a quadratic convergence yielding iterative method for the implementation of lavrentiev regularization method for ill posed equations george and elmahdy 2012 considered an iterative method which converges quadratically to the unique solution is a nonlinear monotone operator defined on a real hilbert space x the convergence analysis of the method was based on a majorizing sequence in this paper we are concerned with the problem of expanding the applicability of the method considered by george and elmahdy 2012 by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular lipschitz type hypotheses considered in earlier studies such as george and elmahdy 2012 mahale and nair 2009 mathe and perverzev 2003 nair and ravishankar 2008 semenova 2010 and tautanhahn 2002 we show that the adaptive scheme considered by perverzev and schock 2005 for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate in the concluding section the method is applied to numerical solution of the inverse gravimetry problem

non polynomial spline solution of singularly perturbed boundary value problems a generalized scheme based on quartic non polynomial spline functions is proposed this scheme is designed for numerical solution of singularly perturbed two point boundary value problems arising in the study of science and engineering the scheme leads to a five diagonal linear system of equations convergence analysis of the method is briefly discussed two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method

ill posed problems with unbounded operators abstract let a be a linear closed densely defined unbounded operator in a hilbert space assume that a is not boundedly invertible if eq 1 a u f is solvable and f f then the following results are provided problem f u a u f 2 u 2 has a unique global minimizer u for any f u a a a i 1 f there is a function lim 0 0 such that lim 0 u y 0 where y is the unique minimal norm solution to 1 a priori and a posteriori choices of are given dynamical systems method dsm is justified for eq 1

generalized exponents of primitive two colored digraphs abstract a l colored digraph d l is primitive if there exists a nonnegative integer vector such that for each ordered pair of vertices x and y not necessarily distinct there exists an walk in d l from x to y the exponent of the primitive l colored digraph d l is defined to be the minimum value of the sum of all coordinates of taken over all such in this paper we generalize the concept of exponent of a primitive l colored digraph by introducing three types of generalized exponents further we study the generalized exponents of primitive two colored wielandt digraphs

on four dimensional terminal quotient singularities we report on an investigation of four dimensional terminal cyclic quotient singularities which are not gorenstein for simplicity we focus on quotients by cyclic groups of prime order an enumeration using a computer of all such singularities for primes 1600 led us to conjecture a structure theorem for these singularities which is rather more complicated than the known structure theorem in dimension three we discuss this conjecture and our evidence for it we also discuss properties of the anticanonical and antibicanonical linear systems of these singularities introduction the recent successes in understanding the birational geometry of algebraic varieties of dimension greater than two have focused attention on a class of singularities called terminal singularities which appear on the birational models which the theory selects in dimension three the structure of these terminal singularities is known in some detail the terminal quotient singularities were classified in what is now called the terminal lemma by several groups of people working independently cf 1 2 5 8 and all other three dimensional terminal singularities were subsequently classified by work of the first author 6 and of kollar and shepherd barron 4 both of these classifications were based on key results of reid 10 11 who reduced the problem to an analysis of quotients of smooth points and double points by cyclic group actions a detailed account of the classification can be found in 12 a consequence of this classification apparently first observed by reid 12 is that for any three dimensional terminal singularity t the general element of the anticanonical linear system i kt has only canonical singularities this in turn implies that if we form the double cover of t branched on the general antibicanonical divisor d e i 2kt i that double cover will also have only canonical singularities at first glance this second property appears to have no advantages over the first but looking at things in these terms proved decisive in a slightly more global context kawamata 3 showed that if t is an extremal neighborhood of a rational curve c on a threefold then the existence of such a divisor d globally on t is sufficient to conclude the existence of a certain kind of birational modification a directed flip centered on c the first author 7 showed that such divisors always exist completing the proof of the minimal model theorem for threefolds see 7 for more details received august 26 1987 revised december 17 1987 1980 mathematics subject classification 1985 revision primary 14b05 secondary 14 04 14j10 14l32 research partially supported by the national science foundation current addresses s m department of mathematics faculty of science nagoya university nagoya 464 japan d r m department of mathematics duke university durham north carolina 27706 i m department of mathematics fordham university bronx new york 10458 1988 american mathematical society 0025 5718 88 1 00 25 per page

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

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

dynamical systems method for solving nonlinear equations with monotone operators a version of the dynamical systems method dsm for solving ill posed nonlinear equations with monotone operators in a hilbert space is studied in this paper an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically the results of two numerical experiments are presented they show that the proposed version of dsm is numerically efficient the numerical experiments consist of solving nonlinear integral equations

numerical solutions of integral and integro differential equations using chebyshev polynomials of the third kind abstract our purpose in this study is to construct an algorithm based on the use of a finite expansion in chebyshev polynomials of the third kind to solve singularly perturbed volterra integral equations first order integro differential equations of volterra type arising in fluid dynamics and volterra delay integro differential equations the convergence of the method is investigated finally some numerical experiments which confirm the theoretical results are shown and comparisons with other methods in literature are given

groetsch s representation of moore penrose inverses and ill posed problems in hilbert c modules abstract the explicit representation lim 0 1 t t 1 t for the moore penrose inverse of an operator t between hilbert spaces has been given by c w groetsch 1975 6 we obtain his formula for the moore penrose inverse of an unbounded operator between hilbert c modules ill posed problems with unbounded operator between hilbert c modules are also discussed

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

exponents of tuples of nonnegative matrices abstract the notions of irreducibility primitivity and exponent of a nonegative matrix are generalized to k tuples of non negative matrices of the same order it is shown that for each positive integer k the maximum exponent of a primitive k tuple of n by n nonnegative matrices is n k 1

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

a globally optimal tri vector method to solve an ill posed linear system in the present paper a future cone in the minkowski space defined in terms of the square norm of the residual vector for an ill posed linear system to be solved is used to derive a nonlinear system of ordinary differential equations then the forward euler scheme is used to generate an iterative algorithm two critical values in the critical descent tri vector are derived which lead to the largest convergence rate of the resultant iterative algorithm namely the globally optimal tri vector method gotvm some numerical examples are used to reveal the superior performance of the gotvm than the famous methods of conjugate gradient cgm and generalized minimal residual gmres through the numerical tests we also set forth the rationale by assuming the tri vector as being a better descent direction

the product of operators with closed range in hilbert c modules abstract suppose t and s are bounded adjointable operators with close range between hilbert c modules then ts has closed range if and only if ker t ran s is an orthogonal summand if and only if ker s ran t is an orthogonal summand moreover if the dixmier or minimal angle between ran s and ker t ker t ran s is positive and ker s ran t is an orthogonal summand then ts has closed range

dynamical systems method dsm for selfadjoint operators abstract let a be a selfadjoint linear operator in a hilbert space h the dsm dynamical systems method for solving equation a v f consists of solving the cauchy problem u t u u 0 u 0 where is a suitable operator and proving that i u t t 0 ii u and iii a u f it is proved that if equation a v f is solvable and u solves the problem u i a i a u i f u 0 u 0 where a 0 is a parameter and u 0 is arbitrary then lim a 0 lim t u t a y where y is the unique minimal norm solution of the equation a v f stable solution of the equation a v f is constructed when the data are noisy i e f is given in place of f f f the case when a a t 0 0 a t d t a t 0 as t is considered it is proved that in this case lim t u t y and if f is given in place of f then lim t u t y where t is properly chosen

an efficient dynamical systems method for solving singularly perturbed integral equations with noise in this paper we apply the dynamical systems method dsm proposed by a g ramm and the variational regularization method vrm to obtain numerical solution to some singularly perturbed ill posed problems contaminated by noise the results obtained by these methods are compared to the exact solution for the model problems it is found that the dynamical systems method is preferable because it is easier to apply highly stable robust and it always converges to the solution even for large size models

directed graphs 2d state models and characteristic polynomials of irreducible matrix pairs abstract the definition and main properties of a 2d digraph namely a directed graph with two kinds of arcs are introduced under the assumption of strong connectedness the analysis of its paths and cycles is performed based on an integer matrix whose rows represent the compositions of all circuits and on the corresponding row module natural constraints on the composition of the paths connecting each pair of vertices lead to the definition of a 2d strongly connected digraph for a 2d digraph of this kind the set of vertices can be partitioned into disjoint 2d imprimitivity classes whose number and composition are strictly related to the structure of the row module irreducible matrix pairs i e pairs endowed with a 2d strongly connected digraph are subsequently discussed equivalent descriptions of irreducibility naturally extending those available for a single irreducible matrix are obtained these refer to the free evolution of the 2d state models described by the pairs and to their characteristic polynomials finally primitivity is viewed as a special case of irreducibility and completely characterized in terms of 2d digraphs characteristic polynomials and 2d system dynamics

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

positive semi definite matrices of adjointable operators on hilbert c modules abstract let a be a c algebra h k be two hilbert a modules and b be an adjointable operator from h to k in this paper we prove that the moore penrose inverse of b exists if and only if b has closed range in addition some known results about the moore penrose inverses acting on hilbert spaces as well as in c algebras are extended in the context of hilbert c modules as an application a characterization of positive semi definite matrices of adjointable operators with respect to an orthogonal pair of a hilbert c module is given

on a new notion of the solution to an ill posed problem a new understanding of the notion of the stable solution to ill posed problems is proposed the new notion is more realistic than the old one and better fits the practical computational needs a method for constructing stable solutions in the new sense is proposed and justified the basic point is in the traditional definition of the stable solution to an ill posed problem au f where a is a linear or nonlinear operator in a hilbert space h it is assumed that the noisy data f d d are given f f email protected email protected email protected and a stable solution u d r df d is defined by the relation lim d email protected r df email protected 0 where y solves the equation au f i e ay f in this definition y and f are unknown any email protected b f d d can be the exact data where b f d d f f f email protected email protected email protected the new notion of the stable solution excludes the unknown y and f from the definition of the solution the solution is defined only in terms of the noisy data noise level and an a priori information about a compactum to which the solution belongs

iterative solution of linear equations with unbounded operators a convergent iterative process is constructed for solving any solvable linear equation in a hilbert space including equations with unbounded closed densely defined linear operators the method is proved to be stable towards small perturbation of the data some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators the dynamical systems method dsm is justified for unbounded closed densely defined linear operators the stopping time is chosen by a discrepancy principle equations with selfadjoint operators are considered separately numerical examples illustrating the efficiency of the proposed method are given

a discrepancy principle for equations with monotone continuous operators a discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated the existence and uniqueness of the corresponding regularization parameter a are proved convergence of the solution obtained by the discrepancy principle is justified the results are obtained under natural assumptions on the nonlinear operator

descriptions of partial isometries on hilbert c modules abstract partial isometries and moore penrose inverses of bounded adjointable operators on hilbert c modules are investigated it is shown that a contraction between hilbert c modules is a partial isometry if and only if it possesses a contractive moore penrose inverse a bounded adjointable operator t is a partial isometry if and only if its bounded transform is 2 1 2 t

singularly perturbed integral equations we study singularly perturbed fredholm equations of the second kind we give sufficient conditions for existence and uniqueness of solutions and describe the asymptotic behavior of the solutions we examine the relationship between the solutions of the perturbed and unperturbed equations exhibiting the degeneration of the boundary layer to delta functions the results are applied to several examples including the volterra equations

dynamical systems method dsm for solving equations with monotone operators without smoothness assumptions on f u abstract a version of the dynamical systems method dsm for solving ill posed nonlinear equations f u f with monotone operators f in a hilbert space is studied in this paper under less restrictive assumptions on the nonlinear operators f than the assumptions used earlier a new method of proof of the basic results is used an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically under weaker assumptions on the nonlinear operator f than the assumptions used earlier

exponents of two colored digraphs with two cycles abstract for two nonnegative matrices a and b of the same order the pair a b is primitive if there exist nonnegative integers h and k such that the sum of all products formed by words consisting of h a s and k b s is a positive matrix the exponent of the pair a b is the smallest value of h k over all such h and k there is a natural correspondence between two colored digraphs and nonnegative matrix pairs in this paper we consider the exponent of a especial nonnegative matrix pair whose associated digraph consists of two cycles

a note on k primitive directed graphs abstract we consider the problem of which primitive directed graphs can be k colored to yield a k primitive directed graph if such a k coloring exists then certainly such a graph must have at least k cycles we prove that any primitive directed graph admits a 2 coloring that is 2 primitive by contrast for each k 4 we construct examples of primitive directed graphs having k cycles for which no k coloring is k primitive we also give some partial results for the case that k 3

global convergence for ill posed equations with monotone operators the dynamical systems method consider an operator equation f u 0 in a real hilbert space let us call this equation ill posed if the operator f u is not boundedly invertible and well posed otherwise if f is monotone c2loc h operator then we construct a cauchy problem which has the following properties 1 it has a global solution for an arbitrary initial data 2 this solution tends to a limit as time tends to infinity and 3 the limit is the minimum norm solution to the equation f u 0 an example of applications to linear ill posed operator equation is given

ep modular operators and their products abstract we study first ep modular operators on hilbert c modules and then we provide necessary and sufficient conditions for the product of two ep modular operators to be ep these enable us to extend some results of koliha 2000 13 for an arbitrary c algebra and the c algebras of compact operators

a new discrepancy principle the aim of this note is to prove a new discrepancy principle the advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem approximately rather than exactly and in the proof of a stability result

exponents of nonnegative matrix pairs the notions of primitivity and exponent of a square nonnegative matrix a are classical a is primitive provided there is a nonnegative integer k such that a k is entrywise positive and in the case a is primitive the exponent of a is the smallest such k fornasini and valcher have extended the notion of primitivity to pairs a b of square nonnegative matrices of the same order the pair a b is primitive provided there exist nonnegative integers h and k such that the sum of all products formed by words consisting of ha s and kb s is entrywise positive this paper defines the exponent of a nonnegative matrix pair to be the smallest value of h k over all such h and k it is then shown that the largest exponent of a primitive pair of n by n nonnegative matrices lies in the interval n 3 5n 2 2 3n 3 2n 2 2n 2 in addition the exponent of a pair of nonnegative matrices is related to properties of an associated twodimensional dynamical system 2001 elsevier science inc all rights reserved