the bishop phelps bollobas version of lindenstrauss properties a and b we study a bishop phelps bollobas version of lindenstrauss properties a and b for domain spaces we study banach spaces x such that x y has the bishopphelps bollobas property bpbp for every banach space y we show that in this case there exists a universal function x e such that for every y the pair x y has the bpbp with this function this allows us to prove some necessary isometric conditions for x to have the property we also prove that if x has this property in every equivalent norm then x is one dimensional for range spaces we study banach spaces y such that x y has the bishop phelps bollobas property for every banach space x in this case we show that there is a universal function y e such that for every x the pair x y has the bpbp with this function this implies that this property of y is strictly stronger than lindenstrauss property b the main tool to get these results is the study of the bishop phelps bollobas property for c0 1and sums of banach spaces

a bishop phelps bollobas type theorem for uniform algebras abstract this paper is devoted to showing that asplund operators with range in a uniform banach algebra have the bishop phelps bollobas property i e they are approximated by norm attaining asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it to prove this result we use the weak to norm fragmentability of weak compact subsets of the dual of asplund spaces and we need to observe a urysohn type result producing peak complex valued functions in uniform algebras that are small outside a given open set and whose image is inside a stolz region

a urysohn type theorem and the bishop phelps bollobas theorem for holomorphic functions abstract a urysohn type theorem is introduced for a subalgebra of the algebra c b of all bounded complex valued continuous functions on a hausdorff topological space with use of this theorem it is shown that a type of the bishop phelps bollobas theorem holds for certain classes of holomorphic functions on the unit ball of a complex banach space x if x is either a locally uniformly convex space or a locally c uniformly convex order continuous sequence space

characterization of banach spaces y satisfying that the pair l 4 y has the bishop phelps bollobas property for operators abstract we study the bishop phelps bollobas property for operators from l 4 to a banach space for this reason we introduce an appropriate geometric property namely the ahsp l 4 we prove that spaces y satisfying ahsp l 4 are precisely those spaces y such that l 4 y has the bishop phelps bollobas property we also provide classes of banach spaces satisfying this condition for instance finite dimensional spaces uniformly convex spaces c 0 l and l 1 satisfy ahsp l 4

emerging notions of norm attainment for lipschitz maps between banach spaces abstract we classify several notions of norm attaining lipschitz maps which were introduced previously and present the relations among them in order to verify proper inclusions we also analyze some results for the sets of lipschitz maps satisfying each of these properties to be dense or not in lip 0 x y for instance we characterize a banach space y with the radon nikod m property in terms of the denseness of norm attaining lipschitz maps with values in y further we introduce a property called the local directional bishop phelps bollobas property for lipschitz compact maps which extends the one studied previously for scalar valued functions and provide some new positive results

polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers it is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences given sufficiently many initial bits of an algebraic number its minimal polynomial can be reconstructed and therefore the further bits of the algebraic number can be computed this also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients all algorithms work in polynomial time

the bishop phelps bollobas theorem for operators we prove the bishop phelps bollobas theorem for operators from an arbitrary banach space x into a banach space y whenever the range space has property of lindenstrauss we also characterize those banach spaces y for which the bishop phelps bollobas theorem holds for operators from 1 into y s everal examples of classes of such spaces are provided for instance the bishop phelps bollobas theorem holds when the range space is finite dimensional an l1 space for a finite measure a c k space for a compact hausdorff space k or a uniformly convex banach space

korteweg de vries surfaces abstract we consider 2 surfaces arising from the korteweg de vries kdv hierarchy and the kdv equation the surfaces corresponding to the kdv equation are in a three dimensional minkowski m 3 space they contain a family of quadratic weingarten and willmore like surfaces we show that some kdv surfaces can be obtained from a variational principle where the lagrange function is a polynomial function of the gaussian and mean curvatures we also give a method for constructing the surfaces explicitly i e finding their parameterizations or finding their position vectors

