fixed points of uniformly lipschitzian mappings abstract two fixed point theorems for uniformly lipschitzian mappings in metric spaces due respectively to e lifsic and to t c lim and h k xu are compared within the framework of the so called cat 0 spaces it is shown that both results apply in this setting and that lifsic s theorem gives a sharper result also a new property is introduced that yields a fixed point theorem for uniformly lipschitzian mappings in a class of hyperconvex spaces a class which includes those possessing property p of lim and xu

super lie groups global topology and local structure a general mathematical framework for the super lie groups of supersymmetric theories is presented the definition of super lie group is given in terms of supermanifolds and two theorems analogous to theorems in classical lie group theory are proved the relationship of the super lie groups defined here to the formal groups of berezin and kac and the graded lie groups of kostant is analyzed

p adic quantum mechanics with p adic valued functions an extension of the formalism of quantum mechanics to the case where the canonical variables and functions are valued in a field of p adic numbers is considered a new p adic integral calculus is used for the realization of the gauss representation in p adic quantum mechanics

demiclosedness principle with applications for asymptotically pseudo contractions in hilbert spaces abstract in the present paper we establish a demiclosedness principle for asymptotically pseudo contractions in hilbert spaces as its applications we obtain a new fixed point theorem some weak and strong convergence theorems for asymptotically pseudo contractions the results of this paper are interesting extensions of those known results

nonlocality as well as rejection of realism are only sufficient but non necessary conditions for violation of bell s inequality in this review we remind the viewpoint that violation of bell s inequality might be interpreted not only as an evidence of the alternative either nonlocality or death of reality under the assumption the quantum mechanics is incomplete violation of bell s type inequalities is a well known sufficient condition of probabilistic incompatibility of random variables impossibility to realize them on a single probability space thus in fact we should take into account an additional interpretation of violation of bell s inequality a few pairs of random variables two dimensional vector variables involved in the epr bohm experiment are incompatible they could not be realized on a single kolmogorov probability space thus one can choose between a completeness of quantum mechanics b nonlocality c death of reality d non kolmogorovness in any event violation of bell s inequality has a variety of possible interpretations hence it could not be used to obtain the definite conclusion on the relation between quantum and classical models

superspaces and supersymmetries a theory of graded banach modules over a banach grassmann algebra is developed and applied to differential geometry of super manifolds the explicit structure of superspaces carrying poincare supersymmetry and extended supersymmetry including central charges is described

functional equations generating space densifying curves in this paper infinite dimensional vector spaces of dense curves are generated by means of the functional equations f x f 2x 0 and f x f 2x f 3x 0 for both equations a basis of continuous solutions has been constructed obtaining in this way the main achievement which is the fact of that these bases may densify a large class of compact sets of the plane for arbitrary small finally a constructive method to find the noncontinuous solutions of the second functional equation is also settled

inequalities in metric spaces with applications abstract analogues of the parallelogram identity and the cn inequality of bruhat and tits in uniformly convex metric spaces are established as an application of the new inequalities we prove two fixed point results for single valued and multi valued lipschitzian mappings

analysis on generalized superspaces the analysis over commutative algebras generalized supercommutative algebras that is differentiation and integration for functions defined on superspace over a commutative algebra is studied

on mean nonexpansive mappings and the lifshitz constant abstract we present how the theory of polynomials can be used to describe the asymptotic behaviour of the sequence of lipschitz constants for iterates of mean nonexpansive mappings we find as consequences using the lifshitz theorem some new fixed point theorems we also prove that the fixed point set of every mean nonexpansive mapping is closed and convex provided the banach space is strictly convex

cat k spaces weak convergence and fixed points abstract in this paper we show that some of the recent results on fixed point for cat 0 spaces still hold true for cat 1 spaces and so for any cat k space under natural boundedness conditions we also introduce a new notion of convergence in geodesic spaces which is related to the convergence and applied to study some aspects on the geometry of cat 0 spaces at this point two recently posed questions in w a kirk b panyanak a concept of convergence in geodesic spaces nonlinear anal 68 12 2008 3689 3696 are answered in the negative the work finishes with the study of the lifsic characteristic and property p of lim xu to derive fixed point results for uniformly lipschitzian mappings in cat k spaces a conjecture raised in s dhompongsa w a kirk b sims fixed points of uniformly lipschitzian mappings nonlinear anal 65 2006 762 772 on the lifsic characteristic function of cat k spaces is solved in the positive

