lyapunov exponents and the natural invariant density determination of chaotic maps an iterative maximum entropy ansatz we apply the maximum entropy principle to construct the natural invariant density and the lyapunov exponent of one dimensional chaotic maps using a novel function reconstruction technique that is based on the solution of the hausdorff moment problem via maximizing shannon entropy we estimate the invariant density and the lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments the accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available furthermore we also consider a very complex example for which no exact analytical result for the invariant density is available a comparison of our results to those available in the literature is also discussed

periodic orbits near bifurcations of codimension two classical mechanics semiclassics and stokes transitions we investigate classical and semiclassical aspects of codimension two bifurcations of periodic orbits in hamiltonian systems a classification of these bifurcations in autonomous systems with two degrees of freedom or time periodic systems with one degree of freedom is presented we derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurcations in conjunction with stokes transitions that become important in the ensuing diffraction catastrophe integrals

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

smoothing of stokes discontinuities when the asymptotics of an analytic function f z are given in terms of two asymptotic forms s 1 z s 2 z alternately dominant in alternating sectors of the z plane stokes observed that the coefficients of s 1 s 2 while constant in any one sector can change from sector to sector and that to a first approximation the change appears to be discontinuous recently berry has given an interesting formal argument which shows in more detail how the change takes place and the present paper gives a rigorous treatment of berry s result

solving differential equations by a maximum entropy minimum norm method with applications to fokker planck equations the method of maximum entropy minimum norm is utilized to produce a general method of solving differential equations the technique is a generalization and extension of previous work performed by baker jarvis j math phys 30 302 1989 it is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis a number of simplifications occur in this extended method the lagrange multipliers and solution vector can be solved for explicitly thus eliminating the necessity of solving systems of nonlinear equations for the lagrange multipliers as was required in the previous approach it is shown that the solution obtained is equivalent to a minimum norm approximation the maximum entropy solution of differential equations with fourier moments is shown to be identical to a fourier series solution additionally the new method is applied to solving the random walk and fokker planck equations

error bounds for the method of steepest descents new error bounds are found for the asymptotic expansions which arise in the method of steepest descents these bounds are found by exploiting the reformulation of the method of steepest descents given in recent work by berry howls a major difference between this and previous work is that the global properties of the integrand and not just those on the path of steepest descent play an essential role the bounds are significantly easier to compute than those in the current literature applications of the error bounds are given for asymptotic expansions arising from the pearcey integral the airy function the modified bessel function and the integral of the airy function

a maximum entropy method for solving frobenius perron operator equations let s 0 1 0 1 be a nonsingular transformation that preserves an absolutely continuous invariant measure with density f in this paper we develop a general algorithm for approximating f using the maximum entropy approach

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

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

maximum entropy approach with higher moments for solving fokker planck equation abstract the maximum entropy method with higher number of moments is used to solve the fokker planck equation an adopted newton method is used to iterate the maximum entropy set of equations the method is used to calculate the probability density function of the fokker planck equation the calculations are carried out for three examples 1 the bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems 2 the malthus verhulst model which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light 3 the black scholes equation used in financial market option pricing although the maximum entropy approach has several advantages it is not convergent at large times and so cannot be used to calculate the steady state solution

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

solutions for fredholm equations through nonlinear iterative processes by adjunction of one or two well chosen functionals taking account simultaneously of the unknown u x and of the inhomogeneous term f x every fredholm equation of the second kind can easily be transformed into new nonlinear integral equations for which the solution of the primary equation remains obviously valid however when usual iterative solving methods are tested the convergence of the various sequences now available are very different and one needs criteria to select the best ones fredholm equations of the first kind can also be solved using the new processes described which are particularly efficient after a preliminary transform pointing out the first iterated operator which is necessarily positive optimisation techniques are detailed in order to work out nonlinear equations of particular interest i e they are very suitable to perform numerically an accurate iterative solution

applications of catastrophe theory to the physical sciences abstract catastrophe theory was introduced by rene thom in the late 1960 s as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations such as differential and difference equations and in particular discussed the special case of elementary catastrophe theory singularities of smooth real valued functions popular expositions have tended to overemphasize this special case but it remains the major source of ideas and methods here we survey the applications of catastrophe theory to the physical sciences physics chemistry engineering fluid mechanics etc for brevity we confine attention to an area lying between elementary and general catastrophe theory usually known as singularity theory this is the theory of singularities of smooth vector valued functions which mathematically is a straightforward though non trivial generalization of the real valued case in the last few years it has developed into a powerful and useful technique in several areas of theoretical physics notably optics and bifurcation theory equivariant catastrophe theory taking account of symmetry is likely to prove especially interesting

