fast evaluation of gaunt coefficients recursive approach solution of electromagnetic multisphere scattering problems involves the determination of a large number of gaunt coefficients that are connected with definite integrals of products of three associated legendre functions this paper presents general recursion formulae cycling a single index for the gaunt coefficient with the use of the recursive scheme gaunt coefficients of either low or high degree and order can be evaluated accurately and expeditiously

convergence properties of the sbar d transformation and a fast and accurate numerical evaluation of molecular integrals the present work concerns the study of the convergence properties of the nonlinear s transformation a fast and accurate numerical evaluation of highly oscillatory semi infinite integrals involved in the analytic expressions of molecular integrals over slater type orbitals is obtained using the s transformation these semi infinite integrals involve spherical and reduced bessel functions the convergence properties are analysed and they show that the approximations obtained using the s approach converge to the exact values of the semi infinite integrals without any constraint the numerical tables show that the s method gives unprecedented accuracy with speed up by a factor 2 over the h method and a factor 10 over the transformation this illustrates the superiority of this new approach

stability of semiclassical bound states of nonlinear schrodinger equations with potentials in this paper we study the lyapunov stabilities of some semiclassical bound states of the nonhomogeneous nonlinear schrodinger equation we prove that among those bound states those which are concentrated near local minima respectively maxima of the potentialv are stable respectively unstable we also prove that those bound states are positive if is sufficiently small

a general method for approximating tail probabilities abstract a new transformation referred to as the g m n transformation is introduced to establish a general methodology for finding functions that are easy to evaluate and give very good approximations to tail probabilities the transformation is based only on a general class of differential equations that include the specified density function in the solution set at least asymptotically as a result this new method applies to a broad class of distributions not only is the method general it is also accurate unlike many other approximation methods the approximation functions produced maintain their high degree of accuracy even in the extreme tails making them also suitable for extreme tail probability approximation in this article the method is applied to the pearson family members the inverse gaussian and the ratio of 2 and lognormal that do not belong to the pearson family furthermore it is applicable in the case of all of the standard noncentral distributions tables are included that sho

convergence of translation formulas for the computation of multicenter integrals over slater orbitals abstract multicenter electron repulsion integrals are calculated using auxiliary functions and two kinds of translation formulas for slater type orbitals stos obtained from the expansion of stos in terms of exponential type orbitals at a displaced center that form complete orthonormal sets and are represented by linear combinations of stos the convergence of the series for real stos is tested by calculating concrete cases accuracy of the results is quite high for quantum numbers screening constants and location of stos

analytical and numerical development for the two centre overlap like quantum similarity integrals over slater type functions in molecular quantum similarity measure between two molecules using the molecular electron density the major task involves the accurate numerical evaluation of overlap between the two electron densities in the integral measure by expanding the electron densities in terms of atomic orbitals using the linear combination of atomic orbitals approach a large number of overlap like quantum similarity integrals over the basis functions in the two molecules will be required accurately for the calculation of a meaningful quantum similarity measure improvement of the computational methods of these integrals would be indispensable to a further development in computational studies of large molecular systems analytic expressions were obtained for these overlap like quantum similarity integrals over slater type functions in terms of the usual two centre overlap integrals over b functions different approaches were developed for the numerical evaluation of these two centre overlap integrals over b functions which can be used for the numerical evaluation of the two centre overlap like quantum similarity integrals over slater type functions in this work we present our approach which is based on the use of nonlinear transformations for improving convergence of highly oscillatory integrals in the case of more complicated multicentre integrals this approach is shown to be able to produce remarkably good results we also present different representations which were obtained by steinborn group and which can be used for a fast and accurate computation of the two centre overlap like quantum similarity integrals over slater type functions

asymptotic expansions of legendre series coefficients for functions with interior and endpoint singularities abstract let n 0 en f pn x be the legendre expansion of a function f x on 1 1 in an earlier work a sidi asymptot anal 65 2009 pp 175 190 we derived asymptotic expansions as n for en f assuming that f c 1 1 but may have arbitrary algebraic logarithmic singularities at one or both endpoints x 1 in the present work we extend this study to functions f x that are infinitely differentiable on 0 1 except at finitely many points x1 xm in 1 1 and possibly at one or both of the endpoints x0 1 and xm 1 1 where they may have arbitrary algebraic singularities including finite jump discontinuities specifically we assume that for each r f x has asymptotic expansions of the form

a generalization of the bernstein polynomials abstract we give an interesting generalization of the bernstein polynomials we find sufficient and necessary conditions for uniform convergence by the new polynomials and we generalize the bernstein theorem

