a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

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

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

semiclassical asymptotics of perturbed cat maps we derive an exact representation for tr u n where u is the quantum propagator associated with an anosov perturbed cat map this takes the form of a sum over the fixed points of the n th iterate of the classical transformation the contribution of each one being given by an n fold multiple integral we focus in particular on the case when n 1 an asymptotic evaluation of the integral in question then leads to a complete semiclassical series expansion the first term of which corresponds to the gutzwiller tabor trace formula it is demonstrated that this series diverges but that summing it down to its least term provides an approximation to the quantum trace that is exponentially accurate in 1 a simple universal approximation to the late terms is then derived this explains the divergence of the semiclassical expansion in terms of complex tunnelling periodic orbits and implies the existence of unusual relations between different orbit actions it also allows us to recover the semiclassical contributions from the complex orbits explicitly using borel resummation these exponentially subdominant terms are shown to exhibit the stokes phenomenon which causes them to depend sensitively on the size of the perturbation parameter finally we develop an alternative expansion based on the orbits of the unperturbed cat map rather than diverging this is shown to converge absolutely thus making possible an exact calculation of the quantum trace using only classical mechanics its properties are however distinctly anti semiclassical

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

superadiabatic tracking of quantum evolution the adiabatic quantum evolution of a two state system without energy level crossings is an example of the stokes phenomenon in the latter a small subdominant exponential is an asymptotic expansion appears when a stokes line is crossed truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth compact and universal manner across the stokes line in quantum evolution this corresponds to a smooth transition universal in form between superadiabatic basis states high order wkb approximate solutions of the time dependent schrodinger equation the authors give a numerical demonstration of this previously predicted universality by constructing for two hamiltonians the superadiabatic quantum bases asymptotic to the actual evolving state universality when a stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term

extremal and optimal properties of b bases collocation matrices totally positive matrices are related with the shape preserving representations of a space of functions the normalized b basis of the space has optimal shape preserving properties bernstein polynomials b splines and rational bernstein bases are examples of normalized b bases it is proven that the minimal eigenvalue and singular value of a collocation matrix of a normalized b basis is bounded below by the minimal eigenvalue and singular value of the corresponding collocation matrix of any normalized totally positive basis of the same space the optimal conditioning for the infty norm of a collocation matrix of a normalized b basis among all the normalized totally positive bases of a space of functions is also shown numerical examples confirm the theoretical results and answer related questions

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

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

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

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

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

bubble simulations with an interface tracking technique based on a partitioned fluid structure interaction algorithm numerical techniques frequently used for the simulation of one bubble can be classified as interface tracking techniques and interface capturing techniques most of these techniques calculate both the flow around the bubble and the shape of the interface between the gas and the liquid with one code in this paper a rising axisymmetric bubble is simulated with an interface tracking technique that uses separate codes to determine the position of the gas liquid interface and to calculate the flow around the bubble the grid converged results correspond well with the experimental data the gas liquid interface is conceived as a zero mass zero thickness structure whose position is determined by the liquid forces a uniform gas pressure and surface tension iterations between the two codes are necessary to obtain the coupled solution of both problems and these iterations are stabilized with a fluid structure interaction fsi algorithm the flow around the bubble is calculated on a moving mesh in a reference frame that rises at the same speed as the bubble the flow solver first updates the mesh throughout the liquid domain given a position of the gas liquid interface and then calculates the flow around the bubble it is considered as a black box with the position of the gas liquid interface as input and the liquid forces on the interface as output during the iterations a reduced order model of the flow solver is generated from the inputs and outputs of the solver the solver that calculates the interface position uses this model to adapt the liquid forces on the gas liquid interface during the calculation of the interface position

stokes s phenomenon for superfactorial asymptotic series superfactorial series depending on a parameter z are those whose terms a n z grow faster than any power of n if the terms get smaller before they increase the function f z represented by sum 0 infty a n z will exhibit a stokes phenomenon similar to that occurring in asymptotic series whose divergence is merely factorial across stokes lines in the z plane where the late terms all have the same phase a small exponential switches on in the remainder when the series is truncated near its least term the jump is smooth and described by an error function whose argument has a slightly more general form than in the factorial case this result is obtained by a method which is heuristic but applies to superfactorial series where borel summation fails several examples are given including an analytical interpretation of the sum and a numerical test of the error function formula for the function represented by f z sum 0 infty exp n 2 a 2nz where a gg 1