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

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

theory of axially symmetric cusped focusing numerical evaluation of a bessoid integral by an adaptive contour algorithm a numerical procedure for the evaluation of the bessoid canonical integral j x y is described j x y is defined for x and y real by eq1 where j0 is a bessel function of order zero j x y plays an important role in the description of cusped focusing when there is axial symmetry present it arises in the diffraction theory of aberrations in the design of optical instruments and of highly directional microwave antennas and in the theory of image formation for high resolution electron microscopes the numerical procedure replaces the integration path along the real t axis with a more convenient contour in the complex t plane thereby rendering the oscillatory integrand more amenable to numerical quadrature the computations use a modified version of the cuspint computer code kirk et al 2000 comput phys commun at press which evaluates the cuspoid canonical integrals and their first order partial derivatives plots and tables of j x y and its zeros are presented for the grid 8 0 x 8 0 and 8 0 y 8 0 some useful series expansions of j x y are also derived

the pearcey integral in the highly oscillatory region we consider the pearcey integral p x y for large values of y and bounded values of x the integrand of the pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated then we consider here the modified saddle point method introduced in lopez perez and pagola 2009 4 with this method the analysis is simpler and it is possible to derive a complete asymptotic expansion of p x y for large y the asymptotic analysis requires the study of three different regions for arg y separately in the three regions the expansion is given in terms of inverse powers of y2 3 and the coefficients are elementary functions of x the accuracy of the approximation is illustrated with some numerical experiments

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

asymptotic solutions of second order linear differential equations having almost coalescent turning points with an application to the incomplete gamma function uniform asymptotic expansions are derived for solutions of the differential equation d 2 w d 2 u 2 2 u u w which are uniformly valid for u real and large bounded real or complex and lying in a well defined bounded or unbounded complex domain which contains the origin the function u is assumed to be holomorphic in this domain and is o u ln u uniformly as u oo the approximations involve parabolic cylinder functions and include explicit and realistic error bounds the new theory is then applied to the complementary incomplete gamma function x furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying arg x oo

properties of maximum entropy approximate solutions to fredholm integral equations the properties of the maximum entropy approximate solutions to fredholm integral equations of the second kind are discussed it is proved that the approximate solutions exist and converge to the exact one for a wider class of problems than previously determined the correspondence between the maximum entropy technique and the galerkin method is established and on its basis an upper a posterior bound for a solution is found

the aharonov bohm wave and the cornu spiral for flux the aharonov bohm wavefunction ab r is represented exactly in terms of fresnel integrals numerical evidence shows that in the complex ab plane the wave clings to the cornu spiral for all positions r except very close to the flux line that is the argument w r of the fresnel integrals is almost real this is confirmed by deriving an asymptotic expansion for w in powers of 1 r

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

deconvolution and regularization for numerical solutions of incorrectly posed problems in this paper we have converted the laplace transform into fredholm integral equation of the first kind of convolution type which is an ill posed problem and used deconvolution and regularization method to solve it the method is applied to various test examples taken from the literature the method gives a good approximation to the true solution our results confirm that our method is competitive with other recently presented methods the results are shown in the table and the diagrams

borel summable solutions to one dimensional schrodinger equation it is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be borel summable to the solutions themselves appear also to be the unique solutions to the one dimensional 1d schrodinger equation having this property namely it is shown in this paper that for the polynomial potentials the borel function defined by the fundamental solutions can be considered as the canonical one the latter means that any borel summable solution can be obtained by the borel transformation of this unique canonical borel function multiplied by some dependent and borel summable constant this justifies the exceptional role the fundamental solutions play in 1d quantum mechanics and completes the relevant semiclassical theory relied on the borel resummation technique and developed in our other papers

uniform asymptotic expansions of integrals a selection of problems abstract on the occasion of the conference we mention examples of stieltjes work on asymptotics of special functions the remaining part of the paper gives a selection of asymptotic methods for integrals in particular on uniform approximations we discuss several standard problems and examples in which known special functions error functions airy functions bessel functions etc are needed to construct uniform approximations finally we discuss the recent interest and new insights in the stokes phenomenon an extensive bibliography on uniform asymptotic methods for integrals is given together with references to recent papers on the stokes phenomenon for integrals and related topics

matrix scaling and explicit doubly stochastic limits abstract the process of alternately row scaling and column scaling a positive n n matrix a converges to a doubly stochastic positive n n matrix s a often called the sinkhorn limit of a the main result in this paper is the computation of exact formulae for the sinkhorn limits of certain symmetric positive 3 3 matrices

solution to boundary value problems using the method of maximum entropy the method of maximum entropy is used to solve a class of linear boundary value problems the method is based on using various moments of the differential equation as constraints when maximizing the entropy various examples are presented and compared to exact solutions for varying numbers of moments it is found that the maximum entropy approximation is in many cases better than a fourier series solution for a given number of expansion terms and moments the method is very general and will find applications in many areas of physics a comparison of the amount of work necessary for the maximum entropy solution versus finite difference techniques is presented and it is found that the maximum entropy technique shows promise as an alternative solution technique

differential equations for the cuspoid canonical integrals differential equations satisfied by the cuspoid canonical integrals i n a are obtained for arbitrary values of n 2 where n 1 is the codimension of the singularity and a 1 2 n 1 a set of linear coupled ordinary differential equations is derived for each step in the sequence i n 0 0 0 0 i n 0 0 0 n 1 i n 0 0 n 2 n 1 i n 0 2 n 2 n 1 i n 1 2 n 2 n 1 the initial conditions for a given step are obtained from the solutions of the previous step as examples of the formalism the differential equations for n 2 fold n 3 cusp n 4 swallowtail and n 5 butterfly are given explicitly in addition iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points the results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

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

the stokes phenomenon associated with the hurwitz zeta function ζ s a we examine the exponentially improved asymptotic expansion of the hurwitz zeta function s a for large complex values of a with s regarded as a parameter it is shown that an infinite number of subdominant exponential terms switch on across the stokes lines arg a

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

higher order airy scaling in deformed dyck paths we introduce a deformed version of dyck paths ddp where additional to the steps allowed for dyck paths jumps orthogonal to the preferred direction of the path are permitted we consider the generating function of ddp weighted with respect to their half length area and number of jumps this represents the first example of an exactly solvable two dimensional lattice vesicle model showing a higher order multicritical point applying the generalized method of steepest descents we see that the associated two variable scaling function is given by the logarithmic derivative of a generalized higher order airy integral

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

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

maximum entropy approximation for lyapunov exponents of chaotic maps let s 0 1 0 1 be a nonsingular chaotic map that preserves an integrable density f that describes the statistics of the orbits in this article we use the maximum entropy approach to approximate the density f and the corresponding lyapunov exponent

semiclassical quantization of sawtooth map with inclusion of next leading order and its failure the semiclassical quantization of sawtooth maps is reconsidered from the viewpoint of the higher order terms in the asymptotic series we carry out the semiclassical quantization in the leading and next leading orders however we fail in both cases the reasons are discussed in detail

on the complexity of nonnegative matrix scaling an n n nonnegative matrix a is said to be doubly stochastic scalable if there exist two positive diagonal matrices x and y such that xay is doubly stochastic we derive an upper bound on the norms of the scaling factors x and y and give a polynomial time complexity bound on the problem of computing the scaling factors to a prescribed accuracy

theory and numerical evaluation of oddoids and evenoids oscillatory cuspoid integrals with odd and even polynomial phase functions the properties of oscillating cuspoid integrals whose phase functions are odd and even polynomials are investigated these integrals are called oddoids and evenoids respectively and collectively oddenoids we have studied in detail oddenoids whose phase functions contain up to three real parameters for each oddenoid we have obtained its maclaurin series representation and investigated its relation to airy hardy integrals and bessel functions of fractional orders we have used techniques from singularity theory to characterise the caustic or bifurcation set associated with each oddenoid including the occurrence of complex whiskers plots and short tables of numerical values for the oddenoids are presented the numerical calculations used the software package cuspint n p kirk j n l connor c a hobbs an adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives comput phys commun 132 2000 142 165

the uniform asymptotic swallowtail approximation practical methods for oscillating integrals with four coalescing saddle points practical methods for the numerical implementation of the uniform swallowtail approximation have been developed this approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points a complex contour quadrature technique has been used to evaluate the swallowtail canonical integral s x y z and its partial derivatives delta s delta x delta s delta y delta s delta z this method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained a comparison is made with other methods that have been reported in the literature for the evaluation of s x y z isometric plots of mod s x y z mod mod delta s delta x mod mod delta s delta y mod mod delta s delta z mod are presented and some properties of the zeros of s 0 y z that lie on the line y 0 are also discussed two methods for the evaluation of the mapping parameters x y z are described an iterative method that is valid when x y z is not close to the swallowtail caustic and an algebraic method valid for x y z on the caustic and for y 0 symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations in practice both methods for determining x y z are complementary an application of the uniform swallowtail approximation to the butterfly canonical integral has been made the uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

