performance comparison of numerical inversion methods for laplace and hankel integral transforms in engineering problems abstract different methods for the numerical evaluations of the inverse laplace and inverse of joint laplace hankel integral transforms are applied to solve a wide range of initial boundary value problems often arising in engineering and applied mathematics the aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half space to concentrated impulse or those related to viscoelastic wave motion in layered media these classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations theoretical seismology soil structure interaction in time domain and so on therefore their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems some different methods are also investigated in detail and compared for the inversion of the joint hankel laplace transforms where more sophisticated integrand functions are encountered it is shown that durbin crump d amore fixed talbot gaver whyn rho gwr and direct integration methods have excellent performance and produce good results when applied to the same problems on the contrary gaver stehfest and piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for simple transformed functions particularly the performance of gwr algorithm is very good even for transformed functions with infinite number of singularities where the other methods fail in addition in case of double integral transforms only the fixed talbot durbin and weeks methods are recommended

choice of specific interpolation points in quadrature formula for integrals containing bessel functions a method for calculation of integrals containing bessel functions is suggested which based on linear interpolation applied to the function entering the integral along with the bessel function a specific choice of interpolation points allows excluding the contribution into integral of all interpolation points besides the first one the study of errors leads to determination of bounds of the considered integral

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

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

computation of infinite integrals involving bessel functions of arbitrary order by the d transformation abstract the d transformation due to the author is an effective extrapolation method for computing infinite oscillatory integrals of various kinds in this work two new variants of this transformation are designed for computing integrals of the form a g t c v t d t where g x is a nonoscillatory function and c v x may be an arbitrary linear combination of the bessel functions of the first and second kinds jv x and yv x of arbitrary real order v when applied to such integrals the d transformation and its new variants are observed to produce very accurate results it is also seen that their performance is very similar to that of the modified w transformation due to the author as extended in a recent work by lucas and stone with c v x j v x the present paper is concluded by stating the relevant convergence and stability results and by appending a numerical example

simple expressions for radial functions appearing in the expansions of ym1l1 f m2l2 r and 2nym1l1 f m2l2 r in this paper simple analytic expressions for the radial functions r ll1l2 f r and glnl1l2 f r which appear in the expressions for ym1l1 fl2 r ym2l2 r and 2nym1l1 fl2 r ym2l2 r when expressed as linear combinations of yml r where ym1l1 is obtained from the solid harmonic ym1l1 r r l1ym1l1 r are derived by replacing x y and z in its representation as a polynomial of degree l1 by x y and z respectively writing r ll1l2 f r r ll1l20 r f r and glnl1l2 f r glnl1l20 r f r the expression for r ll1l20 r glnl1l20 r is found to be 2 r l1 2 r l1 2n times a product of commuting factors containing the single operator 1 2 r d dr and not two noncommuting operators r and 1 r d dr in the many equivalent previously obtained results which are thus synthesized in this approach also expressions for these operators manifest the symmetries in the problem in addition a simple connection glnl1l20 r r ll1 2nl20 r between these two operators and the c

mahler measures short walks and log sine integrals the mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years very few closed forms are proven but more are conjectured in the case of multiple mahler measures more tractable but interesting families exist using values of log sine integrals we provide systematic evaluations of various higher and multiple mahler measures the evaluations in terms of log sine integrals become particularly useful in light of the fact that log sine integrals may be automatically reexpressed as polylogarithmic values we present this correspondence along with related generating functions for log sine integrals our initial interest in considering mahler measures stems from a study of uniform random walks in the plane as first introduced by pearson the main results on the moments of the distance traveled by an n step walk as well as the corresponding probability density functions are reviewed it is the derivative values of the moments that are mahler measures this work would be impossible without very extensive symbolic and numeric computations it also makes frequent use of the new nist handbook of mathematical functions and similar tools our intention is to show off the interplay between numeric and symbolic computing while exploring the three mathematical topics in the title

on the complexity of fixed parameter clique and dominating set we provide simple faster algorithms for the detection of cliques and dominating sets of fixed order our algorithms are based on reductions to rectangular matrix multiplication we also describe an improved algorithm for diamonds detection

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

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 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

weakly convergent expansions of a plane wave and their use in fourier integrals the fourier transform of an irreducible spherical tensor is normally computed with the help of the rayleigh expansion of a plane wave in terms of spherical bessel functions and spherical harmonics the angular integrations are then trivial however the remaining radial integral containing a spherical bessel function may be so complicated that the applicability of fourier transformation is severely restricted as an alternative the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested the weakly convergent expansions of a plane wave are constructed in such a way that their application in fourier integrals leads to expansions of the fourier or inverse fourier transform that converge with respect to the norm of either the hilbert space l2 r3 or the sobolev space w 1 2 r3 accordingly these weakly convergent expansions may be viewed as distributions that are defined on either l2 r3 or w 1 2 r3 the properties of some complete orthonormal se

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

on approximating the number of k cliques in sublinear time we study the problem of approximating the number of k cliques in a graph when given query access to the graph we consider the standard query model for general graphs via 1 degree queries 2

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

approximating the weight of the euclidean minimum spanning tree in sublinear time we consider the problem of computing the weight of a euclidean minimum spanning tree for a set of n points in mathbb r d we focus on the setting where the input point set is supported by certain basic and commonly used geometric data structures that can provide efficient access to the input in a structured way we present an algorithm that estimates with high probability the weight of a euclidean minimum spanning tree of a set of points to within 1 eps using only widetilde o sqrt n text poly 1 eps queries for constant d the algorithm assumes that the input is supported by a minimal bounding cube enclosing it by orthogonal range queries and by cone approximate nearest neighbor queries

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

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

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

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

approximating the minimum spanning tree weight in sublinear time we present a probabilistic algorithm that given a connected graph g represented by adjacency lists of average degree d with edge weights in the set 1 w and given a parameter 0 eps 1 2 estimates in time o dw varepsilon 2 log frac dw varepsilon the weight of the minimum spanning tree mst of g with a relative error of at most eps note that the running time does not depend on the number of vertices in g we also prove a nearly matching lower bound of omega dw varepsilon 2 on the probe and time complexity of any approximation algorithm for mst weight the essential component of our algorithm is a procedure for estimating in time o d eps 2 log frac d varepsilon the number of connected components of an unweighted graph to within an additive error of varepsilon n this becomes o eps 2 log frac 1 varepsilon for d o 1 the time bound is shown to be tight up to within the log frac d varepsilon factor our connected components algorithm picks o 1 varepsilon 2 vertices in the graph and then grows local spanning trees whose sizes are specified by a stochastic process from the local information collected in this way the algorithm is able to infer with high confidence an estimate of the number of connected components we then show how estimates on the number of components in various subgraphs of g can be used to estimate the weight of its mst

asymptotic analysis of a generalized richardson extrapolation process on linear sequences in this work we give a detailed convergence and stability analysis for the author s generalized richardson extrapolation process grep m as this is being applied to linearly convergent or divergent infinite sequences a n where a n a m k 1 n k i 0 ki n k i as n k 1 being distinct the quantity we would like to compute is a whether it is the limit or antilimit of a n such sequences arise for example as partial sums of power series and of fourier series of functions that have algebraic and or logarithmic branch singularities specifically we define the grep m approximation a n m j to a with n n 1 n m and 0 via the linear systems a l a n m j m k 1 l k n i 0 1 ki l k 1 j l j m k 1 n k where ki are additional unknowns we study the convergence and stability properties of a n m j as j we show in particular that a n m j a m k 1 o j k j k 2nk as j when compared with a j a m k 1 o j k j k as j this result shows that grep m is a true convergence acceleration method for the sequences considered in addition we show that grep m is stable for the case being studied and we also quantify its stability properties the results of this work are the first ones pertaining to grep m with m 1

the numerical inversion of two classes of kontorovich lebedev transform by direct quadrature abstract the inversion of the conventional kontorovich lebedev transform with the macdonald function as kernel and also with a kelvin function as kernel is considered strategies are presented first for computing values of these functions for purely imaginary order the interior strategy and these are tested in the subsequent inversion routines the exterior strategy which are developed for use where high speed mainframe or workstation computing is available both the interior and exterior strategies can exploit the w transformation and the w algorithm pioneered by sidi it is shown that conventional nag quadrature software is often incapable of carrying out the inversion without an implementation of this type examples considered show that relative error can generally be kept to o 1 d 8 although for work requiring hundreds of full inversion computations this would have to be relaxed somewhat to obviate the need for overnight computer runs application to a divergent inversion integral summable in the sense of abel is made with satisfactory results a case with weaker summability is also considered with somewhat inferior results

large time and distance asymptotics of field correlation function of impenetrable bosons at finite temperature abstract asymptotics of the temperature time dependent correlation function of two fields one dimensional bosons with infinite point like repulsion is calculated

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

tolerating the community detection resolution limit with edge weighting communities of vertices within a giant network such as the world wide web are likely to be vastly smaller than the network itself however fortunato and barthelemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than l 2 edges where l is the number of edges in the entire network this resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks fortunato and barthelemy s argument can be extended to networks with weighted edges as well and we derive this corollary argument we conclude that weighted modularity algorithms may fail to resolve communities with less than we 2 total edge weight where w is the total edge weight in the network and e is the maximum weight of an intercommunity edge if e is small then small communities can be resolved given a weighted or unweighted network we describe how to derive new edge weights in order to achieve a low e we modify the clauset newman and moore cnm community detection algorithm to maximize weighted modularity and we show that the resulting algorithm has greatly improved accuracy in experiments with an emerging community standard benchmark we find that our simple cnm variant is competitive with the most accurate community detection methods yet proposed

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

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

community structure and scale free collections of erdős renyi graphs community structure plays a significant role in the analysis of social networks and similar graphs yet this structure is little understood and not well captured by most models we formally define a community to be a subgraph that is internally highly connected and has no deeper substructure we use tools of combinatorics to show that any such community must contain a dense erd ifmmode mbox h o else h o fi s r enyi er subgraph based on mathematical arguments we hypothesize that any graph with a heavy tailed degree distribution and community structure must contain a scale free collection of dense er subgraphs these theoretical observations corroborate well with empirical evidence from this we propose the block two level erd ifmmode mbox h o else h o fi s r enyi bter model and demonstrate that it accurately captures the observable properties of many real world social networks

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

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

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

reexamination of the calculation of two center two electron integrals over slater type orbitals i coulomb and hybrid integrals in this paper which constitutes the first part of the series we consider calculation of two center coulomb and hybrid integrals over slater type orbitals general formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods the first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand in both methods simple one dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas it is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well behaved functions calculated recursively with decent speed special attention is paid to the numerical stability of the algorithms applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including at most 7i type functions l 6

convergence accelerators in the computation of molecular integrals over slater type basis functions in the two range one center expansion method the so called nonlinear transformations may be of substantial help in many practical problems including the so called multicenter integrals in this work wynn s epsilon algorithm and the levin s ital u transformation have been used to improve the convergence of the series representation of three center two electron integrals obtained via the so called two range one center expansion method

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

indefinite integration of oscillatory functions by the chebyshev series expansion abstract an automatic quadrature scheme is presented for evaluating the indefinite integral of oscillatory function x 0 t e i t dt 0 x 1 of a given function t which is usually assumed to be smooth the function t is expanded in the chebyshev series to make an efficient evaluation of the indefinite integral combining the automatic quadrature method obtained and sidi s extrapolation method makes an effective quadrature scheme for oscillatory infinite integral a x cos x d x for which numerical examples are also presented

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

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

arboricity and subgraph listing algorithms in this paper we introduce a new simple strategy into edge searching of a graph which is useful to the various subgraph listing problems applying the strategy we obtain the following four algorithms the first one lists all the triangles in a graph g in o a g m time where m is the number of edges of g and a g the arboricity of g the second finds all the quadrangles in o a g m time since a g is at most three for a planar graph g both run in linear time for a planar graph the third lists all the complete subgraphs k l of order l in o la g l 2 m time the fourth lists all the cliques in o a g m time per clique all the algorithms require linear space we also establish an upper bound on a g for a graph g a g leqq lceil 2m n 1 2 rceil where n is the number of vertices in g

perturbation theory for the decay rate of eigenfunctions in the generalized n body problem simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe we develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights we show that generically in a baire category sense eigenfunction decay rates are governed by the lowest threshold

an algorithm for a generalization of the richardson extrapolation process in this paper we present a recursive method designated the w m algorithm for implementing a generalization of the richardson extrapolation process that has been ntroduced in 8 compared to the direct solution of the linear systems of equations defining the extrapolation procedure this method requires a small number of arithmetic operations and very little storage the technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d transformation of 6 13 to power series in the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced which is similar to that given in 2 4 for solving the same problem a fortran program for the w m algorithm is also appended

calculation of the addition coefficients in electromagnetic multisphere scattering theory one of the most intractable problems in electromagnetic multisphere scattering theory is the formulation and evaluation of vector addition coefficients introduced by the addition theorems for vector spherical harmonics this paper presents an efficient approach for the calculation of both scalar and vector translational addition coefficients which is based on fast evaluation of the gaunt coefficients the paper also rederives the analytical expressions for the vector translational addition coefficients and discusses the strengths and limitations of other formulations and numerical techniques found in the literature numerical results from the formulation derived in this paper agree with those of a previously published recursion scheme that completely avoids the use of the gaunt coefficients but the method of direct calculation proposed here reduces the computing time by a factor of 4 6

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

approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms for a given graph g over n vertices let opt g denote the size of an optimal solution in g of a particular minimization problem e g the size of a minimum vertex cover a randomized algorithm will be called an a approximation algorithm with an additive error for this minimization problem if for any given additive error parameter e 0 it computes a value opt such that with probability at least 2 3 it holds that opt g opt a opt g en assume that the maximum degree or average degree of g is bounded in this case we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem this reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms such as the dominating set problem we also show that for the minimum vertex cover problem the query complexity of such approximation algorithms must grow at least linearly with the average degree d of the graph this lower bound holds for every multiplicative factor a and small constant e as long as d o n a in particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o n

higher order analogues of tracy widom distributions via the lax method we study the distribution of the largest eigenvalue in hermitian one matrix models when the spectral density acquires an extra number of k 1 zeros at the edge the distributions are directly expressed through the norms of orthogonal polynomials on a semi infinite interval as an alternative to using fredholm determinants they satisfy nonlinear recurrence relations which we show form a lax pair making contact to the string literature in the early 1990s the technique of pseudo differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the painleve xxxiv hierarchy these are the higher order analogues of the tracy widom distribution using known backlund transformations we show how to simplify earlier equivalent results that are derived from fredholm determinant theory valid for even k in terms of the painleve ii hierarchy

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

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

on sums of independent random variables with unbounded variance and estimating the average degree in a graph we prove the following inequality for every positive integer n and every collection x 1 ldots x n of nonnegative independent random variables each with expectation 1 the probability that their sum remains below n 1 is at least alpha 0 our proof produces a value of alpha 1 13 simeq 0 077 but we conjecture that the inequality also holds with alpha 1 e simeq 0 368 as an example for the use of the new inequality we consider the problem of estimating the average degree of a graph by querying the degrees of some of its vertices we show the following threshold behavior approximation factors above 2 require far fewer queries than approximation factors below 2 the new inequality is used in order to get tight up to multiplicative constant factors relations between the number of queries and the quality of the approximation we show how the degree approximation algorithm can be used in order to quickly find those edges in a network that belong to many shortest paths

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

summability experiments with a class of divergent inverse kontorovich lebedev transforms abstract the computation of inverse kontorovich lebedev transforms is examined with the help of the extrapolative m w transformation first presented by sidi in user friendly form in sidi 1988 recently published improved values for the macdonald function of purely imaginary order have enabled this strategy to give 10 digit accuracy in the inversion including an example with competing oscillations having different phase functions moreover the method is extended with similar accuracy to cases where the integral diverges but is summable in the sense of abel further probing with divergent integrals that are not abel summable but instead summable in the sense defined by jones 1980 is also yielding encouraging results and some analysis culminating in a formal theorem of convergence in that sense is presented to support the interpretation and computation of these

exact solvability and asymptotic aspects of generalized xx0 spin chains abstract building on our earlier work saeedian et al 2018 we introduce and study generalized xx0 models we explicitly construct a long range interacting spin chain referred to as the selberg model and study the correlation functions of the selberg and xx0 models using a matrix integral representation of the generalized xx0 model and applying asymptotic analysis in non intersecting brownian motion the phase structure of the selberg model is determined we find that tails of the tracy widom distribution of gaussian unitary ensemble govern a discrete to continuous third order phase transition in selberg model the same method also reproduces the gross witten phase transition of the original xx0 model finally we conjecture universal features for the phase structure of the generalized xx0 model

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

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

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

extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability in a recent work by the author an extrapolation method the w transformation was developed by which a large class of oscillatory infinite integrals can be computed very efficiently the results of this work are extended to a class of divergent oscillatory infinite integrals in the present paper it is shown in particular that these divergent integrals exist in the sense of abel summability and that the w transformation can be applied to them without any modifications convergence results are stated and numerical examples given

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

extrapolation methods for sommerfeld integral tails a review is presented of the extrapolation methods for accelerating the convergence of sommerfeld type integrals i e semi infinite range integrals with bessel function kernels which arise in problems involving antennas or scatterers embedded in planar multilayered media attention is limited to partition extrapolation procedures in which the sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real axis tail segment spl alpha spl infin of the integration path where spl alpha 0 is selected to ensure that the integrand is well behaved an analytical form of the asymptotic truncation error or the remainder which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods is derived several extrapolation algorithms deemed to be the most suitable for the sommerfeld integrals are described and their performance is compared it is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points spl rho and by the choice of the subinterval break points furthermore it is found that some well known extrapolation techniques may fail for a number of values of spl rho and ways to remedy this are suggested finally the most effective extrapolation methods for accelerating sommerfeld integral tails are recommended

temperature correlators of the impenetrable bose gas as an integrable system it is shown that the temperature equal time correlators of impenetrable bosons in one space dimension are described by a classical integrable system partial differential equations for two point as well as for multipoint correlators are obtained the short distance and low density expansions are constructed

application of the w transformation to compute the linearised 2 d beach problem potentials the computation of the 2 d beach problem is reconsidered a method which allows both regular and singular components to be evaluated for arbitrary values of beach slopes through direct application of the w transformation to a divergent form of the solution integral is established previous models were restricted to the very simple beach angles 2k k n results indicate that the procedure is sufficiently fast that these potentials so computed may be used as kernels in future green function computations

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

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

a user friendly extrapolation method for oscillatory infinite integrals developpement d une methode d extrapolation flexible pour les integrales infinies oscillatoires

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

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

evaluating infinite integrals involving bessel functions of arbitrary order abstract the evaluation of intergrals of the form i n 0 x j n x d x is considered in the past the method of dividing an oscillatory integral at its zeros forming a sequence of partial sums and using extrapolation to accelerate convergence has been found to be the most efficient technique available where the oscillation is due to a trigonometric function or a bessel function of order n 0 1 here we compare various extrapolation techniques as well as choices of endpoints in dividing the integral and establish the most efficient method for evaluating infinite integrals involving bessel functions of any order n not just zero or one we also outline a simple but very effective technique for calculating bessel function zeros