dirichlet problem for demi coercive functionals on considere des fonctionnelles integrales du type f u f du ou f r n 0 est une fonction convexe ayant la propriete suivante il existe a 0 b 0 r n tel que f z a z b z r n

compatibility conditions for nonlinear neumann problems abstract results we recall an abstract definition introduced in the general framework of the minimization of non coercive functionals see bbgt definition 2 1 let v a be a topological vector space and let f v co co be any mapping we call a sequential recession functional the mapping f u inf lim inf i f tnun t al u p u n m t i where t and u are sequences notice that when f is convex and a lower semicontinuous f coincides with the usual recession functional see for instance r f tu u f v lim t r r to2 where u is any point in dom f in the following we set ker f u e i f u o remark that in general ker f is not a subspace but is a possibly nonconvex set of half lines starting from the origin the following necessary condition for existence was proved in bbgt proposition 2 2 assume that the problem min f u ue v 2 1

elliptic surfaces and linear systems with fat points we investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces for the projectivization of the atiyah bundle over an elliptic curve with a certain polarization we observe that the special case of only one fat point implies the general case of arbitrarily many fat points as well as results concerning other surfaces we conjecture that this special case holds in characteristic 0 but prove that it fails in any positive characteristic

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

a simple proof that rational curves on k3 are nodal the picard group of s is generated by two effective divisors c and f with c 2 f 2 0 and c f 1 it can be realized as an elliptic fibration over p with a unique section c fibers f and 2 here c1 is the first chern class of the hodge bundle of the fibration s p see h m it is a standard result that the number of nodal fibers of an elliptic fibration are given by 12 h m p 158 so there are exactly 24 rational nodal curves in the linear series f for s general this is the same special k3 surface used by bryan and leung in their counting of curves on k3 surfaces b l it is actually the attempt to understand their method that leads us to our proof we will call a k3 surface with picard lattice 1 1 a bl k3 surface a bl k3 surface s lies on the boundary of the moduli space of k3 surfaces of genus g with c gf as the corresponding primitive divisor every curve in the linear series os c gf is totally reducible i e it consists of the 2 curve c and g elliptic tails attached to c a curve d os c gf is the image of a stable rational map only if d c m1f1 m2f2 m24f24 where f1 f2 f24 are 24

noncoercive variational inequalities some applications ever since the fundamental paper of lions and stampacchia io the subject of variational inequalities has become a very active area of study and become an important tool in the study of nonlinear partial differential equations which are subject to various types of constraints see e g l 5 8 111 in lo the authors consider first the case of coercive forms and then using elliptic regularization problems involving noncoercive but nonnegative forms we shall use a similar approach to study variational inequalities involving noncoercive bilinear forms and nonlinear functionals which are nice perturbations of lower semi continuous convex functionals furthermore our bilinear forms need not necessarily have a finite dimensional kernel our investigation has been motivated by some work of baiocchi gastaldi and tomarelli see 3 61 where variational inequalities involving so called compact coercive bilinear forms are studied we shall introduce here a somewhat more general concept that of a p coercive bilinear form and study variational inequalities involving such forms and nonlinear functionals which are compact and bounded perturbations of lower semi continuous convex functionals in the next section we shall introduce the necessary terminology prove some preliminary results and give the statement of the abstract problem we then shall state and prove our main abstract result and conclude the paper with applications including one to the study of a flat plate on an elastic foundation with unilateral boundary conditions see e g l for the terminology and one to the study of unilateral problems in semipermeable media see 5

on reduced stable pair invariants let x s times e be the product of a k3 surface s and an elliptic curve e reduced stable pair invariants of x can be defined via 1 cutting down the reduced virtual class with incidence conditions or 2 the behrend function weighted euler characteristic of the quotient of the moduli space by the translation action of e we show that 2 arises naturally as the degree of a virtual class and that the invariants 1 and 2 agree this has applications to deformation invariance rationality and a dt pt correspondence for reduced invariants of s times e

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

the hopf bifurcation with symmetry for the navier stokes equations in l p ω n with application to plane poiseuille flow the existence of periodic solutions of the navier stokes equations in function spaces based upon lp nis proved the paper has three parts a a proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of lp n b the evolution equation is restricted to a space of time periodic functions and a fredholm integral equation on this space is formed the lyapunov schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry c the theory is applied to the bifurcation of periodic solutions from planar poiseuille flow in the presence of symmetry so 2 x o 2 x s1 yielding new results for this classic problem the o 2 invariance is in the spanwise direction with the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction there are however points of degeneracy on the neutral surface a numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi periodic solutions