solving fredholm integral equations via a piecewise linear maximum entropy method we propose a piecewise linear approximation method based on the maximum entropy principle to approximate a nonnegative solution of a fredholm integral equation numerically the theoretical analysis and numerical examples show that our method has a convergence rate of order 2 and it can get more accurate approximations with more moments used without ill condition of the classic maximum entropy approach the method can also be applied to solve fredholm integral equations with singular kernels

on the asymptotics of some pearcey type integrals the author discusses the asymptotic behaviour of the pearcey type integrali alpha x y 2 sup approximately 0 ualpha 1 exp i u4 xu2 jalpha yu du for 1 alpha 5 2 where jalpha is a bessel function as x to or infinity y fixed as y to infinity x fixed and as y rho 2 3 mod x mod 32 x to infinity rho fixed the case alpha 1 2 gives the classical pearcey integral whose asymptotics has been investigated recently by kaminski 1989 and paris 1991 in the case alpha 0 i alpha x y as a function of y or 0 represents the radial part of the impulse response function describing the image formation in high resolution electron microscopes at normalized defocus x he uses the approach of paris by representing i alpha x y in terms of weber parabolic cylinder functions and he augments this approach by invoking the chester friedman ursell method 1957 to obtain the leading asymptotics of i alpha x y around the caustic y2 2 3 mod x mod 3 x to infinity

error bounds in maximum entropy approximations a useful technique in underdetermined inverse problems is that of maximum entropy a simple error bound for averages over a distribution approximated by the maximum entropy method in the case of the undetermined hausdorff moment problem was devised under the conditions specified the error bound for averages over such an approximate distribution can be very tight numerical examples to illustrate are presented

levy flights exact results and asymptotics beyond all orders a comprehensive study of the symmetric levy stable probability density function is presented this is performed for orders both less than 2 and greater than 2 the latter class of functions are traditionally neglected because of a failure to satisfy non negativity the complete asymptotic expansions of the symmetric levy stable densities of order greater than 2 are constructed and shown to exhibit intricate series of transcendentally small terms asymptotics beyond all orders it is demonstrated that the symmetric levy stable densities of any arbitrary rational order can be written in terms of generalized hypergeometric functions and a number of new special cases are given representations in terms of special functions a link is shown between the symmetric levy stable density of order 4 and pearcey s integral which is used widely in problems of optical diffraction and wave propagation this suggests the existence of applications for the symmetric levy stable densities of order greater than 2 despite their failure to define a probability density function

on an airy function of two variables abstract let z c be a function of x y c 2 defined by z c x y c exp t 4 4 xt 2 2 yt d t here c is a path of integration such that the integrand vanishes at its terminal points this is called the pearcey integral and may be regarded as an airy function of two variables it is known that z c x y satisfies a system of partial differential equations of the form x 2 u x 2 x u y 4 y u 1 4 u x y u x 2 y u y 4 u y 2 u 2 x u which possesses the singularities x and y of irregular type in the present paper the asymptotic behavior of suitably chosen linearly independent solutions of near y is studied

structure of avoided crossings for eigenvalues related to equations of heun s class we study the phenomenon of avoided crossings of eigenvalue curves for boundary value problems related to differential equations of heun s class the eigenvalues are given explicitly in asymptotic form taking into account power type as well as exponentially small terms it is exhibited that the phenomenon of avoided crossings of eigenvalue curves show a periodical structure in the sense that at any integer value of the additional controlling parameter an infinite in the sense of a large parameter number of avoided crossings take place simultaneously some relations to other phenomena of the asymptotics of exponentially small terms are discussed at the end of the article

the maximum entropy method applied to stationary density computation abstract the maximum entropy method maxent is widely used in the context of the moment problem which appears naturally in many branches of physics and engineering it is used to numerically recover the density with least bias from finitely many known moments we introduce the basic idea behind this method and apply this method to approximating fixed densities of markov operators in stochastic analysis and frobenius perron operators in ergodic theory of chaotic dynamics

boolean matrices with prescribed row column sums and stable homogeneous polynomials we prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications our main application is the first poly time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor this new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles

error bounds for maximum entropy approximate solutions to fredholm integral equations a new method of calculating a posteriori error bounds for maximum entropy approximate solutions to fredholm integral equations is introduced a numerical example confirming the efficacy of the method is also presented