compactness and the fixed point property in l1 abstract in this paper we prove that compactness can be characterized by means of the existence of a fixed point for some classes of mappings defined on convex closed subsets of the space l 1 nominally our result involves nonexpansive mappings uniformly lipschitzian mappings and cascading nonexpansive mappings we also extend the results to some more general classes of banach spaces

on the fourier transform and the spectral properties of the p adic momentum and schrodinger operators a momentum representation for p adic quantum mechanics is constructed it uses a fourier transform in the space with respect to the p adic lebesgue distribution and a study of its properties the momentum representation is used to investigate the spectral properties of the p adic momentum operator and to study the p adic valued schrodinger equation for the p adic harmonic oscillator

reducing transformation and global optimization abstract in this paper we give new results on the alienor method of dimension reduction this technique is used to solve multidimensional global optimization problems of type min x x f x where f is a non convex lipschitz function and x a compact set of r n n 2 defined by lipschitz constraints the idea is to construct an dense curve h in the feasible set x the global minimum of f on x is then approximated by the global minimum of f on the curve h that is our problem has become a one dimensional problem which can be solved by the piyavskii shubert method examples of these curves and numerical implementations on several test functions are given

on uniformly lipschitzian multivalued mappings in banach and metric spaces abstract let x d be a metric space a mapping t x x is said to be uniformly lipschitzian if there exists a constant k such that d t n x t n y k d x y for all x y x and n 1 it is known that such mappings always have fixed points in certain metric spaces for k 1 provided k is sufficiently near 1 these spaces include uniformly convex metric and banach spaces as well as metric spaces having lifsic characteristic greater than 1 a uniformly lipschitzian concept for multivalued mappings is introduced in this paper and multivalued analogues of these results are obtained

on uniformly convex functions abstract in this paper we study uniformly convex functions and uniformly convex functions at a point giving some properties and characterizations of them further we give some examples and applications of these types of functions

fixed point results for a generalized nonexpansive map in uniformly convex metric spaces abstract in this paper the existence of a unique fixed point of a map satisfying a very general contractive condition on a suitable subset of a uniformly convex metric space is proved this fixed point is approximated by averaging krasnosel skii iterations of a generalized nonexpansive map our results substantially improve and extend several known results existing in the literature

on some lp inequalities in best approximation theory h x jxp p ax p is strictly increasing in the interval p j o thus h x 0 has a unique solution x p note that x l 2 1 x p is strictly increasing in p and x 0 is the unique solution of the equation p 2 xp p l f2 1 in the interval 0 6 x 1 1 1 in 11 we proved the inequality

geometrical coefficients and the structure of the fixed point set of asymptotically regular mappings in banach spaces abstract it is shown that if e is a separable and uniformly convex banach space with opial s property and c is a nonempty bounded closed convex subset of e then for some asymptotically regular self mappings of c the set of fixed points is not only connected but even a retract of c our results qualitatively complement in the case of a uniformly convex banach space a corresponding result presented in t dominguez m a japon g lopez metric fixed point results concerning measures of noncompactness mappings in w a kirk b sims eds handbook of metric fixed point theory kluwer acad publishers dordrecht 2001 pp 239 268

a revised cut peak function method for box constrained continuous global optimization in this paper we modify the concept of cut peak function given in wang et al y wang w fang t wu a deterministic algorithm of global optimization using cut peak functions technique report in the conference of mathematical programming of china 2006 and then propose a revised cut peak function algorithm for solving box constrained continuous optimization problems the smoothing technique is used to overcome the difficulty arising from the non smoothness of the constructed function by using the exterior penalty function method we find iteratively a better minimizer from the current local minimizer till a global minimizer of the concerned problem is found some preliminary numerical results are reported

a new extension of piyavskii s method to holder functions of several variables in this paper we suggest a new extension of the method of piyavskii for global optimization of a holder function with exponent 1 1 in the one dimensional case a modification of piyavskii s algorithm is introduced and it is based on the construction of sub estimators which are piecewise linear the algorithm is then some what easily moreover the results we obtain seem interesting in the higher dimension a new variant of the alienor reducing transformation which has been devised for exploiting one dimensional global optimization techniques known for their great efficiency is used the method consists in reducing a multidimensional problem to a one dimensional one using the so called dense curves the convergence of the methods is also studied

analysis on the p adic superspace i generalized functions gaussian distribution the theory of generalized functions on the p adic superspace qn mp over the super commutative banach superalgebra 0 1 with the trivial 1 annihilator is proposed the p adic gaussian supermeasure is defined as a generalized function on the p adic superspace the connection between the p adic gaussian integration and the p adic gamma function of morita is considered

consistent superspace integration the standard berezin method for integration over odd variables is combined in a new way with de witt s contour method for integration over even grassmann variables to give a new method of superspace integration it is shown that this integral unlike the standard superspace integral is invariant under coordinate transformations in superspace the relation between the new method and the standard method is discussed

fixed points of uniformly lipschitzian mappings in spaces with uniformly normal structure dans un espace de banach x on etudie quand reflexivite et structure normale sont suffisantes pour assurer pour de bon 1 la propriete de point fixe pour des auto applications uniformement lipschitziennes

on the structure of fixed point sets of asymptotically regular semigroups abstract we show that the set of fixed points of an asymptotically regular mapping acting on a convex and weakly compact subset of a banach space is in some cases a holder continuous retract of its domain our results qualitatively complement the corresponding fixed point existence theorems and extend a few recent results of gornicki 15 16 17 we also characterize bynum s coefficients and the opial modulus in terms of nets

generalized complex superspace involutions of superfields distinguished involutions called transposition and conjugation in the algebra of complex supernumbers are introduced on the base of these involutions an analysis is developed over complex superspaces consistency of integration under supertransformations between two complex superspaces is shown

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

mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces abstract let m d be a complete 2 uniformly convex metric space let c be a nonempty bounded closed and convex subset of m and let t c c be an asymptotic pointwise nonexpansive mapping in this paper we prove that the modified mann iteration process defined by x n 1 t n t n x n 1 t n x n converges in a weaker sense to a fixed point of t

p adic superanalysis i infinitely generated banach superalgebras and algebraic groups a sequence of infinitely generated p adic banach superalgebras and their locally convex projective and inductive limits were constructed their spectral properties are characterized herein and by studying the algebras of matrices constructed with these superalgebras an exponential map and its inverse can be defined some examples of superalgebra with pathological spectral properties are discussed

normal structure coefficients of l p ω let x be a uniformly convex banach space and n x the normal structure coefficient of x in this paper it is proved that n x can be calculated by considering only sets whose points are equidistant from their chebyshev centre this result is applied to prove that n l p min 2 1 1 p 2 1 p being a finite measure space the computation of n l p lets us also calculate some other coefficients related to the normal structure

fixed point theorems for mappings of asymptotically nonexpansive type the purpose of this paper is to provide fixed point theorems for asymptotically nonexpansive type mappings in a banach space with uniform normal structure

stopping rules for box constrained stochastic global optimization we present three new stopping rules for multistart based methods the first uses a device that enables the determination of the coverage of the bounded search domain the second is based on the comparison of asymptotic expectation values of observable quantities to the actually measured ones the third offers a probabilistic estimate for the number of local minima inside the search domain their performance is tested and compared to that of other widely used rules on a host of test problems in the framework of multistart

fixed point theorems for uniformly lipschitzian semigroups in uniformly convex spaces for all x y in c and each n 1 2 goebel and kirk 2 first studied these mappings and they proved that such t has a fixed point if c is a closed bounded convex subset of a uniformly convex banach space x and k k k is the unique solution of the equation 1 a l k k 1 with 6 being the modulus of convexity of x lifschitz s showed that in a hilbert space i a uniformly k lipschitzian mapping t with k c 21j2 has a fixed point lim 6 7 and lim et al 8 extended goebel kirk s theorem in lp spaces recently downing and ray l and ishihara and takahashi 3 verified that lifschitz s theorem is valid for uniformly lipschitzian semigroups in hifbert spaces the purpose of this paper is to prove fixed point theorems for a uniformly k lipschitzian semigroup in uniformly convex banach spaces which is left reversible or for which the space of right uniformly continuous functions on the semigroup has a left invariant mean the results generalize those of downing and ray l lau 4 lim 6 7 and ishihara and takahashi 3

improved versions of the lede algorithm for constrained global optimization two modified versions of the authors recent differential evolution algorithm for constrained global optimization are proposed they incorporate a filter set which results in more efficient implementations of the original algorithm numerical results are presented which suggest that the new algorithms are competitive

analysis on the p adic superspace ii differential equations on p adic superspace the cauchy problem for differential equations on the p adic superspace is considered the application of this mathematical theory to the model of the supersymmetric quantum mechanics on the p adic riemannian surface is proposed the non archimedean superdiffusion is also considered

continuous global optimization through the generation of parametric curves in this paper we develop a new approach for solving a large class of global optimization problems the objective function is only continuous non smooth and non lipschitzian defined on a rectangle of r n this approach is based on the generation in the feasible set of a family of parametrized curves satisfying certain properties combined with the one dimensional evtushenko algorithm to accelerate the corresponding mixed algorithm we have incorporated in a variant a pattern search type deterministic local optimization method and in another variant a new stochastic local optimization method both variants have been applied to several typical test problems a comparison with some well known methods is highlighted through numerical experiments

barnes type multiple q zeta functions and q euler polynomials the purpose of this paper is to present a systemic study of some families of multiple q euler numbers and polynomials and we construct multiple q zeta functions which interpolate multiple q euler numbers at a negative integer this is a partial answer to the open question in a previous publication see kim et al 2001 j phys a math gen 34 7633 8

a quadratic rate of asymptotic regularity for cat 0 spaces abstract in this paper we obtain a quadratic bound on the rate of asymptotic regularity for the krasnoselski mann iterations of nonexpansive mappings in cat 0 spaces whereas previous results guarantee only exponential bounds the method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of groetsch s theorem obtained by kohlenbach using methods from mathematical logic so called proof mining

statistical interpretation of p adic quantum theories with p adic valued wave functions the development of p adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events are p adic numbers the foundations of this theory are developed here the frequency definition of probability is used a general principle of statistical stabilization of relative frequencies is formulated by virtue of this principle statistical stabilization of relative frequencies which are like all experimental data rational numbers can be considered not only in the real topology but also in p adic topologies

strong convergence theorems for a family of quasi asymptotic pseudo contractions in hilbert spaces abstract in the present paper we propose two kinds of new algorithms for a family of quasi asymptotic pseudo contractions in real hilbert spaces by using the proposed algorithms we prove several strong convergence theorems for a family of quasi asymptotic pseudo contractions the results of this paper are interesting extensions of those known results

margin error bounds for support vector machines on reproducing kernel banach spaces support vector machines which maximize the margin from patterns to the separation hyperplane subject to correct classification have received remarkable success in machine learning margin error bounds based on hilbert spaces have been introduced in the literature to justify the strategy of maximizing the margin in svm recently there has been much interest in developing banach space methods for machine learning large margin classification in banach spaces is a focus of such attempts in this letter we establish a margin error bound for the svm on reproducing kernel banach spaces thus supplying statistical justification for large margin classification in banach spaces

an lp inequality and its applications to fixed point theory and approximation theory on etablit une inegalite l p pour 1 p 2 et on donne des applications aux points fixes des applications uniformement lipschitziennes et aux meilleures approximations fortement uniques

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

uniformly normal structure and uniformly lipschitzian semigroups abstract assume that x is a real banach space with uniformly normal structure and c is a nonempty closed convex subset of x we show that a uniformly lipschitzian semigroup of nonlinear self mappings of c admits a common fixed point if the semigroup has a bounded orbit and if is appropriately greater than one this result applies in particular to the framework of uniformly convex banach spaces

semigroup of nonexpansive mappings on a hilbert space soit s un semi groupe semi topologique soit h un espace de hilbert et s t s s s une representation continue de s comme applications non dilatation sur un sous ensemble convexe clos c de h dans c soit x c on etudie la convergence de t s x s s vers un point fixe commun z de s dans c

characterization and generation of α dense curves abstract if a curve ranges through a cube h n i 1 a i b i so that for every point x h there is at least a point in the curve at a distance not greater than a positive number the curve is called dense in h it is very important in the construction of such curves to find algorithms of approximation for global optimization and in fact there are several methods to do it e g 1 3 in this paper we give a characterization of such curves in terms of stochastically independent functions as well as a constructive method to generate them by means of only one function called uniformly distributed finally in the above class i e in the set of the uniformly distributed functions we will choose one of them will construct the corresponding curve and will compute its theoretic calculation time

global optimization of generalized linear fractional programming with nonlinear constraints this paper presents a branch and bound algorithm for globally solving a wide class of generalized linear fractional programming problems glfp this class includes such problems as minimizing a sum or error for product of a finite number of ratios of linear functions linear multiplicative programming polynomial programming etc over nonconvex feasible region first a problem q is derived which is equivalent to problem glfp in the algorithm lower bounds are derived by solving a sequence of linear relaxation programming problems which is based on the construction of the linear lower bounding functions for the objective function and constraint functions of problem q over the feasible region convergent property of the presented algorithm is proved and numerical results are given to show the feasibility of the proposed algorithm

global optimization a new variant of the alienor method abstract the multidimensional alienor global optimization method has been elaborated in the 1980s by cherruault and guillez it has been used among other possibilities for identification of mathematical models many interesting results have been obtained 1 3 in this paper we present a new variant of alienor it uses an dense curve with a small length and it has a quite simple parametrical representation this variant has been coupled with the evtushenko algorithm for decreasing the time required to obtain the global minimum

casimir invariants characteristic identities and young diagrams for color algebras and superalgebras the generalized commutation relations satisfied by generators of the general linear special linear and orthosymplectic color super algebras are presented in matrix form tensor operators including casimir invariants are constructed in the enveloping algebra for the general special linear and orthosymplectic cases eigenvalues of the quadratic and higher casimir invariants are given in terms of the highest weight vector correspondingly characteristic polynomial identities satisfied by the matrix of generators are obtained in factorized form classes of finite dimensional representations are identified using young diagram techniques and dimension branching and product rules for these are given finally the connection between color super algebras and generalized particle statistics is elucidated

existence and approximation of fixed points of nonlinear mappings in spaces with weak uniform normal structure it is shown that in a banach space x with weak uniform normal structure every demicontinuous asymptotically regular nearly lipschitzian self mapping t with lim sup n h t n

lie groups and lie algebras with generalized supersymmetric parameters matrices with symmetric parameters the most general extension of supersymmetric parameters are investigated the superdeterminants of such matrices are defined lie groups consisting of these matrices and their lie algebras are studied

random fixed points of uniformly lipschitzian mappings abstract let be a measurable space x a banach space c a weakly convex subset of x and t c c a random operator we prove the random version of a deterministic fixed point theorem when t is an asymptotically nonexpansive mapping and the characteristic of convexity e0 x is less than 1 let n x be the normal structure coefficient of x and 0 x its lifschitz constant if t is k uniformly lipschitzian and there exists a constant c such that k c 1 1 4n x 0 x 1 2 we prove that t has a random fixed point

a class of p convex spaces lacking normal structure abstract we prove that for any 1 the space e l 2 where max 2 is p convex it is known that for 2 e lacks normal structure

on an inequality of bynum and drew an extension of the bynum drew inequality in the abstract lp space 1 p 2 is presented the extended inequality is applied to show the strong uniqueness of best approximations and existence of fixed points for uniformly lipschitzian mappings and to prove an estimate for the self jung constants

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

generalized supermanifolds ii analysis on superspaces the analysis on generalized superspaces is studied g differentiability of functions and indetermination of partial g derivatives are treated in detail a canonical expansion for supersmooth functions and a particular choice of partial g derivatives are given and their properties studied existence of nonanalytic g functions is proved

an lp inequality we obtain an inequality for l p spaces 1 p 2 which corrects an inequality claimed by xu and xu 6 and has connections with some quantities of interest in fixed point theory

the cohomology of the generalized lie algebras in this paper we develop the cohomology theory of the generalized lie algebras using the identification of h2 g v with extf g h we discuss some of the extended lie algebras useful in physics for instance the supersymmetry quantized field and su 2 u 1 algebras finally we show that for a certain class of representations of some generalized lie algebras the cohomology is trivial in so doing we generalize the result obtained in our earlier investigation

generalized supermanifolds iii ρ supermanifolds supermanifolds are characterized as supermanifolds modeled in a superspace s having a well behaved body tangent cotangent and tensor bundles to supermanifolds are discussed the elements of the theory of lie supergroups and principal superfiber bundles as supermanifolds are given