topological expansion and exponential asymptotics in 1d quantum mechanics borel summable semiclassical expansions in 1d quantum mechanics are considered these are the borel summable expansions of fundamental solutions and of quantities constructed with their help an expansion called topological is constructed for the corresponding borel functions this allows us to study the borel plane singularity structure in a systematic way examples of such structures are considered for linear harmonic and anharmonic potentials together with the best approximation provided by the semiclassical series the exponentially small contributions completing the approximation are considered a natural method of constructing such exponential asymptotics based on the borel plane singularity structures provided by the topological expansion is developed the method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator

gevrey asymptotics and stieltjes transforms of algebraically decaying functions the development of asymptotic expansions of stieltjes transforms of exponentially decaying functions has been well established in this paper we are concerned with the more difficult case in which the functions decay only algebraically at infinity by using a gevrey type condition we obtain an exponentially improved asymptotic expansion and give three representation theorems to show that the stieltjes transform of algebraically decaying functions can be written as the difference of two integral transforms with exponentially decaying kernels thus making the asymptotic theory developed for integral transforms with exponentially decaying kernels relevant to stieltjes transforms of algebraically decaying functions including the smoothing of the 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

thin layer solutions of the helmholtz and related equations this paper concerns a certain class of two dimensional solutions to four generic partial differential equations the helmholtz modified helmholtz and convection diffusion equations and the heat conduction equation in the frequency domain and the connections between these equations for this particular class of solutions specifically we consider thin layer solutions valid in narrow regions across which there is rapid variation in the singularly perturbed limit as the coefficient of the laplacian tends to zero for the well studied helmholtz equation this is the high frequency limit and the solutions in question underpin the conventional ray theory wkb approach in that they provide descriptions valid in some of the regions where these classical techniques fail examples are caustics shadow boundaries whispering gallery and creeping waves and focusing and bouncing ball modes it transpires that virtually all such thin layer models reduce to a class of generalized parabolic wave equations of which the heat conduction equation is a special case moreover in most situations we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the helmholtz equation we also show how reasonably well understood thin layer phenomena associated with any one of the four generic equations may translate into less well known effects associated with the others in addition our considerations also shed some light on the relationship between the methods of matched asymptotic wkb and multiple scales expansions

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

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

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

normalized b basis of the space of trigonometric polynomials and curve design a normalized b basis of the space of trigonometric polynomials of degree n is presented some interesting properties of the basis functions are given based on the basis symmetric trigonometric polynomial curves like bezier curves are constructed the trigonometric polynomial curves present the shape of their control polygons well thus the theoretics and methods are proposed for curve representation of the trigonometric polynomial space by adding additional control points the given graph examples show that the trigonometric polynomial curves are nearer their control polygons than the bezier curves for the same parametric variable and the same degree

on the asymptotic and numerical solution of linear ordinary differential equations an investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank we introduce a classification of the solutions into two types explicit and implicit for the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached no such sector exists for implicit solutions in consequence the two types of solution have different uniqueness properties another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions we also investigate the computation of the solutions by numerical integration of the differential equation it is shown that it is the exception rather than the rule for the integration process to be stable particularly so for implicit solutions to overcome these instabilities we develop boundary value methods complete with error analysis numerical examples illustrate the computation of both explicit and implicit solutions and also the associated stokes multipliers

a stable recurrence for the incomplete gamma function with imaginary second argument even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction for an imaginary second argument there can be a considerable loss of correct digits before stability sets in we present an approach to compute the recurrence relation to full precision also for small values of the arguments when the first argument is negative and the second one is purely imaginary a detailed analysis shows that this approach works well for all values considered

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

accurate eigenvalues and exact zero jordan blocks of totally nonnegative matrices we present a new algorithm which given a bidiagonal decomposition of a totally nonnegative matrix computes all its eigenvalues to high relative accuracy in floating point arithmetic in o n 3 time it also computes exactly the jordan blocks corresponding to zero eigenvalues in up to o n 4 time