numerical integration of functions with a sharp peak at or near one boundary using mo bius transformations abstract a class of coordinate transformations depending on a single parameter is studied as quadrature tool working rules for the choice of the parameter are proposed numerical tests for the method are presented they show that these coordinate transformations when combined with gauss legendre quadrature rules are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration a method to combine the transformations with automatic quadrature routines is also proposed it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior

the numerical evaluation of very oscillatory infinite integrals by extrapolation recently the author has given two modifications of a nonlinear extrapolation method due to levin and sidi which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity in this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity the new method is applied to a number of complicated integrals among them the solution to a problem in viscoelasticity some convergence results for this method are presented

energy growth in schrodinger s equation with markovian forcing schrodinger s equation is considered on a one dimensional torus with time dependent potential v t v x t where v is an even trigonometric polynomial and x t is a stationary markov process it is shown that when the coupling constant is sufficiently small the average kinetic energy grows as the square root of time more generally the h s norm of the wave function is shown to behave as t s 4

the hd and hd methods for accelerating the convergence of three center nuclear attraction and four center two electron coulomb integrals over b functions and their convergence properties abstract three center nuclear attraction and four center two electron coulomb integrals over slater type orbitals are required for ab initio and density functional theory dft molecular structure calculations they occur in many millions of terms even for small molecules and require rapid and accurate evaluation the b functions are used as a basis set of atomic orbitals these functions are well adapted to the fourier transform method that allowed analytical expressions for the integrals of interest to be developed rapid and accurate evaluation of these analytical expressions is now made possible by applying the hd and hd methods for accelerating the convergence of the semi infinite oscillatory integrals the convergence properties of the new methods are analysed the numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods

extended procedures for extrapolation to the limit in this paper new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed they are based on the notions of error estimates and annihilation operators we obtain generalizations of the discrete and confluent e transformation which are the most general sequence and function transformations known so far it is shown that many transformations studied in the literature are included in our formalism particular cases of these procedures are discussed then several extensions of the e algorithm are given finally the procedure q is applied to our procedures to produce new procedures for extrapolation to the limit

sharp estimates for dirichlet eigenfunctions in horn shaped regions we prove sharp exponential decay for the dirichlet eigenfunctions in horn shaped regions in the plane the estimate is obtained using a method of carleman which is widely used in the study of harmonic measure such estimates can be applied to study the intrinsic ultracontractivity properties of the heat semigroup for such regions

boundary behaviour of dirichlet eigenfunctions of second order elliptic operators we investigate the rate of decay of solutions of a second order elliptic equation tf f in an arbitrary domain r n which satisfy zero boundary conditions and then apply the results to a study of the behaviour of the associated heat semigroup in various weighted l 2 spaces our results concern the behaviour at infinity and at points on the boundary of especially irregular ones like corners and cusps the decay rates we obtain are expressed in terms of weighted l 2 estimates which are governed by the sharpness of the cusps and the properties of the coefficients of also the meaning of the zero boundary condition at a cusp is addressed

two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series two new classes of nonlinear transformations the d transformation to accelerate the convergence of infinite integrals and the d transformation to accelerate the convergence of infinite series are presented in the course of the development of these transformations two interesting asymptotic expansions one for infinite integrals and the other for infinite series are derived the transformations d and d can easily be applied to infinite integrals 0f t dt whose integrands f t satisfy linear differential equations of the form f t s m k 1p k t f k t and to infinite series s r 1f r w terms f r satisfy a linear difference equation of the form f r s m k 1p k r d kf r such that in both cases the p k have asymptotic expansions in inverse powers of their arguments in order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy mere knowledge of the existence of such a differential or difference equation and its order m is enough this broadens the areas to which these methods can be applied the connection between the d and d transformations with some known transformations in shown the use and the remarkable efficiency of the d and d transformations are demonstrated through several numerical examples the computational aspects of these transformations are described in detail

a kohn sham equation solver based on hexahedral finite elements we design a kohn sham equation solver based on hexahedral finite element discretizations the solver integrates three schemes proposed in this paper the first scheme arranges one a priori locally refined hexahedral mesh with appropriate multiresolution the second one is a modified mass lumping procedure which accelerates the diagonalization in the self consistent field iteration the third one is a finite element recovery method which enhances the eigenpair approximations with small extra work we carry out numerical tests on each scheme to investigate the validity and efficiency and then apply them to calculate the ground state total energies of nanosystems c 6 0 c 1 2 0 and c 2 7 5h 1 7 2 it is shown that our solver appears to be computationally attractive for finite element applications in electronic structure study

