on bounding the number of generators for fat point ideals on the projective plane abstract let x be the surface obtained by blowing up general points p1 pn of the projective plane over an algebraically closed ground field k and let l be the pullback to x of a line on the plane if c is a rational curve on x with c l d then for every t there is a natural map c o c t x o x l c o c t d given by multiplication on simple tensors the ranks of such maps are determined as a function of t d and m where m is the largest multiplicity of c at any of the points pi if i is the ideal defining the fat point subscheme z m1p1 mnpn p2 and is the least degree in which i has generators then the ranks of the maps c o c t x o x l c o c t d can be used for bounding the number of generators of i in degrees t 1

polarized endomorphisms of complex normal varieties it is shown that a complex normal projective variety has non positive kodaira dimension if it admits a non isomorphic quasi polarized endomorphism the geometric structure of the variety is described by methods of equivariant lifting and fibrations

zariski decomposition of b divisors based on a recent work of thomas bauer s j algebr geom to appear reproving the existence of zariski decompositions for surfaces we construct a b divisorial analogue of zariski decomposition in all dimensions

a divisorial valuation with irrational volume of an m primary ideal a r in fact if qm a for a fixed ideal a then it is evident that e a vol v the volume of a valuation has implicitly been studied already in 4 but it was first explicitly defined in 5 the terminology is intended to emphasize the relation with global invariants of linear series on projective varieties a natural question is to what extent the properties of vol v mirror those of the samuel multiplicity results of 5 9 assert that

characterizing volume via cone duality for divisors over smooth projective varieties we show that the volume can be characterized by the duality between the pseudo effective cone of divisors and the movable cone of curves inspired by this result we define and study a natural intersection theoretic volume functional for 1 cycles over compact kahler manifolds in particular for numerical equivalence classes of curves over projective varieties it is closely related to the mobility functional studied by lehmann

threefolds with vanishing hodge cohomology we consider algebraic manifolds y of dimension 3 over c with h i y j y 0 for all j 0 and i 0 let x be a smooth completion of y with d x y an effective divisor on x with normal crossings if the d dimension of x is not zero then y is a fibre space over a smooth affine curve c i e we have a surjective morphism from y to c such that the general fibre is smooth and irreducible such that every fibre satisfies the same vanishing condition if an irreducible smooth fibre is not affine then the kodaira dimension of x is oc and the d dimension of x is 1 we also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of hodge cohomology and the affineness of y

volume functions of linear series the volume of a cartier divisor is an asymptotic invariant which measures the rate of growth of sections of powers of the divisor it extends to a continuous homogeneous and log concave function on the whole neron severi space thus giving rise to a basic invariant of the underlying projective variety analogously one can also define the volume function of a possibly non complete multigraded linear series in this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties in the multigraded setting inspired by the work of lazarsfeld and musta ann inst fourier grenoble 56 6 1701 1734 2006 on okounkov bodies we show that any continuous homogeneous and log concave function appears as the volume function of a multigraded linear series by contrast we show that there exists countably many functions which arise as the volume functions of projective varieties we end the paper with an example where the volume function of a projective variety is given by a transcendental formula emphasizing the complicated nature of the volume in the classical case

local positivity in terms of newton okounkov bodies abstract in recent years the study of newton okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series after its introduction by lazarsfeld musta and kaveh khovanskii one reason for this is that they encode all numerical equivalence information of divisor classes by work of jow at the same time they can be seen as local positivity invariants and kuronya lozovanu have studied them in depth from this point of view we determine what information is encoded by the set of all newton okounkov bodies of a big divisor with respect to flags centered at a fixed point of a surface by showing that it determines and is determined by the numerical equivalence class of the divisor up to negative components in the zariski decomposition that do not go through the fixed point

on the solution of the inverse scattering problem at fixed energy for the class of yukawian potentials in this paper we prove that if in the fourier legendre expansion of the scattering amplitude the partial waves fl satisfy a hausdorff type condition and are exponentially bounded in l then this expansion converges uniformly to a function f u iv is the complexified scattering angle analytic in a strip contained in the plane furthermore we prove that this function has a holomorphic extension to a cut domain from the discontinuity of f across the cuts we can reconstruct the spectral density k 2 cosh v 1 1 2 associated with the class of yukawian potentials we give also an inversion algorithm which is able to reconstruct the density via an iterative procedure starting from the partial waves fl 0 since the data fl are necessarily affected by noise and we know only a finite number of them then the data set is given by f e l n0 where e characterizes the noise it follows that there is a certain degree of irreversibility in the inverse problem due to its ill posedne

nef dimension of minimal models we reduce the abundance conjecture in dimension 4 to the following numerical statement if the canonical divisor k is nef and has maximal nef dimension then k is big from this point of view we classify in dimension 2 nef divisors which have maximal nef dimension but which are not big

polytope finite elements in this paper finite elements based on arbitrary convex and non convex polytopes are introduced polytopes in combination with natural element coordinates necs permit a uniform element formulation of interpolation functions that are independent of the dimension of space localization and the number of vertices necs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes the differentiation and integration of these interpolation functions on the basis of necs is essential for finite element approximations the accuracy of the finite element interpolation or approximation can be controlled by either applying the h version or by utilizing the p version of the finite element method fem advantages in the handling of hanging nodes are discussed furthermore we present construction methods for lagrangian as well as for hierarchical interpolation functions based on necs numerical experiments on different convex and non convex decompositions will show the usability accuracy and convergence of the developed polytope fem copyright 2007 john wiley sons ltd

positivity and complexity of ideal sheaves the problem of bounding the complexity of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years results in this direction go back at least as far as the classical work 17 of hermann on the ideal membership problem and the effective nullstellensatz of brownawell 4 and kollar 21 marks a major recent advance with the development of computational algebraic geometry the question has taken on increasing importance and it came into particularly clear focus through the influential paper 3 of bayer andmumford more recently the theorem of 8 and 20 concerning regularity of powers raises the question of bounding the complexity of powers of an ideal and suggests that asymptotically the picture should become very clean the aim of the present paper is to examine some of the results and questions of 3 30 and 7 from a geometric perspective in the spirit of 12 our thesis is that much of this material is clarified and parts rendered transparent when viewed through the lenses of vanishing theorems and intersection theory specifically motivated by thework 25 26 of paolettiwe introduce an invariant s j thatmeasures in effect howmuchonehas to twist an idealj in order tomake

extended finite element method on polygonal and quadtree meshes in this paper we present mesh independent modeling of discontinuous flelds on polygonal and quadtree flnite element meshes this approach falls within the class of extended and generalized flnite element methods where the partition of unity framework is used to introduce additional enrichment functions within the classical displacement based flnite element approximation for crack modeling a discontinuous function and the two dimensional asymptotic crack tip flelds are used as enrichment functions linearly complete partition of unity approximations are adopted on polygonal convex and nonconvex elements and quadtree meshes excellent agreement with reference solution results is obtained for mixed mode stress intensity factors on benchmark crack problems and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics

a new computational approach to contact mechanics using variable node finite elements in this paper a new computational strategy for two dimensional contact problems is developed with the aid of variable node finite elements within the range of infinitesimal deformations the variable node elements which are among mls moving least square based finite elements enable us to transform node to surface contact problems into node to node contact problems this contact formulation with variable node elements leads to an accurate and effective solution procedure needless to mention that the contact patch test is passed without any additional treatment through several numerical examples we demonstrate its simplicity and the effectiveness of the proposed scheme

hausdorff moments hardy spaces and power series in this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the hausdorff conditions which play a relevant role in the moment problem theory we prove that these series converge to functions analytic in cut domains we are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the pollaczek polynomials we can thus furnish a solution for a class of cauchy integral equations

a high order mixed polygonal finite element for incompressible stokes flow analysis abstract this paper presents a new high order mixed scheme of polygonal finite element method pfem for incompressible flow computation the system of polygonal basis functions with bubble enriched functions is applied to represent the fluid velocity field while the pressure field is approximated by the set of polygonal basis functions the new element is named as minipe from a practical point of view the present element naturally satisfies the well known inf sup condition the numerical tests of the incompressible fluid flow problems governed by stokes equation systems are presented to verify the performance quality of proposed method with respect to the stability and convergence rate

volterra algebra and the bethe salpeter equation the diagonalization of the bethe salpeter equation for the absorptive part of the amplitude is reconsidered in particular the mathematical tools required by the diagonalization like the volterra algebra and the related spherical laplace transform are investigated in detail

potentially non klt locus and its applications we introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor the potentially non klt locus is a subset of x which is birationally transformed precisely into the non klt locus on a k x minimal model of x we prove basic properties of potentially non klt locus in comparison with those of classical non klt locus as applications we give a new characterization of varieties of fano type and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor

uniform approximation of abhyankar valuation ideals in smooth function fields fix a rank one valuation centered at a smooth point x on an algebraic variety over a field of characteristic zero assume that is abhyankar that is that its rational rank plus its transcendence degree equal the dimension of the variety let a m denote the ideal of elements in the local ring of x whose valuations are at least m our main theorem is that there exists k 0 such that a mn is contained in a m k n for all m and n this can be viewed as a greatly strengthened form of izumi s theorem for abhyankar valuations centered on smooth complex varieties the proof uses the theory of asymptotic multiplier ideals

a polytree based adaptive approach to limit analysis of cracked structures abstract we in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures the key idea is to develop a general refinement algorithm based on a so called polytree mesh structure the method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones which are considered as special cases also polytree meshes are conforming and can be regarded as a generalization of quadtree meshes for the aim of this paper we restrict our main interest in plane strain limit analysis to von mises type materials yet its extension to a wide class of other solid mechanics problems and materials is completely possible to avoid volumetric locking we propose an approximate velocity field enriched with bubble functions using wachspress coordinates on a primal mesh and design carefully strain rates on a dual mesh level an adaptive mesh refinement process is guided by an l 2 norm based indicator of strain rates through numerical validations we show that the present method reaches high accuracy with low computational cost this allows us to perform large scale limit analysis problems favorably

shape functions for polygonal domains with interior nodes the presented formulation follows in a series of publications which outline a method for constructing test functions which satisfy essential edge conditions exactly the method promises a complete solution satisfying allof the requirements of a ritz coordinate function the influence of interior points on the domain solution is included in this construction similar to conformal bubble functions the test functions are zero along the boundary and single valued only at the points they describe unlike the bubble function construction the interior points can be located at any desired point in the domain the resulting set of trial functions can satisfy the required global conditions including the exact reproduction of constant and linear fields

asymptotic cohomological functions on projective varieties we consider certain cohomological invariants called asymptotic cohomological functions which are associated to irreducible projective varieties asymptotic cohomological functions are generalizations of the concept of the volume of a line bundle the asymptotic growth of the number of global sections to higher cohomology we establish that they give a notion invariant under the numerical equivalence of divisors and extend uniquely to continuous functions on the real neron severi space to illustrate the theory we work out these invariants for abelian varieties smooth surfaces and certain homogeneous spaces

