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

simply connected 4 manifolds with a given boundary let m be a closed oriented connected 3 manifold for each bilinear symmetric pairing zn l our goal is to calculate the set vl m of all oriented homeomorphism types of compact 1 connected oriented 4manifolds with boundary m and intersection pairing isomorphic to zn l for each pair zn l which presents h m we construct a double coset space bl m and a function ct vl m bl m the set bt m is the quotient of the group of all link pairing preserving isomorphisms of the torsion subgroup of h1 m by two naturally occuring subgroups when zn l is an even pairing we construct another double coset space bl m a function 6l vl m b fl m and a projection p2 bl m bl m such that p2 c l ct our main result states that when zn l is even the function c l is injective as is the function c x a vl m bl m x z 2 when z l is odd here a is a kirby siebenmann obstruction to smoothing it follows that the sets vl m are finite and of an order bounded above by a constant depending only on h1 m we also show that when hi m q 0 and z l is even ct p2 c 6l is injective it seems likely that via the functions cl x a and 6l the sets bl m x z 2 and bl m calculate vl m when zn l is respectively odd and even we verify this in several cases most notably when h1 m is free abelian the results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1connected 4 manifolds extending a given homeomorphism of their boundaries the theory developed is then applied to show that there is an m 0 depending only on h1 m such that for any self homeomorphism f of m fm extends to a self homeomorphism of any 1 connected compact 4 manifold with boundary m introduction in m freedman s fundamental paper fr he classified closed simply connected oriented 4 manifolds up to orientation preserving homeomorphism he showed that modulo the kirby siebenmann invariant these manifolds are in bijective correspondence through their intersection pairings with the set of unimodular bilinear symmetric pairings over z see theorem 1 5 of fr and corollary 2 2 3 of q in this paper we begin a classification of simply connected 4 manifolds with connected boundary our results with regard to this problem are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between 1 connected 4 manifolds extending a given homeomorphism of their boundaries received by the editors september 19 1985 1980 mathematics subject classification 1985 revision primary 57n15 research supported by an nserc canada post doctoral fellowship 1986 american mathematical society 0002 9947 86 1 00 25 per page

integral action in output feedback for multi input multi output nonlinear systems we address a particular problem of output regulation for multi input multi output nonlinear systems specifically we are interested in making the stability of an equilibrium point and the regulation to zero of an output robust to small unmodelled discrepancies between design model and actual system in particular those introducing an offset we propose a novel procedure which is intended to be relevant to realistic scenarios as illustrated by a non academic example

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

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

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

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

gaussian random measures generated by berry s nodal sets we consider vectors of random variables obtained by restricting the length of the nodal set of berry s random wave model to a finite collection of possibly overlapping smooth compact subsets of mathbb r 2 our main result shows that as the energy diverges to infinity and after an adequate normalisation these random elements converge in distribution to a gaussian vector whose covariance structure reproduces that of a homogeneous independently scattered random measure a by product of our analysis is that when restricted to rectangles the dominant chaotic projection of the nodal length field weakly converges to a standard wiener sheet in the banach space of real valued continuous mappings over a fixed compact set an analogous study is performed for complex valued random waves in which case the nodal set is a locally finite collection of random points

non universality for the variance of the number of real roots of random trigonometric polynomials in this article we consider the following family of random trigonometric polynomials p n t y sum k 1 n y k 1 cos kt y k 2 sin kt for a given sequence of i i d random variables y i k i in 1 2 k ge 1 which are centered and standardized we set mathcal n 0 pi y the number of real roots over 0 pi and mathcal n 0 pi g the corresponding quantity when the coefficients follow a standard gaussian distribution we prove under a doeblin s condition on the distribution of the coefficients that begin aligned lim n rightarrow infty frac text var left mathcal n n 0 pi y right n lim n rightarrow infty frac text var left mathcal n n 0 pi g right n frac 1 30 left mathbb e left left y 1 1 right 4 right 3 right end aligned the latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis actually a more general result is proven in this article which does not require that the coefficients are identically distributed the proof mixes a recent result regarding edgeworth expansions for distribution norms established in bally et al electron j probab 23 45 1 51 2018 with the celebrated kac rice formula

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

surgery in dimension four and noncompact 5 manifolds this paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact 5 manifolds two outstanding problems of low dimensional topology are completing 4 dimensional surgery and finding boundaries for noncompact 5 dimensional manifolds although no complete theory is known for either problem some interesting results have been found see 1 4 5 10 and addendum in this paper we describe a complete relationship between these two problems consequently knowledge of 4 dimensional surgery yields information about noncompact 5 manifolds and conversely all manifolds and maps in this paper are assumed to be smooth each manifold is provided with a basepoint and when appropriate basepaths will be chosen to given 2 spheres without explicit mention i wish to thank gerald a anderson and frank quinn for helpful discussions also i wish to thank patrick gilmer for his advice and encouragement during this project 1 statement of results let f b m am y x be a degree one normal map in the sense of wall 11 where m is a compact connected 4 manifold and y x is a poincare pair not necessarily simple if x 0 then we require that f jam am x induces an isomorphism of homology groups with a z vi y coefficients also we assume that f is 2 connected k2 m is a free a module and the surgery obstruction a f b e lh 7 c is zero if h c k2 m is a subkernel we will say that h is representable in m if we can find disjoint framed embedded 2 spheres si sr c int m such that the inclusion m ui si m induces an isomorphism of fundamental groups and the classes of sl sr in h2 m a 7t2 m form a basis over a for h the subkernel h is stably representable if for some integer k 0 h ed kei ek is representable in m k s2 x s2 here ei denotes the class of s2 x 1 c ith summand s2 x s2 and e1 ek is the submodule generated by el ek we regard h ed ke ek as received by the editors august 10 1982 1980 mathematics subject classification primary 57r65 secondary 57m99 1985 american mathematical society 0002 9947 85 1 00 25 per page

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

the global structure of simple space times according to a standard definition of penrose a space time admitting well defined future and past null infinitiesi andi is asymptotically simple if it has no closed timelike curves and all its endless null geodesics originate fromi and terminate ati the global structure of such space times has previously been successfully investigated only in the presence of additional constraints the present paper deals with the general case it is shown thati is diffeomorphic to the complement of a point in some contractible open 3 manifold the strongly causal regioni0 ofi is diffeomorphic to mathbb s 2 times mathbb r and every compact connected spacelike 2 surface ini is contained ini0 and is a strong deformation retract of bothi0 andi moreover the space time must be globally hyperbolic with cauchy surfaces which subject to the truth of the poincare conjecture are diffeomorphic to 3

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

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

a family of freely slice good boundary links we show that every good boundary link with a pair of derivative links on a seifert surface satisfying a homotopically trivial plus assumption is freely slice this subsumes all previously known methods for freely slicing good boundary links with two or more components and provides new freely slice links

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

expected volume and euler characteristic of random submanifolds abstract in a closed manifold of positive dimension n we estimate the expected volume and euler characteristic for random submanifolds of codimension r 1 n in two different settings on one hand we consider a closed riemannian manifold and some positive then we take r independent random functions in the direct sum of the eigenspaces of the laplace beltrami operator associated to eigenvalues less than and consider the random submanifold defined as the common zero set of these r functions we compute asymptotics for the mean volume and euler characteristic of this random submanifold as goes to infinity on the other hand we consider a complex projective manifold defined over the reals equipped with an ample line bundle l and a rank r holomorphic vector bundle e that are also defined over the reals then we get asymptotics for the expected volume and euler characteristic of the real vanishing locus of a random real holomorphic section of e l d as d goes to infinity the same techniques apply to both settings

stable mapping class groups of 4 manifolds with boundary we give a complete algebraic description of the mapping class groups of compact simply connected 4 manifolds with boundary up to connected sum with copies of s 2 s 2

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

locally flat 2 knots in with the same fundamental group we consider a locally flat 2 sphere in s x s2 representing a primitive homology class 4 which is referred to as a 2 knot in s x s2 representing 4 then for any given primitive class there exists a 2 knot in s2 x s2 representing 4 with simply connected complement in this paper we consider the classification of 2 knots in s2 x s2 whose complements have a fixed fundamental group we show that if the complement of a 2 knot s in s2 x s2 is simply connected then the ambient isotopy type of s is determined in the case of nontrivial 7r1 however we show that the ambient isotopy type of a 2 knot in s2 x s2 with nontrivial 71 is not always determined by 71

construction of group actions on four manifolds it is shown that any cyclic group of odd prime order acts on any closed simply connected topological 4 manifold inducing the identity on integral homology the action is locally linear except perhaps at one isolated fixed point in the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear introduction in this paper we shall show that every closed simply connected topological 4 manifold m4 admits an action of any cyclic group zp of odd prime order p the action will be homologically trivial that is induce the identity on integral homology be pseudofree that is have only isolated fixed points except in certain cases when p 3 and be locally linear except perhaps at one isolated fixed point when p 2 and the intersection form of m4 has even type then the same conclusion holds this is also due to s kwasik and p vogel 1985 by a somewhat different proof if p 2 and the intersection form of m has odd type however then an action of z2 which is homologically trivial must have a fixed point set containing two dimensional components see 7 further it is harder to control the kirby siebenmann triangulation obstruction which is then not determined by the intersection form it is interesting to ask whether these actions can be locally linear or smooth when m4 is smooth the existence of a locally linear z2 action implies the vanishing of the kirby siebenmann obstruction see kwasik and vogel 1984 we shall show however that for p 3 in general these actions can be constructed to be locally linear in a recent preprint kwasik has shown that this is the case for the fake cp2 when p is odd by a rather different proof these actions are homologically trivial and have only isolated fixed points except sometimes when p 3 the g signature theorem shows that not every simply connected 4 manifold admits pseudofree locally linear actions when p 3 we shall see for example that neither the e8 manifold the kummer surface nor a nontrivial connected sum of copies of cp2 admits such an action for p 3 in broad outline our construction goes as follows let a closed oriented simply connected 4 manifold m be given by studying equivariant framed links we construct a compact smooth 4 manifold with boundary having the intersection form of m and admitting a smooth zp action which is pseudofree and homologically trivial the action is free on the boundary homology sphere e in a paper primarily focused on equivariant plumbing diagrams for high dimensional highly connected manifolds with even intersection forms weintraub 1975 carried out most of this we give an independent development because we need to be able to better control received by the editors october 16 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 57s17 57s25 57n15 research supported in part by a grant from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

a surgery sequence in dimension four the relations with knot concordance we present a systematic treatment of the classification problem for compact smooth 4 manifolds m it is modeled on the surgery exact sequence the central theorem in the classification of n manifolds n 5 the price for the extension to dimension 4 is a hole in m where a homotopy 1 skeleton should be there is no homotopy theoretic or surgical obstruction to complet ing m with a wedge of circles so that the completion has the topology of a compact smooth manifold this point set problem is all that stands between 4 manifolds and the tranquility that prevails in higher dimensions when m is simply connected only a point is missing from the model the applications of this are discussed in f and fq 1 the general theory has application to knot and link theory in particular knots with alexander polynomial 1 are characterized geometrically as knots admitting a certain

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

linking and causality in globally hyperbolic space times the classical linking number lk is defined when link components are zero homologous in 15 we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes here we apply alk to the study of causality in lorentzian manifolds

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