an extremely efficient approach for accurate and rapid evaluation of three centre two electron coulomb and hybrid integrals over b functions analytic expressions of three centre two electron coulomb and hybrid integrals over b functions are obtained using the fourier transform method thoroughly explored by steinborn s group these analytic expressions involve semi infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical bessel functions in the integrands we have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the h approach which greatly simplifies the application of the nonlinear transformation this work presents a rapid and accurate evaluation of these integrals obtained by using a new approach which we called s this new method is based on the h and methods and some practical properties of spherical bessel reduced bessel and sine functions the s method has greatly simplified the calculations avoiding the long and difficult implementation of the successive zeros of the spherical bessel function and a method for solving linear systems which are required by h and

error estimates for hermite and even tempered gaussian approximations in quantum chemistry atomic like basis functions provide a natural physically motivated description of electronic states among which gaussian type orbitals are the most widely used basis functions in molecular simulations this paper aims at developing a systematic analysis of numerical approximations based on linear combinations of gaussian type orbitals we derive a priori error estimates for hermite type gaussian bases as well as for even tempered gaussian bases numerical results are presented to support the theory

new representations for the spherical tensor gradient and the spherical delta function in this article we analyze representations for the product ym1l1 fm2l2 r with yml specifying a solid harmonic whose argument is the nabla operator r instead of the vector r since both ym1l1 and fm2l2 r are irreducible spherical tensors we can use angular momentum algebra for evaluating the product accordingly the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product analytical expressions for these radial functions are derived by direct differentiation and with the help of fourier transforms closely related to the spherical tensor gradient yml is the spherical delta function ml r we derive new representations for ml by considering convolution integrals involving b functions these functions are closely related to the modified bessel functions and also to the yukawa potential e r r we show that the definition of the b functions can be extended to include a large class of derivatives of the delta func

efficient evaluation of coulomb integrals the nonlinear d and transformations the two electron multicentre coulomb integrals constitute the rate limiting step of ab initio and density functional theory dft molecular structure calculations speed up can be achieved by limiting the number of integrals to evaluate analytically but these analytical evaluations remain rate limiting for large molecules here we apply the nonlinear d and transformations to evaluate coulomb integrals over b functions more rapidly than the alternative transformation methods to a given predetermined high accuracy

efficient evaluation of vector translation coefficients in multiparticle light scattering theories vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle scattering problems these theorems include a large number of vector addition coefficients there exist three basic types of analytical expressions for vector translation coefficients stein s quart appl math 19 15 1961 cruzan s quart appl math 20 33 1962 and xu s j comput phys 127 285 1996 stein s formulation relates vector translation coefficients with scalar translation coefficients cruzan s formulas use the wigner 3jm symbol xu s expressions are based on the gaunt coefficient since the scalar translation coefficient can also be expressed in terms of the gaunt coefficient the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the wigner 3jm symbol or the gaunt coefficient we present highly efficient recursive approaches to accurately evaluating wigner 3jm symbols and gaunt coefficients armed with these recursive approaches we discuss several schemes for the calculation of the vector translation coefficients using the three general types of formulation respectively our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of stein s type is less accurate in some particular cases these extensive test calculations also show that the scheme using the formulation based on the gaunt coefficient is the most efficient in practical computations

sparse pca optimal rates and adaptive estimation principal component analysis pca is one of the most commonly used statistical procedures with a wide range of applications this paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting under mild technical conditions we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate the lower bound is obtained by calculating the local metric entropy and an application of fano s lemma the rate optimal estimator is constructed using aggregation which however might not be computationally feasible we then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently it is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces a key idea in our construction is a reduction scheme which reduces the sparse pca problem to a high dimensional multivariate regression problem this method is potentially also useful for other related problems

q bernstein polynomials of the cauchy kernel due to the fact that in the case q 1 q bernstein polynomials are not positive linear operators on c 0 1 the study of their approximation properties is essentially more difficult than that for 0 1 is still open in this paper the q bernstein polynomials bn q fa z of the cauchy kernel fa 1 z a a c 0 1 are found explicitly and their properties are investigated in particular it is proved that if q 1 then polynomials bn q fa z converge to fa uniformly on any compact set k z z a the sequence bn q fa z is not even uniformly bounded

