existence of solutions for variational inequalities on riemannian manifolds abstract we establish the existence and uniqueness results for variational inequality problems on riemannian manifolds and solve completely the open problem proposed in s z nemeth variational inequalities on hadamard manifolds nonlinear anal 52 2003 1491 1498 also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on riemannian manifolds are studied

on the second greedy weight for linear codes of dimension at least 4 the maximum of g sub 2 d sub 2 for linear n k d q codes c is studied here d sub 2 is the smallest size of the support of a two dimensional subcode of c and g sub 2 is the smallest size of the support of a two dimensional subcode of c which contains a codeword of weight d for codes of dimension 4 or more upper and lower bounds on the maximum of g sub 2 d sub 2 are given

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

a fast fourier galerkin method solving a boundary integral equation for the biharmonic equation we develop a fast fourier galerkin method for solving a boundary integral equation which is a reformulation of the dirichlet problem of the biharmonic equation the proposed method is based on a splitting of the resulting boundary integral operator that is we write the operator as a sum of two integral operators one having the fourier basis functions as eigenfunctions and the other whose representation matrix in the fourier basis can be compressed to a sparse matrix having only mathcal o n log n number of nonzero entries where n is the order of the fourier basis functions used in the method we then project the solution of the boundary integral equation onto the space spanned by the fourier basis functions this leads to a system of linear equations a fast solver for the system is based on a compression of its coefficient matrix we show that the method has the optimal convergence order mathcal o n q where q denotes the degree of regularity of the exact solution of the boundary int

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

approximation methods for the muskhelishvili equation on smooth curves we investigate the possibility of applying approximation methods to the famous muskhelishvili equation on a simple closed smooth curve since the corresponding integral operator is not invertible the initial equation has to be corrected in a special way it is shown that the spline galerkin spline collocation and spline qualocation methods for the corrected equation are stable and the corresponding approximate solutions converge to an exact solution of the muskhelishvili equation in appropriate norms numerical experiments confirm the effectiveness of the proposed methods

porosity results on fixed points for nonexpansive set valued maps in hyperbolic spaces abstract we establish the porosity results on the existence and the topological structure of fixed points for nonexpansive set valued maps in hyperbolic spaces which in particular extend and or improve some known results in this direction

on stability of approximation methods for the muskhelishvili equation we study approximation methods for the muskhelishvili integral equations on curves with comer points and establish necessary and sufficient conditions for their stability

variational inequalities for set valued vector fields on riemannian manifolds convexity of the solution set and the proximal point algorithm we consider variational inequality problems for set valued vector fields on general riemannian manifolds the existence results of the solution convexity of the solution set and the convergence property of the proximal point algorithm for the variational inequality problems for set valued mappings on riemannian manifolds are established applications to convex optimization problems on riemannian manifolds are provided

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

characterization of compact geodesic spaces abstract driven by klee s well known work about the topological properties of convex sets in locally convex linear spaces we give a complete characterization of compact geodesic spaces with curvature bounded below in terms of the fixed point property for continuous functions furthermore we provide an example which highlights the role of the sectional curvature in our result

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

spline galerkin methods for the sherman lauricella equation on contours with corners spline galerkin approximation methods for the sherman lauricella integral equation on simple closed piecewise smooth contours are studied and necessary and sufficient conditions for their stability are obtained it is shown that the method under consideration is stable if and only if certain operators associated with the corner points of the contour are invertible numerical experiments demonstrate a good convergence of the spline galerkin methods and validate theoretical results moreover it is shown that if all corners of the contour have opening angles located in interval 0 1 pi 1 9 pi then the corresponding galerkin method based on splines of order 0 1 or 2 is always stable these results are in strong contrast with the behavior of the nystrom method which has a number of instability angles in the interval mentioned

the schauder fixed point theorem in geodesic spaces abstract we give a direct proof of schauder s fixed point theorem in the setting of geodesic metric spaces generalizing the classical schauder s theorem and improving a recent version of this theorem in cat spaces as an application we prove an existence result for a variational inequality in the setting of cat spaces

variational inequalities on hadamard manifolds the notion of variational inequalities is extended to hadamard manifolds and related to geodesic convex optimization problems existence and uniqueness theorems for variational inequalities on hadamard manifolds are proved a convexity property of the solution set of a variational inequality on a hadamard manifold is presented

on the stability of the nystrom method for the muskhelishvili equation on contours with corners the stability of the nystrom method for the muskhelishvili equation on piecewise smooth simple contours is studied it is shown that in the space l 2 the method is stable if and only if certain operators a j from an algebra of toeplitz operators are invertible the operators a j depend on the parameters of the equation considered on the opening angles j of the corner points j and on parameters of the approximation method mentioned numerical experiments show that there are opening angles where the operators a j are noninvertible therefore for contours with such corners the method under consideration is not stable otherwise the method is always stable numerical examples show an excellent convergence of the method

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

on the second relative greedy weight of 4 dimensional codes based on applications to the wire tap channel of type ii with the coset coding scheme we will define optimal codes which are described as the ones achieving upper bounds on e 2 m 2 where e 2 and m 2 are the second relative greedy weight and the second relative generalized hamming weight respectively by using the finite geometry method we will first classify all the 4 dimensional codes with respect to their k 1 dimensional subcodes for k 1 1 and k 1 2 and determine the upper bounds on e 2 m 2 for each class of codes and then obtain optimal codes by explicitly constructing the value assignments which achieve upper bounds on e 2 m 2

a fast and stable solver for singular integral equations on piecewise smooth curves a scheme for the numerical solution of singular integral equations on piecewise smooth curves is presented it relies on several techniques reduction nystrom discretization composite quadrature recursive compressed inverse preconditioning and multipole acceleration the scheme is fast and stable its computational cost grows roughly logarithmically with the precision sought and linearly with overall system size when the integral equation models a boundary value problem the achievable accuracy may be close to the condition number of that problem times machine epsilon this is illustrated by application to elastostatic problems involving zigzag shaped cracks with up to twenty thousand corners and branched cracks with hundreds of triple junctions

the weight hierarchies of q ary codes of dimension 4 the weight hierarchy of a linear n k q code c over gf q is the sequence d sub 1 d sub 2 d sub k where d sub spl tau is the smallest support of an spl tau dimensional subcode of c the possible weight hierarchies of n 4 q codes are studied in particular the possible weight hierarchies of n 4 3 codes are determined

stability of the nystrom method for the sherman lauricella equation the stability of the nystrom method for the sherman lauricella equation on piecewise smooth closed simple contour gamma is studied it is shown that in the space l 2 the method is stable if and only if certain operators associated with the corner points of gamma are invertible if gamma does not have corner points the method is always stable numerical experiments show the transformation of solutions when the unit circle is continuously transformed into the unit square and then into various rhombuses examples also show an excellent convergence of the method

firmly nonexpansive mappings in classes of geodesic spaces firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators in this paper we do a thorough study of fixed point theory and the asymptotic behaviour of picard iterates of these mappings in different classes of geodesic spaces such as uniformly convex w hyperbolic spaces busemann spaces and cat 0 spaces furthermore we apply methods of proof mining to obtain effective rates of asymptotic regularity for the picard iterations msc primary 47h09 47h10 53c22 secondary 03f10 47h05 90c25 52a41