms of elementary transcendental functions a new generating function for jacobi functions of nonintegral degree off the cut is obtained a special case of which yields a generating function for q n cos this is used to check the numerical results as is a casoratian relation we show that the recurrence for q n cos is stable in the forward direction with errors growing like o n we also present some numerics demonstrating the success of the method

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

computation of integrals with oscillatory singular factors of algebraic and logarithmic type in this paper we present the clenshaw curtis filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration the methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in n 1 clenshaw curtis points then the required modified moments can be accurately and efficiently computed by constructing some recurrence relations moreover for these quadrature rules their absolute errors in inverse powers of the frequency are given the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

periodic solutions for 2kth order ordinary differential equations with resonance in this paper a non variational version of a max min principle is extended and some unique existence results are obtained for the periodic boundary value problem of the higher order ordinary differential equations under a resonant condition

convergence acceleration for the numerical solution of second order linear recurrence relations the concept of modification used for accelerating the convergence of ordinary continued fractions is adapted to the case of the gautschi aggarwal burgmeier algorithm for the computation of nondominant solutions of nonhomogeneous second order linear recurrence relations

strichartz estimates for long range perturbations we study local in time strichartz estimates for the schroedinger equation associated to long range perturbations of the flat laplacian on the euclidean space we prove that in such a geometric situation outside of a large ball centered at the origin the solutions of the schroedinger equation enjoy the same strichartz estimates as in the non perturbed situation the proof is based on the isozaki kitada parametrix construction if in addition the metric is non trapping we prove that the strichartz estimates hold in the whole space

an analytic method for convergence acceleration of certain hypergeometric series a method is presented for convergence acceleration of the generalized hypergeometric series 3 f 2 with the argument 1 using analytic properties of their terms iterated transformation of the series is performed analytically which results in obtaining new fast converging expansions for some special functions and mathematical constants

numerical computation of the incomplete lipschitz hankel integral je o a z abstract two factorial neumann series expansions are derived for the incomplete lipschitz hankel integral je 0 a z these expansions are used together with the neumann series expansion given by agrest in an algorithm which efficiently computes je 0 a z to a user defined number of significant digits other expansions for je 0 a z which are found in the literature are also discussed but these expansions are found to offer no significant computational advantages when compared with the expansions used in the algorithm

algorithms to numerically evaluate the hankel transform abstract the efficient and accurate evaluation of the hankel transform is required in a number of applications this paper reviews a number of algorithms that have only recently been exposed in the literature results of a comparative study of some useful algorithms are presented it is found that the performance of all algorithms depends on the type of function to be transformed the projection based methods provide acceptable accuracy with better efficiency than numerical quadrature

generalized associated polynomials and functions of second kind let pv v n0 pv v v 1 be a sequence of polynomials generated by a three term recurrence relation shifting the recurrence coefficients of the elements of pv v n0 we get a sequence of so called associated polynomials which play an important role in the theory of orthogonal polynomials we generalize this concept of associating for arbitrary polynomials vn n especially if vn is expanded in terms of pv v 0 n their associated polynomials are clenshaw polynomials which are used in numerical mathematics as a consequence of it we present some results from the viewpoint of associated polynomials and from the viewpoint of clenshaw polynomials analogously as for orthogonal polynomials we define functions of second kind for vn we prove some properties of them which depend on the generalized associated polynomials and functions of second kind for orthogonal polynomials

computation of multiple eigenvalues of infinite tridiagonal matrices in this paper it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues the computation of such double eigenvalues is enabled by the newton method of two variables the three term recurrence relations obtained from its eigenvalue problem evp subsume the well known relations of a the zeros of j v z b the zeros of zj f v z hj v z c the evp of the mathieu differential equation and d the evp of the spheroidal wave equation the results of experiments are shown for the three cases a c for the computation of their double pairs

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

coulomb and bessel functions of complex arguments and order abstract the coulomb wavefunctions originally constructed for real 0 real and integer 0 are defined for and all complex we examine the complex continuation of a variety of series and continued fraction expansions for the coulomb functions and their logarithmic derivatives and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives the resulting algorithm 46 is a complex generalisation of steed s method 6 7 as it appears in the real procedure coulfg 10 complex whittaker confluent hypergeometric and bessel functions can also be calculated

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

difference equations in statistical mechanics ii solid on solid models in two dimensions a review and some new results are presented for the solid on solid models of wetting in two dimensions i e line interfaces with an emphasis on the difference equations arising in the transfer matrix calculations for these models methods for solving the appropriate difference equations exactly or approximately are surveyed including specific results for short range long range power law and applied field like binding external potentials

periodic solutions for 2kth boundary value problem with resonance abstract in this paper a non variational version of a max min principle is extended and some unique existence results for the periodic boundary value problem of the higher order nonconservative systems with resonance u 2 k t j 1 k a j u 2 j 1 t 1 k 1 g u t e t are obtained

constrained multi degree reduction of triangular bezier surfaces using dual bernstein polynomials we propose a novel approach to the problem of multi degree reduction of bezier triangular patches with prescribed boundary control points we observe that the solution can be given in terms of bivariate dual discrete bernstein polynomials the algorithm is very efficient thanks to using the recursive properties of these polynomials the complexity of the method is o n 2m 2 n and m being the degrees of the input and output bezier surfaces respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined triangular bezier surfaces the result is a composite surface of global c r continuity with a prescribed order r some illustrative examples are given

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

on the periodic solution to the newtonian equation of motion on considere l equation du mouvement newtonienne d un systeme mecanique soumis a des forces internes conservatives et a des forces externes periodiques u t grad g u t p t ou p r r 11 est continue et de periode 2 et g a des derivees partielles d ordre 2 continues

on the computation of parameter derivatives of solutions of linear difference equations a method is given to compute the parameter derivatives of recessive solutions of second order inhomogeneous linear difference equations the case of difference equations in which all solutions have the same rate of growth is also discussed the method is illustrated by numerical computations of parameter derivatives of incomplete gamma functions and confluent hypergeometric functions

asymptotic expansions for second order linear difference equations abstract formal series solutions are obtained for the difference equation y n 2 a n y n 1 b n y n 0 where a n and b n have asymptotic expansions of the form a n s 0 a s n s and b n s 0 b s n s for large values of n and b0 0 these solutions are characterized by the roots of the characteristic equation 2 a0 b0 0 our discussion is divided into three cases according to whether the roots are distinct or equal and do not satisfy the auxiliary equation a1 b1 0 or equal and do satisfy the auxiliary equation the last case is further divided into three subcases according to whether the roots of the indicial equation 1 2 a1 a2 b2 0 do not differ by a nonnegative integer or differ by a positive integer or are equal in all cases the formal series solutions will be shown to be asymptotic our approach is based on the method of successive approximations

existence of periodic solutions for second order duffing equations with p laplacian like operators the existence of periodic solutions for second order duffing equations with p laplacian like operator is studied by applying a new generalization of polar coordinates

a new algorithm for computing orthogonal polynomials the aim of this paper is to present an algorithm for computing orthogonal polynomials the fast fourier transform completes the connection given by draux between the three term recurrence relationship and the euclidian algorithm applications to hankel systems and numerical examples illustrate our purpose

dispersion and strichartz estimates for the liouville equation we consider the liouville equation associated with a metric g of class c2 and we prove dispersion and strichartz estimates for the solution of this equation in terms of geodesics associated with g we introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the euclidean case holds to deal with the case of non trapped long range perturbation of the euclidean metric we prove a global velocity moments effect on the solution in particular we obtain global in time strichartz estimates for metrics such that the dispersion estimate is not satisfied

the chebyshev fast gauss and nonuniform fast fourier transforms and their application to the evaluation of distributed heat potentials we present a method for the fast and accurate computation of distributed heat potentials in two dimensions the distributed source is assumed to be given in terms of piecewise space time chebyshev polynomials we discretize uniformly in time whereas in space the polynomials are defined on the leaf nodes of a quadtree the quadtree can vary at each time step we combine a product integration rule with fast algorithms fast heat potentials nonuniform fft fast gauss transform to obtain a high order accurate method with optimal complexity if n is the number of time steps m is the maximum number of leaf nodes over all the time steps and the input contains a qth order polynomial representation of f then our method requires o q 3mnlogm work to evaluate the heat potential at arbitrary mn space time target locations the overall convergence rate of the method is of order q we present numerical experiments for q 4 8 and 16 and we verify the theoretical convergence rate of the method when the solution is sufficiently smooth the 16th order variant results in significant computational savings even in the case in which we require only a few digits of accuracy