γ flatness and bishop phelps bollobas type theorems for operators the bishop phelps bollobas property deals with simultaneous approximation of an operator t and a vector x at which t nearly attains its norm by an operator t 0 t 0 and a vector x 0 x 0 respectively such that t 0 t 0 attains its norm at x 0 x 0 in this note we extend the already known results about the bishop phelps bollobas property for asplund operators to a wider class of banach spaces and to a wider class of operators instead of proving a bpb type theorem for each space separately we isolate two main notions flat operators and banach spaces with ack ack structure in particular we prove a general bpb type theorem for flat operators acting to a space with ack ack structure and show that uniform algebras and spaces with the property have ack ack structure we also study the stability of the ack ack structure under some natural banach space theory operations as a consequence we discover many new examples of spaces y such that the bishop phelps bollobas property for asplund operators is valid for all pairs of the form x y x y

modified korteweg de vries surfaces in this work we consider 2 surfaces in r3 arising from the modified korteweg de vries mkdv equation we give a method for constructing the position vector of the mkdv surface explicitly for a given solution of the mkdv equation mkdv surfaces contain willmore like and weingarten surfaces we show that some mkdv surfaces can be obtained from a variational principle where the lagrange function is a polynomial of the gaussian and mean curvatures

local bishop phelps bollobas properties abstract in this paper we introduce some local versions of bishop phelps bollobas type property for operators that is the function which appears in their definitions depends not only on a given e 0 but also on either a fixed norm one operator t or a fixed norm one vector x we investigate those properties and show differences between local and uniform versions

monotonicity and complex convexity in banach lattices abstract the goal of this article is to study the relations among monotonicity properties of real banach lattices and the corresponding convexity properties in the complex banach lattices we introduce the moduli of monotonicity of banach lattices we show that a banach lattice e is uniformly monotone if and only if its complexification e c is uniformly complex convex we also prove that a uniformly monotone banach lattice has finite cotype in particular we show that a banach lattice is of cotype q for some 2 q if and only if there is an equivalent lattice norm under which it is uniformly monotone and its complexification is q uniformly pl convex we also show that a real kothe function space e is strictly respectively uniformly monotone and a complex banach space x is strictly respectively uniformly complex convex if and only if kothe bochner function space e x is strictly respectively uniformly complex convex

a generalized formula for integrable classes of surfaces in lie algebras we discuss relations between the approach of fokas and gelfand to immersions on lie algebras and the theory of soliton surfaces of sym we show that many results concerning immersions on lie algebras can be reduced to or interpreted within the soliton surfaces approach we present also some new results including a generalization of the fokas gelfand formula for integrable classes of surfaces in lie algebras and in particular in pseudo euclidean n dim spaces the generalized formula is used to formulate a method of constructing integrable classes of surfaces as an example we discuss the class of linear weingarten surfaces defined by the linear relationship between gaussian and mean curvatures we construct explicitly a one parameter family of linear weingarten surfaces parallel equidistant to a given pseudospherical surface

multiple summing operators on banach spaces abstract in this paper we improve some previous results about multiple p summing multilinear operators by showing that every multilinear form from l 1 spaces is multiple p summing for 1 p 2 the proof is based on the existence of a predual for the banach space of multiple p summing multilinear forms we also show the failure of the inclusion theorem in this class of operators and improve some results of y melendez and a tonge about dominated multilinear operators

approximating generalized distance functions on weighted triangulated surfaces with applications given p a simple connected possibly non convex polyhedral surface composed of positively weighted triangular faces we consider paths from generalized sources points segments polygonal chains or polygonal regions to points on p that stay on p and avoid obstacles segments polygonal chains or polygonal regions the distance function defined by a generalized source is a function that assigns to each point of p the cost of the shortest path from the source to the point in this paper we present an algorithm for computing approximate generalized distance functions we also provide an algorithm that computes a discrete representation of the approximate distance function and as applications algorithms for computing discrete order k voronoi diagrams and for approximately solving facility location problems finally we present experimental results obtained with our implementation of the provided algorithms

norm or numerical radius attaining polynomials on c k abstract let c k c be the banach space of all complex valued continuous functions on a compact hausdorff space k we study when the following statement holds every norm attaining n homogeneous complex polynomial on c k c attains its norm at extreme points we prove that this property is true whenever k is a compact hausdorff space of dimension less than or equal to one in the case of a compact metric space a characterization is obtained as a consequence we show that for a scattered compact hausdorff space k every continuous n homogeneous complex polynomial on c k c can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of c k c is a norming set for every continuous complex polynomial similar results can be obtained if norm is replaced by numerical radius

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

bishop phelps bollobas property for certain spaces of operators abstract we characterize the banach spaces y for which certain subspaces of operators from l 1 into y have the bishop phelps bollobas property in terms of a geometric property of y namely ahsp this characterization applies to the spaces of compact and weakly compact operators new examples of banach spaces y with ahsp are provided we also obtain that certain ideals of asplund operators satisfy the bishop phelps bollobas property