convergence analysis for a generalized richardson extrapolation process with an application to the d 1 transformation on convergent and divergent logarithmic sequences in an earlier work by the author the generalized richardson extrapolation process grep was introduced and some of its convergence and stability properties were discussed in a more recent work by the author a special case of grep which we now call grep 1 was considered and its properties were reviewed with emphasis on oscillatory sequences in the first part of the present work we give a detailed convergence and stability analysis of grep 1 as it applies to a large class of logarithmic sequences both convergent and divergent in particular we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table these theorems are very realistic in the sense that they explain the remarkable efficiency of grep 1 in a very precise manner in the second part we apply this analysis to the levin sidi d 1 transformation as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically or to sum the divergent extensions of such series this is made possible by the observation that when the proper analogy is drawn the d 1 transformation is in fact a grep 1 we append numerical examples that demonstrate the theory

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

a new algorithm for accurate and fast numerical evaluation of hybrid and three centre two electron coulomb integrals over slater type functions recently we developed a new algorithm for a fast and accurate numerical evaluation of three centre nuclear attraction integrals over slater type functions the results obtained were very satisfactory now it is shown that this new algorithm can also be applied to hybrid and three centre two electron coulomb integrals over slater type functions these integrals which are numerous are very difficult to evaluate to a high accuracy because of the presence of spherical bessel functions and hypergeometric series in the integrands we have proved that the integrands that occur in the analytic expressions of the integrals under consideration satisfy all the conditions to apply the approach the hypergeometric functions which occur in the semi infinite integrals can be expressed as a finite expansion and the semi infinite integrals involving the spherical bessel functions can be transformed into semi infinite integrals involving the simple sine function the numerical results obtained with linear and non linear systems illustrate clearly a further improvement of accuracy and a substantial reduction in calculation times comparisons with existing codes stop developed by bouferguene et al 1996 int j quantum chem 57 801 and adggstngint developed by rico et al 1997 comp phys commun 105 216 are listed

nonparametric empirical bayes and compound decision approaches to estimation of a high dimensional vector of normal means we consider the classical problem of estimating a vector 1 n based on independent observations y i n i 1 i 1 n suppose i i 1 n are independent realizations from a completely unknown g we suggest an easily computed estimator such that the ratio of its risk e 2 with that of the bayes procedure approaches 1 a related compound decision result is also obtained our asymptotics is of a triangular array that is we allow the distribution g to depend on n thus our theoretical asymptotic results are also meaningful in situations where the vector is sparse and the proportion of zero coordinates approaches 1 we demonstrate the performance of our estimator in simulations emphasizing sparse setups in moderately sparse situations our procedure performs very well compared to known procedures tailored for sparse setups it also adapts well to nonsparse situations

positive solutions to schrodinger equations and geometric applications a variant of li tam theory which associates to each end of a complete riemannian manifold a positive solution of a given schrodinger equation on the manifold is developed it is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant sobolev inequality consequently a finiteness result for the number of ends follows in the case when the sobolev inequality is of particular type the finiteness result is proven directly as an application an estimate on the number of ends for shrinking gradient ricci solitons and submanifolds of euclidean space is obtained

a complexity analysis of the gauss bessel quadrature as applied to the evaluation of multi centre integrals over stfs in a previous work bouferguene 2005 j phys a math gen 38 3923 we have shown that in the framework of the gaussian integral transform multi centre integrals over slater type functions can be evaluated to an acceptable accuracy using a tailored gauss quadrature in which the weight function has the form w z z exp z z to be considered a solution worth implementing within a software for routine use in ab initio molecular simulations the method must also prove to be at least as efficient as those methods previously published in the literature two major results are provided in this paper firstly an improvement of the procedure used to generate the roots and weights of the gauss bessel quadrature is proposed secondly a computational cost analysis of the present method and the safouhi 2001 j phys a math gen 34 2801 based approach are compared hence proving the equivalence of the two from a complexity point of view

harmonic approximation of difference operators for a general class of difference operators he te ve on l2 ez d where ve is a multi well potential and e is a small parameter we analyze the asymptotic behavior as e 0 of the low lying eigenvalues and eigenfunctions we show that the first n eigenvalues of he converge to the first n eigenvalues of the direct sum of harmonic oscillators on rd located at the several wells our proof is microlocal

fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices abstract we present a unified treatment of the fourier spectra of spherically symmetric nonlocal diffusion operators we develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source and target tends to 0 rapid algorithms are derived for their fourier spectra with the computation of each eigenvalue independent of all others the algorithms are trivially parallelizable capable of leveraging more powerful compute environments and the accuracy of the eigenvalues is individually controllable the algorithms include a maclaurin series and a full divergent asymptotic series valid for any d spatial dimensions using drummond s sequence transformation we prove linear complexity recurrence relations for degree graded sequences of numerators and denominators in the rational approximations to the divergent asymptotic series these relations are important to ensure that the algorithms are efficient and also increase the numerical stability compared with the conventional algorithm with quadratic complexity

addition theorems for b functions and other exponentially declining functions in this paper addition theorems are derived for a special class of exponentially declining functions the so called b functions bmn l r e filter and e o steinborn phys rev a 18 1 1978 although these b functions have a relatively complicated analytical structure they nevertheless have some mathematical properties that are particularly advantageous in connection with multicenter problems also all the commonly occurring exponentially declining functions like for instance bound state hydrogen eigenfunctions and slater type functions can be expressed by simple finite sums of b functions consequently addition theorems for these functions can also be written down immediately the various addition theorems for b functions are derived by applying suitable generating differential operators to the well known addition theorem of the special b function bm l l which is that solution of the modified helmholtz equation that is irregular at the origin and regular at infinity e j weniger and e o

a numerical evaluation of very oscillatory infinite integrals by asymptotics recently levin and sidi have given some nonlinear methods for the accurate evaluation of slowly converging infinite integrals but one needs to solve some very instable linear systems when for example the integrand have an increasingly rapid oscillatory behavior at infinity in this work we present a direct method which avoids the instable linear systems

a new gauss quadrature for multicentre integrals over stos in the gaussian integral transform approach when computing multicentre integrals over slater type orbitals stos by means of the shavitt and karplus gaussian integral transforms shavitt and karplus 1962 j chem phys 36 550 one usually ends up with a multiple integral of the form shavitt and karplus 1965 j chem phys 43 398 in which all the integrals are inter related the most widely used approach for computing such an integral is to apply a product of gauss legendre quadratures for the integrals over 0 1 while the semi infinite term is evaluated by a special procedure although numerous approaches have been developed to accurately perform the integration over 0 efficiently it is the aim of this work to add a new tool that could be of some benefit in carrying out the hard task of multicentre integrals over stos the new approach relies on a special gauss quadrature referred to as gauss bessel to accurately evaluate the semi infinite integral of interest in this work emphasis is put on accuracy rather than efficiency since its aim is essentially to bring a proof of concept showing that gauss bessel quadrature can successfully be applied in the context of multicentre integrals over stos the obtained accuracy is comparable to that obtained with other methods available in the literature

an extremely efficient and rapid algorithm for numerical evaluation of three centre nuclear attraction integrals over slater type functions the present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of the three centre nuclear attraction integrals over slater type functions these integrals are numerous they occur in many millions of terms even for small molecules and they require rapid and accurate evaluation the new algorithm is based on nonlinear transformation methods on numerical quadrature and on properties of the sine and bessel functions the section with numerical results shows the high accuracy and the substantial gain in calculation time realized using the new algorithm the complete expressions of the three centre nuclear attraction integrals over b functions and over slater type functions are evaluated for different values of the quantum numbers to show the efficiency of the new approach numerical results obtained with linear and nonlinear systems and comparisons with numerical results from the literature are listed

on the analyticity of functions approximated by their q bernstein polynomials when q 1 abstract since in the case q 1 the q bernstein polynomials bn q are not positive linear operators on c 0 1 the investigation of their convergence properties for q 1 turns out to be much harder than the one for 0 1 create a serious obstacle for the numerical experiments with the q bernstein polynomials despite the intensive research conducted in the area lately the class of functions which are uniformly approximated by their q bernstein polynomials on 0 1 is yet to be described in this paper we prove that if f 0 1 c is analytic at 0 and can be uniformly approximated by its q bernstein polynomials q 1 on 0 1 then f admits an analytic continuation from 0 1 into z z

vector sequence transformations for the acceleration of logarithmic convergence logarithmic convergence of scalar sequences is generalized to vector sequences some sequence transformations for logarithmically convergent scalar sequences are also extended to the vector case theoretical results and numerical examples are given

the shape resonance for a class of schrodinger operatorsh 2 2m v onl 2 n with potentials having minima embedded in the continuum of the spectrum and non trapping tails we show the existence of shape resonances exponentially close to the real axis as 0 the resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling

multicentre two electron coulomb and exchange integrals over slater functions evaluated using a generalized algorithm based on nonlinear transformations when calculating molecular electronic energies the contributions involving the coulomb operator for bielectronic terms are required rapidly and to high chemically significant accuracy the atomic orbital basis functions chosen in the present work are slater type functions stfs these functions can be expressed as finite linear combinations of b functions which are suitable to apply the fourier transform method the difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of two or three dimensional integral representations in this work we present a generalized algorithm for a precise and fast numerical evaluation of molecular integrals over stfs numerical results obtained with c2h2 c2h4 and ch4 molecules show the efficiency of the approach presented in this work comparisons with the existing codes are also listed

ground states in non relativistic quantum electrodynamics the excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay but such a particle should have a true ground state one that minimizes the energy and satisfies the schrodinger equation we prove quite generally that this state exists for all values of the fine structure constant and the ultraviolet cutoff we also show the same thing for a many particle system under physically natural conditions

the norm estimates for the q bernstein operator in the case q 1 the q bernstein basis with 0 1 the behavior of the q bernstein basic polynomials on 0 1 combines the fast increase in magnitude with sign oscillations this seriously complicates the study of q bernstein polynomials in the case of q 1 the aim of this paper is to present norm estimates in c 0 1 for the q bernstein basic polynomials and the q bernstein operator b n q in the case q 1 while for 0 1 the norm b n q increases rather rapidly as n oo we prove here that b n q c q q n n 1 2 n n oo with c q 2 q 2 q 2 e such a fast growth of norms provides an explanation for the unpredictable behavior of q bernstein polynomials q 1 with respect to convergence

fast evaluation of the gaunt coefficients addition theorems for vector spherical harmonics require the determination of the gaunt coefficients that appear in a linearization expansion of the product of two associated legendre functions this paper presents an algorithm for the efficient calculation of these coefficients through solving the most appropriate lower triangular linear system and derives all relevant recurrence relations needed in the calculation this algorithm is also applicable to the calculation of the clebsch gordan coefficients that are closely related to the gaunt coefficients and are frequently encountered in the quantum theory of angular momentum the new method described in this paper reduces the computing time to 1 compared to the existing formulation that is widely used this new method can be applied to the calculation of both low and high degree coefficients while the existing formulation works well only for low degrees

optimal estimation of bacterial growth rates based on a permuted monotone matrix motivated by the problem of estimating the bacterial growth rates for genome assemblies from shotgun metagenomic data we consider the permuted monotone matrix model y theta pi z where y in mathbb r n times p is observed theta in mathbb r n times p is an unknown approximately rank one signal matrix with monotone rows pi in mathbb r p times p is an unknown permutation matrix and z in mathbb r n times p is the noise matrix this paper studies the estimation of the extreme values associated to the signal matrix theta including its first and last columns as well as their difference treating these estimation problems as compound decision problems minimax rate optimal estimators are constructed using the spectral column sorting method numerical experiments through simulated and synthetic microbiome metagenomic data are presented showing the superiority of the proposed methods over the alternatives the methods are illustrated by comparing the growth rates of gut bacteria between inflammatory bowel disease patients and normal controls

the properties of sine spherical bessel and reduced bessel functions for improving convergence of semi infinite very oscillatory integrals the evaluation of three centre nuclear attraction integrals over b functions this paper focuses on the use of useful properties of sine spherical bessel and reduced bessel functions to simplify the application of the nonlinear d and transformations for accelerating the convergence of semi infinite very oscillatory integrals and to reduce the calculation times keeping a high predetermined accuracy three centre nuclear attraction integrals which are one of the most difficult type involved in density functional theory methods when using a basis set of b functions are evaluated using the new approach the numerical results show the efficiency of the new method compared with other alternatives

the hyperbolic cross space approximation of electronic wavefunctions the electronic schrodinger equation describes the motion of electrons under coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry the present article continues the author s recent work numer math 98 731 759 2004 on the regularity of its solutions the electronic wavefunctions it was shown in the mentioned article that these wavefunctions possess square integrable high order mixed weak derivatives as they are needed to justify and underpin sparse grid and hyperbolic cross space approximation techniques theoretically how fast the norms of these derivatives can increase with the number of electrons is studied in the present article provided that all quantities are properly related to a characteristic lengthscale of the considered atomar or molecular system the norms of the derivatives remain bounded independent of the number of electrons and the number the positions and the charges of the nuclei

the fourier transform method and the sd approach for the analytical and numerical treatment of multicenter overlap like quantum similarity integrals molecular overlap like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by dirac delta function when the electronic densities are expanded over atomic orbitals using the usual lcao mo approach linear combination of atomic orbitals overlap like quantum similarity integrals could be expressed in terms of four center overlap integrals it is shown that by introducing the fourier transform of delta dirac function in the integrals and using the fourier transform approach combined with the so called b functions one can obtain analytic expressions of the integrals under consideration these analytic expressions involve highly oscillatory semi infinite spherical bessel functions which are the principal source of severe numerical and computational difficulties in this work we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap like quantum similarity integrals over slater type functions this algorithm is based on the s d approach due to safouhi recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm the numerical result section shows the efficiency of our algorithm compared with the alternatives using the one center two range expansion method which led to very complicated analytic expressions the epsilon algorithm and the nonlinear d transformation

the saturation of convergence on the interval 0 1 for the q bernstein polynomials in the case q 1 abstract in the note we consider saturation of convergence on the interval 0 1 for the q bernstein polynomials of a continuous function f for arbitrary fixed q 1 we show that the rate of uniform convergence on 0 1 is o q n if and only if f is linear the result is sharp in the following sense it ceases to be true if we replace o by o

sparse grid spaces for the numerical solution of the electronic schrodinger equation this article complements the author s recent work numer math 98 731 759 2004 on the regularity of the electronic schrodinger equation in hilbert spaces of mixed derivatives it has been shown there that the solutions of this equation are surprisingly smooth and possess square integrable mixed weak derivatives of order up to n 1 with n the number of electrons across the singularities of the interaction potentials and it has been claimed that this result can help to break the complexity barriers in computational quantum mechanics using correspondingly antisymmetrized sparse grid trial functions a construction of this kind that can be interpreted as a sparse grid sampling theorem is sketched here

new formulae for higher order derivatives and applications we present new formulae the slevinsky safouhi formulae i and ii for the analytical development of higher order derivatives these formulae which are analytic and exact represent the kth derivative as a discrete sum of only k 1 terms involved in the expression for the kth derivative are coefficients of the terms in the summation these coefficients can be computed recursively and they are not subject to any computational instability as examples of applications we develop higher order derivatives of legendre functions chebyshev polynomials of the first kind hermite functions and bessel functions we also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations we also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical bessel integral functions

three centre two electron coulomb and hybrid integrals evaluated using nonlinear d and bard transformations of the two electron integrals occuring in the molecular context the three centre coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work accelerates the convergence of these integrals by first manipulating the indices of the hypergeometric function and exploiting relationships to express this function as a finite expansion and exploiting the properties of bessel functions which satisfy second order linear differential equations this is a suitable form of the integrand to apply the nonlinear d due to d levin and a sidi and due to a sidi transformations the extensive numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals

on condition numbers of the shanks transformation abstract quasi linear functions generate sequence transformation methods whose conditioning depends upon the nature of the sequence to be accelerated these methods are often well conditioned when they are applied to alternating sequences however they are relatively ill conditioned in case of monotonic convergence the condition numbers of the shanks transformation e k s n are given in order to prove that the closely related e algorithm to a such transformation is ill conditioned when performed on the set of totally monotonic sequences in the same way we show that this algorithm is well conditioned on the set of totally oscillating sequences

on the asymptotics of nodes ofl2 solutions of schrodinger equations in dimensions 3 let r n br wheren 3 andbr x n x r we investigate the asymptotics of real valued solutions l2 r of the schrodinger equation v e 0 wheree 0 begin gathered mathop lim inf limits r to infty r gamma int limits s r psi 2 d sigma int limits s r psi 2 d sigma 0 and hfill mathop lim inf limits r to infty ln volume d cap b r ln r geqq n 1 2 hfill end gathered results of this type are already non trivial for radial problems with satisfying non radial boundary conditions on r or for excited states of the hydrogen atom if one considers linear combinations of differentl waves

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

optimal hardy weight for second order elliptic operator an answer to a problem of agmon abstract for a general subcritical second order elliptic operator p in a domain r n or noncompact manifold we construct hardy weight w which is optimal in the following sense the operator p w is subcritical in for all 1 null critical in for 1 and supercritical near any neighborhood of infinity in for any 1 moreover if p is symmetric and w 0 then the spectrum and the essential spectrum of w 1 p are equal to 1 and the corresponding agmon metric is complete our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators the constructed hardy weight is given by an explicit simple formula involving two distinct positive solutions of the equation p u 0 the existence of which depends on the subcriticality of p in