compactness in kinetic transport equations and hypoellipticity abstract we establish improved hypoelliptic estimates on the solutions of kinetic transport equations using a suitable decomposition of the phase space our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions f x v l 1 satisfying some appropriate transport relation v x f 1 x 2 1 v 2 g may be inferred solely from additional integrability and compactness with respect to v in a forthcoming work the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the boltzmann equation with a rough force field arsenio and saint raymond in preparation 4

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

evaluation of bessel function integrals with algebraic singularities abstract a new method for the numerical evaluation of 1 0 1 x x j v ax dx is proposed where a are given constants and j is the bessel function of the first kind and of order it is assumed that f is a smooth function the method is based on approximation of the function f by a polynomial given in the form of a linear combination of the shifted jacobi polynomials r k and on the use of chebyshev series expansion for the function j ax

a user friendly method for computing indefinite integrals of oscillatory functions for indefinite integrals q f x 1 x f t e i t d t x 1 1 torii and the first author hasegawa and torii 1987 developed a quadrature method of clenshaw curtis c c type its improvement was made and combined with sidi s m w transformation by sidi and the first author hasegawa and sidi 1996 to compute infinite oscillatory integrals the improved method per se however has not been elucidated in its attractive features which here we reveal with new results and its detailed algorithm a comparison with a method of c c type for definite integrals q f 1 due to dominguez et al 2011 suggests that a smaller number of computations is required in our method this is achieved by exploiting recurrence and normalization relations and their associated linear system we show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension for f z analytic on and inside an ellipse in the complex plane z the error of the approximation to q f x of the improved method is shown to be bounded uniformly numerical examples illustrate the stability and performance of the method

microlocal dispersive smoothing for the schrodinger equation this paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for schrodinger s equation and the global behavior of bicharacteristics of the principal symbol in particular localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow the origin of the phenomenon is in the dispersive nature of the equation the results imply microlocal regularity properties of the fundamental solution 1995 john wiley sons inc

completely monotonic functions associated with the gamma function and its q analogues several functions involving the gamma function x and the q gamma function q x are proved to be completely monotonic some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions

an application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations abstract in this paper a class of systems of nonlinear differential equations at resonance is considered with the use of a global inversion theorem which is an extended form of a non variational version of a max min principle we prove that this class of equations possesses a unique 2 periodic solution under a rather weaker condition for existence and uniqueness than those given in papers j chen w li periodic solution for 2 k th boundary value problem with resonance j math anal appl 314 2006 661 671 f cong periodic solutions for 2 k th order ordinary differential equations with nonresonance nonlinear anal 32 1998 787 793 f cong periodic solutions for second order differential equations appl math lett 18 2005 957 961 w li periodic solutions for 2 k th order ordinary differential equations with resonance j math anal appl 259 2001 157 167 w li h li a min max theorem and its applications to nonconservative systems int j math math sci 17 2003 1101 1110 w li z shen a constructive proof of existence and uniqueness of 2 periodic solution to duffing equation nonlinear anal 42 2000 1209 1220 this result extends the results known so far

besov estimates in the high frequency helmholtz equation for a non trapping and c 2 potential abstract we study the high frequency helmholtz equation for a potential having c 2 smoothness and satisfying the non trapping condition we prove optimal morrey campanato estimates that are both homogeneous in space and uniform in the frequency parameter the homogeneity of the obtained bounds together with the weak assumptions we require on the potential constitute the main new result in the present text our result extends previous bounds obtained by perthame and vega in that we do not assume the potential satisfies the virial condition a strong form of non trapping

an algorithm for nondominant solutions of linear second order inhomogeneous difference equations an algorithm is given for computing a weighted sum of a nondominant solution of a linear second order inhomogeneous difference equation to a prescribed accuracy by estimating the truncation error the present method is an extension of both the stable numerical method due to olver and sookne and a summation technique due to deuflhard for computing minimal solutions of a homogeneous difference equation the method is illustrated by numerical examples