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

numerical radius attaining compact linear operators abstract we show that there are compact linear operators on banach spaces which cannot be approximated by numerical radius attaining operators

the bishop phelps bollobas theorem fails for bilinear forms on l1 l1 abstract in this paper we show that the bishop phelps bollobas theorem fails for bilinear forms on l 1 l 1 while it holds for linear operators from l 1 to l

the bishop phelps bollobas theorem for l l1 μ l 0 1 abstract we show that the bishop phelps bollobas theorem holds for all bounded operators from l 1 into l 0 1 where is a finite measure

a lindenstrauss theorem for some classes of multilinear mappings abstract under some natural hypotheses we show that if a multilinear mapping belongs to some banach multilinear ideal then it can be approximated by multilinear mappings belonging to the same ideal all whose arens extensions attain their norms at the same point we prove a similar result for the class of symmetric multilinear mappings we see that the quantitative bollobas type version of these results fails in every multilinear ideal

soliton surfaces associated with generalized symmetries of integrable equations in this paper based on the fokas et al approach fokas and gel fand 1996 commun math phys 177 203 20 fokas et al 2000 sel math 6 347 75 we provide a symmetry characterization of continuous deformations of soliton surfaces immersed in a lie algebra using the formalism of generalized vector fields their prolongation structure and links with the frechet derivatives we express the necessary and sufficient condition for the existence of such surfaces in terms of the invariance criterion for generalized symmetries and identify additional sufficient conditions which admit an explicit integration of the immersion functions of 2d surfaces in lie algebras we discuss in detail the su n valued immersion functions generated by conformal symmetries of the cpn 1 sigma model defined on either the minkowski or euclidean space we further show that the sufficient conditions for explicit integration of such immersion functions impose additional restrictions on the admissible conformal symmetries of the model defined on minkowski space on the other hand the sufficient conditions are identically satisfied for arbitrary conformal symmetries of finite action solutions of the cpn 1 sigma model defined on euclidean space

a quadratically convergent method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum nolimits i 1 m r i x here the residuals r i x are affine functions fromr n tor 1 n 1 2 m 2 this arises in a number of applications including single and multi facility location problems the functionf is in general not differentiable atx if at least oner i x is zero

simultaneously continuous retraction and bishop phelps bollobas type theorem abstract the dual space x of a banach space x is said to admit a uniformly simultaneously continuous retraction if there is a retraction r from x onto its unit ball b x which is uniformly continuous in norm topology and continuous in weak topology we prove that if a banach space resp complex banach space x has a normalized unconditional schauder basis with unconditional basis constant 1 and if x is uniformly monotone resp uniformly complex convex then x admits a uniformly simultaneously continuous retraction it is also shown that x admits such a retraction if x x i c 0 or x x i l 1 where x i is a family of separable banach spaces whose duals are uniformly convex with moduli of convexity i e with inf i i e 0 for all 0 e 1 let k be a locally compact hausdorff space and let c 0 k be the real banach space consisting of all real valued continuous functions vanishing at infinity as an application of simultaneously continuous retractions we show that a pair x c 0 k has the bishop phelps bollobas property for operators if x admits a uniformly simultaneously continuous retraction as a corollary c 0 s c 0 k has the bishop phelps bollobas property for operators for every locally compact metric space s

the geometric median on riemannian manifolds with application to robust atlas estimation one of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images the study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space which map one anatomy to another it is now well established that representing the geometry of shapes or images in euclidian spaces undermines our ability to represent natural variability in populations in our previous work we have extended classical statistical analysis techniques such as averaging principal components analysis and regression to riemannian manifolds which are more appropriate representations for describing anatomical variability in this paper we extend the notion of robust estimation a well established and powerful tool in traditional statistical analysis of euclidian data to manifold valued representations of anatomical variability in particular we extend the geometric median a classic robust estimator of centrality for data in euclidean spaces we formulate the geometric median of data on a riemannian manifold as the minimizer of the sum of geodesic distances to the data points we prove existence and uniqueness of the geometric median on manifolds with non positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds generalizing the weiszfeld procedure for finding the geometric median of euclidean data we present an algorithm for computing the geometric median on an arbitrary manifold we show that this algorithm converges to the unique solution when it exists in this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images using this approach we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images