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

existence theorems for generalized noncoercive equilibrium problems the quasi convex case we provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems an extension of generalized quasi variational inequalities defined in reflexive banach spaces in the quasi convex case in addition several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems in fact we find some particular cones as estimates for the recession cone of the solution set these cones for the ones containing the latter set are proved to be sharp enough to encompass several special situations found in the literature

holomorphic anomaly equations and the igusa cusp form conjecture let s be a k3 surface and let e be an elliptic curve we solve the reduced gromov witten theory of the calabi yau threefold s times e for all curve classes which are primitive in the k3 factor in particular we deduce the igusa cusp form conjecture the proof relies on new results in the gromov witten theory of elliptic curves and k3 surfaces we show the generating series of gromov witten classes of an elliptic curve are cycle valued quasimodular forms and satisfy a holomorphic anomaly equation the quasimodularity generalizes a result by okounkov and pandharipande and the holomorphic anomaly equation proves a conjecture of milanov ruan and shen we further conjecture quasimodularity and holomorphic anomaly equations for the cycle valued gromov witten theory of every elliptic fibration with section the conjecture generalizes the holomorphic anomaly equations for elliptic calabi yau threefolds predicted by bershadsky cecotti ooguri and vafa we show a modified conjecture holds numerically for the reduced gromov witten theory of k3 surfaces in primitive classes

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

semilinear evolution equations in banach spaces with application to parabolic partial differential equations a theory for a class of semilinear evolution equations in banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data

existence of optimal solutions and duality results under weak conditions in this paper we consider an ordinary convex program with no qualification conditions such as slater s condition and for which the optimal set is neither required to be compact nor to be equal to the sum of a compact set and a linear space it is supposed only that the infimum is finite a very wide class of convex functions is exhibited for which the optimum is always attained and is equal to the supremum of the ordinary dual program additional results concerning the existence of optimal solutions in the non convex case are also given

integrals of ψ classes over double ramification cycles a double ramification cycle or dr cycle is a codimension g cycle in the moduli space overline mathcal m g n of stable curves roughly speaking given a list of integers a 1 ldots a n the dr cycle rm dr g a 1 ldots a n subset overline mathcal m g n is the locus of curves c x 1 ldots x n such that the divisor sum a ix i is principal we compute the intersection numbers of dr cycles with all monomials in psi classes

open gromov witten invariants on elliptic k3 surfaces and wall crossing in this paper we study holomorphic discs in k3 surfaces and defined the open gromov witten invariants using this new invariant we can establish a version of correspondence between tropical discs and holomorphic discs with non trivial invariants we give an example of wall crossing phenomenon of the invariant and expect it satisfies kontsevich soibelman wall crossing formula

peristaltic transport of a heat conducting viscous fluid as an application of abstract differential equations and semigroup of operators abstract a recent abstract result of rankin trans amer math soc to appear is applied to the peristaltic transport of a heat conducting fluid in a flexible tube the oberbeck boussinesq equations are used as the governing equations a pressure drop over a wavelength of the tube is prescribed to ensure uniqueness and newton s cooling law for the temperature is imposed on the boundary with the notion of a weak solution introduced local existence uniqueness regularity and continuation of solutions are considered in particular w r 2 regularity of the solutions is established

contact between nonlinearly elastic bodies we study the contact between nonlinearly elastic bodies by variational methods after the formulation of the mechanical problem we provide existence results based on polyconvexity and on quasiconvexity then we derive the euler lagrange equation as a necessary condition for minimizers here clarke s generalized gradients are the essential tool to treat the nonsmooth obstacle condition

rational curves in holomorphic symplectic varieties and gromov witten invariants abstract we use gromov witten theory to study rational curves in holomorphic symplectic varieties we present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of k 3 n type we also classify all rational curves in the primitive curve class of the fano variety of lines in a very general cubic 4 fold and prove the irreducibility of the corresponding moduli space our proofs rely on gromov witten calculations by the first author and in the fano case on a geometric construction of voisin in the fano case a second proof via classical geometry is sketched

relaxed formulation and existence result of the degenerated elliptic small disturbance model abstract we consider an elliptic parabolic pde governing the flow of a steady state two dimensional irrotational compressible fluid in a channel or around a thin profile the model is formulated in term of a noncoercive variational problem with an integral functional in a truncated domain we propose a relaxed formulation in the bounded variation functions space and we prove the existence and the uniqueness of the relaxed optimal solution we consider also the asymptotic behavior of the solution when the domain grows to infinity

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

virasoro constraints for target curves we prove generalized virasoro constraints for the relative gromov witten theories of all nonsingular target curves descendents of the even cohomology classes are studied first by localization degeneration and completed cycle methods descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations as an outcome of our results the relative theories of target curves are completely and explicitly determined

density of rational curves on k3 surfaces using the dynamics of self rational maps of elliptic k3 surfaces together with deformation theory we prove that the union of rational curves is dense on a very general k3 surface and that the union of elliptic curves is dense in the 1st jet space of a very general k3 surface both in the strong topology

existence and regularity for a class of integrodifferential equations of parabolic type the method we use rests on the inversion of the formal laplace transform of r and allows one to analyze the regularity of the solution u problem 1 1 has been studied by several authors using various methods see for instance l 18 211 the laplace transform approach presented here has been used in s 7 9 14 153 in our previous papers s 7 93 we have in fact analyzed a more general situation while here we take full 36 0022 247x 85 3 00

a semilinear parabolic volterra integrodifferential equation abstract we consider the semilinear volterra integrodifferential equation u t a t u t t 0 t a t s g s u s ds t u t t t 0 u t 0 u 0 we assume that for each t 0 the linear operator a t is the negative generator of an analytic semigroup in a banach space x the nonlinear operator g t is lipschitz continuous on the domain of a 0 into x and the forcing term t u satisfies a holder condition in both variables we prove existence of solutions in the banach space x and uniqueness of solutions when x is a hilbert space we also consider an example where a t a a t s a t s and show that the corresponding solutions u t x converge as t to a solution u x of a related semilinear elliptic boundary value problem

weak solutions for a class of parabolic volterra integrodifferential equations abstract we define weak solutions for a class of volterra integrodifferential equations of the form u t au t 0t a t s g s u s ds f t u t t 0 u 0 0 the operator a is the negative infinitesimal generator of an analytic semigroup in a banach space x the operator g t u is related to a by a special form g t u a 1 2 q t u where q t u is an appropriate lower order operator we show the existence and uniqueness of weak solutions and their continuability to infinity under suitable conditions using our results we study the asymptotic behavior as time goes to infinity of strong solutions of a second order initial boundary value problem

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

convex analysis can be helpful for the asymptotic analysis of monotone operators asymptotic analysis of monotone operators we use representations of maximal monotone operators for studying recession or asymptotic operators associated to maximal monotone operators such a concept is useful for dealing with unboundedness

existence theorems for contact with friction of two linear elastic bodies abstract in this article i give an iteration method for solving the signorini problem by the idea of concentration cancellation the nonlinear quantity in the mathematical formulation of the problem caused by contact friction effect on the boundary is relaxed to a convex functional therefore the relaxed problem can be solved by using convex minimization techniques which are numerically stable finally good convergence property of the proposed method is proved

gopakumar vafa bps invariants hilbert schemes and quasimodular forms i we prove a closed formula for leading gopakumar vafa bps invariants of local calabi yau geometries given by the canonical line bundles of toric fano surfaces it shares some sim ilar features with gottsche yau zaslow formula connection with hilbert schemes connection with quasimodular forms and qua dratic property after suitable transformation in part i of this paper we will present the case of projective plane more general cases will be presented in part ii the problem of counting curves in algebraic varieties dates back to the nineteenth century classical examples include the famous 27 lines on a cubic surface and the 2875 lines on a quintic 3 fold through the interaction with string theory spectacular progresses in this classi cal branch of algebraic geometry have been made over the years since 1990 s first of all gromov witten theory and its various variants have laid the foundation for the modern treatment of many classical enumerative problems and greatly expanded the range of enumerative problems being considered see pt for an introduction to some of the exciting developments the study of the problems of counting curves in surfaces and calabi yau 3 folds have served both as motivations and applications of gromov witten theory we will recall some results in these directions in 2 and 3 to serve as motivations for this work we emphasize on the following three salient features for the curving counting problems for algebraic surfaces connection with hilbert schemes connection with quasimod ular forms and quadratic properties of the node polynomials after suit able transformation our main results will show that these features are also shared in the curve counting problems of some noncompact calabi yau 3 folds arising as the the total space of the canonical line bundles of toric fano surfaces 1

fully nonlinear integrodifferential equations in general banach space on etudie des equations integrodifferentielles non lineaires dans un espace de banach qui peuvent etre considerees comme la version abstraite d equations integrodifferentielles partielles non lineaires on considere un probleme aux valeurs limites et initiales

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

gromov witten pairs correspondence for the quintic 3 fold we use the gromov witten pairs descendent correspondence for toric 3 folds and degeneration arguments to establish the gw p correspondence for several compact calabi yau 3 folds including all cy complete intersections in products of projective spaces a crucial aspect of the proof is the study of the gw p correspondence for descendents in relative geometries projective bundles over surfaces relative to a section play a special role the gw p correspondence for calabi yau complete intersections provides a structure result for the gromov witten invariants in a fixed curve class after change of variables the gromov witten series is a rational function in the variable q eiu invariant under q q 1

genus zero gopakumar vafa type invariants for calabi yau 4 folds abstract in analogy with the gopakumar vafa conjecture on cy 3 folds klemm and pandharipande defined gv type invariants on calabi yau 4 folds using gromov witten theory and conjectured their integrality in this paper we propose a sheaf theoretic interpretation of their genus zero invariants using donaldson thomas theory on cy 4 folds more specifically we conjecture genus zero gv type invariants are d t 4 invariants for one dimensional stable sheaves on cy 4 folds some examples are computed for both compact and non compact cy 4 folds to support our conjectures we also propose an equivariant version of the conjectures for local curves and verify them in certain cases

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

gromov witten theory and cycle valued modular forms in this paper we proved generating functions of gromov witten cycles of the elliptic orbifold lines with weights 3 3 3 4 4 2 and 6 3 2 are cycle valued quasi modular forms this is a generalization of milanov and ruan s work on cycle valued level first we construct a global cohomology field theory cohft for simple elliptic singularities modulo an extension problem and prove its modularity then we apply teleman s reconstruction theorem to prove mirror theorems on cycled valued level and match it with a cohft from gromov witten theory of a corresponding orbifold this solves the extension property as well as inducing the modularity for a gromov witten cohft

compactness properties for families of quasistationary solutions of some evolution equations the following typical problem occurs in passing to the limit in some phase field models for two sequences of space time dependent functions n x n representing e g suitable approximations of the temperature and the phase variable we know that the sum n x n converges in some l p space as n and that the time integrals of a suitable space functional evaluated on n x n are uniformly bounded with respect to n can we deduce that n and x n converge separately luckhaus 1990 gave a positive answer to this question in the framework of the two phase stefan problem with gibbs thompson law for the melting temperature plotnikov 1993 proposed an abstract result employing the original idea of luckhaus and arguments of compactness and reflexivity type we present a general setting for this and other related problems providing necessary and sufficient conditions for their solvability these conditions rely on general topological and coercivity properties of the functionals and the norms involved and do not require reflexivity

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

noncoercive optimization problems this paper modifies a recent sufficient condition for the existence of optimal solutions proposed by buttazzo and tomarelli to obtain a condition that is necessary and sufficient applications to mathematical programming are given and stability results established

asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications a class of semilinear hyperbolic volterra integrodifferential equations in hilbert spaces is considered the main results in this paper are the global existence of solutions to the initial problem of the equation for large data and an asymptotic estimation of the solution an application to a system relating to three dimensional viscoelastic dynamics is presented

how to deal with the unbounded in optimization theory and algorithms the aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions but also for constructing new methods

a conjectural generating function for numbers of curves on surfaces i give a conjectural generating function for the numbers of nodal curves in a linear system of dimension on an algebraic surface it reproduces the results of vainsencher v for the case le 6 and kleiman piene k p for the case le 8 the numbers of curves are expressed in terms of five universal power series three of which i give explicitly as quasimodular forms this gives in particular the numbers of curves of arbitrary genus on a k3 surface and an abelian surface in terms of quasimodular forms generalizing the formula of yau zaslow for rational curves on k3 surfaces the coefficients of the other two power series can be determined by comparing with the recursive formulas of caporaso harris for the severi degrees in 2 we verify the conjecture for genus 2 curves on an abelian surface we also discuss a link of this problem with hilbert schemes of points

unilateral problems in nonlinear three dimensional elasticity in 1977 john ball obtained a series of very important existence results in hyperelasticity the objective of the present paper is to extend his results in two directions first by considering unilateral boundary conditions corresponding to contact without friction secondly by allowing locking constraints in the set of admissible deformations

macmahon s sum of divisors functions chebyshev polynomials and quasi modular forms we investigate a relationship between macmahon s generalized sum of divisors functions and chebyshev polynomials of the rst kind this determines a recurrence relation to compute these functions as well as proving a conjecture of macmahon about their general form by relating them to quasimodular forms these functions arise as solutions to a curve counting problem on abelian surfaces

rational curves and points on k3 surfaces we study the distribution of algebraic points on k3 surfaces 1 introduction let k be a field and k a fixed algebraic closure of k we are interested in connections between geometric properties of algebraic varieties and their arithmetic properties over k over its finite extensions k k or over k here we study certain varieties of intermediate type namely k3 surfaces and their higher dimensional generalizations calabi yau varieties to motivate the following discussion let 5 be a k3 surface over k in positive characteristic 5 may be unirational and covered by rational curves examples are supersingular k3 surfaces over fields of characteristic two or the surface