recurrence plots revisited abstract we show that recurrence plots rps give detailed characterizations of time series generated by dynamical systems driven by slowly varying external forces for deterministic systems we show that rps of the time series can be used to reconstruct the rp of the driving force if it varies sufficiently slowly if the driving force is one dimensional its functional form can then be inferred up to an invertible coordinate transformation the same results hold for stochastic systems if the rp of the time series is suitably averaged and transformed these results are used to investigate the nonlinear prediction of time series generated by dynamical systems driven by slowly varying external forces we also consider the problem of detecting a small change in the driving force and propose a surrogate data technique for assessing statistical significance numerically simulated time series and a time series of respiration rates recorded from a subject with sleep apnea are used as illustrative examples

generalized convexities and generalized gradients based on algebraic operations abstract in this paper we investigate properties of generalized convexities based on algebraic operations introduced by ben tal a ben tal on generalized means and generalized convex functions j optim theory appl 21 1977 1 13 and relations between these generalized convexities and generalized monotonicities we also discuss the h generalized directional derivative and gradient and explore the relation between this gradient and the clarke generalized gradient definitions of some generalized averages of the values of a generalized convex function at n equally spaced points based on the algebraic operations are also presented and corresponding results are obtained finally the convexity is defined and some properties of convex functions are derived

optimality conditions for multiobjective programming with generalized i ρ θ convex set functions abstract necessary conditions for pareto optimality in multiobjective programming with subdifferentiable set functions are established in theorem 12 of h c lai and l j lin j math anal appl 132 1988 558 571 in this paper we establish some sufficient conditions under which a feasible solution of such a problem will be pareto optimal provided that a weaker convexity requirement is satisfied for instance i convexity is assumed for both objective and constraint set functions some duality models are also discussed wolfe type and mond weir type duality theorems are proved

optimization theory for n set functions abstract an optimization theory is developed for functions of n sets optimality conditions are established and a lagrangian duality is obtained

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

saddle points and lagrangian type duality for discrete minmax fractional subset programming problems with generalized convex functions abstract in this paper we present four sets of saddle point type optimality conditions construct two lagrangian type dual problems and prove weak and strong duality theorems for a discrete minmax fractional subset programming problem we establish these optimality and duality results under appropriate b convexity hypotheses

mixed type duality for multiobjective variational problems with generalized f ρ convexity a mixed type dual for multiobjective variational problems is formulated several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized f convexity static mixed type dual multiobjective problems are particular cases of these problems

optimality and duality for a multi objective programming problem involving generalized d type i and related n set functions in this paper we introduce several generalized convexity for a real valued set function and establish optimality and duality results for a multi objective programming problem involving generalized d type i and related n set functions

duality for nondifferentiable static multiobjective variational problems involving generalized f ϱ convex functions using the generalized schwarz inequality we establish some weak duality theorems for nondifferentiable static multiobjective variational problems involving generalized f convex functions later in the sequel we introduce three dual models for the nondifferentiable static multiobjective fractional variational problems and derive weak duality results for such programs

on the optimality conditions of vector valued n set functions abstract in a finite atomless measure space x the optimization problem of vector valued n set functions defined on a convex subfamily s of n is investigated the necessary and sufficient conditions of pareto optimal solution or proper r p solution of optimization problem with differentiable vector valued n set functions are given

interpolation of nonlinear partial differential operators and generation of differentiale evolutions abstract by interpolating between sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain hence some techniques from the theory of ordinary differential equations can be applied to some p d e s using these ideas we study a class of nonlinear evolutions in a banach space we obtain some very simple existence and continuous dependence results the theory is applicable to reaction diffusion equations dispersion equations and hyperbolic equations before shocks develop

new metric properties for prox regular sets in this paper we present diverse new metric properties that prox regular sets shared with convex ones at the heart of our work lie the legendre fenchel transform and complements of balls first we show that a connected prox regular set is completely determined by the legendre fenchel transform of a suitable perturbation of its indicator function then we prove that such a function is also the right tool to extend to the context of prox regular sets the famous connection between the distance function and the support function of a convex set on the other hand given a prox regular set we examine the intersection of complements of open balls containing the set we establish that the distance of a point to a prox regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point this is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes to the best of our knowledge these results are new in the literature and show that the class of prox regular sets have good properties known in convex analysis mathematics subject classification 2010 49j52 49j53

