conditions imposed by tacnodes and cusps the study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress the horace method introduced by a hirschowitz has been successfully exploited to prove many h1 vanishing theorems even in higher dimension other specialization techniques which include degenerations of the plane are due to z ran and c ciliberto and r miranda g m greuel c lossen and e shustin use a local specialization procedure together with the horace method to give the first asymptotically proper general existence criterion for singular curves of low degree in this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by greuel lossen and shustin for curves with tacnodes and cusps

an analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports abstract a general analytical solution is obtained for the free vibration of rectangular plates supported on their lateral surface by symmetrically distributed point supports highly accurate eigenvalues are provided for a square plate supported at various points along the diagonal results are compared with earlier findings obtained by a finite element method it is shown that the method may be extended to solve various other problems the tabulation of eigenvalues appears to be the most comprehensive yet made for a square plate with point supports on the diagonals

the minkowski problem for polytopes abstract the traditional solution to the minkowski problem for polytopes involves two steps first the existence of a polytope satisfying given boundary data is demonstrated in the second step the uniqueness of that polytope up to translation is then shown to follow from the equality conditions of minkowski s inequality a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context in this article we adapt the classical argument to prove both the existence theorem of minkowski and his mixed volume inequality simultaneously thereby providing a new proof of minkowski s inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

on use of the dirac delta function in the vibration analysis of elastic structures the dirac delta function has often been employed to represent the amplitude of concentrated harmonic forces in the analysis of vibration of elastic structures such as beams and plates it is known that this function as represented by a truncated fourier series does not provide a true representation of a concentrated force nevertheless it is frequently employed and good convergence is usually though not always encountered in solutions thereby obtained in this paper the nature of the function is discussed and for illustrative purposes it is used to obtain series solutions for some selected beam and plate free vibration problems in some cases problems are chosen for which exact solutions are already obtainable by analytical means this permits powerful checks to be made on rates of convergence experienced when the series solutions are investigated rates of convergence are discussed in detail and it is explained why convergence is to be expected when analyzing certain families of problems when employing this function and a lack of convergence is to be expected in others

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

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

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

handling convexity like constraints in variational problems we provide a general framework to construct finite dimensional approximations of the space of convex functions which also applies to the space of c convex functions and to the space of support functions of convex bodies we give precise estimates of the distance between the approximation space and the admissible set this framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite dimensional spaces such as tensor product splines we show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints our implementation relies on proximal algorithms and can be easily parallelized thus making it applicable to large scale problems in dimension two and three we illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation three dimensional denoising the principal agent problem and optimizat

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

bending of a rectangular plate with rotationally restrained edges under a concentrated force the bending problem of a rectangular plate with rotationally restrained edges under a concentrated force is studied an emphasis is placed on the determination of the corner forces and deflection the problem is solved by superposition of classical navier s and levy s solutions analytic solutions in terms of series are obtained for rsss rrss and rsrs plates respectively where r and s stand for rotational restraint and simple support respectively some important design parameters such as maximum deflection and corner force are evaluated the effect of edge restraint on these parameters is illustrated

the spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

embedding sn into rn 1 with given integral gauss curvature and optimal mass transport on sn in a d aleksandrov convex polyhedra gittl moscow ussr 1950 in russian english translation a d aleksandrov convex polyhedra springer berlin new york 2005 a d aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data the first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in euclidean space with prescribed integral gauss curvature our solution includes the case of a convex polytope this problem was also first considered by aleksandrov and below it is referred to as aleksandrov s problem the second goal of this paper is to show that in variational form the aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

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

moduli of nodal curves on k3 surfaces abstract we consider modular properties of nodal curves on general k3 surfaces let k p be the moduli space of primitively polarized k3 surfaces s l of genus p 3 and v p m k p be the universal severi variety of nodal irreducible curves in m l on s l k p we find conditions on p m for the existence of an irreducible component v of v p m on which the moduli map v m g with g m 2 p 1 1 has generically maximal rank differential our results which for any p leave only finitely many cases unsolved and are optimal for m 5 except for very low values of p are summarized in theorem 1 1 in the introduction

degenerations of planar linear systems fixing n general points pi in the plane what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i in this article we propose an approach to attack this problem and demonstrate it by successfully computing this dimension for all n and for mi constant at most 3 this application while previously known see hi1 demonstrates the utility of our approach which is based on an analysis of the corresponding linear system on a degeneration of the plane itself leading to a simple recursion for these dimensions we also obtain results in the quasi homogeneous case when all the multiplicities are equal except one this is the natural family to consider in the recursion

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

monge s transport problem on a riemannian manifold monge s problem refers to the classical problem of optimally transporting mass given borel probability measures find the measurepreserving map s m m between them which minimizes the average distance transported set on a complete connected riemannian manifold m and assuming absolute continuity of an optimal map will be shown to exist aspects of its uniqueness are also established

free in plane vibration analysis of rectangular plates by the method of superposition abstract the superposition method is introduced as a means for obtaining analytical type solutions for free in plane vibration of rectangular plates the governing differential equations and boundary conditions are expressed in dimensionless form the problem of free in plane vibration of the completely free rectangular plate is resolved for illustrative purposes convergence is found to be rapid and excellent agreement between computed results and those obtained by previous authors utilizing the rayleigh ritz energy method is obtained it is pointed out that following procedures analogous to those utilized in resolving lateral plate vibration problems in plane free vibration problems related to point supported plates plates with in plane elastic boundary support etc are now amenable to solution by this method

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

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

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

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

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159