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions

optimal permutation recovery in permuted monotone matrix model motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies we consider a permuted monotone matrix model y z where the rows represent different samples th

comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences abstract let log be the set of logarithmic sequences i e of convergent sequences sn satisfying lim e n 1 e n 1 where en sn s and s limsn this paper is devoted to the comparison of four nonlinear sequence transforms on some subsets lf r of log for any positive integer r lf r denotes the set of fixed point sequences sn whose associated error sequences en sn s s limsn have an asymptotic expansion of the following type en 1 en i 1 ien1 ir where 1 c 0 r 1 2 r 1 2 1 2 is different from zero for sn lf r there holds e n a 1 n 1 r o n 1 1 r log n with a 1 r 1 1 r the four are respectively modifications of the algorithm and of the iterated 2 transform the iterated lubkin s transform and the algorithm of brezinski all of them accelerate the convergence of sequences in lf r we give accurate results concerning the asymptotic behaviour of transformed sequences we also study two algorithms combining the second and third transforms numerical examples are given to illustrate theoretical results

efficient and rapid numerical evaluation of the two electron four center coulomb integrals using nonlinear transformations and useful properties of sine and bessel functions two electron four center coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations millions of such integrals are required for molecules of interest therefore rapidity is the primordial criterion when the precision has been reached this work presents an extremely efficient approach for improving convergence of semi infinite very oscillatory integrals based on the nonlinear d transformation and some useful properties of spherical bessel reduced bessel and sine functions the new method is now shown to be applicable to evaluating the two electron four center coulomb integrals over b functions the section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest

analysis of radiative scattering for multiple sphere configurations an analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented the analysis builds upon the previously developed superposition solution in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble the addition theorems for vector spherical harmonics which transform harmonics from one coordinate system into another are rederived and simple recurrence relations for the addition coefficients are developed the relations allow for a very efficient implementation of the order of scattering solution technique for determining the scattered field coefficients for each sphere prediction of the radiative absorption and scattering characteristics of small particles is important to researchers in a number of fields e g atmospheric modelling analysis of radiative transfer from flames and development of nonintrusive laser based optical diagnostic methods computation of the radiative characteristics of spherical particles from lorenz mie theory is practically a trivial matter due to the existence of efficient computer codes bohren huffman 1983 however it is not unusual to encounter situations in which the individual particles while spherical in shape are so close together that the isolated sphere assumption inherent in lorenz mie theory is questionable a common example is soot formed in combustion processes electron micrographs of the individual soot particles reveal them to be agglomerates of a large number of primary spherical particles dobbins

regular article efficient and rapid evaluation of three center two electron coulomb and hybrid integrals using nonlinear transformations among the two electron integrals occurring in the molecular context the three center coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work pursues the acceleration of convergence for three center two electron coulomb and hybrid integrals we have proven that the hypergeometric function can be expressed as a finite expansion and that the integrand involving this series and a product of bessel functions satisfies a linear differential equation with coefficients having a power series expansion in the reciprocal of the variable suitable for application of the nonlinear d and d transformations these transformations depend strongly on the order of the differential equation that the integrand of interest satisfies this work concentrates on reduction of this order to two exploiting properties of spherical and reduced bessel functions leading to greatly simplified calculations to evaluate the integrals precisely by reducing the order of linear systems to be solved it also avoids the long and difficult implementations of successive derivatives of the integrands the numerical results section illustrates clearly the reduction of the calculation time we obtained for a high predetermined accuracy

unified analytical treatment of multicentre electron attraction electric field and electric field gradient integrals over slater orbitals the new central and noncentral potential functions cpfs and ncpfs of a molecule depending on the coordinates of the nuclei are introduced using complete orthonormal sets of exponential type orbitals etos introduced by the author the series expansion formulae for the multicentre electronic attraction ea electric field ef and electric field gradient efg integrals over slater type orbitals stos in terms of cpfs and ncpfs are derived the relationships obtained are valid for the arbitrary location quantum numbers and screening constants of stos

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

acceleration of convergence of a family of logarithmically convergent sequences the asymptotic behavior of several sequence transformations is in vestigated as n oo when applied to a certain family of logarithmically convergent sequences the transformations considered are the iterations of the transformations e an of shanks and wn of lubkin the algorithm of brezinski the levin u and u transforms and generalizations of the p algorithm and the neville table computational results are given for both real and complex sequences

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method

numerical analysis 2000 vol ii interpolation and extrapolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the rst 75 years of this century including a pioneering paper by kronecker in the 19th century c 2000 elsevier science b v all rights reserved