a characteristic number for links of surface singularities milnor and thurston mt define a characteristic number of a closed orientable 3 manifold m to be a real valued topological invariant r m such that if p m is defined and m is a k sheeted covering of m then p m is defined and equal to k p m this multiplicative property is clear for the euler characteristic but of course this is 0 for a 3 manifold if m has an appropriate riemannian metric e g hyperbolic or with all sectional curvatures 1 then the volume is a characteristic number milnor and thurston introduce several characteristic numbers one is defined for all 3 manifolds is positive on a fixed hyperbolic manifold and has the additional property that if m n is a general map of degree k then m i kk i n another characteristic number is the simplicial volume introduced by gromov g in this paper we introduce a characteristic number for each link of a complex surface singularity it is a nonnegative rational number is 0 only in wellunderstood cases is computable from any resolution dual graph plumbing diagram and has in addition the submultiplicative property above for degree k maps arising from morphisms of the singularities themselves this is the only characteristic number for links which we know of that can be computed from the graph we call this invariant p p since it is the negative of the self intersection number of a rational cycle on a complex surface a resolution of the singularity the definition arose from our work on the generalized miyaoka inequality for normal surfaces w3 this term appeared in the inequality another ingredient is the notion of zariski decomposition of a line bundle or divisor on a surface especially on a resolution of a surface singularity as in sakai s let x o be the germ of a normal complex surface singularity necessarily isolated with x contractible and ax m the link of x m is a closed connected orientable 3 manifold let x e x o be a good resolution hence the inverse image e u ei of o is the union of nonsingular curves intersecting transversally no three through a point the resolution dual graph

scaled boundary polygons with application to fracture analysis of functionally graded materials summary this study presents a framework for the development of polygon elements based on the scaled boundary fem underpinning this study is the development of generalized scaled boundary shape functions valid for any n sided polygon these shape functions are continuous inside each polygon and across adjacent polygons for uncracked polygons the shape functions are linearly complete for cracked polygons the shape functions reproduce the square root singularity and the higher order terms in the williams eigenfunction expansion this allows the singular stress field in the vicinity of the crack tip to be represented accurately using these shape functions a novel scaled boundary polygon formulation that captures the heterogeneous material response observed in functionally graded materials is developed the stiffness matrix in each polygon is derived from the principle of virtual work using the scaled boundary shape functions the material heterogeneity is approximated in each polygon by a polynomial surface in scaled boundary coordinates the intrinsic properties of the scaled boundary shape functions enable accurate computation of stress intensity factors in cracked functionally graded materials directly from their definitions the new formulation is validated and its salient features are demonstrated using five numerical benchmarks copyright 2014 john wiley sons ltd

the asymptotic behavior of plurigenera for a normal isolated singularity we study here some invariants of normal isolated singularities which look like the kodaira dimensions and thereby prepare for the possible localization of the rough classification theory in algebraic geometry let x x be a normal isolated singularity on an analytic space x of dimension n 2 kn 511er and watanabe independently defined m genera y x x and 6re x x of x x by considering the space of meromorphic m pie n forms divided by those with natural growth conditions at the singularity kn w1 in this paper we introduce a third m genus d x x me 1 and investigate the basic properties of these m genera it turns out that all these three m genera ym x x 6m x x d x x have integral growth orders in m which are denoted by xr x x x6 x x and xa x x respectively the asymptotic behavior of the above three m genera are summarized as follows

canonical singular hermitian metrics on relative canonical bundles we introduce a new class of canonical analytic zariski decompositions azd s in short called the supercanonical azd s on the canonical bundles of smooth projective varieties with pseudoeffective canonical classes we study the variation of the supercanonical azd hat h rm can under projective deformations and give a new proof of the invariance of plurigenera

the volume of an isolated singularity we introduce a notion of volume of a normal isolated singularity that gener alizes wahl s characteristic number of surface singularities to arbitrary dimensions we prove a basic monotonicity property of this volume under finite morphisms we draw sev eral consequences regarding the existence of non invertible finite endomorphisms fixing an isolated singularity using a cone construction we deduce that the anticanonical di visor of any smooth projective variety carrying a non invertible polarized endomorphism is pseudoeffective our techniques build on shokurov s b divisors we define the notion of nef weil b divisors and of nef envelopes of b divisors we relate the latter notion to the pull back of weil divisors introduced by de fernex and hacon using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by takagi we carry over to the isolated singularity case the intersection theory of nef weil b divisors formerly developed by boucksom favre and jonsson in the smooth case