gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

a class of asymmetric simplicial finite element methods for solving finite incompressible elasticity problems abstract we consider the numerical solution of finite incompressible elasticity problems for mooney rivlin materials in n dimensional space n 2 3 in representing the displacement vector field with standard lagrange finite elements in general one faces the difficulty of the total number of constraints exceeding the total number of displacement degrees of freedom in the discrete problem if the nonlinear incompressibility condition is to be approximated within satisfactory accuracy in this paper we introduce a new class of piecewise reduced quadratic conforming simplicial finite elements that eliminates the above inconvenience with almost no extra implementational cost thereby allowing highly satisfactory approximate solutions this statement is illustrated by numerical examples in which our method is combined with an algorithm of augmented lagrangian type proposed by glowinski and le tallec for the continuous problem

an almost symmetric strang splitting scheme for the construction of high order composition methods in this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the partial flows that results from the splitting procedure cannot be computed exactly instead we insert a well chosen state y into the corresponding nonlinearity b y y which results in a linear term b y y whose exact flow can be determined efficiently therefore in the spirit of splitting methods it is still possible for the numerical simulation to satisfy certain properties of the exact flow however strang splitting is no longer symmetric even though it is still a second order method and thus high order composition methods are not easily attainable we will show that an iterated strang splitting scheme can be constructed which yields a method that is symmetric up to a given order this method can then be used to attain high order composition schemes we will illustrate our theoretical results up to order six by conducting numerical experiments for a charged particle in an inhomogeneous electric field a post newtonian computation in celestial mechanics and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to strang splitting for the first example we also perform a comparison with the standard fourth order runge kutta methods and find significant gains in efficiency as well better conservation properties

convergence analysis of strang splitting for vlasov type equations a rigorous convergence analysis of the strang splitting algorithm for vlasov type equations in the setting of abstract evolution equations is provided it is shown that under suitable assumptions the convergence is of second order in the time step tau as an example it is verified that the vlasov poisson equations in 1 1 dimensions fit into the framework of this analysis further numerical experiments for the latter case are presented

descendant invariants and characteristic numbers on a stack of stable maps the cotangent line classes are modified by subtracting certain boundary divisors these modified cotangent line classes are compatible with forgetful morphisms and are well suited to enumerative geometry tangency conditions allow simple expressions in terms of modified cotangent line classes topological recursion relations are established among their top products in genus 0 yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety in higher genus the obtained numbers are only virtual due to contributions from spurious components of the space of maps for the projective plane the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases

an almost symmetric strang splitting scheme for nonlinear evolution equations in this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations where one partial flow cannot be computed exactly instead we use a numerical approximation based on the linearization of the vector field this is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences however in the situation described the classic strang splitting scheme while still being a method of second order is not longer symmetric this in turn implies that the construction of higher order methods by composition is limited to order three only to remedy this situation based on previous work in the context of ordinary differential equations we construct a class of strang splitting schemes that are symmetric up to a desired order we show rigorously that under suitable assumptions on the nonlinearity these methods are of second order and can then be used to construct higher order methods by composition in addition we illustrate the theoretical results by conducting numerical experiments for the brusselator system and the kdv equation

singular schemes of hypersurfaces let y be the singular locus of a hypersurface x in a smooth variety m with the scheme structure defined by the jacobian ideal of x we will say then that y is the singular scheme of x to emphasize that the scheme structure of y is important for our considerations in this note we consider a class in the chow group of y which arises naturally in this setup and which captures much intersection theoretic information about the situation the guiding question we have in mind is which schemes y can arise as singular schemes of hypersurfaces we will obtain strong constraints showing for instance that the only hypersurfaces in p whose singular schemes are positive dimensional linear subspaces of p are quadrics and that no reduced nodal curve can be the singular scheme of a hypersurface in a non singular variety many more statements of this sort can be found in section 3 a different type of application is in section 2 we show how our class relates to other invariants of the singularity of a hypersurface the class can be used to recover results of holme and parusi ski on degree and multiplicity of dual varieties and leads naturally to a generalization of the notion of ranks of a smooth projective variety also we obtain a strengthening of landman s parity result and a new proof of a result of zak on the dimension of the dual of a smooth variety the duality results follow by applying the framework to hyperplane sections of m the singular scheme of a section is supported on the locus of contact of the hyperplane with m and the class can be used to measure this contact for example the class measures how general a given section is we show corollary 2 6 that if the contact scheme is a linear subspace pr 1 then the corresponding hyperplane is a smooth point of the dual variety of m and the dual variety has codimension r the main general results are in section 1 where we prove corollary 1 7 that the class we introduce depends in fact only on y and on the line bundle l o x y and not on the ambient variety m provided that y is the singular scheme of a

two characteristic numbers for smooth plane curves of any degree we use a sequence of blow ups over the projective space parametrizing plane curves of degree d to obtain some enumerative results concerning smooth plane curves of arbitrary degree for d 4 this gives a first modern verification of results of h g zeuthen

the degree of the variety of rational ruled surfaces and gromov witten invariants we compute the degree of the variety parametrizing rational ruled surfaces of degree d in p 3 by relating the problem to gromov witten invariants and quantum cohomology

tautological classes on the moduli spaces of stable maps to p r via torus actions we present a localization proof of the fact that the cohomology of the moduli spaces of genus zero stable maps to projective spaces is entirely tautological in addition we obtain a description of a bialynicki birula decomposition of the stack of stable maps in the context of gathmann s moduli spaces of maps with prescribed contact order to a fixed hyperplane we show that the decomposition is filterable

the geometry of stable quotients in genus one stable quotients provide an alternative to stable maps for compactifying spaces of maps when n ge 2 the space overline q g mathbb p n 1 d overline q g g 1 n d compactifies the space of degree d maps of smooth genus g curves to mathbb p n 1 while overline q g g 1 1 d simeq overline m 1 d cdot epsilon s d is a quotient of a hassett weighted pointed space in this paper we study the coarse moduli schemes associated to the smooth proper deligne mumford stacks overline q 1 mathbb p n 1 d for all n ge 1 we show these schemes are projective unirational and have picard number 2 then we give generators for the picard group compute the canonical divisor the cones of ample divisors and in the case n 1 the cones of effective divisors we conclude that overline q 1 mathbb p n 1 d is fano if and only if n d 1 d 2 20 moreover we show that overline q 1 mathbb p n 1 d is a mori fiber space for all n d hence always minimal in the sense of the minimal model program in the case n 1 we write in addition a closed formula for the poincare polynomial

numerical solution of problems in incompressible finite elasticity by augmented lagrangian methods ii three dimensional problems in a first paper siam j appl math 42 1982 pp 400 429 we introduced an augmented lagrangian methodology for solving equilibrium problems in incompressible finite elasticity and then applied it to the solution of several two dimensional and axisymmetric test problems in this second paper we extend our methods to the solution of genuine three dimensional problems actually using these augmented lagrangian techniques we can reduce the solution of these complicated nonlinear nonconvex problems to the solution of a sequence of linear problems and also to the solution of a sequence of minimization problems for convex functionals on convex sets the capabilities of our methods are illustrated by the results of numerical experiments

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

universal relations on stable map spaces in genus zero we introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point this leads to a study of universal relations in the cohomology of stable map spaces in genus zero

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

a reconstruction theorem in quantum cohomology and quantum k theory a reconstruction theorem for genus 0 gravitational quantum cohomology and quantum k theory is proved a new linear equivalence in the picard group of the moduli space of genus 0 stable maps relating the pull backs of line bundles from the target via different markings is used for the reconstruction result examples of calculations in quantum cohomology and quantum k theory are given

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

the enumerative geometry of plane cubics i smooth cubics we construct a variety of complete plane cubics by a sequence of five blow ups over p9 this enables us to translate the problem of comput ing characteristic numbers for a family of plane cubics into one of computing five segre classes and to recover classic enumerative results of zeuthen and maillard

une methode numerique pour les problemes d equilibre de corps hyperelastiques compressibles en grandes deformations cet article applique les techniques de lagrangien augmente a la resolution numerique des problemes d equilibre de corps hyperelastiques compressibles soumis a de grandes deformations la methode obtenue est illustree par plusieurs exemples numeriques this paper uses augmented lagrangian techniques for the numerical solution of equilibrium problems of compressible hyperelastic bodies subjected to large deformations the resulting method is illustrated by several numerical examples