the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

c semigroups and the cauchy problem abstract we extend the definition of generator to c semigroups that may not be exponentially bounded where the range of c may not be dense we then characterize linear operators a for which the associated abstract cauchy problem has a unique solution for every initial value in the domain of another operator b without assuming that the domain of a is dense or the solutions are exponentially bounded we also give hille yosida type characterizations of generators that may not be densely defined

efficiency scalability and robustness of parallel multilevel methods for nonlinear partial differential equations in this paper we compare the performance scalability and robustness of different parallel algorithms for the numerical solution of nonlinear boundary value problems arising in magnetic field computation and solid mechanics these problems are discretized by using the finite element method with triangular meshes and piecewise linear functions the nonlinearity is handled by a nested newton solver and the linear systems of algebraic equations within each newton step are solved by means of various iterative solvers namely multigrid methods and conjugate gradient methods with preconditioners based on domain decomposition multigrid or bpx techniques respectively the basis of the implementation of all solvers is a nonoverlapping domain decomposition data structure such that they are well suited for parallel machines with mimd architecture

incomplete iterated cauchy problems abstract we treat numerous higher order incomplete abstract cauchy problems by writing them in the form 1 1 k 1 n d dt a k u t 0 t 0 2 2 ui 1 0 xi 1 i j 3 3a ui 1 t bounded 1 i n j or 4 3b limt u i 1 t 0 1 i n j the abstract cauchy problem 1 2 and 3a is wellposed in a certain sense when a1 aj generate bounded c semigroups and aj 1 an generate uniformly stable c semigroups for 1 2 and 3b we need a1 aj and aj 1 an to generate stable c semigroups

analysis of a fully discrete finite element method for a nonlinear magnetic field problem in this paper nonlinear stationary magnetic field problems and their approximation by finite elements are studied as a new aspect estimates for the error caused by the approximation of the material function are given a technique for the interpolation of a pointwise given material function that preserves monotonicity is described and error estimates are presented

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

a new regularized method for two dimensional nonhomogeneous backward heat problem we consider the problem of finding from the final data u x y t g x y the initial data u x y 0 of the temperature function u x y t x y i 0 p x 0 p t 0 t satisfying the following systemu t u x x u y y f x y t x y t ix 0 t u 0 y t u p y t u x 0 t u x p t 0 x y t ix 0 t the problem is severely ill posed in this paper a simple and convenient new regularization method for solving this problem is considered meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

c semigroups and the abstract cauchy problem where by a solution u x we mean that u x e c 0 co x u t x e d u for t 2 0 and acp u x is satisfied it is well known that the acp is closely related to semigroup theory and covers a number of problems of interest to the applications see s 6 9 141 recently davies and pang 2 introduced exponentially bounded c semigroups and studied their connection with the acp other relating results are in tanaka ll later delaubenfels 3 investigated c semigroups that may not be exponentially bounded and their connection with the acp roughly speaking the results in 2 3 111 are as follows

fano n folds with nef tangent bundle and picard number greater than n 5 we prove that fano n folds with nef tangent bundle and picard number greater than n 5 are rational homogeneous manifolds

continuous dependence results for inhomogeneous ill posed problems in banach space abstract we apply semigroup theory and other operator theoretic methods to prove holder continuous dependence on modeling for the inhomogeneous ill posed cauchy problem in banach space the inhomogeneous ill posed cauchy problem is given by d u d t a u t h t u 0 0 t t where a is the infinitesimal generator of a holomorphic semigroup on a banach space x x and h 0 t x for a suitable function f the approximate problem is given by d v d t f a v t h t v 0 under certain stabilizing conditions we prove that u t v t c 1 t m t for a related norm where c and m are computable constants independent of 0 1 and t is a harmonic function these results extend earlier work of ames and hughes on the homogeneous ill posed problem

a bound of lengths of chains of minimal rational curves on fano manifolds of picard number 1 abstract in this paper we investigate the minimal length of chains of minimal rational curves needed to join two general points on a fano manifold of picard number 1 under mild assumptions in particular we give a sharp bound of the length by a fundamental argument as an application we compute the length for fano manifolds of dimension 7

perturbation of local c semigroups abstract let c be a bounded linear injective operator and a be the generator of a local c semigroup s t 0 t t on a banach space x it is proved that if b is a bounded operator such that r b r c and bcx cbx for all x d a then a b also generates a local c semigroup

column continuous transition functions abstract a column continuous transition function is by definition a standard transition function p t whose every column is continuous for t 0 in the norm topology of bounded sequence space l we will prove that it has a stable q matrix and that there exists a one to one relationship between column continuous transition functions and increasing integrated semigroups on l using the theory of integrated semigroups we give some necessary and sufficient conditions under which the minimal q function is column continuous in terms of its generator of the markov semigroup as well as its q matrix furthermore we will construct all column continuous q functions for a conservative single exit and column bounded q matrix q as applications we find that many interesting continuous time markov chains ctmcs say feller reuter riley processes monotone processes birth death processes and branching processes etc have column continuity

the boundedness of degree of fano varieties with picard number one contents 0 introduction 1 rational curves 1 1 deformation theory of morphisms 1 2 free rational curves 1 3 the invariant d m 2 special subvarieties in fano varieties of large degree 2 1 the statement 2 2 a siegel type lemma 2 3 multiplicity subspaces 2 4 the product theorem part i 2 5 the product theorem part ii 2 6 the proof 3 the covering lemma 4 collections of subsets 4 1 base change and push forward 4 2 an example 4 3 universally open morphisms 4 4 flat families 5 proof of theorem a

lengths of chains of minimal rational curves on fano manifolds abstract in this paper we consider a natural question how many minimal rational curves are needed to join two general points on a fano manifold x of picard number 1 in particular we study the minimal length of such chains in the cases where the dimension of x is at most 5 the coindex of x is at most 3 and x equips with a structure of a double cover as an application we give a better bound on the degree of fano 5 folds of picard number 1

rational curves of minimal degree and characterizations of projective spaces in this paper we investigate complex uniruled varieties x whose rational curves of minimal degree satisfy a special property namely we assume that the tangent directions to such curves at a general point x x form a linear subspace of txx as a first application of our main result we give a unified geometric proof of mori s wahl s campana peternell s and andreatta wi niewski s characterizations of open image in new window we also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree

matrices of operators and regularized semigroups where p is a matr ix of polynomials in n variables and ia1 ia is a family of c o m m u t i n g generators of bounded linear strongly cont inuous groups of operators intuitively one wants to take the exponent ia l of the matr ix of opera to r s p a since the spec t rum of ak is real for all k this means we want the complex matr ix pi j x to be contained in a left half plane in some sense for all x in r the strongest sense is to have the numerical range of the complex matr ix conta ined in a half plane r e z 0 xer a sense that is easier to calculate or at least estimate is to have the spectrum of p d x conta ined in a left half plane for all x r the system 1 1 is then said to be petrowsky correct see 9 the intuit ion we just described is not quite true in general wi thout some