a bie method for a nonlinear bvp abstract a nonlinear boundary value problem bvp governed by laplace s equation with a nonlinear boundary condition is considered this paper is concerned with the numerical solvability of a nonlinear boundary integral equation bie obtained by reformulating the nonlinear bvp we give a simple alternative to the standard collocation method for the nonlinear bie which requires less computational effort the method is a bie analogue of the product approximation method used for solving some nonlinear problems by the finite element method we obtain superconvergence of approximate solutions introduced in this paper

from real affine geometry to complex geometry we construct from a real ane manifold with singularities a tropical manifold a degeneration of calabi yau manifolds this solves a fundamental problem in mirror symmetry furthermore a striking feature of our approach is that it yields an explicit and canonical order by order description of the degeneration via families of tropical trees this gives complete control of the b model side of mirror symmetry in terms of tropical geometry for example we expect that our deformation parameter is a canonical coordinate and expect period calculations to be expressible in terms of tropical curves we anticipate this will lead to a proof of mirror symmetry via tropical methods

unobstructedness of calabi yau orbi kleinfolds we show that calabi yau spaces with certain types of hypersurface quotient singularities have unobstructed deformations this applies in particular to all calabi yau orbifolds nonsingular in codimension 2

a generalization of bieberbach s theorem in 1912 bieberbach proved that every compact flat riemannian manifold m is finitely covered by a flat torus more precisely m has the form f g h where g is a group of translations of euclidean space f c g is a discrete subgroup and h is a finite group of isometries of the space of right cosets f g for a proof see e g wolf 18 the condition that m has a flat riemannian metric can be separated into two conditions first m has an affine structure a distinguished covering by coordinate charts whose coordinate changes are affine second m has a riemannian metric whose coefficients in the affine charts are constants in this paper we relax the second condition a polynomial riemannian metric on the affine manifold m is a riemannian metric whose local expression in affine coordinates has the form gi x dxidx where the gij are polynomial functions on euclidean space e by letting the gij be rational functions on e we arrive at the more general notion of a rational riemannian metric it is not assumed that these expressions define riemannian metrics on all of e the object of this paper is to determine which affine manifolds have polynomial riemannian metrics and to give examples of affine manifold having rational riemannian metrics

ergodic theory group representations and rigidity in these lectures we discuss some topics concerning the relationship of ergodic theory representation theory and the structure of lie groups and their discrete subgroups

on the reconstruction problem in mirror symmetry abstract let m b be a lagrangian torus fibration with singularities such that the fibers are of maslov index zero and unobstructed the paper constructs a rigid analytic space m 0 over the novikov field which is a deformation of the semi flat complex structure of the dual torus fibration over the smooth locus b 0 b of transition functions of m 0 are obtained via a homomorphisms which capture the wall crossing phenomenon of moduli spaces of holomorphic disks

tropical counting from asymptotic analysis on maurer cartan equations let be a toric surface and let be its landau ginzburg lg mirror where is the hori vafa potential as shown in their preprint we apply asymptotic analysis to study the extended deformation theory of the lg model and prove that semi classical limits of fourier modes of a specific class of maurer cartan solutions naturally give rise to tropical disks in with maslov index 0 or 2 the latter of which produces a universal unfolding of for our construction reproduces gross perturbed potential adv math 224 2010 pp 169 245 which was proven to be the universal unfolding of written in canonical coordinates we also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of across walls of the scattering diagram formed by maslov index 0 tropical disks originally observed by gross in the same work in the case of

the rate of convergence of a matrix power series abstract to estimate the truncation error of a matrix power series we need information about the magnitude of high powers of a matrix inequalities bearing on this question are surveyed and their use is exemplified by calculations of bounds for the truncation error of the geometric series in matrices

functorial tropicalization of logarithmic schemes the case of constant coefficients the purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients our approach crucially uses the theory of fans in the sense of kato and generalizes thuillier s retraction map onto the non archimedean skeleton in the toroidal case for the convenience of the reader many examples as well as an introductory treatment of the theory of kato fans are included

fast multilevel augmentation methods for nonlinear boundary integral equations we propose a fast algorithm for the solution of the nonlinear boundary integral equation resulting from a reformulation of a boundary value problem of the laplace equation with nonlinear boundary conditions the fast algorithm is developed by using the multilevel augmentation method introduced recently by chen wu and xu for general nonlinear integral equations in conjunction with a matrix truncation strategy and an error control technique of numerical integrations for integrals appeared in the process of solving the equation we prove that the proposed algorithm has an optimal convergence order up to a logarithmic factor and a nearly linear computational complexity order measured in the number of multiplications and functional evaluations numerical experiments are presented to demonstrate its approximation accuracy and computational efficiency verifying the theoretical estimates and to compare performance of the proposed algorithm with that of the atkinson and chandler algorithm

strong formulations for quadratic optimization with m matrices and indicator variables we study quadratic optimization with indicator variables and an m matrix i e a psd matrix with non positive off diagonal entries which arises directly in image segmentation and portfolio optimization with transaction costs as well as a substructure of general quadratic optimization problems we prove under mild assumptions that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem to strengthen the formulation we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex hull descriptions of these sets we also describe strong conic quadratic valid inequalities preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation

a mesh independence principle for operator equations and their discretizations the mesh independence principle asserts that when newton s method is applied to a nonlinear equation between some banach spaces as well as to some finite dimensional discretization of that equation then the behavior of the discretized process is asymptotically the same as that for the original iteration and as a consequence the number of steps required by the two processes to converge to within a given tolerance is essentially the same so far this result has been proved only for certain classes of boundary value problems in this paper a proof is presented for a general class of operator equations and discretizations it covers the earlier results and extends them well beyond the cases that have been considered before

gluing semi orthogonal decompositions abstract we introduce preordered semi orthogonal decompositions psod s of dg categories we show that homotopy limits of dg categories equipped with compatible psod s carry a natural psod this gives a way to glue semi orthogonal decompositions along faithfully flat covers extending the main result of 4 as applications we will construct semi orthogonal decompositions for root stacks of log pairs x d where d is a not necessarily simple normal crossing divisor generalizing results from 17 and 3 further we will compute the kummer flat k theory of general log pairs x d generalizing earlier results of hagihara and niziol in the simple normal crossing case 15 23

the versal deformation of an isolated toric gorenstein singularity given a lattice polytope q n we define an affine scheme open image in new window that reflects the possibilities of splitting q into a minkowski sum denoting by y the toric gorenstein singularity induced by q we construct a flat family over open image in new window with y as special fiber in case y has an isolated singularity this family is versal

the superconvergence of a galerkin collocation method for first kind boundary integral equations on smooth curves here we present a galerkin collocation method for the solution of first kind boundary integral equations with logarithmic kernel as principal part by using piecewise constant trial functions with uniform meshes we discover the phenomenon of superconvergence at midpoints for the numerical solution we shall prove this result in this paper

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

an extrapolation method for a class of boundary integral equations boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using green s formulas for solving the induced boundary integral equations a nystrom scheme and its extrapolation method are derived for periodic fredholm integral equations of the second kind with logarithmic singularity asymptotic expansions for the approximate solutions obtained by the nystrom scheme are developed to analyze the extrapolation method some computational aspects of the methods are considered and two numerical examples are given to illustrate the acceleration of convergence

mesh duality and legendre duality we describe a sense in which mesh duality is equivalent to legendre duality that is a general pair of meshes which satisfy a definition of duality for meshes are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a legendre dual transformation in applications the latter functions can be a tangent plane approximation to a smoother function and a chordal plane approximation to its legendre dual convex examples include one from meteorology and also the relation between the delaunay mesh and the voronoi tessellation the latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian

a meshless galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions abstract this paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations these types of integral equations arise from boundary value problems of laplace s equations with nonlinear robin boundary conditions the discrete galerkin method together with the inverse multiquadric radial basis functions established on scattered points is utilized to approximate the solution the discrete galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method the proposed scheme uses a special accurate quadrature formula via the nonuniform gauss legendre integration rule to compute logarithm like singular integrals appeared in the scheme since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain it is meshless and does not depend to the domain form we also investigate the error analysis of the proposed method illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates

constructing integrable systems of semitoric type let m be a connected symplectic 4 manifold a semitoric integrable system on m essentially consists of a pair of independent real valued smooth functions j and h on m for which j generates a hamiltonian circle action under which h is invariant in this paper we give a general method to construct starting from a collection of five ingredients a symplectic 4 manifold equipped a semitoric integrable system then we show that every semitoric integrable system on a symplectic 4 manifold is obtained in this fashion in conjunction with the uniqueness theorem proved recently by the authors this gives a classification of semitoric integrable systems on 4 manifolds in terms of five invariants

convergence estimates for a series approximation of dynamic response of a perturbed system abstract this paper presents a new method for efficiently computing the response of a perturbed dynamic system the method is based on predicting the convergence of a neumann series expansion that approximates the displacement response of a perturbed system the neumann series is often faster than a direct solve but only converges when the spectral radius of the system is less than unity however direct computation of the spectral radius is often expensive the proposed method avoids this expense by predicting convergence using a ratio of adjacent terms early in the evaluation of the series stochastic simulations are presented for determining the convergence bound on this ratio as a function of term index these simulations demonstrate that the resulting convergence predictions are very accurate after only 5 20 terms and that the proposed method is significantly faster than the computation of the spectral radius the proposed method is also inherently faster than other published methods as it requires fewer terms in the series to assess convergence numerical examples involving the steady state vibrations of an elastic bar and an elastic plate illustrate the accuracy and efficiency of the method

boundary integral equation methods for solving laplace s equation with nonlinear boundary conditions the smooth boundary case a nonlinear boundary value problem for laplace s equation is solved numerically by using a reformulation as a nonlinear boundary integral equation two numerical methods are proposed and analyzed for discretizing the integral equation both using product integration to approximate the singu lar integrals in the equation the first method uses the product simpson s rule and the second is based on trigonometric interpolation iterative methods in cluding two grid methods for solving the resulting nonlinear systems are also discussed extensively numerical examples are included

mirror symmetry via logarithmic degeneration data i this paper lays the foundations of a program to study mirror symmetry by studying the log structures of illusie fontaine and kato on degenerations of calabi yau manifolds the basic idea is that one can associate to certain sorts of degenerations of calabi yau manifolds a log calabi yau space which is a log structure on the degenerate fibre the log cy space captures essentially all the information of the degeneration and hence all mirror statements for the large complex structure limit given by the degeneration can already be derived from the log cy space in this paper we begin by discussing affine manifolds with singularities given such an affine manifold along with a polyhedral decomposition we show how to construct a scheme consisting of a union of toric varieties in certain non degenerate cases we can also construct log structures on these schemes conversely given certain sorts of degenerations one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration mirror symmetry is then obtained as a discrete legendre transform on these affine manifolds thus providing an algebro geometrization of the strominger yau zaslow conjecture the deepest result of this paper shows an isomorphism between log complex moduli of a log cy space and log kahler moduli of its mirror

on the birationality of complete intersections associated to nef partitions we prove that generic complete intersections associated to multiple mirror nef partitions are all birational this result solves a conjecture by batyrev and nill in 6 under some mild assumptions

the radiance obstruction and parallel forms on affine manifolds a manifold m is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine when they are locally linear m is called radiant the obstruction to radiance is a one dimensional class cm with coefficients in the flat tangent bundle of m exterior powers of cm give information on the existence of parallel forms on m especially parallel volume forms as applications various kinds of restrictions are found on the holonomy and topology of compact affine manifolds introduction an affine manifold m is a manifold with a distinguished maximal atlas of charts all of whose coordinate changes are locally affine on such a manifold there is an intrinsic notion of a parallel tensor one whose components in any affine chart are constants more generally there is the notion of a polynomial tensor field of given degree in 1962 l markus conjectured in mk that a compact orientable affine n dimensional manifold has parallel volume form if and only if it is complete meaning that its universal covering is affinely isomorphic to euclidean n space rn the problem of constructing a parallel volume form determines an n dimensional twisted real cohomology class originally studied by j smillie sm2 in this paper we express this class as the nth exterior power of a twisted one dimensional real class which we call the radiance obstruction cm by computing cm in various cohomology theories cech singular de rham and others we are able to exploit cm in several ways to yield more information on the structure of affine manifolds some of these results will appear in a subsequent paper gh3 the basic tool used is a formula proved in 2 6 which expresses the cohomology class of a parallel exterior k form in terms of the k th exterior power of the radiance obstruction akcvf min 2 7 the special case of a parallel volume form on a compact n dimensional manifold m is examined the existence of such a form implies that ancm 0 in 2 8 we show that the affine holonomy group f of such an m cannot preserve a proper affine subspace in gh3 this result will be improved by showing that r preserves no proper semialgebraic subset of rn in 2 10 we show that there tend to be plenty of parallel forms on a compact affine manifold with nilpotent affine holonomy received by the editors may 26 1983 1980 mathematics subject classification primary 57r99 53c05 secondary 53c10 55r25 4c1984 amencan mathematical society 0002 9947 84 1 00 25 per page

semistable degenerations and period spaces for polarized k3 surfaces modular compactifications of moduli spaces for polarized k3 surfaces are constructed using the tools of logarithmic geometry in the sense of fontaine and illusie the relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces are discussed and it is explained how the techniques of this paper yield models for the latter spaces over number fields the paper also contains a discussion of picard functors for log schemes and a logarithmic version of artin s method for proving representability by an algebraic stack

a bound for norms of functions of matrices abstract if a is an m m and is an analytic function then a depends only on the values of and its first m 1 derivatives on the spectrum of a in this paper we estimate a for certain matrix norms in terms of the maximum moduli of these derivatives on the convex hull of the spectrum of a

a wavelet collocation method for boundary integral equations of the modified helmholtz equation abstract a wavelet collocation method is to proposed for solving the linear boundary integral equations reformulated from the modified helmholtz equation with robin boundary conditions to deal with the difficulties caused by robin boundary conditions we provide an improved version of wavelet collocation method by employing a matrix compression strategy and augmentation method we obtain fully discrete system and solve efficiently the resulting systems at last we point out that the proposed method employs an optimal convergence order and a nearly linear computational complexity numerical experiments are presented to demonstrate its approximation accuracy and computational efficiency

mirror symmetry for p 2 and tropical geometry abstract this paper explores the relationship between mirror symmetry for p 2 at the level of big quantum cohomology and tropical geometry the mirror of p 2 is typically taken to be c 2 w where w is a landau ginzburg potential of the form x y 1 x y the complex moduli space of the mirror is the universal unfolding of w and oscillatory integrals produce a frobenius manifold structure on this universal unfolding we show that w can be deformed by counting maslov index two tropical disks and the natural parameters appearing in this deformation are then the flat coordinates on the moduli space furthermore the oscillatory integrals are shown to read off directly tropical curve counts from the potential thus we show in fact that mirror symmetry for p 2 is equivalent in a strong sense to tropical curve counting formulas including tropical formulas for gravitational descendent invariants

new calabi yau orbifolds with mirror hodge diamonds abstract we prove a representation theoretic version of borisov batyrev mirror symmetry and use it to construct infinitely many new pairs of orbifolds with mirror hodge diamonds with respect to the usual hodge structure on singular complex cohomology we conjecture that the corresponding orbifold hodge diamonds are also mirror when x is the fermat quintic in p 4 and x is a sym 5 equivariant toric resolution of its mirror x we deduce that for any subgroup of the alternating group a 5 the hilbert schemes hilb x and hilb x are smooth calabi yau threefolds with explicitly computed mirror hodge diamonds

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions

boundary element methods for potential problems with nonlinear boundary conditions galerkin boundary element methods for the solution of novel first kind steklov poincare and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two and three dimensional lipschitz domains with nonlinear boundary conditions for the numerical solution of the resulting newton iterate linear boundary integral equations we propose practical variants of the galerkin scheme and give corresponding error estimates we also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry