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

broccoli curves and the tropical invariance of welschinger numbers abstract in this paper we introduce broccoli curves certain plane tropical curves of genus zero related to real algebraic curves the numbers of these broccoli curves through given points are independent of the chosen points for arbitrary choices of the directions of the ends of the curves possibly with higher weights and also if some of the ends are fixed in the toric del pezzo case we show that these broccoli invariants are equal to the welschinger invariants with real and complex conjugate point conditions thus providing a proof of the independence of welschinger invariants of the point conditions within tropical geometry the general case gives rise to a tropical caporaso harris formula for broccoli curves which suffices to compute all welschinger invariants of the plane

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

floor diagrams relative to a conic and gw w invariants of del pezzo surfaces abstract we enumerate via floor diagrams complex and real curves in c p 2 blown up in n points on a conic as an application we deduce gromov witten and welschinger invariants of del pezzo surfaces these results are mainly obtained using li s degeneration formula and its real counterpart

normal crossings singularities for symplectic topology abstract we introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent in a suitable sense to the desired geometric notions our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology this partially answers gromov s question on the feasibility of defining singular symplectic sub varieties and lays foundation for rich developments in the future in subsequent papers we establish a smoothability criterion for symplectic normal crossings varieties in the process providing the multifold symplectic sum envisioned by gromov and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors

intersection numbers and rank one cohomological field theories in genus one we obtain a simple recursive presentation of the tautological and classes on the moduli space of curves in genus 0 and 1 in terms of boundary strata graphs we derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials as an application we describe the moduli space of normalized restricted rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products our results simplify and generalize those of kaufmann manin and zagier

a fock space approach to severi degrees the classical severi degree counts the number of algebraic curves of fixed genus and class passing through points in a surface we express the severi degrees of p1 p1 as matrix elements of the exponential of a single operator ms on fock space the formalism puts severi degrees on a similar footing as the more developed study of hurwitz numbers of coverings of curves the pure genus 1 invariants of the product e p1 with e an elliptic curve are solved via an exact formula for the eigenvalues of ms to initial order the severi degrees of p2 are also determined by ms via the 1 d 1 d2 disk multiple cover formula for calabi yau threefold geometries

approximation properties of bezier summation integral type operators based on polya bernstein functions in this article we introduce the bezier variant of summation integral type operators having polya and bernstein basis functions we give a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation

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

enumerative geometry of singular plane curves general points in the plane the answer is easy for 6 1 namely 3 d l 2 but otherwise was to my knowledge unknown until now we are going to develop a recursive procedure for solving such problems to make the recursion possible it will be necessary to consider a generalization of independent interest of the above problem let s be the blow up of ip 2 at a point p with exceptional divisor e1 and a fixed line section eo we call s an irreducible fan a curve c c s is said to be of type d e if it is the proper transform of a curve of degree d in z having multiplicity e at p one then asks how many curves c c s of given type d e having only smooth branches of given configuration along e0 w e1 and otherwise having a given number of nodes as their only singularities pass through the appropriate number of points of s some of which may be on e o w e we will solve this problem by recursion on d e the recursion is based on degenerating the surface s to a surface called a 2 fan of the form so 1 w s 2 with si s i i 2 using and suitably enhancing methods of 3 we can describe a suitable part of the limit on so of the relevant families of curves of type d e on s d e 1 in terms of similar families of curves of type d e and e e e e d on 1 and 2 respectively

on the existence regularity and decay of solitary waves to a generalized benjamin ono equation in r2 where 0 and 1 2 is the operator de2ned by f 1 2u u f or represent the fourier transform eq 1 describes the dynamics of three dimensional slightly nonlinear disturbances in boundary layer shear ows without the assumption of a di erence in their scales along and across the ow see 1 6 the solitary waves of 1 are solutions of the form a x y t v x ct y where c is the speed of the solitary wave it seems that solitary waves play an important role in the evolution of 1 such a solution must satisfy the equation

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

the caporaso harris formula and plane relative gromov witten invariants in tropical geometry some years ago caporaso and harris have found a nice way to compute the numbers n d g of complex plane curves of degree d and genus g through 3d g 1 general points with the help of relative gromov witten invariants recently mikhalkin has found a way to reinterpret the numbers n d g in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes we relate these two results by defining an analogue of the relative gromov witten invariants and rederiving the caporaso harris formula in terms of both tropical geometry and lattice paths

szasz durrmeyer type operators based on charlier polynomials in the present paper we introduce a durrmeyer type modification of szasz operators based on charlier polynomials and study a voronovskaja type asymptotic theorem local approximation theorem weighted approximation statistical convergence and approximation of functions with derivatives of bounded variation for these operators

rate of convergence of bernstein polynomials for functions with derivatives of bounded variation abstract for any x 0 1 we first prove that if x t t x on 0 1 then the bernstein polynomials of x satisfy the asymptotic relation k 0 n k n x k n x k 1 x n k 2x 1 x 1 2 1 n o 1 n this asymptotic relation is then used to study the rate of convergence of bernstein polynomials for functions with derivatives of bounded variation an estimate of the rate of convergence is given this estimate is asymptotically the best possible at points where is continuous

welschinger invariants of real del pezzo surfaces of degree 3 we give a recursive formula for purely real welschinger invariants of real del pezzo surfaces of degree k 2 3 where in the case of surfaces of degree 3 with two real components we introduce a certain modification of welschinger invariants and enumerate exclusively the curves traced on the non orientable component as an application we prove the positivity of the invariants under consideration and their logarithmic asymptotic equivalence as well as congruence modulo 4 to genus zero gromov witten invariants

uniruledness of strata of holomorphic differentials in small genus abstract we address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials we show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on k3 and del pezzo surfaces respectively restricting to genus 3 g 6 we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g 1 hence showing in addition that these strata are unirational

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

a numerical study of the whitham equation as a model for steady surface water waves the object of this article is the comparison of numerical solutions of the so called whitham equation to numerical approximations of solutions of the full euler free surface water wave problem the whitham equation t 3 2 c 0 h 0 x k h 0 x 0 was proposed by whitham 1967 as an alternative to the kdv equation for the description of wave motion at the surface of a perfect fluid by simplified evolution equations but the accuracy of this equation as a water wave model has not been investigated to date in order to understand whether the whitham equation is a viable water wave model numerical approximations of periodic solutions of the kdv and whitham equation are compared to numerical solutions of the surface water wave problem given by the full euler equations with free surface boundary conditions computed by a novel spectral element method technique the bifurcation curves for these three models are compared in the phase velocity waveheight parameter space and wave profiles are compared for different wavelengths and waveheights it is found that for small wavelengths the steady whitham waves compare more favorably to the euler waves than the kdv waves for larger wavelengths the kdv waves appear to be a better approximation of the euler waves

symmetry and decay of traveling wave solutions to the whitham equation abstract this paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model an exponential decay result for supercritical solitary wave solutions is given moreover it is shown that all such solitary wave solutions are symmetric and monotone on either side of the crest the proof is based on the method of moving planes furthermore a close relation between symmetric and traveling wave solutions is established

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

statistical approximation of a kind of kantorovich type q szasz mirakjan operators abstract in the present paper we introduce a kantorovich type generalization of the q szasz mirakjan operators and investigate their a statistical approximation properties also the rate of a statistical convergence of the proposed sequence of linear positive operators is obtained by a weighted modulus of continuity

on statistical approximation of a general class of positive linear operators extended in q calculus in this paper we present a general class of positive linear operators of discrete type based on q calculus and we investigate their weighted statistical approximation properties by using a bohman korovkin type theorem we also mark out two particular cases of this general class of operators

the kodaira dimension of moduli spaces of curves with marked points the purpose of this paper is to generalize the methods and results due to eisenbud harris and mumford on the moduli space of curves of genus g to the moduli space of curves of genus g with n marked points the main result is the determination for each g with 4 g 23 of an n such that the moduli space is of general type 1 introduction in the early 1980s eisenbud harris and mumford stud ied mg the moduli space of curves of a given genus and succeeded in proving that if the genus is at least 24 the moduli space is of general type this was contrary to many people s expectations as it had been known for some time that moduli spaces of curves of small genus were rational or unirational briefly what is involved in their proof is the computation of the canonical divisor of mg the exhibition of certain effective divisors and the determination of their classes and the verification that the canonical divisor is the sum of an ample and an effective divisor in the other direction it is known that mg has kodaira dimension when g 13 or g 15 one can generalize the definition of mg by asking for a parametrization of the curves of genus g together with n distinct nonsingular points on them this amounts to studying instead of families of curves families of curves with n disjoint sections as before it turns out that one obtains a noncompact moduli space which is compactified by replacing nonsingular curve by stable curve in the definitions in order to be considered stable an n pointed curve is required only to have finitely many automorphisms that fix all the marked points the space can still be compact even when the marked points are required to be distinct and nonsingular because coincident or singular marked points can be blown up on the total space of the family

the whitham equation as a model for surface water waves the whitham equation was proposed as an alternate model equation for the simplified description of uni directional wave motion at the surface of an inviscid fluid as the whitham equation incorporates the full linear dispersion relation of the water wave problem it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the kdv equation in this work we identify a scaling regime in which the whitham equation can be derived from the hamiltonian theory of surface water waves the whitham equation is integrated numerically and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the euler equations the performance of the whitham equation as a model for free surface dynamics is also compared to two standard free surface models the kdv and the bbm equation it is found that in a wide parameter range of amplitudes and wavelengths the whitham equation performs on par with or better than both the kdv and bbm equations

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

weighted statistical approximation by kantorovich type q szasz mirakjan operators abstract in the present paper we introduce a kantorovich type modification of q szasz mirakjan operators and obtain weighted statistical approximation properties of these operators also for introduced operators we give a voronovskaja type theorem related to q derivatives

rate of convergence of lupas kantorovich operators based on polya distribution in the present paper we consider the kantorovich modification of lupas operators based on polya distribution we estimate the rate of convergence for absolutely continuous functions having a derivative coinciding a e with a function of bounded variation

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

on the integrated baskakov type operators the present paper deals with the study of the rate of convergence of the integrated baskakov durrmeyer type operators in simultaneous approximation we also mention some of the improvements on the recent paper of deo n deo direct results on exponential type operators applied mathematics and computation 204 1 2008 109 115

generalized baskakov szasz type operators abstract in the present paper we introduce generalized baskakov szasz type operators and study some approximation properties of these operators e g rate of convergence in ordinary and simultaneous approximation statistical convergence and the estimate of the rate of convergence for absolutely continuous functions having a derivative coinciding a e with a function of bounded variation

a combinatorial analysis of severi degrees abstract based on results by brugalle and mikhalkin fomin and mikhalkin give formulas for computing classical severi degrees n d using long edge graphs in 2012 block colley and kennedy considered the logarithmic version of a special function associated to long edge graphs which appeared in fomin mikhalkin s formula and conjectured it to be linear they have since proved their conjecture at the same time motivated by their conjecture we consider a special multivariate function associated to long edge graphs that generalizes their function the main result of this paper is that the multivariate function we define is always linear a special case of our result gives an independent proof of block colley kennedy s conjecture the first application of our linearity result is that by applying it to classical severi degrees we recover quadraticity of q d and a bound for the threshold of polynomiality of n d next in separate joint work with osserman we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the gottsche yau zaslow formula as a result we provide combinatorial formulas for the two unidentified power series b 1 q and b 2 q appearing in the gottsche yau zaslow formula the proof of our linearity result is completely combinatorial we define graphs which generalize long edge graphs and a closely related family of combinatorial objects we call n words by introducing height functions and a concept of irreducibility we describe ways to decompose certain families of n words into irreducible words which leads to the desired results

durrmeyer type modification of generalized baskakov operators abstract in the present paper durrmeyer type modification of generalized baskakov operators is introduced and some direct results are established for these operators

weighted approximation by a certain family of summation integral type operators in this study we investigate the weighted approximation properties of a general sequence of summation integral type operators introduced by srivastava and gupta 1 we estimate the rate of convergence of these operators for functions of polynomial growth in terms of weighted modulus of continuity on the interval 0

enumeration of rational plane curves tangent to a smooth cubic abstract we compute the number of rational degree d plane curves having prescribed fixed and moving contacts to a smooth plane cubic e we use twisted stable maps to the stack p e r 2 for r large where p e r 2 is the rth root of p 2 along e we prove that certain gromov witten invariants of this stack are enumerative and establish recursive formulas for these numbers

approximation of baskakov type polya durrmeyer operators in the present paper we propose the durrmeyer type modification of baskakov operators based on inverse polya eggenberger distribution first we estimate a recurrence relation by using hypergeometric series we give a global approximation theorem in terms of second order modulus of continuity a direct approximation theorem by means of the ditzian totik modulus of smoothness and a voronovskaja type theorem some approximation results in weighted space are obtained also we show the rate of convergence of these operators to certain functions by illustrative graphics using the maple algorithms

enumeration of rational curves with cross ratio constraints abstract in this paper we prove the algebraic tropical correspondence for stable maps of rational curves with marked points to toric varieties such that the marked points are mapped to given orbits in the big torus and in the boundary divisor the map has prescribed tangency to the boundary divisor and certain quadruples of marked points have prescribed cross ratios in particular our results generalize the results of nishinou siebert 14 the proof is very short involves only the standard theory of schemes and works in arbitrary characteristic including the mixed characteristic case

exponential decay and symmetry of solitary waves to degasperis procesi equation abstract we improve the decay argument by bona and li 1997 5 for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady degasperis procesi equation in the nonlocal formulation in addition we give a method which confirms the symmetry of solitary waves including those of the maximum height finally we discover how the symmetric structure is connected to the steady structure of solutions to the degasperis procesi equation and give a more intuitive proof for symmetric solutions to be traveling waves the improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

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

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k

severi degrees on toric surfaces ardila and block used tropical results of brugalle and mikhalkin to count nodal curves on a certain family of toric surfaces building on a linearity result of the first author we revisit their work in the context of the goettsche yau zaslow formula for counting nodal curves on arbitrary smooth surfaces addressing several questions they raised by proving stronger versions of their main theorems in the process we give new combinatorial formulas for the coefficients arising in the goettsche yau zaslow formulas and give correction terms arising from rational double points in the relevant family of toric surfaces