weakly convex sets and modulus of nonconvexity we consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of vial and a proximally smooth set in the sense of clarke from the case of the hilbert space to a class of banach spaces with the modulus of convexity of the second order using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set valued mappings one of which has nonconvex images and new affirmative solutions of the splitting problem for selections we also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set

sufficiency and duality in multiobjective programming involving generalized f ρ convexity abstract new classes of generalized f convexity are introduced for vector valued functions examples are given to show their relations with f pseudoconvex f quasiconvex and strictly f pseudoconvex vector valued functions the sufficient optimality conditions and duality results are obtained for multiobjective programming involving generalized f convex vector valued functions

on several results about convex set functions in 1979 in an interesting paper r j morris introduced the notion of convex set function defined on an atomless finite measure space after a short period this notion as well as generalizations of it began to be studied in several papers the aim was to obtain results similar to those known for usual convex or generalized convex functions unfortunately several notions are ambiguous and the arguments used in the proofs of several results are not clear or not correct in this way there were stated even false results the aim of this paper is to point out that using some simple ideas it is possible on one hand to deduce the correct results by means of convex analysis and on the other hand to emphasize the reasons for which there are problems with other results

on multiple objective programming problems with set functions abstract the convexity of a subset of a algebra and the convexity of a set function on a convex subset are defined related properties are also examined a farkas minkowski theorem for set functions is then proved these results are used to characterize properly efficient solutions for multiple objective programming problems with set functions by associated scalar problems

anti deterministic behaviour of discrete systems that are less predictable than noise abstract we present a new type of deterministic dynamical behaviour that is less predictable than white noise we call it anti deterministic ad because time series corresponding to the dynamics of such systems do not generate deterministic lines in recurrence plots for small thresholds we show that although the dynamics is chaotic in the sense of exponential divergence of nearby initial conditions and although some properties of ad data are similar to white noise the ad dynamics is in fact less predictable than noise and hence is different from pseudo random number generators

evolution generated by semilinear dissipative plus compact operators existence results and sharp continuous dependence results are given for an evolution equation in an arbitrary banach space the right hand side of the equation consists of a linear dissipative term plus a continuous dissipative term plus a compact term

duality for generalized fractional programs involving n set functions abstract making use of a strong duality theorem for nonlinear programs involving n set functions we first establish a gordan type transposition theorem which is then utilized to construct a dual problem for a class of generalized fractional programming problems with n set functions as special cases of the main result we also obtain dual problems for discrete minmax and fractional programming problems

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

about the lipschitz property of the metric projection in the hilbert space abstract we consider the metric projection operator from the real hilbert space onto a strongly convex set we prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is lipschitz continuous with the lipschitz constant strictly less than 1 this property characterizes the class of strongly convex sets and to a certain degree the hilbert space we apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set

efficiency and generalized convex duality for multiobjective programs abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective non linear programs the results are obtained for convex functions convex i e weakly convex and strongly convex functions and pseudoconvex functions for the convex and convex functions a wolfe type of dual is formulated while for the pseudoconvex and convex functions a mond weir type dual is proposed

vector variational inequality as a tool for studying vector optimization problems the paper aims to show that a vector variational inequality can be an useful tool for studying a vector optimization problem

on optimality and duality for multiobjective programming problems involving generalized d type i and related n set functions in this paper we establish some optimality and duality results under generalized convexity assumptions for a multiobjective programming problem involving generalized d type i and related n set functions

duality for a minimax programming problem containingn set functions abstract parameter necessary and sufficient optimality conditions for the discrete minimax programming problem with differentiable n set functions are established in zalmai optimization 20 1989 377 395 in this paper we establish nonparameter necessary and sufficient optimality conditions the nonparameter necessary optimality conditions suggest the establishment of two parameter free duality models for the discrete minimax programming problem with differentiable n set functions and the weak strong and strict converse duality theorems have been derived

second order f α ρ d convexity and duality in multiobjective programming a class of second order f a r d convex functions and their generalizations is introduced using the assumptions on the functions involved weak strong and strict converse duality theorems are established for a second order mond weir type multiobjective dual

role of ρ η a invexity to ϵ optimality conditions for multiple objective fractional programming abstract based on a general framework for a class of a invex n set functions some results on the optimality conditions for multiple objective fractional programming problems are established the obtained results are not only general in nature but also unify results on generalized invexity to the context of multiple fractional programming

optimality and duality for multiobjective fractional programming involvingn set functions abstract we consider a multiobjective fractional programming problem mfp involving vector valued objective n set functions in which their numerators are different from each other but their denominators are the same by using the concept of proper efficiency we establish optimality conditions and duality relations for our problem mfp under convexity assumptions on objective and constrained functions

generalized convexity and concavity of the optimal value function in nonlinear programming in this paper we consider generalized convexity and concavity properties of the optimal value functionf for the general parametric optimization problemp of the form minxf x s t x r many results on convexity and concavity characterizations off were presented by the authors in a previous paper such properties off and the solution set maps form an important part of the theoretical basis for sensitivity stability and parametric analysis in mathematical optimization we give sufficient conditions for several types of generalized convexity and concavity off in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point to set mapr specializations of these results to the parametric inequality equality constrained nonlinear programming problem are provided

on minimax fractional optimality conditions with f ρ convexity abstract we establish the sufficient conditions for generalized fractional programming in the framework of f convex functions when the sufficient conditions are utilized one parametric dual problem and two parametric free dual problems may be formulated and duality results are derived

epigraphs of convex set functions in this paper we characterize a convex set function by its epigraph the w semicontinuities for set functions are defined we identify a convex set function with a functional in l and show that the w closure of such a functional is a convex functional a fenchel duality theorem for set functions is then derived

sufficiency criteria and duality for nonlinear programs involving n set functions abstract several sufficiency criteria and duality results are established under generalized convexity conditions for a class of nonlinear programming problems involving differentiable n set functions

generalized convex set functions abstract concepts of quasiconvexity and pseudoconvexity of set functions are introduced properties and relations between these generalized convex set functions are investigated and optimality criteria for differentiable and convex set functions are extended to these larger classes of set functions

a note on fractional minmax programs containingn set functions abstract generalized convexity assumptions are imposed on a few of the functions involved in the objective and the constraints to obtain sufficient optimality conditions and duality results for a generalized minmax fractional programming problem with n set functions

recurrence plot based measures of complexity and their application to heart rate variability data the knowledge of transitions between regular laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems while several linear approaches are often insufficient to describe such processes there are several nonlinear methods that however require rather long time observations to overcome these difficulties we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart rate variability data for the logistic map these measures enable us not only to detect transitions between chaotic and periodic states but also to identify laminar states i e chaos chaos transitions the traditional recurrence quantification analysis fails to detect the latter transitions applying our measures to the heart rate variability data we are able to detect and quantify the laminar phases before a life threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event our findings could be of importance for the therapy of malignant cardiac arrhythmias

moreau rockafellar type theorem for convex set functions abstract let x be an atomless finite measure space and s a convex subfamily it is proved that the moreau rockafellar theorem f 1 f n f 1 f n holds for proper convex set functions f 1 f n and s if all set functions f i except possibly one are w lower semicontinuous on s as applications the kuhn tucker type condition for an optimal solution of convex programming problem with set functions and the fritz john type condition for an optimal solution of vector valued minimization problem for set functions are obtained

nonconvex sparse logistic regression with weakly convex regularization in this paper we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem the idea is based on the finding that a weakly convex function as an approximation of the l 0 pseudo norm is able to better induce sparsity than the commonly used l 1 norm for a class of weakly convex sparsity inducing functions we prove the nonconvexity of the corresponding problem and study its local optimality conditions and the choice of the regularization parameter despite the nonconvexity a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression and its convergence behavior is studied theoretically then the general framework is applied to a specific weakly convex function and a local optimality condition and a bound on the logistic loss at a local optimum are provided the solution method is instantiated in this case as an iterative firm shrinkage algorithm and a nesterov acceleration is used with a convergence guarantee its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets

sufficiency and duality in nondifferentiable multiobjective programming involving generalized type i functions abstract recently hachimi and aghezzaf defined generalized f d type i functions a new class of functions that unifies several concepts of generalized type i functions in this paper the generalized f d type i functions are extended to nondifferentiable functions by utilizing the new concepts we obtain several sufficient optimality conditions and prove mixed type and mond weir type duality results for the nondifferentiable multiobjective programming problem

mixed type duality in multiobjective programming problems two mixed type duals are introduced for multiobjective programming and for multiobjective fractional programming respectively results about weak duality strong duality and strictly reverse duality are presented under more suitable conditions

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

one sided continuous dependence of maximal solutions abstract we consider the maximal solution of u s s u s where satisfies a one sided variant of caratheodory s conditions a best possible condition is proved for the dependence of u on also we show that a function v satisfies v t v r r t s v s ds if and only if v is dominated by the maximal solution u

duality for minmaxb vex programming involvingn set functions abstract a minmax programming problem involving several b vex n set functions is considered necessary and sufficient optimality theorems and wolfe type duality results are established duality for the n set generalized minmax fractional programming problem is derived as a special case of the main problem

duality in mathematical programming of set functions on fenchel duality theorem abstract the optimization problem of a set function defined on a family a of measurable subsets in an atomless finite measure space x a m is investigated the generalized fenchel theorem is formulated and proved in this note

noise level estimation of time series using coarse grained entropy we present a method of noise level estimation that is valid even for high noise levels the method makes use of the functional dependence of coarse grained correlation entropy k 2 ensuremath varepsilon on the threshold parameter ensuremath varepsilon we show that the function k 2 ensuremath varepsilon depends in a characteristic way on the noise standard deviation ensuremath sigma it follows that observing k 2 ensuremath varepsilon one can estimate the noise level ensuremath sigma although the theory has been developed for the gaussian noise added to the observed variable we have checked numerically that the method is also valid for the uniform noise distribution and for the case of langevin equation corresponding to the dynamical noise we have verified the validity of our method by applying it to estimate the noise level in several chaotic systems and in the chua electronic circuit contaminated by noise

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

efficiency conditions and duality models for multiobjective fractional subset programming problems with generalized ϝ α ϱ θ v convex functions abstract in this paper we first introduce a new class of generalized convex n set functions called v convex functions and then present numerous sets of parametric and semiparametric sufficient efficiency conditions under generalized v convexity assumptions for a multiobjective fractional subset programming problem moreover we construct three parametric and three semiparametric duality models and prove appropriate duality theorems

parameter free sufficient optimality conditions and duality models for minmax fractional subset programming problems with generalized f ϱ θ convex functions abstract in this paper we present numerous sets of nonparametric sufficient optimality conditions and duality results under various generalized f convexity assumptions for a minmax fractional subset programming problem

on minimax fractional programming of generalized convex set functions abstract we derive necessary and sufficient optimality conditions for the discrete minimax programming problem with subdifferentiable i convex set functions then we apply these optimality criteria to construct two models of parameter free dual problems and a third model of another dual problem we also establish weak strong and strictly converse duality theorems

optimality and duality for generalized fractional programming involving nonsmooth f ρ convex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing f r convex functions

duality theorems of vector valued n set functions abstract the wolfe and mond weir types of dual problems for convex vector valued n set functions are formulated and we prove the weak duality duality and converse duality theorems for these types of problems

generalized convex duality for multiobjective fractional programs abstract egudo derived some duality theorems for multi objective programs using the concept of efficiency coupled with some generalized convexity assumptions on the objective and constraint functions the main results of the present work can be thought of as extensions of the results of egudo in the context of multi objective fractional programs

lagrange duality in multiobjective fractional programming problems with n set functions in this paper lagrange multiplier theorems are developed for the multiobjective fractional programming problem involving n set functions a lagrange dual is introduced and duality results in terms of efficient solutions are established

proximal methods for cohypomonotone operators conditions are given for the viability and the weak convergence of an inexact relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a hilbert space in turn new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming

currents carried by the graphs of semi monotone maps in this paper we study the structure weak continuity and approximability properties for the integer multiplicity rectifiable currents carried by the graphs of maximal semi monotone set valued maps on an n dimensional convex domain especially we give an enhanced version of approximation theorem for the subgradients of semi convex functions

nonsmooth minimax programming under locally lipschitz φ ρ invexity abstract minimax programming problems involving locally lipschitz invex functions are considered the parametric and non parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable invexity assumption imposed on objective and constraint functions when the sufficient conditions are utilized parametric and non parametric dual problems in the sense of mond weir and wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem with the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems

necessary and sufficient conditions for convergence of temporally irregular evolutions on considere une suite d equations d evolution u n t a n t u n t et on montre que u n u si et seulement si a u s ds a s ds dans un sens faible approprie

duality for multiobjectiveb vex programming involvingn set functions abstract in the present paper we consider a class of multiobjectiveb vex programming problems involving differentiableb vexn set functions and establish duality results in terms of properly efficient solutions further we relate the problem to a certain saddle point of a lagrangian and show multiobjective fractional program as a special case of the main problem

area theorem and smoothness of compact cauchy horizons we obtain an improved version of the area theorem for not necessarily differentiable horizons which in conjunction with a recent result on the completeness of generators allows us to prove that under the null energy condition every compactly generated cauchy horizon is smooth and compact we explore the consequences of this result for time machines topology change black holes and cosmic censorship for instance it is shown that compact cauchy horizons cannot form in a non empty spacetime which satisfies the stable dominant energy condition wherever there is some source content

optimality and mixed saddle point criteria in multiobjective optimization in this paper higher order strong convexity for lipschitz functions is introduced and is utilized to derive the optimality conditions for the new concept of strict minimizer of higher order for a multiobjective optimization problem variational inequality problem is introduced and its solutions are related to the strict minimizers of higher order for a multiobjective optimization problem the notion of vector valued partial lagrangian is also introduced and equivalence of the mixed saddle points of higher order and higher order minima are provided

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

continuous dependence of nonmonotonic discontinuous differential equations continuous dependence of solutions for a class of nonmonotonic discontinuous differential equations is studied first a local existence theorem due to z wu is extended to a larger class then a result concerning continuous dependence for this larger class is proven this employs a type of convergence similar to gihman s convergence criterion witch is defined to be for all a b and y formule the significance of gihman s convergence criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence finally examples are presented to motivate and clarify this continuous dependence result

on the optimality of differentiable nonconvex n set functions abstract our main contribution is the extension of the concepts of quasiconvexity and pseudoconvexity to n set functions some properties of differentiable nonconvex n set functions are established necessary and sufficient conditions for the existence of an optimal solution of the nonconvex program with n set functions are characterized by derivatives of the n set functions involved a duality theorem for the nonconvex program with n set functions is also developed in this paper

optimality of differentiable vector valued n set functions abstract the necessary and sufficient conditions for the existence of an optimal solution of a vector valued n set functions optimization problem is obtained in this paper

on the degree and separability of nonconvexity and applications to optimization problems we study qualitative indications for d c representations of closed sets in and functions on hilbert spaces the first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity we show that a closed set is weakly closed if this indication is finite using this result we can prove the solvability of nonconvex minimization problems by duality a minimization problem on a feasible set in which this indication is low can be reduced to a quasi concave minimization over a convex set in a low dimensional space the second indication is the separability which can be incorporated in solving dual problems both the index of nonconvexity and the separability can be characteristics to good d c representations for practical computation we present a notion of clouds which enables us to obtain a good d c representation for a class of nonconvex sets using a generalized caratheodory s theorem we present various applications of clouds

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

uniform prox regularity of functions and epigraphs in hilbert spaces abstract we study the local uniform prox regularity of functions make the connection with the lower c 2 property and essentially with the prox regularity of epigraphs

a new method to estimate the kolmogorov entropy from recurrence plots its application to neuronal signals recurrence plots are a useful graphic tool for studying dynamical systems a new variable is proposed to quantify their deterministic structure it allows an easy computation of the k2 entropy since the distance between points needs to be calculated only once instead of every time for each new embedding dimension and it bypasses some of the limitations inherent to the classical grassberger and procaccia method it is valuable for biological data as illustrated here by tests with various dynamical models and its application to experimental signals recorded from neurons