the bishop phelps bollobas theorem for operators from l1 sums of banach spaces abstract we introduce a generalized approximate hyperplane series property for a pair x y of banach spaces to characterize when l 1 x y has the bishop phelps bollobas property in particular we show that x y has this property if x y are finite dimensional if x is a c k space and y is a hilbert space or if x is asplund and y c 0 l where k is a compact hausdorff space and l is a locally compact hausdorff space

banach spaces with a large semigroup of contractive automorphisms abstract transitivity almost transitivity and convex transitivity are classical notions in the isometric theory of banach spaces all of them related to mazur s problem asking whether transitive separable banach spaces are hilbert spaces 24 33 in this paper we study near versions of the above notions roughly speaking nearness means that contractive automorphisms play the role played by surjective linear isometries in the corresponding classical notion we also consider asymptotic transitivity as defined in 35 which is implied by almost transitivity and implies near almost transitivity and introduce the corresponding notion of asymptotic convex transitivity we show that nearly convex transitive respectively asymptotically convex transitive banach spaces are nearly almost transitive respectively asymptotically transitive uniformly smooth and uniformly convex as soon as they be asplund spaces or have the radon nikod m property we also deal with a property stronger than the near transitivity and weaker than the uniform micro semitransitivity 18 we prove that banach spaces with this intermediate property are uniformly smooth and uniformly convex finally we show that nearly almost transitive norm unital complete normed possibly non associative algebras are hilbert spaces

fuzzy disk for covering fuzzy points abstract in this paper we consider an important fuzzy version of the well known smallest covering circle problem which is also called the euclidean 1 center problem here we are given a set of points in the plane whose locations are crisp however the requirement for coverage of each point is fuzzy as represented by its grade of membership as such we qualify the points as fuzzy in the above framework we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points after developing a general model certain special cases are investigated in detail polynomial algorithms are presented for the special cases

new nonlinear evolution equations from surface theory we point out that the connection between surfaces in three dimensional flat space and the inverse scattering problem provides a systematic way for constructing new nonlinear evolution equations in particular we study the imbedding for guichard surfaces which gives rise to the calapso guichard equations generalizing the sine gordon sg equation further we investigate the geometry of surfaces and their imbedding which results in the korteweg devries kdv equation then by constructing a family of applicable surfaces we obtain a generalization of the kdv equation to a compressible fluid

the bishop phelps bollobas property for numerical radius on l1 abstract guirao and kozhushkina recently introduced the bishop phelps bollobas property for numerical radius geometrically speaking the bishop phelps bollobas property says that if we have a state and an operator that almost attains its numerical radius at this state then there exist another state close to the original state and another operator close to the original operator such that the new operator attains its numerical radius at this new state in this paper we study the bishop phelps bollobas property for numerical radius in the context of the banach space of lebesgue integrable functions over the real line

the bishop phelps bollobas property for operators from c k to uniformly convex spaces abstract we show that the pair c k x has the bishop phelps bollobas property for operators if k is a compact hausdorff space and x is a uniformly convex space

extreme polynomials and multilinear forms onl1 abstract given a 2 homogeneous polynomialp x y ax2 by2 cxywith real coefficients let p rand p cdenote the norms ofpon the real and complex banach spacel21 respectively we show p r p c and obtain a sufficient and necessary condition on the coefficientsa b andcforpto have norm 1 applying these results we characterize extreme points of the unit ball of p 2l1 for the real banach spacel21and examine them for the complex banach spacel21 we apply them to find extreme points and strongly extreme points of the unit ball of p 2l1 and get an extremal 2 homogeneous polynomial onl1that is not an extreme point we also characterize extreme points and strongly extreme points of the unit ball of l ml1 or l ml1 0 1

the bishop phelps bollobas property for operators from c0 into some banach spaces abstract we exhibit a new class of banach spaces y such that the pair c 0 y has the bishop phelps bollobas property for operators this class contains uniformly convex banach spaces and spaces with the property of lindenstrauss we also provide new examples of spaces in this class

on the polynomial lindenstrauss theorem abstract under certain hypotheses on the banach space x we show that the set of n homogeneous polynomials from x to any dual space whose aron berner extensions are norm attaining is dense in the space of all continuous n homogeneous polynomials to this end we prove an integral formula for the duality between tensor products and polynomials we also exhibit examples of lorentz sequence spaces for which there is no polynomial bishop phelps theorem but our results apply finally we address quantitative versions in the sense of bollobas of these results

the complex convexity of quasi normed linear spaces abstract a notion of uniform convexity is defined for quasi normed complex spaces by replacing norms of midpoints of segments in the space by norms of centers of complex discs in the space complex uniformly convex spaces c u c spaces always have cotype in banach lattices possessing cotype and being complex uniformly convex are the same duals of c algebras are c u c and in fact have cotype 2 the behavior of certain martingale difference sequences in c u c spaces is examined this leads to an isomorphic description of the spaces having the property

a strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on two terminal series parallel networks we present strongly polynomial algorithms to find rational and integer flow vectors that minimize a convex separable quadratic cost function on two terminal series parallel graphs

on the continuous fermat weber problem we give the firstexact algorithmic study of facility location problems that deal with finding a median for acontinuum of demand points in particular we consider versions of the continuousk median fermat weber problem where the goal is to select one or more center points that minimize the average distance to a set of points in a demandregion in such problems the average is computed as an integral over the relevant region versus the usual discrete sum of distances the resulting facility location problems are inherently geometric requiring analysis techniques of computational geometry we provide polynomial time algorithms for various versions of thel1 1 median fermat weber problem we also consider the multiple center version of thel1k median problem which we prove is np hard for largek

symmetric multilinear forms on hilbert spaces where do they attain their norm abstract we characterize the sets of norm one vectors x 1 x k in a hilbert space h such that there exists a k linear symmetric form attaining its norm at x 1 x k we prove that in the bilinear case any two vectors satisfy this property however for k 3 only collinear vectors satisfy this property in the complex case while in the real case this is equivalent to x 1 x k spanning a subspace of dimension at most 2 we use these results to obtain some applications to symmetric multilinear forms symmetric tensor products and the exposed points of the unit ball of l s h k

soliton surfaces and generalized symmetries of integrable systems in this paper we discuss some specific features of symmetries of integrable systems which can be used to construct the fokas gel fand formula for the immersion of 2d soliton surfaces associated with such systems in lie algebras we establish a sufficient condition for the applicability of this formula this condition requires the existence of two vector fields which generate a common symmetry of the initial system and its corresponding linear spectral problem this means that these two fields have to be group related and we determine an explicit form of this relation it provides a criterion for the selection of symmetries suitable for use in the fokas gel fand formula we include some examples illustrating its application

on the bishop phelps bollobas theorem for multilinear mappings abstract we study the bishop phelps bollobas property and the bishop phelps bollobas property for numerical radius our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases we characterize the pair l 1 x y to have the bpbp for bilinear forms and prove that on l 1 the numerical radius and the norm of a multilinear mapping are the same we also show that l 1 fails the bpbp nu for multilinear mappings although l 1 satisfies it in the operator case for every measure

an efficient algorithm for minimizing a sum of euclidean norms with applications in recent years rich theories on polynomial time interior point algorithms have been developed these theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications in this paper the problem of minimizing a sum of euclidean norms is studied this problem is convex but not everywhere differentiable by transforming the problem into a standard convex programming problem in conic form we show that an epsilon optimal solution can be computed efficiently using interior point algorithms as applications to this problem polynomial time algorithms are derived for the euclidean single facility location problem the euclidean multifacility location problem and the shortest network under a given tree topology in particular by solving the newton equation in linear time using em gaussian elimination on leaves of a tree we present an algorithm which computes an epsilon optimal solution to the shortest network under a given full steiner topology interconnecting n regular points in o n sqrt n log bar c epsilon log n arithmetic operations where bar c is the largest pairwise distance among the given points the previous best known result on this problem is a graphical algorithm which requires o n2 arithmetic operations under certain conditions

the bishop phelps bollobas point property abstract in this article we study a version of the bishop phelps bollobas property we investigate a pair of banach spaces x y such that every operator from x into y is approximated by operators which attain their norm at the same point where the original operator almost attains its norm in this case we say that such a pair has the bishop phelps bollobas point property bpbpp we characterize uniform smoothness in terms of bpbpp and we give some examples of pairs x y which have and fail this property some stability results are obtained about l 1 and l sums of banach spaces and we also study this property for bilinear mappings

the bishop phelps bollobas theorem for operators from l1 μ to banach spaces with the radon nikodým property abstract let y be a banach space and be a finite measure space where is an infinite algebra of measurable subsets of we show that if the couple l 1 y has the bishop phelps bollobas property for operators then y has the ahsp further for a banach space y with the radon nikod m property we prove that the couple l 1 y has the bishop phelps bollobas property for operators if and only if y has the ahsp

computing generalized higher order voronoi diagrams on triangulated surfaces we present an algorithm for computing exact shortest paths and consequently distance functions from a generalized source point segment polygonal chain or polygonal region on a possibly non convex triangulated polyhedral surface the algorithm is generalized to the case when a set of generalized sites is considered providing their distance field that implicitly represents the voronoi diagram of the sites next we present an algorithm to compute a discrete representation of the distance function and the distance field then by using the discrete distance field we obtain the voronoi diagram of a set of generalized sites points segments polygonal chains or polygons and visualize it on the triangulated surface we also provide algorithms that by using the discrete distance functions provide the closest furthest and k order voronoi diagrams and an approximate 1 center and 1 median

deformation of surfaces integrable systems and chern simons theory a few years ago some of us devised a method to obtain integrable systems in 2 1 dimensions from the classical non abelian pure chern simons action via the reduction of the gauge connection in hermitian symmetric spaces in this article we show that the methods developed in studying classical non abelian pure chern simons actions can be naturally implemented by means of a geometrical interpretation of such systems the chern simons equation of motion turns out to be related to time evolving two dimensional surfaces in such a way that these deformations are both locally compatible with the gauss mainardi codazzi equations and completely integrable the properties of these relationships are investigated together with the most relevant consequences explicit examples of integrable surface deformations are displayed and discussed

on integrability of weingarten surfaces a forgotten class rediscovered by a systematic search a forgotten class of integrable surfaces is shown to disprove the finkel wu conjecture the associated integrable nonlinear partial differential equation possesses a zero curvature representation a third order symmetry and a nonlocal transformation to the sine gordon equation sin we leave open the problem of finding a backlund autotransformation and a recursion operator that would produce a local hierarchy

some geometric properties of disk algebras abstract in this paper we study some geometrical properties of certain classes of uniform algebras in particular the ball algebra a u b x of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex banach space x we prove that a u b x has k numerical index 1 for every k the lushness and also the ahsp moreover the disk algebra a d and more in general any uniform algebra whose choquet boundary has no isolated points is proved to have the polynomial daugavet property most of those properties are extended to the vector valued version a x of a uniform algebra a

two sided polynomial ideals on banach spaces abstract classes of homogeneous polynomials between banach spaces have been studied in the last three decades from the perspective of the stability with respect to the composition of a homogeneous polynomial with linear operators in an attempt to explore the underlying nonlinearity in a more consistent way and to bring the subject closer to its roots in complex analysis and also taking into account recent results in the field in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials

on the intrinsic and the spatial numerical range for a bounded function f from the unit sphere of a closed subspace x of a banach space y w e study when the closed convex hull of its spatial numerical range w f is equal to its intrinsic numerical range v f we show that for every infinite dimensional banach space x there is a superspace y and a bounded linear operator t x y such that co w t v t we also show that up to renormig for every non reflexive banach space y one can find a closed subspace x and a bounded linear operator t l x y such that co w t v t finally we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range which we call the bishop phelps bollobas property and which is weaker than the uniform smoothness and the finite dimensionality we characterize strong subdifferentiability and uniform smoothness in terms of this property

the bishop phelps bollobas property and lush spaces abstract we prove that for every lush space x the couple l 1 x has the bishop phelps bollobas property for operators that is every lush space has the ahsp standing for the approximate hyperplane series property while every lush space has the alternative daugavet property there exists a space with the alternative daugavet property that does not have the ahsp we also show that there is a banach space with both the ahsp and the alternative daugavet property which is not lush

on banach spaces whose group of isometries acts micro transitively on the unit sphere abstract we study banach spaces whose group of isometries acts micro transitively on the unit sphere we introduce a weaker property inherited by one complemented subspaces that we call uniform micro semitransitivity we prove a number of results about both micro transitive and uniformly micro semitransitive spaces in particular they are uniformly convex and uniformly smooth and form a self dual class to this end we relate the fact that the group of isometries acts micro transitively with a property of operators called the pointwise bishop phelps bollobas property and use some known results on it besides we show that if there is a non hilbertian non separable banach space with uniform micro semitransitive or micro transitive norm then there is a non hilbertian separable one finally we show that an l p space is micro transitive or uniformly micro semitransitive only when p 2

operators on separable l1 predual spaces abstract we give an extension of the classical bartle dunford schwartz theorem for weakly compact operators on a c k space to weakly compact operators on a separable l 1 predual space using this we show that for operators on these spaces the set of weakly compact operators that attain their norm is dense in the space of weakly compact operators for operators from the space of affine continuous functions on a metrizable choquet simplex with values in an uniformly convex space we show that the operator theoretic version of the bishop phelps bollobas property is valid this gives an extension of some recent work of kim and lee

optimization and differentiation in banach spaces abstract we show that some old ideas of smulian can be used to give another proof of a theorem of bourgain we characterize subsets of banach spaces having the radon nikodym property by means of optimization results

an efficient approximate algorithm for the 1 median problem in metric spaces we propose a simple and natural linear randomized algorithm for the approximate 1 median selection problem in metric spaces the 1 median of a finite subset s of a metric space is the element of s which minimizes the average distance from the remaining points in s this problem is extremely important in most applications using clustering of metric spaces but also in connection with several algorithms in bioinformatics the only linear approximation algorithm for the 1 median problem which provably works in any metric space without going through any euclidean space has been proposed by indyk in proceedings of the 31st annual acm symposium on theory of computing atlanta 1999 pp 428 432 however indyk s algorithm which is based on sufficiently large sampling turns out not to be a practical solution the same holds true even for its heuristic variants which use samplings of smaller size the algorithm we propose has a simple and efficient implementation which performs better than indyk s algorithm in practice on the other hand while the performance of indyk s algorithm is guaranteed by an approximation factor in the case of our algorithm we are only able to produce experimental evidence of its precision extensive experimentation has been performed on both synthetic and real input datasets synthetic datasets were generated with uniform and skewed distributions using various metrics real datasets have been extrapolated from real world official databases available on the web successful results of the proposed algorithm are reported for several applications in bioinformatics and various classes of approximate search queries

hypercyclic convolution operators on frechet spaces of analytic functions a result of godefroy and shapiro states that the convolution operators on the space of entire functions on c n which are not multiples of identity are hypercyclic analogues of this result have appeared for some spaces of holomorphic functions on a banach space in this work we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators some known results come out as particular cases of this setting we also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the schatten von neumann class

stock cutting to minimize cutting length in this paper we investigate the following problem given two convex p in and p out where p in is completely contained in p out we wish to find a sequence of guillotine cuts to cut out p in from p out such that the total length of the cutting sequence is minimized this problem has applications in stock cutting where a particular shape or design in this case the polygon p in needs to be cut out of a given piece of parent material the polygon pout using only guillotine cuts and where it is desired to minimize the cutting sequence length to improve the cutting time required per piece we first prove some properties of the optimal solution to the problem and then give an approximation scheme for the problem that given an error range x03b4 produces a cutting sequence whose total length is at most x03b4 more than that of the optimal cutting sequence then it is shown that this problem has optimal solutions that lie in the algebraic extension of the field that the input data belongs to hence due to this algebraic nature of the problem an approximation scheme is the best that can be achieved extensions of these results are also studied in the case where the polygons p in and p out are non convex

bishop phelps bollobas property for bilinear forms on spaces of continuous functions it is shown that the bishop phelps bollobas theorem holds for bilinear forms on the complex c 0 l 1 times c 0 l 2 for arbitrary locally compact topological hausdorff spaces l 1 and l 2

algebraic optimization the fermat weber location problem the fermat weber location problem is to find a point in n that minimizes the sum of the weighted euclidean distances fromm given points in n in this work we discuss some relevant complexity and algorithmic issues first using tarski s theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one dimensional polynomial secondly we exhibit an explicit solution to the strong separation problem associated with the fermat weber model this separation result shows that ane approximation solution can be constructed in polynomial time using the standard ellipsoid method

on the construction of integrable surfaces on lie groups the problem of the immersion of a two dimensional surface into a three dimensional euclidean space can be formulated in terms of the immersion of surfaces in lie groups and lie algebras a general formalism for this problem is developed as well as an equivalent mauer cartan system of differential forms the particular case of the lie group su 2 is examined and it is shown to be useful for studying integrable surfaces some examples of such surfaces and their equations are presented at the end in particular the cases of constant mean curvature and of zero gaussian curvature

the bishop phelps bollobas theorem for operators on l1 μ in this paper we show that the bishop phelps bollobas theorem holds for l l1 l1 for all measures and and also holds for l l1 l for every arbitrary measure and every localizable measure finally we show that the bishop phelps bollobas theorem holds for two classes of bounded linear operators from a real l1 into a real c k if is a finite measure and k is a compact hausdorff space in particular one of the classes includes all bochner representable operators and all weakly compact operators

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

deformations of surfaces associated with integrable gauss mainardi codazzi equations using the formulation of the immersion of a two dimensional surface into the three dimensional euclidean space proposed recently a mapping from each symmetry of integrable equations to surfaces in r3 can be established we show that among these surfaces the sphere plays a unique role indeed under the rigid su 2 rotations all integrable equations are mapped to a sphere furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine gordon equation are homeomorphic to a sphere we also find some new weingarten surfaces arising from the deformations of the modified kurteweg de vries and of the nonlinear schrodinger equations surfaces can also be associated with the motion of curves we study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity

the bishop phelps bollobas property for lipschitz maps abstract in this paper we introduce and study a lipschitz version of the bishop phelps bollobas property lip bpb property this property deals with the possibility of making a uniformly simultaneous approximation of a lipschitz map f and a pair of points at which f almost attains its norm by a lipschitz map g and a pair of points such that g strongly attains its norm at the new pair of points we first show that if m is a finite pointed metric space and y is a finite dimensional banach space then the pair m y has the lip bpb property and that both finiteness assumptions are needed next we show that if m is a uniformly gromov concave pointed metric space i e the molecules of m form a set of uniformly strongly exposed points then m y has the lip bpb property for every banach space y we further prove that this is the case for finite concave metric spaces ultrametric spaces and holder metric spaces the extension of the lip bpb property from m r to some banach spaces y and some results for compact lipschitz maps are also discussed

the bishop phelps bollobas property for operators between spaces of continuous functions abstract we show that the space of bounded linear operators between spaces of continuous functions on compact hausdorff topological spaces has the bishop phelps bollobas property a similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and hausdorff topological space into a uniformly convex space and for the class of compact operators from a banach space into a predual of an l 1 space

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

norm attaining bilinear forms onl1 0 1 abstract we show that the set of norm attaining bilinear forms onl1 0 1 is not dense in the space of all continuous bilinear forms

a smoothing newton method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum i 1 m b i a i tx this problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero in this paper we present a smoothing newton method for this problem by applying the smoothing newton method proposed by qi sun and zhou math programming 87 2000 pp 1 35 directly to a system of strongly semismooth equations derived from primal and dual feasibility and a complementarity condition this method is globally and quadratically convergent as applications to this problem smoothing newton methods are presented for the euclidean facilities location problem and the steiner minimal tree problem under a given topology preliminary numerical results indicate that this method is extremely promising

polynomial time approximation schemes for geometric min sum median clustering the johnson lindenstrauss lemma states that n points in ahigh dimensional hilbert space can be embedded with smalldistortion of the distances into an o log n dimensional space by applying a random linear transformation weshow that similar though weaker properties hold for certainrandom linear transformations over the hamming cube we use thesetransformations to solve np hard clustering problems in the cube aswell as in geometric settings more specifically we address thefollowing clustering problem given n points in a larger set e g d endowed with a distance function e g l2 distance we would like to partition the dataset into k disjoint clusters each with a cluster center so as to minimize the sum over all data points of the distancebetween the point and the center of the cluster containing thepoint the problem is provably np hard in some high dimensionalgeometric settings even for k 2 we give polynomial timeapproximation schemes for this problem in several settings including the binary cube 0 1 d with hamming distance and d either with l1 distance or with l2 distance or with the square ofl2 distance in all these settings the bestprevious results were constant factor approximation guarantees wenote that our problem is similar in flavor to the k medianproblem and the related facility location problem which has beenconsidered in graph theoretic and fixed dimensional geometricsettings where it becomes hard when k is part of the input in contrast we study the problem when k is fixed but thedimension is part of the input

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

a multilinear lindenstrauss theorem abstract we show that the set of n linear mappings on a product of n banach spaces such that all their arens extensions attain their norms at the same element is norm dense in the space of all bounded n linear mappings

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed