the thick thin decomposition and the bilipschitz classification of normal surface singularities we describe a natural decomposition of a normal complex surface singularity x 0 into its thick and thin parts the former is essentially metrically conical while the latter shrinks rapidly in thickness as it approaches the origin the thin part is empty if and only if the singularity is metrically conical the link of the singularity is then seifert fibered in general the thin part will not be empty in which case it always carries essential topology our decomposition has some analogy with the margulis thick thin decomposition for a negatively curved manifold however the geometric behavior is very different for example often most of the topology of a normal surface singularity is concentrated in the thin parts

accelerated monotone scheme for finite difference equations concerning steady state prey predator interactions abstract in previous work by adapting a suitable finite difference method to a particular monotone scheme the authors and a lazer have studied the numerical solution of a system of semilinear elliptic partial differential equations which determines the equilibria of the volterra lotka equations describing prey predator interactions with diffusion in this paper in order to improve the efficiency of the method we show how newton s method can be successfully combined with the previous scheme to greatly accelerate the convergence in some particularly difficult problems the new method reduces the average number of iterations necessary to generate each element of the monotone sequences from 15 to about 3

smoothness of equisingular families of curves francesco severi 1921 showed that equisingular families of plane nodal curves are t smooth i e smooth of the expected dimension whenever they are non empty for families with more complicated singularities this is no longer true given a divisor d on a smooth projective surface it thus makes sense to look for conditions which ensure that the family v irr d s 1 s r of irreducible curves in the linear system d l with precisely r singular points of types s 1 s r is t smooth considering different surfaces including the projective plane general surfaces in p 3 c products of curves and geometrically ruled surfaces we produce a sufficient condition of the type e t i 1 s i d k 2 where is some invariant of the singularity type and is some constant this generalises the results of greuel lossen and shustin 2001 for the plane case combining their methods and the method of bogomolov instability for many singularity types the invariant leads to essentially better conditions than the invariants used by greuel lossen and shustin 1997 and for most classes of geometrically ruled surfaces our results are the first known for t smoothness at all

monotone iterative methods for the adaptive finite element solution of semiconductor equations picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive finite element solution of semiconductor equations in terms of the slotboom variables the adaptive meshes are generated by the 1 irregular mesh refinement scheme based on these unstructured meshes and a corresponding modification of the scharfetter gummel discretization scheme it is shown that the resulting finite element stiffness matrix is an m matrix which together with the shockley read hall model for the generation recombination rate leads to an existence uniqueness comparison theorem with simple upper and lower solutions as initial iterates numerical results of simulations on a mosfet device model are given to illustrate the accuracy and efficiency of the adaptive and monotone properties of the present methods

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

monotone iterative methods for finite difference system of reaction diffusion equations this paper presents an existence comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary value problems the basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system using upper and lower solutions as two distinct initial iterations two monotone sequences from a suitable linear system are constructed it is shown that these two sequences converge monotonically from above and below respectively to a unique solution of the nonlinear discrete equations this formulation leads to a well posed problem for the nonlinear discrete system applications are given to several models arising from physical chemical and biological systems numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences

fault tolerant compact routing schemes for general graphs this paper considers compact fault tolerant routing schemes for weighted general graphs namely routing schemes that avoid a set of failed or forbidden edges we present a compact routing scheme capable of handling multiple edge failures assume a source node s contains a message m designated to a destination target t and assume a set f of edges crashes unknown to s our scheme routes the message to t provided that s and t are still connected in g f over a path whose length is proportional to the distance between s and t in g f to f 3 and to some poly log factor the routing table required at a node v is of size proportional to the degree of v in g and some poly log factor this improves on the previously known fault tolerant compact routing scheme for general graphs which was capable of overcoming at most 2 edge failures

an inexact monotone method for solving semilinear parabolic problems abstract the paper deals with numerical solving semilinear parabolic problems based on the method of upper and lower solutions an inexact monotone iterative method is constructed where monotone linear systems are solved by the jacobi or gauss seidel methods only approximately an analysis of convergence rates of the inexact monotone iterative method based on different stopping tests is given results of numerical experiments implemented in the framework of an inexact newton method are presented where iteration counts of the inexact monotone method are compared with a monotone iterative method whose convergence rate is linear

oracles for distances avoiding a failed node or link we consider the problem of preprocessing an edge weighted directed graph g to answer queries that ask for the length and first hop of a shortest path from any given vertex x to any given vertex y avoiding any given vertex or edge as a natural application this problem models routing in networks subject to node or link failures we describe a deterministic oracle with constant query time for this problem that uses o n 2 log n space where n is the number of vertices in g the construction time for our oracle is o mn 2 n 3 log n however if one is willing to settle for theta n 2 5 space we can improve the preprocessing time to o mn 1 5 n 2 5 log n while maintaining the constant query time our algorithms can find the shortest path avoiding a failed node or link in time proportional to the length of the path

stabilized explicit implicit domain decomposition methods for the numerical solution of parabolic equations we report a class of stabilized explicit implicit domain decomposition seidd methods for the numerical solution of parabolic equations explicit implicit domain decomposition eidd methods are globally noniterative nonoverlapping domain decomposition methods which when compared with schwarz algorithm based parabolic solvers are computationally and communicationally efficient for each simulation time step but suffer from small time step size restrictions by adding a stabilization step to eidd the seidd methods retain the time stepwise efficiency in computation and communication of the eidd methods but exhibit much better numerical stability three seidd algorithms are presented in this paper which are experimentally tested to show excellent stability computation and communication efficiencies and high parallel speedup and scalability

block monotone iterative method for semilinear parabolic equations with nonlinear boundary conditions two block monotone iterative schemes called jacobi and gauss seidel monotone iterations are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method these iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations including a comparison result between them it is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system some error estimates and a convergence theorem are given and various sufficient conditions for the construction of upper and lower solutions are obtained numerical results are presented for some physical model problems including some problems with known continuous solutions and two problems with l shaped and trapezoidal domains

monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given numerical experiments are presented

on linear monotone iteration and schwarz methods for nonlinear elliptic pdes the schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains the solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain in this paper proofs of convergence of some schwarz alternating methods for nonlinear elliptic problems which are known to have solutions by the monotone method also known as the method of subsolutions and supersolutions are given in particular an additive schwarz method for scalar as well some coupled nonlinear pdes are shown to converge to some solution on finitely many subdomains even when multiple solutions are possible in the coupled system case each subdomain pde is linear decoupled and can be solved concurrently with other subdomain pdes these results are applicable to several models in population biology

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

fault tolerant subgraph for single source reachability general and optimal let g be a directed graph with n vertices m edges and a designated source vertex s we address the problem of single source reachability ssr from s in the presence of failures of vertices edges we show that for every k geq1 there is a subgraph h of g with at most 2 kn edges that preserves the reachability from s even after the failure of any k edges formally given a set f of k edges a vertex v in v g is reachable from s in g setminus f if and only if v is reachable from s in h setminus f we call h a k fault tolerant reachability subgraph textsc k ftrs we also prove a matching lower bound of omega 2 kn edges for such subgraphs that holds for all n k with 2 k leq n our results extend to vertex failures without any extra overhead the construction of k ftrs is interesting from several different perspectives from the graph theory perspective it reveals a separation between ssr and single source shortest paths sssp in directed grap

linear time algorithms for dominators and other path evaluation problems we present linear time algorithms for the classic problem of finding dominators in a flowgraph and for several other problems whose solutions require evaluating a function defined on paths in a tree although all these problems had linear time solutions previously our algorithms are simpler in some cases substantially our improvements come from three new ideas a refined analysis of path compression that gives a linear bound if the compressions favor certain nodes replacement of random access table look up by a radix sort and a more careful partitioning of a tree into easily managed parts in addition to finding dominators our algorithms find nearest common ancestors off line verify and construct minimum spanning trees do interval analysis of a flowgraph and build the component tree of a weighted tree our algorithms do not require the power of a random access machine they run in linear time on a pointer machine the genesis of our work was the discovery of a subtle error in the analysis of a previous allegedly linear time algorithm for finding dominators that algorithm was an attempt to simplify a more complicated algorithm which itself was intended to correct errors in a yet earlier algorithm our work provides a systematic study of the subtleties in the dominators problem the techniques needed to solve it in linear time and the range of application of the resulting methods we have tried to make our techniques as simple and as general as possible and to understand exactly how earlier approaches to the dominators problem were either incorrect or overly complicated

monodromy jordan blocks b functions and poles of zeta functions for germs of plane curves abstract we study the poles of several local zeta functions the igusa topological and motivic zeta function associated to a germ of a holomorphic function in two variables it was known that there is at most one double pole for any of these zeta functions which is then given by the log canonical threshold of the function at the singular point if the germ is reduced loeser showed that such a double pole always induces a monodromy eigenvalue with a jordan block of size 2 here we settle the non reduced situation describing precisely in which case such a jordan block of maximal size 2 occurs we also provide detailed information about the bernstein sato polynomial in the relevant non reduced situation confirming a conjecture of igusa denef and loeser

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

the abel map for surface singularities i generalities and examples let x o be a complex normal surface singularity we fix one of its good resolutions widetilde x rightarrow x an effective cycle z supported on the reduced exceptional curve and any possible first chern class l in h 2 widetilde x mathbb z with these data we define the variety mathrm eca l z of those effective cartier divisors d supported on z which determine a line bundles mathcal o z d with first chern class l furthermore we consider the affine space mathrm pic l z subset h 1 mathcal o z of isomorphism classes of holomorphic line bundles with chern class l and the abel map c l z mathrm eca l z rightarrow mathrm pic l z the present manuscript develops the major properties of this map and links them with the determination of the cohomology groups h 1 z mathcal l where we might vary the analytic structure x o supported on a fixed topological type resolution graph and we also vary the possible line bundles mathcal l in mathrm pic l z the case of generic line bundles of mathrm pic l z and generic line bundles of the image of the abel map will have priority roles rewriting the abel map via laufer duality based on integration of forms on divisors we can make explicit the abel map and its tangent map the case of superisolated and weighted homogeneous singularities are exemplified with several details the theory has similar goals but rather different techniques as the theory of abel map or brill noether theory of reduced smooth projective curves

a block monotone domain decomposition algorithm for a semilinear convection diffusion problem this paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed convection diffusion problem a block monotone domain decomposition algorithm based on a schwarz alternating method and on block iterative scheme is constructed this monotone algorithm solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the nonlinear problem the rate of convergence of the block monotone domain decomposition algorithm is estimated numerical experiments are presented

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

torelli theorems for singularities in this paper we study how const deformations of an isolated hypersurface singularity affect the mixed hodge structure mhs on the cohomology of the milnor fiber using scherk and steenbrink sd description of this mhs and the related approach of varchenko we give a necessary and sufficient condition for the infinitesimal torelli theorem to hold and identify the const stratum in the discriminant of the miniversal deformation after examining the geometric reasons causing torelli to fail for certain deformations we define a local moduli space and prove a local torelli theorem for singularities in the process we complete the work of m saito and malgrange on k saito s conjecture about primitive forms