a theory of algebraic cocycles we introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an eilenberg maclane space using these cocycles we develop a cohomology theory for complex algebraic varieties the theory is bigraded functorial and admits gysin maps it carries a natural cup product and a pairing to l homology chern classes of algebraic bundles are defined in the theory there is a natural transformation to singular integral cohomology theory that preserves cup products computations in special cases are carried out on a smooth variety it is proved that there are algebraic cocycles in each algebraic rational p p cohomology class in this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle details will appear in fl our cohomology is a companion to the l homology theory recently studied in l f l f1 l f2 fm this homology is a bigraded theory based directly on the structure of the space of algebraic cycles it admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles it carries a ring structure coming from the complex join an elementary construction of projective geometry and it admits a natural transformation to integral cohomology chern classes are defined in the theory and transform under to the usual ones our definition of cohomology is very far from a duality construction on l homology nonetheless there is a natural and geometrically defined kronecker pairing between our morphic cohomology and l homology the foundation stone of our theory is the notion of an effective algebraic cocycle which is of some independent interest roughly speaking such a cocycle on a variety x with values in a projective variety y is a morphism from x to the space of cycles on y when x is normal this is equivalent by graphing to a cycle on x y with equidimensional fibres over x such cocycles abound in algebraic geometry and arise naturally in many circumstances the simplest perhaps is that of a flat morphism f x y whose corresponding cocycle associates to x x the pullback cycle f x many more arise naturally from synthetic constructions in projective geometry we show that every variety is rich in cocycles indeed if 1991 mathematics subject classification primary 14f99 14c05

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

grothendieck standard conjectures morphic cohomology and hodge index theorem using morphic cohomology we produce a sequence of conjectures called morphic conjectures which terminates at the grothendieck standard conjecture a a refinement of hodge structures is given and with the assumption of morphic conjectures we prove a hodge index theorem we answer a question of friedlander and lawson by assuming the grothendieck standard conjecture b and prove that the topological filtration from morphic cohomology is equal to the grothendieck arithmetic filtration for some cases

algebraic cycles and infinite loop spaces in this paper we use recent results about the topology of chow varieties to answer an open question in infinite loop space theory that is we construct an infinite loop space structure on a certain product of eilenberg maclane spaces so that the total chern map is an infinite loop map an analogous result for the total stiefel whitney map is also proved further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined

milnor k theory of rings higher chow groups and applications if r is a smooth semi local algebra of geometric type over an infinite field we prove that the milnor k group kmn r surjects onto the higher chow group chn r n for all n 0 our proof shows moreover that there is an algorithmic way to represent any admissible cycle in chn r n modulo equivalence as a linear combination of symbolic elements defined as graphs of units in r as a byproduct we get a new and entirely geometric proof of results of gabber kato and rost related to the gersten resolution for the milnor k sheaf furthermore it is also shown that in the semi local pid case we have under some mild assumptions an isomorphism some applications are also given

on a new type of mixed interpolation abstract we approximate every function f by a function f n x of the form a cos kx b sin kx n 2 i 0 c i x i so that f jh f n jh for the n 1 equidistant points jh j 0 n that interpolation function f n x is proved to be unique and can be written as the sum of the n th degree interpolation polynomial based on the same points and two correction terms the error term is also discussed the results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0 this new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations

asymptotic trace formula for the hecke operators given integers m n and k we give an explicit formula with an optimal error term with square root cancelation for the petersson trace formula involving the mth and nth fourier coefficients of an orthonormal basis of s k left n right the weight k newforms with fixed square free level n provided that 4 pi sqrt mn k o left k frac 1 3 right moreover we establish an explicit formula with a power saving error term for the trace of the hecke operator mathcal t n on s k left n right averaged over k in a short interval by bounding the second moment of the trace of mathcal t n over a larger interval we show that the trace of mathcal t n is unusually large in the range 4 pi sqrt n k o left n frac 1 6 right as an application for any fixed prime p coprime to n we show that there exists a sequence k n of weights such that the error term of weyl s law for mathcal t p is unusually large and violates the prediction of arithmetic quantum chaos in particular this generalizes the result of gamburd et al j eur math soc 1 1 51 85 1999 theorem 1 4 with an improved exponent

techniques computations and conjectures for semi topological k theory we establish the existence of an atiyah hirzebruch like spectral sequence relating the morphic cohomology groups of a smooth quasi projective complex variety to its semi topological k groups this spectral sequence is compatible with and indeed is built from the motivic spectral sequence that relates the motivic cohomology and algebraic k theory of varieties and it is also compatible with the classical atiyah hirzebruch spectral sequence in algebraic topology in the second part of this paper we use this spectral sequence in conjunction with another computational tool that we introduce namely a variation on the integral weight filtration of the borel moore singular homology of complex varieties introduced by h gillet and c soule to compute the semi topological k theory of a large class of varieties in particular we prove that for curves surfaces toric varieties projective rational three folds and related varieties the semi topological k groups and topological k groups are isomorphic in all degrees permitted by cohomological considerations we also formulate integral conjectures relating semi topological k theory to topological k theory analogous to more familiar conjectures namely the quillen lichtenbaum and beilinson lichtenbaum conjectures concerning mod n algebraic k theory and motivic cohomology in particular we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by a suslin our computations verify that these conjectures hold for the list of varieties above

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

algebraic cycles and homotopy theory examine its homotopy groups which after a certain idealization of the space turn out to be astonishingly simple in fact for complex projective space pn the structure can be understood completely this yields new information about the topology of the classical chow varieties and establishes an explicit relationship with universal cohomology operations in the general case complete computations are difficult but we shall establish a complex suspension theorem and lay the foundations for a theory based on the homotopy groups of chow varieties to state the results we must give precise meaning to the space of subvarieties we begin with the fundamental case of subvarieties in complex projective n space pln for each pair of integers p and d with d 1 and 0 p n consider the set wp d pn of all finite formal sums

a parallel approach to the modified numerov like eigenvalue determination for sturm liouville problems abstract a modified numerov like eigenvalue algorithm previously introduced is parallelized an inplementation of this algorithm on a helios based parallel processing transputer system is discussed time savings with respect to a sequential approach are commented

resultants and the algebraicity of the join pairing on chow varieties the chow van der waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension the algebraicity of the join pairing on chow varieties then follows the approach implies a more algebraic proof of lawson s complex suspension theorem in characteristic 0 the continuity of the action of the linear isometries operad on the group completion of the stable chow variety is a consequence

intersections via resolutions abstract we investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities for varieties with resolutions having a certain structure including all varieties over a field of characteristic zero we obtain a stratification which reflects the geometry of the centers and the exceptional divisors this stratification is sufficiently fine that divisors can be intersected with r cycles for r 1 and 2 cycles can be intersected on a fourfold provided their incidences with the strata are controlled similar pairings are defined on a variety with one dimensional singular locus

algebraic cycles on real varieties and z 2 equivariant homotopy theory in this paper the spaces of algebraic cycles on a real projective variety x are studied as z 2 spaces under the action of the galois group gal c r in particular the equivariant homotopy type of the group of algebraic p cycles zp pc is computed a version of lawson homology for real varieties is proposed the real lawson homology groups are computed for a class of real varieties

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

accurate computation of higher sturm liouville eigenvalues the computation of eigenvalues of regular sturm liouville problems is considered it is shown that a simple step dependent linear multistep method can be used to reduce the error of the orderk 4 h 2 of the centered finite difference estimate of thek th eigenvalue with uniform step lengthh to an error of orderkh 2 by an appropriate minimization of the local error term of the method one can obtain even more accurate results a comparison of the simple correction techniques of paine de hoog and anderssen and of andrew and paine is given numerical examples demonstrate the usefulness of this correction even for low values ofk

on the error estimation for a mixed type of interpolation we analyse the error induced by the approximation of a function by an interpolation function which is a combination of algebraic and first order trigonometric polynomials we prove that under certain conditions this error can be expressed in a similar form as in the purely polynomial case as an application we establish in closed form the local truncation error for a class of extended linear multistep methods of the adams type

harnack thom theorem for higher cycle groups and picard varieties we generalize the harnack thom theorem to relate the ranks of the lawson homology groups with 22 coefficients of a real quasiprojective variety with the ranks of its reduced real lawson homology groups in the case of zero cycle group we recover the classical harnack thom theorem and generalize the classical version to include real quasiprojective varieties we use weil s construction of picard varieties to construct reduced real picard groups and milnor s construction of universal bundles to construct some weak models of classifying spaces of some cycle groups these weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors we obtain some congruences involving the picard number of a nonsingular real projective variety and the rank of its reduced real lawson homology groups of divisors

algebraic cycles representing cohomology operations in this paper we will show that certain universal homology classes which are fundamental in topology are algebraic to be specific the products of eilenberg maclane spaces mathcal k 2q equiv k mathbf z 2 times k mathbf z 4 times cdots times k mathbf z 2q have models which are limits of complex projective varieties precisely we have mathcal k 2q varinjlim nolimits d rightarrow infty mathcal c q d mathbf p n where mathcal c q d mathbf p n denotes the chow variety of effective cycles of codimension q and degree d on mathbf p mathbf c n it is natural to ask which elements in the homology of mathcal k 2q are represented by algebraic cycles in these approximations in this paper we find such representations for the even dimensional classes which are known as steenrod squares as well as their pontrjagin and join products these classes are dual to the cohomology classes which correspond to the basic cohomology operations also known as the steenrod squares

completions and fibrations for topological monoids we show that for a certain class of topological monoids there is a homotopy equivalence between the homotopy theoretic group completion m of a monoid m in that class and the topologized grothendieck group m associated to m the class under study is broad enough to include the chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite cw complexes we associate principal fibrations to the completions of pairs of monoids showing the existence of long exact sequences for the naive approach to lawson homology fri91 lf91 a another proof of the eilenberg steenrod axioms for the functors x sp x in the category of finite cw complexes dold thom theorem dt56 is obtained

vanishing theorems for real algebraic cycles we establish the analogue of the friedlander mazur conjecture for teh s reduced lawson homology groups of real varieties which says that the reduced lawson homology of a real quasi projective variety x vanishes in homological degrees larger than the dimension of x in all weights as an application we obtain a vanishing of homotopy groups of the mod 2 topological groups of averaged cycles and a characterization in a range of indices of dos santos real lawson homology as the homotopy groups of the topological group of averaged cycles we also establish an equivariant poincare duality between equivariant friedlander walker real morphic cohomology and dos santos real lawson homology we use this together with an equivariant extension of the mod 2 beilinson lichtenbaum conjecture to compute some real lawson homology groups in terms of bredon cohomology

lawson homology for varieties with small chow groups and the induced filtration on the griffiths groups these groups form an intriguing set of invariants despite the vast literature on this subject the groups have been calculated only in very few cases such as for the projective spaces one of the goals of this note is to understand these invariants in case the variety has few cycles in the sense that rational and homological equivalence coincide for all m cycles up to a certain rank m s actually the results only deal with that part of lawson homology that is not visible in ordinary homology in that it is in the kernel of the cycle class maps cm k 2m lmhk 2m x hk 2m x

optimal strong approximation for quadratic forms for a nondegenerate integral quadratic form f x1 xd in d 5 variables we prove an optimal strong approximation theorem let be a fixed compact subset of the affine quadric f x1 xd 1 over the real numbers take a small ball b of radius 0 0 finally assume that an integral vector 1 d mod m is given then we show that there exists an integral solution x x1 xd of f x n such that xi imodm and xn b provided that all the local conditions are satisfied we also show that 4 is the best possible exponent moreover for a nondegenerate integral quadratic form in four variables we prove the same result if n is odd and n r 1m 6 based on our numerical experiments on the diameter of lps ramanujan graphs and the expected square root cancellation in a particular sum that appears in remark 6 8 we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent 4

spectral corrections for sturm liouville problems abstract the numerical solution of the sturm liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction in this paper we use the shooting method both for eigenvalues and eigenfunctions in integrating the corresponding initial value problems we resort to the boundary value method the technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems a technique to estimate the eigenvalue errors is also suggested and seems to be particularly effective for the higher index eigenvalues numerical experiments on some classical sturm liouville problems are presented

approximations of sturm liouville eigenvalues using differential quadrature dq method the polynomial based differential quadrature pdq and the fourier expansion based differential quadrature fdq methods are applied in this work to compute eigenvalues of the sturm liouville problem it is demonstrated through some examples that accurate results for the first kth eigenvalues of the problem where k 1 2 3 can be obtained by using at least 2k mesh points the results of this work are compared with other published results in the literature

best possible rates of distribution of dense lattice orbits in homogeneous spaces this paper establishes upper and lower bounds on the speed of approximation in a wide range of natural diophantine approximation problems the upper and lower bounds coincide in many cases giving rise to optimal results in diophantine approximation which were inaccessible previously our approach proceeds by establishing more generally upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup in a connected lie or algebraic group g acting on suitable homogeneous spaces g h the upper bound is derived using a quantitative duality principle for homogeneous spaces reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on h and acting on g in particular the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of h in the automorphic representation on l 2 g we show that the rate is best possible when the representation in question is tempered and show that the latter condition holds in a wide range of examples

quaternionic algebraic cycles and reality in this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real brauer severi varieties under the action of the galois group gal c r appropriate stabilizations of these spaces yield two equivariant spectra the first one classifies dupont seymour s quaternionic k theory and the other one classifies an equivariant cohomology theory 3 which is a natural recipient of characteristic classes kh x 3 x for quaternionic bundles over real spaces x

moving algebraic cycles of bounded degree the chow moving lemma is a theorem which asserts that a given algebraic s cycle on a smooth algebraic variety x can be moved within its rational equivalence class to intersect properly a given r cycle on x provided that r s dim x cf chow s2 in the past few years there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation with this in mind it is natural to ask whether one can move a given bounded family of s cycles on the smooth variety x to intersect properly a given bounded family of r cycles the main point of this paper is to formulate and prove just such a result in theorem 3 1 we demonstrate that for any integer e and any smooth projective variety x one can simultaniously and algebraically move all effective s cycles of degree e on x so that each such cycle meets every effective r cycle of degree e on x in proper dimension the primary motivation for this moving lemma for cycles of bounded degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces using theorem 3 1 we have proved such a duality theorem for complex quasi projective varieties in f l2 we prove our moving lemma for varieties over an arbitary infinite field permitting a proof in f v of a duality theorem for motivic cohomology and homology the reader will find that our moving lemma has numerous good properties first of all the move is given as an algebraic move parametrized by a punctured projective line on chow varietes although this move is good only for s cycles of bounded degree it is defined on all effective s cycles moreover the move starts at time 0 by expressing an effective s cycle z as a difference of effective s cycles both of which have intersection properties no worse than z finally our moving lemma is applicable to smooth quasiprojective varieties for it is stated for a possibly singular projective variety x resulting in a conclusion of proper intersection off the singular locus of x the classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence thereby establishing the chow ring a x some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move e g chow r s2 on the other hand cf chev s3 one direct consequence of our moving lemma is a proof for smooth quasi projective varieties that the intersection product is indeed well defined independent of the choice of move theorem 3 4 of course the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to fulton

graph mappings and poincare duality we give a new geometric description for a compact oriented pseduo manifold x of the poincare duality map from the integral cohomology of x to the integral homology of x our construction takes a multi valued lipschitz map on x with values in a sphere sn to its geometric measure theoretic graph in x sn and then to the slice of this graph as an integral cycle on x this construction is compatible with analogous constructions on algebraic cocycles on projective varieties employed by the authors and others

a finite element estimate with trigonometric hat functions for sturm liouville eigenvalues abstract the computation of eigenvalues of regular sturm liouville problems is considered numerical results show that the error of the k th eigenvalue obtained by the finite element method with trigonometric hat functions and mesh length h is of the same order as the one obtained with the linear hat functions together with a simple asymptotic correction technique of paine de hoog and anderssen 1981

rational isomorphisms between k theories and cohomology theories the well known isomorphism relating the rational algebraic k theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map the segre map of infinite loop spaces moreover the associated chern character map on rational homotopy groups is shown to be a ring isomorphism a technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi projective complex varieties to induce a homotopy equivalence of semi topological singular complexes since semi topological k theory and morphic cohomology can be formulated as the semi topological singular complexes associated to algebraic k theory and motivic cohomology this criterion provides a rational isomorphism between the semi topological k theory groups and the morphic cohomology groups of a smooth complex variety consequences include a riemann roch theorem for the chern character on semi topological k theory and an interpretation of the topological filtration on singular cohomology groups in k theoretic terms

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

generalized abel jacobi map on lawson homology we construct an abel jacobi type map on the homologically trivial part of lawson homology groups it generalizes the abel jacobi map constructed by griffiths by using a result of h clemens we answer affirmatively the question whether there exists a smooth projective complex variety with infinitely generated lawson homology groups l ph 2p k x bbb q when k0 as a corollary we find for any nonnegative integer j a smooth complex projective variety x carrying infinitely generated semi topological k groups k sst j x bbb q

chow flag variety and lawson homology the chow flag varieties are introduced and studied in a family of pure dimensional pro jective complex varieties defined via chow flag varieties the isomorphism type of lawson homology of a general member in the family is discussed applications to morphic cohomology and motivic cohomology on smooth projective complex varieties are discussed 1 introduction by variety we mean a quasi projective variety defined as a zariski open subset of the zero locus of a family of homogeneous polynomials in some projective space in particular varieties may not be irreducible if x is a projective complex variety the lawson homology of x denoted as lrhi x where r and i are appropriate nonnegative integers was first considered by lawson in l and then systematically formulated and studied by friedlander in f91 properties of lawson homology have been studied and relations to other important theories have been understood however it is hard to compute lawson homology groups for a general complex variety friedlander and mazur conjectured in fm that lrhi x 0w herex is smooth projective and i is greater than twice the dimension of x this conjecture has been verified in a few situations in this paper we consider the isomorphism type of lawson homology groups of members in a family of projective complex varieties here is what we mean by a family of complex varieties let p n be the usual complex projective space and let cr d p n denote the chow variety of effective r cycles of degree d in p n for any point x in cr d p n l etz x denote the corresponding cycle a family of projective varieties defined in p n of pure dimension r is by definition a subvariety y of cr d p n a member of this family is the support of a cycle z y for some point y y the connection between the notion of families of varieties used in this paper and the usual notion of families of varieties defined via morphisms is explained in f91 theorem 14 according to f91 if y is smooth then a family of projective varieties in p n indexed by y corresponds to a morphism f v y such that the cycle associated to the scheme theoretical fiber over y y is the cycle z y a nd thatv is a subvariety of p n y and f is the restriction of the projection map p n y y if y is a family of projective varieties it is desirable to show certain statements are generically true this means there exists a closed proper subvariety z of y such that the statement is true for any y y z our main result in this paper see