a novel approach for the determination of asymptotic expansions of certain oscillatory integrals abstract a standard method for deriving asymptotic expansion consists of applying integration by parts to standard integral representation in this paper we consider cases in which this method is ineffective to deal with these we derive alternative integral representations which are amenable to this and other standard methods

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

a note on asymptotic evaluation of some hankel transforms asymptotic behavior of the integral if w ejo wx f x2 xdx is investigated where jo x is the bessel function of the first kind and w is a large positive parameter it is shown that 1f w decays exponentially like ewy2 y 0 when f z is an entire function subject to a suitable growth condition a complete asymptotic expansion is obtained when f z is a meromorphic function satisfying the same growth condition similar results are given when f z has some specific branch point singularities

a solvable mixed charge ensemble on the line global results we consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2 1 in the presence of an external field with the total charge fixed and the system held at temperature corresponding to 1 it is proved that the particles form a pfaffian point process when the external field is quadratic the harmonic oscillator potential we produce the explicit family of skew orthogonal polynomials necessary to simplify the related matrix kernels in this setting a variety of limit theorems are proved on the distribution of the number as well as the spatial density of each species of particle as the total charge increases to infinity connections to ginibre s real ensemble of random matrix theory are highlighted throughout

strong and ratio asymptotics for laguerre polynomials revisited abstract in this paper we consider the strong asymptotic behavior of laguerre polynomials in the complex plane the leading behavior is well known from perron and mehler heine formulas but higher order coefficients which are important in the context of krall laguerre or laguerre sobolev type orthogonal polynomials are notoriously difficult to compute in this paper we propose the use of an alternative expansion due to buchholz in terms of bessel functions of the first kind the coefficients in this expansion can be obtained in a straightforward way using symbolic computation as an application we derive extra terms in the asymptotic expansion of ratios of laguerre polynomials in c 0

existence of periodic orbits and horseshoes for mappings in a separable banach space abstract for a smooth map in a separable banach space with a hyperbolic invariant measure we mainly prove that positivity of the measure theoretic entropy of the hyperbolic measure implies the existence of periodic orbits and horseshoes this is an extension of lian and young s work on maps in a hilbert space to the case of systems on a banach space

on the integral representations for the confluent hypergeometric function in this paper we derive a new contour integral representation for the confluent hypergeometric function as well as for its various special cases consequently we derive expansions of the confluent hypergeometric function in terms of functions of the same kind furthermore we obtain a new identity involving integrals and sums of confluent hypergeometric functions our results generalized several well known results in the literature

asymptotic expansions of mellin convolutions by means of analytic continuation we present a method for deriving asymptotic expansions of integrals of the form 0 f t h xt dt for small x based on analytic continuation the expansion is given in terms of two asymptotic sequences the coefficients of both sequences being mellin transforms of h and f many known and unknown asymptotic expansions of important integral transforms are derived trivially from the approach presented here this paper reconsiders earlier work of mcclure and wong explicit error terms for asymptotic expansions of stieltjes transforms j inst math appl 22 1978 129 145 exact remainders for asymptotic expansions of fractional integrals j inst math appl 24 1979 139 147 and asymptotic approximations of integrals academic press new york 1989 chaps 5 where elements of distribution theory are used and wong explicit error terms for asymptotic expansions of mellin convolutions j math anal appl 72 2 1979 740 756 where as in the present paper the asymptotic expansions are obtained without the use of distributions in this paper we re derive the expansions given in wong explicit error terms for asymptotic expansions of mellin convolutions j math anal appl 72 2 1979 740 756 by using a different approach and we obtain new results which are not present in wong explicit error terms for asymptotic expansions of mellin convolutions j math anal appl 72 2 1979 740 756 a proof of the asymptotic character of the expansions and accurate error bounds

exactification of the asymptotics for bessel and hankel functions exactification is the process of showing how a complete asymptotic expansion can be evaluated to yield exact values of the original function it represents because the process does not involve neglecting the remainders of truncated component asymptotic series within a complete asymptotic expansion techniques for evaluating divergent series are required one such technique is borel summation but in many instances it can be computationally slow and consequently lacking in precision another is the numerical technique of mellin barnes regularization which is capable of evaluating divergent series with great precision far more rapidly than borel summation here both techniques are presented in the evaluation of exact values for bessel and hankel functions from their complete asymptotic expansions

exactification of the method of steepest descents the bessel functions of large order and argument the hadamard expansion procedure applied to laplace type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents this mode of expansion is illustrated by the evaluation of the bessel functions j v x and y v v x of large order and argument when x is bounded away from unity the limit x 1 corresponding to the coalescence of the active saddles in the integral representations of the bessel functions translates into a progressive loss of exponential separation between the different levels of the hadamard expansion which renders computation in this limit more difficult it is shown how a simple modification to this procedure can be employed to deal with the coalescence of the active saddles when x 1

a method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes application to pearcey s integral and its derivatives a numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described the method exploits cauchy s theorem and jordan s lemma to write the infinite integration path along different contours in the complex plane the method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques application is made to pearcey s integral p x y and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem tables of p x y delta p x y delta x delta p x y delta y and the real zeros of p x y are presented for the grid 8 0 or x or 8 0 and 0 or y or 8 0

two new asymptotic expansions of the ratio of two gamma functions formal expansions giving as particular cases semiasymptotic expansions of the ratio of two gamma functions are obtained

asymptotic expansions of mellin convolution integrals an oscillatory case in a recent paper j l lopez asymptotic expansions of mellin convolution integrals siam rev 50 2 2008 275 293 we have presented a new very general and simple method for deriving asymptotic expansions of 0 f t h xt dt for small x it contains watson s lemma and other classical methods mellin transform techniques mcclure and wong s distributional approach and the method of analytic continuation used in this approach as particular cases in this paper we generalize that idea to the case of oscillatory kernels that is to integrals of the form 0 e i c tf t h xt dt with c r and we give a method as simple as the one given in the above cited reference for the case c 0 we show that mcclure and wong s distributional approach for oscillatory kernels and the summability method for oscillatory integrals are particular cases of this method some examples are given as illustration

the evaluation of bessel functions via exp arc integrals a standard method for computing values of bessel functions has been to use the well known ascending series for small argument and to use an asymptotic series for large argument with the choice of the series changing at some appropriate argument magnitude depending on the number of digits required in a recent paper d borwein j borwein and r crandall 1 derived a series for an exp arc integral which gave rise to an absolutely convergent series for the j and i bessel functions with integral order such series can be rapidly evaluated via recursion and elementary operations and provide a viable alternative to the conventional ascending asymptotic switching in the present work we extend the method to deal with bessel functions of general non integral order as well as to deal with the y and k bessel functions

on the use of hadamard expansions in hyperasymptotic evaluation differential equations of hypergeometric type we describe how a modication of a common technique for developing asymptotic expansions of solutions of linear di erential equations can be used to derive hadamard expansions of solutions of di erential equations hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions particularly that of having exponentially small remainders when truncated and as a consequence provide a useful computational tool for evaluating special functions the methods we discuss can be applied to linear di erential equations of hypergeometric type and may have wider applicability

convergent and asymptotic expansions of the pearcey integral abstract we consider the pearcey integral p x y for large values of x x y c we can find in the literature several convergent or asymptotic expansions in terms of elementary and special functions with different levels of complexity most of them are based on analytic in particular asymptotic techniques applied to the integral definition of p x y in this paper we consider a different method the iterative technique used for differential equations in lopez 2012 using this technique in a differential equation satisfied by p x y we obtain a new convergent expansion analytically simple that is valid for any complex x and y and has an asymptotic property when x uniformly for y in bounded sets the accuracy of the approximation is illustrated with some numerical experiments and compared with other expansions given in the literature

on the growth rate of periodic orbits for vector fields abstract we establish the relationship between the growth rate of periodic orbits and the topological entropy for c 1 generic vector fields this extends a classical result of katok for c 1 0 surface diffeomorphisms to c 1 generic vector fields of any dimension the main difficulty comes from the existence of singularities and the shear of the flow

the asymptotic behaviour of pearcey s integral for complex variables a deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe generally known as pearcey s integral p prime x y can be obtained by considering its analytic continuation to arbitrary complex variables x and y a new integral representation for p prime x y is given in the form of a contour integral involving a weber parabolic cylinder function whose order is the variable of integration it is shown how the asymptotics of p prime x y may be obtained from this representation for complex x and y when either x or y rightarrow infty without reference to the usual stationary points of the integrand for the case x rightarrow infty y finite the full asymptotic expansion of p prime x y is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the x plane the case y rightarrow infty x finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the y plane and only the first terms in the expansion are given the asymptotic behaviour of p prime x y on the caustic y 2 frac 2 3 x 3 0 is also obtained from the new representation and is shown to agree with recent results of d kaminski the various properties of the weber parabolic cylinder function required in this paper are collected together in the appendix

observing lyapunov exponents of infinite dimensional dynamical systems can lyapunov exponents of infinite dimensional dynamical systems be observed by projecting the dynamics into n using a typical nonlinear projection map we answer this question affirmatively by developing embedding theorems for compact invariant sets associated with c1 maps on hilbert spaces examples of such discrete time dynamical systems include time t maps and poincare return maps generated by the solution semigroups of evolution partial differential equations we make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite dimensional dynamical system in so doing we adopt an empirical approach and formulate checkable conditions under which a lyapunov exponent computed from experimental data will be a lyapunov exponent of the infinite dimensional dynamical system under study provided the nonlinear projection map producing the data is typical in the sense of prevalence

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

a variant of horn s problem and the derivative principle abstract identifying the spectrum of the sum of two given hermitian matrices with fixed eigenvalues is the famous horn s problem in this note we investigate a variant of horn s problem i e we identify the probability density function abbr pdf of the diagonals of the sum of two random hermitian matrices with given spectra we then use it to re derive the pdf of the eigenvalues of the sum of two random hermitian matrices with given eigenvalues via the derivative principle a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries we can also recover jean bernard zuber s recent results moreover as an illustration we derive the analytical expressions of eigenvalues of the sum of two random hermitian matrices from g u e n or wishart ensemble by the derivative principle respectively we also investigate the statistics of exponential of random matrices and connect them with golden thompson inequality and partially answer a question proposed by forrester some potential applications in quantum information theory such as uniform average quantum jensen shannon divergence and average coherence of uniform mixture of two orbits are discussed

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

on the use of hadamard expansions in hyperasymptotic evaluation of laplace type integrals ii complex variable we review and discuss the application of hadamard expansions to the hyperasymptotic evaluation of laplace integrals c ex p t f t dt x where for simplicity in this paper x is restricted to be a positive real variable the integration path c can be taken over both finite and semi infinite intervals in the complex plane in general these expansions take the form of compound expansions each associated with a different exponential level and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor the early terms in each convergent expansion possess a rapid asymptotic like decay when the variable x is large with late terms that can be transformed into a rapid decay comparable with that of the early terms the hadamard expansion of the above integral when the phase function p t is linear is shown to depend significantly on the singularity structure of the amplitude function f t the application of the theory to laplace type integrals with quadratic cubic and nonpolynomial phase functions is considered in addition to the amplitude function the location of the saddle points satisfying p t 0 also plays a role in the detailed structure of the different exponential levels in the resulting hadamard expansion numerical examples are given to illustrate the accuracy that can be achieved with this new procedure

uniform asymptotics and zeros of a system of orthogonal polynomials defined via a difference equation abstract we study a class of sieved pollaczek polynomials defined by a second order difference equation three term recurrence relation the measure of orthogonality is determined by using the markov theorem and the perron stieltjes inversion formula and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point n and a critical point n involving respectively the airy function ai and a 1 2 ai i 2 bi darboux s method the method of steepest descents and various uniform asymptotic techniques such as cubic transformations are used to derive the results asymptotic formulas for the least zeros the largest zeros and the zeros on both sides of n are also obtained several numerical examples are provided to compare the approximate zeros with the true values

high precision evaluation of the bessel functions via hadamard series we present a method of high precision calculation of the bessel functions using hadamard series such series are absolutely convergent expansions involving the normalised incomplete gamma function p a z email protected a z c a and possess early terms that behave like those in an asymptotic expansion in the case of real variables the function p a z acts as a smoothing factor on the terms of the series we show how these series representing the bessel functions of complex argument can be chosen so as to produce rapidly convergent series that possess terms decaying at the geometric rate q k where 0 q 1 and k is the ordinal number of the series we give numerical examples with q 12 13 and 14 the theory is extended to cover the confluent hypergeometric functions f11 a b z and u a b z thereby dealing with many of the special functions arising in mathematical physics

on the zeros of the pearcey integral zeroes for real values of the arguments of the pearcey integral are numerically evaluated and plotted from this numerical examination it is apparent that these zeroes display a high degree of structure the character of which is revealed through asymptotic analysis refinements to the resulting approximations are supplied near the paper s end

crandall s computation of the incomplete gamma function and the hurwitz zeta function with applications to dirichlet l series this paper extends tools developed by crandall 2012 16 to provide robust high precision methods for computation of the incomplete gamma function and the lerch transcendent we then apply these to the corresponding computation of the hurwitz zeta function and so of dirichlet l series and character polylogarithms

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

numerical computation of diffraction catastrophes with codimension eight a computational strategy aimed at evaluating diffraction catastrophes belonging to the x 9 family is presented the approach proposed is based on the use of power series expansions suitably derived for giving meaningful representation of the whole 0 x 9 subfamily jointly with a powerful sequence transformation algorithm the so called weniger transformation the convergence features of the above series expansions are investigated and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the weniger transformation as well as the ease of implementation of the whole approach

exponentially improved asymptotics for the gamma function by expressing the error term in truncation of the asymptotic expansion in terms of a mellin barnes integral we obtain an exponentially improved asymptotic expansion for z as z in arg z we describe the need for such an expansion arising in applications the result is compared with formal results of dingle obtained via borel summation and rigorous methods of olver using laplace integrals for a different class of functions

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

excursions of adaptive algorithms via the poisson clumping heuristic the authors detail the application of the poisson clumping heuristic pch to the least mean square lms adaptive algorithm and its signed variants under certain conditions on the input and disturbance statistics the parameter estimate errors form a markov process the pch asserts that large excursions of the parameter estimates occur in clumps and that these clumps are distributed in a poisson manner with parameter lambda sub b where b is the maximum parameter error expressions are derived for each of the four algorithms in the scalar case which allows calculation of lambda sub b in a relatively straightforward manner these values are compared to simulations of the algorithms given that the results are asymptotic in b the close agreement between simulated and theoretical values is striking even for very modest b the four algorithms are then compared in terms of lambda sub b some observations are made regarding the relative performance of the four variants no single lms variant always outperforms the others suggestions are made as to how this technique might be applied in the vector case and a crucial monotonicity property is verified

estimation of topological entropy in random dynamical systems abstract for c r r 1 random dynamical systems f on a compact smooth riemannian manifold m the fiber topological entropy is bounded above by an integral formula particularly for the c random dynamical systems the integral formula coincides with the fiber topological entropy

exactification of the poincare asymptotic expansion of the hankel integral spectacularly accurate asymptotic expansions and non asymptotic scales we obtain an exactification of the poincare asymptotic expansion pae of the hankel integral as using the distributional approach of mcclure wong we find that for half integer orders of the bessel function the exactified asymptotic series terminates so that it gives an exact finite sum representation of the hankel integral for other orders the asymptotic series does not terminate and is generally divergent but is amenable to superasymptotic summation i e by optimal truncation for specific examples we compare the accuracy of the optimally truncated asymptotic series owing to the mcclure wong distributional method with owing to the mellin barnes integral method we find that the former is spectacularly more accurate than the latter by in some cases more than 70 orders of magnitude for the same moderate value of b moreover the exactification can lead to a resummation of the pae when it is exact with the resummed poincare series exhibiting again the same spectacular accuracy more importantly the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences

continued fractions with applications i introductory examples ii more basics iii convergence criteria iv continued fractions and three term recurrence relations v correspondence of continued fractions vi hypergeometric functions vii moments and orthogonality viii pade approximants ix some applications in number theory x zero free regions xi digital filters and continued fractions xii applications to some differential equations appendix some continued fraction expansions references index

mellin barnes regularization borel summation and the bender wu asymptotics for the anharmonic oscillator we introduce the numerical technique of mellin barnes integral regularization which can be used to evaluate both convergent and divergent series the technique is shown to be numerically equivalent to the corresponding results obtained by borel summation both techniques are then applied to the bender wu formula which represents an asymptotic expansion for the energy levels of the anharmonic oscillator we find that this formula is unable to give accurate values for the ground state energy particularly when the coupling is greater than 0 1 as a consequence the inability of the bender wu formula to yield exact values for the energy level of the anharmonic oscillator cannot be attributed to its asymptotic nature

integrals of the ising class from an experimental mathematical perspective we analyze ising class integrals our experimental results involvedextreme precision multidimensional quadrature on intricate integrands thus highly parallel computation was required

on the use of hadamard expansions in hyperasymptotic evaluation of laplace type integrals iii clusters of saddle points it is shown how the recently developed hadamard expansion procedure can be applied to the hyperasymptotic evaluation of laplace type integrals containing a large variable when the phase function has a cluster of close lying saddle points the modification to this procedure that is required when the saddles in the cluster coalesce to form a single higher order saddle is discussed an example is also considered in which there is both a coalescence of saddles and a stokes phenomenon as the phase of the large variable is allowed to vary numerical examples are given to illustrate the accuracy that can be obtained with this new procedure

asymptotics beyond all orders asymptotology and borel summation asymptotics superasymptotics hyperasymptotics m berry the geometric model of crystal growth numerical analysis of the geometric model of crystal growth j m hammersley dendritic crystal growth an experimental assessment of continuum models of dendritic growth j p gollub directional solidification of solids directional growth of dilute mixtures and lamellar eutectics k kassner et al flow in a hele shaw cell viscous displacement in a hele shaw cell s tanveer the rapidly forced pendulum exponentially small estimate for separatrix splittings j scheurle et al ordinary differential equations singular perturbations of solitons y pomeau solitary water waves in the presence of small surface tension solitary waves with ripples beyond all orders j t beale problems in optics potpourri 23 additional articles index

lyapunov exponents periodic orbits and horseshoes for semiflows on hilbert spaces two settings are considered flows on finite dimensional riemann ian manifolds and semiflows on hilbert spaces with conditions consistent with those in systems defined by dissipative parabolic pdes under certain assump tions on lyapunov exponents and entropy we prove the existence of geometric structures called horseshoes this implies in particular the presence of infinitely many periodic solutions for diffeomorphisms of compact manifolds analogous results are due to a katok here we extend katok s results to i continuous time and ii infinite dimensions courant institute of mathematical sciences new york university 251 mercer street new york new york 10012 e mail address lian cims nyu edu courant institute of mathematical sciences new york university 251 mercer street new york new york 10012 e mail address lsy cims nyu edu

olver s asymptotic method revisited case i abstract we consider the asymptotic method designed by olver f w j olver asymptotics and special functions academic press 1974 for linear differential equations of the second order containing a large asymptotic parameter we only consider here the first case studied by olver differential equations without turning or singular points it is well known that his method gives the poincare type asymptotic expansion of two independent solutions of the equation in inverse powers of in this paper we add two initial conditions to the differential equation and consider the corresponding initial value problem by using the green s function of an auxiliary problem and a fixed point theorem we construct a sequence of functions that converges to the unique solution of the problem this sequence also has the property of being an asymptotic expansion for large not of poincare type of the solution of the problem moreover we show that the idea may be applied to nonlinear differential equations with a large parameter

on the use of hadamard expansions in hyperasymptotic evaluation of laplace type integrals iv poles this paper is one of a series considering the application of hadamard expansions in the hyperasymptotic evaluation of laplace type integrals of the form cexp email protected t f t dt for large values of z it is shown how the procedure can be employed to deal with the case when the amplitude function f t possesses poles which may coalesce with a saddle point of the integrand or approach the integration path c a novel feature introduced here is the reverse expansion procedure this results in contributions at each exponential level after the first of the expansion in the form of rapidly convergent series thereby enabling the high precision evaluation of the above integral in coalescence problems numerical examples are given to illustrate the procedure

a new asymptotic form for the laguerre polynomials a new asymptotic formula in terms of elementary functions is demonstrated for the laguerre polynomials l n x in the limit when all three parameters n x become large this asymptotic limit appears in the problem of obtaining the low field semiclassical approximation from the full quantum mechanical behavior of a two dimensional electron gas in a magnetic field

joint use of the weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle point integrals when the weniger transformation is systematically used in resumming the asymptotic series arising from the application of the steepest descent method can display dramatic numerical instabilities that prevent it from improving the accuracy achievable via superasymptotics in the present paper an explanation of such pathologies through the concept of resurgence introduced by berry and howls several years ago within the context of hyperasymptotics is proposed in particular the way the topology of the whole set of the saddles influences the resummation capabilities of the weniger transformation is here investigated for the integrals defining the lowest order cuspoid diffraction catastrophes eventually a powerful and easily implementable resummation scheme based on a joint use of the weniger transformation and hyperasymptotics is proposed for taking care of the above pathologies such a joint action seems to encompass the main virtues of both approaches and the preliminary numerical results obtained from its application show that relative errors several orders of magnitude smaller than those achievable via superasymptotics can be achieved with modest implementation efforts

a criterion for the triviality of the centralizer for vector fields and applications abstract in this paper we establish a criterion for the triviality of the c 1 centralizer for vector fields and flows in particular we deduce the triviality of the centralizer at homoclinic classes of c r vector fields r 1 furthermore we show that the set of flows whose c 1 centralizer is trivial include i c 1 generic volume preserving flows ii c 2 generic hamiltonian flows on a generic and full lebesgue measure set of energy levels and iii c 1 open set of non hyperbolic vector fields that admit a lorenz attractor we also provide a criterion for the triviality of the c 0 centralizer of continuous flows

hyperasymptotic evaluation of the pearcey integral via hadamard expansions we describe high precision computations of the pearcey integral pe x y for real x and y by means of hadamard expansions numerical results for x y situated in different regions of the x y plane are given to illustrate the levels of precision that can be achieved particular emphasis is given to computation in the neighbourhood of the two cusped curves associated with pe x y across which there is either a coalescence of saddles or a stokes phenomenon

quadratic differentials and asymptotics of laguerre polynomials with varying complex parameters abstract in this paper we study the asymptotics as n of the sequences of laguerre polynomials with varying complex parameters depending on the degree n more precisely we assume that n n a n and lim n a n a c this study has been carried out previously only for n r but complex values of a introduce an asymmetry that makes the problem more difficult the main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane which may have an independent interest while the weak asymptotics is obtained by reduction to the theorem of gonchar rakhmanov stahl the strong asymptotic results are derived via the non commutative steepest descent analysis based on the riemann hilbert characterization of the laguerre polynomials

on the higher order stokes phenomenon during the course of a stokes phenomenon an asymptotic expansion can change its form as a further series prefactored by an exponentially small term and a stokes multiplier appears in the representation the initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters in this paper we introduce the concept of a higher order stokes phenomeno at which a stokes multiplier itself can change value we show that the higher order stokes phenomenon can be used to explain the apparent sudden birth of stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions we provide an example of how the higher order stokes phenomenon can have important effects on the large time behaviour of partial differential equations

the metric entropy of diffeomorphisms soit m une variete de riemann compacte c sans bord soit f un diffeomorphisme c 2 de m et soit m une mesure de probabilite de borel f invariante sur m soit h m f l entropie de f on donne des relations entre h m f et les proprietes geometriques de f et de m

computing the confluent hypergeometric function m a b x the confluent hypergeometric function m a b x arises naturally in both statistics and physics although analytically well behaved extreme but practically useful combinations of parameters create extreme computational difficulties a brief review of known analytic and computational results highlights some difficult regions including b a 0 with x much larger than b existing power series and integral representations may fail to converge numerically while asymptotic series representations may diverge before achieving the accuracy desired continued fraction representations help somewhat variable precision can circumvent the problem but with reductions in speed and convenience in some cases known analytic properties allow transforming a difficult computation into an easier one the combination of existing computational forms and transformations still leaves gaps for b a 0 two new power series in terms of gamma and beta cumulative distribution functions respectively help in some cases numerical evaluations highlight the abilities and limitations of existing and new methods overall a rational approximation due to luke and the new gamma based series provide the best performance

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

fine properties of lp cocycles which allow abundance of simple and trivial spectrum in this paper we generalize 3 and prove that the class of accessible and saddle con servative cocycles a wide class which includes cocycles evolving in glpd rq slpd rq and sppd rq l p densely have a simple spectrum we also generalize 3 1 and prove that for an l p residual subset of accessible cocycles we have a one point spectrum by using a different approach of the one given in 3 finally we show that the linear differential system versions of previous results also hold and give some applications

error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal in 1994 boyd derived a resurgence representation for the gamma function exploiting the 1991 reformulation of the method of steepest descents by berry and howls using this representation he was able to derive a number of properties of the asymptotic expansion for the gamma function including explicit and realistic error bounds the smooth transition of the stokes discontinuities and asymptotics for the late coefficients the main aim of this paper is to modify boyd s resurgence formula making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal we also prove the exponentially improved versions of these expansions complete with error terms finally we provide new formal asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors

asymptotic expansions of some integral transforms by using generalized functions the technique devised by wong to derive the asymptotic expansions of multiple fourier transforms by using the theory of schwartz distributions is extended to a large class of integral transforms the extension requires establishing a general procedure to extend these integral transforms to generalized functions wong s technique is then applied to some of these integral transforms to obtain their asymptotic expansions this class of integral transforms encompasses among others the laplace the airy the k and the hankel transforms

strong asymptotics for laguerre polynomials with varying weights abstract supplementing and extending classical and recent results strong asymptotics for the laguerre polynomials l n n are established as n when the parameter n depends on the degree n suitably a case of particular interest is the one for which n grows faster than n rescaling the argument z appropriately the resulting asymptotic forms are described by elementary functions thereby extending the classical formulae of plancherel rotach type for laguerre polynomials l n z being fixed

on the use of hadamard expansions in hyperasymptotic evaluation ii complex variables this paper extends the method of hyperasymptotic evaluation using absolutely convergent hadamard expansions developed in the first part to the confluent hypergeometric functions of complex argument z it is shown that the representation of this class of functions involves an infinite number of hadamard expansions associated with a decreasing sequence of subdominant exponential levels for large z numerical examples involving the bessel functions are given to illustrate the attainable levels of accuracy

effective laguerre asymptotics it is known that the generalized laguerre polynomials can enjoy subexponential growth for large primary index in particular for certain fixed parameter pairs a z one has the large n asymptotic behavior l n a z sim c a z n a 2 1 4 e 2 sqrt nz we introduce a computationally motivated contour integral that allows efficient numerical laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior we present a fast algorithm for symbolic generation of the rather formidable expansion coefficients along the way we address the difficult problem of establishing effective i e rigorous and explicit error bounds on the general expansion a primary tool for these developments is an exp arc method giving a natural bridge between converging series and effective asymptotics

hyperasymptotics for integrals with finite endpoints berry howls 1991 hereinafter called bh refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp kf z along doubly infinite contours in the complex plane passing over saddlepoints of f z here we derive analogous results for integrals with integrands of a similar form but whose local expansions in powers of 1 k are made about the finite endpoints of semi infinite contours of integration we treat endpoints where f z behaves locally linearly or quadratically generically local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f z we derive resurgence relations which express the original integral exactly as a truncated endpoint expansion plus a remainder involving the global saddle structure of f z via integrals through certain adjacent saddles the saddles adjacent to the endpoint are determined by a topological rule if the least term of the endpoint expansion is the n 0 k th summing to here calculates the endpoint integral up to an error of approximately exp n 0 k we develop a scheme involving iteration of the new resurgence relations with a similar one derived in bh which can reduce this error down to exp 2 386n 0 k this hyperasymptotic formalism parallels that of bh and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane provided that it does not coincide with other saddles we illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc x and a constructed incomplete airy function

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

compound poisson approximation for nonnegative random variables via stein s method the aim of this paper is to extend stein s method to a compound poisson distribution setting the compound poisson distributions of concern here are those of the form pois nu where nu is a finite positive measure on 0 infty a number of results related to these distributions are established these in turn are used in a number of examples to give bounds for the error in the compound poisson approximation to the distribution of a sum of random variables

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

on a novel iterative method to compute polynomial approximations to bessel functions of the first kind and its connection to the solution of fractional diffusion diffusion wave problems we present an iterative method to obtain approximations to bessel functions of the first kind jp x p 1 via the repeated application of an integral operator to an initial seed function f0 x the class of seed functions f0 x leading to sets of increasingly accurate approximations fn x is considerably large and includes any polynomial when the operator is applied once to a polynomial of degree s it yields a polynomial of degree s 2 and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree we focus on the set of polynomial approximations generated from the seed function f0 x 1 this set of polynomials is useful not only for the computation of jp x but also from a physical point of view as it describes the long time decay modes of certain fractional diffusion and diffusion wave problems

poisson approximation for excursions of adaptive algorithms with a lattice state space this paper analyzes excursions of adaptive algorithms such as the lms with a lattice state space under certain conditions on the input and disturbance statistics the parameter estimate error forms a markov chain the approximations are valid if this chain has a strong tendency toward an equilibrium point the distribution of the number of excursions in n units of time is approximated by a poisson distribution the mean and distribution of the time of the occurrence of the first excursion are approximated by those of an exponential distribution expressions for the error in the approximations are also derived the approximations are shown to hold asymptotically as the excursion defining set converges to the empty set all the parameters required for the approximations and all expressions for the error in the approximations are calculable in a relatively straightforward manner

a shadowing lemma for quasi hyperbolic strings of flows abstract in this paper we prove a shadowing lemma for pseudo orbits made by quasi hyperbolic strings we allow singularities in question and hence in particular the quasi hyperbolic strings are formulated by the rescaled linear poincare flow instead of the usual linear poincare flow we also introduce the sectional poincare map and rescaled sectional poincare map for lipschitz vector fields on banach spaces in the article

failure time analysis for lms algorithms few results are available in the literature on failure time in contrast with long time average analysis of the least mean square lms adaptive algorithms filters such analysis is extremely important when failure of the algorithm could cause failure of the entire system in which they are employed a system may fail upon the first occurrence of a large error any value far from 0 or upon staying far from 0 for a period of time a clump of large errors we use a poisson approximation to study excursions failure of the lms algorithm and its three signed variants the distribution and the mean of the first excursion are approximated with the mean and distribution of an exponential distribution the number of excursions in n units of time is approximated by a poisson distribution we also approximate the mean of an excursion length clump size the approximations are derived asymptotically as the excursion defining set converges to the empty set and as the algorithm step size converges to 0