on differentiability of convex operators abstract the main known results on differentiability of continuous convex operators f from a banach space x to an ordered banach space y are due to j m borwein and n k kirov our aim is to prove some supergeneric results i e to show that sometimes the set of g teaux or frechet nondifferentiability points is not only a first category set but also smaller in a stronger sense for example we prove that if y is countably daniell and the space l x y of bounded linear operators is separable then each continuous convex operator f x y is frechet differentiable except for a null angle small set some applications of such supergeneric results are shown

from quasivariational inclusion problems to stampacchia vector quasiequilibrium problems stampacchia set valued vector ekeland s variational principle and caristi s fixed point theorem abstract in this paper we first study existence theorems of solution for quasivariational inclusion problems we apply existence theorems of solution for quasivariational inclusion problem to study the existence theorems of solution for the stampacchia generalized vector quasiequilibrium problems and stampacchia vector quasiequilibrium problems our results and approach to the stampacchia generalized vector quasiequilibrium problem are different from any existence results of this type of problem finally we apply the existence theorems of solution for the stampacchia vector quasiequilibrium problems to study simultaneous solutions of caristi s fixed point theorem and the stampacchia set valued vector ekeland s variational principle our results on stampacchia set valued vector ekeland s variational principle are different from any existence results of these types of problems

on the vectoral ekeland s variational principle and minimal points in product spaces abstract phelps 13 noticed that the scalar ekeland s variational principle evp is equivalent to the existence of a minimal point of the epigraph of the corresponding function with respect to an appropriate order attouch and riahi 1 showed that evp is equivalent to the existence of maximal points with respect to cones satisfying some additional conditions taking these into account gopfert and tammer 6 7 established a maximal point theorem in a product space the aim of this paper is to obtain several minimal point theorems in product spaces and the corresponding variants of the vectorial evp

v subdifferentials of convex operators abstract for any convex operator from a convex set c of a topological vector space e into another one f endowed with a convex cone f a notion of v subdifferential v a of at a c is introduced although it is equivalent to the notion of e sub differential when f r it enjoys many important properties which are not satisfied by the e subdifferential whenever int f o the nonvacuity of v a is proved whenever v is a neighbourhood of zero in f and belongs to a large class of mappings analogous to the class of lower semicontinuous real valued convex functions other properties of v subdifferentials are studied and applications to differentiability of are made

a pre order principle and set valued ekeland variational principle abstract we establish a pre order principle from the principle we obtain a very general set valued ekeland variational principle where the objective function is a set valued map taking values in a quasi ordered linear space and the perturbation contains a family of set valued maps satisfying certain property from this general set valued ekeland variational principle we deduce a number of particular versions of set valued ekeland variational principle which include many known ekeland variational principles their improvements and some new results

convex analysis in groups and semigroups a sampler we define convexity canonically in the setting of monoids we show that many classical results from convex analysis hold for functions defined on such groups and semigroups rather than only vector spaces some examples and counter examples are also discussed

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

vector valued perturbed equilibrium problems the goal of this paper is to prove some general vector valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi continuity we obtain these results by going through a new concept of approximative equilibrium point

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

convergence to equilibrium in the classical model of population genetics abstract in the classical model of population genetics for continuous time fisher s equation for n alleles a stationary point is called regular if the viability of each of the absent alleles is distinct from the average viability of the population it is shown that if the limit set of a trajectory contains a regular point then it contains only one point

existence of equilibria via ekeland s principle abstract in the literature when dealing with equilibrium problems and the existence of their solutions the most used assumptions are the convexity of the domain and the generalized convexity and monotonicity together with some weak continuity assumptions of the function in this paper we focus on conditions that do not involve any convexity concept neither for the domain nor for the function involved starting from the well known ekeland s theorem for minimization problems we find a suitable set of conditions on the function f that lead to an ekeland s variational principle for equilibrium problems via the existence of e solutions we are able to show existence of equilibria on general closed sets for equilibrium problems and systems of equilibrium problems

do wealth distributions follow power laws evidence from rich lists abstract we use data on the wealth of the richest persons taken from the rich lists provided by business magazines like forbes to verify if the upper tails of wealth distributions follow as often claimed a power law behaviour the data sets used cover the world s richest persons over 1996 2012 the richest americans over 1988 2012 the richest chinese over 2006 2012 and the richest russians over 2004 2011 using a recently introduced comprehensive empirical methodology for detecting power laws which allows for testing the goodness of fit as well as for comparing the power law model with rival distributions we find that a power law model is consistent with data only in 35 of the analysed data sets moreover even if wealth data are consistent with the power law model they are usually also consistent with some rivals like the log normal or stretched exponential distributions

on some nonlinear problems induced by an abstract maximal element principle abstract in this paper by applying an abstract maximal element principle on quasi ordered sets established by lin and du we obtain a generalized brezis browder principle system vectorial versions of ekeland s variational principle and maximal element principle and a vectorial version of takahashi s nonconvex minimization theorem moreover we investigate the equivalence between scalar versions and vectorial versions of these results

ekeland type variational principle with applications to quasi variational inclusion problems abstract in this paper we study the ekeland type variational principle a caristi kirk type fixed point theorem and a maximal element theorem in the setting of uniform spaces by using these results we establish some existence results for solutions of quasi variational inclusion problems quasi optimization problems and equilibrium problems defined on separated and sequentially complete uniformly spaces

the ekeland variational principle for henig proper minimizers and super minimizers abstract in this paper we consider for the first time approximate henig proper minimizers and approximate super minimizers of a set valued map f with values in a partially ordered vector space and formulate two versions of the ekeland variational principle for these points involving coderivatives in the sense of ioffe clarke and mordukhovich as applications we obtain sufficient conditions for f to have a henig proper minimizer or a super minimizer under the palais smale type conditions the techniques are essentially based on the characterizations of henig proper efficient points and super efficient points by mean of the henig dilating cones and the hiriart urruty signed distance function

city size distributions for india and china this paper studies the size distributions of urban agglomerations for india and china we have estimated the scaling exponent for zipf s law with the indian census data for the years of 1981 2001 and the chinese census data for 1990 and 2000 along with the biased linear fit estimate the maximum likelihood estimate for the pareto and tsallis q exponential distribution has been computed for india the scaling exponent is in the range of 1 88 2 06 and for china it is in the interval 1 82 2 29 the goodness of fit tests of the estimated distributions are performed using the kolmogorov smirnov statistic

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

h k trichotomies and asymptotics of nonautonomous difference systems abstract discrete nonautonomous nonlinear systems possessing h k trichotomies are considered we specialize to the case of solutions moving inside and in a neighborhood of an invariant manifold of a special kind which we call h k hyperbolic the aulbach coppel knobloch transformation is then developed and used to state conditions for granting asymptotic equivalence of solutions our results generalize some of aulbach s results in which the manifold considered was made of stationary points and the system taken was autonomous

continuous affine support mappings for convex operators abstract we investigate the problem of existence of continuous affine resp linear support mappings for convex resp sublinear partially defined operators we prove that a weakly borel measurable convex operator defined on a closed convex subset of a locally convex frechet space with values in an ordered locally convex vector space with the least upper bound property and an order unit inducing the topology in fact has continuous affine support mappings we give an example indicating that the order unit being present is essential a step towards our result is provided by a variant of the sandwich theorem obtained for the case of weakly borel measurable convex and concave operators finally we obtain necessary and sufficient conditions for the existence of a continuous affine support mapping h for a convex operator such that h x 0 x 0 holds for a fixed point x 0 in terms of the upper directional derivate x 0 for at x 0

an ekeland s variational principle for set valued mappings with applications in this paper we obtain a general ekeland s variational principle for set valued mappings in complete metric space which is different from those in g y chen x x huang ekeland s e variational principle for set valued mapping mathematical methods of operations research 48 1998 181 186 g y chen x x huang s h hou general ekeland s variational principle for set valued mappings journal of optimization theory and applications 106 2000 151 164 s j li w y zhang on ekeland s variational principle for set valued mappings acta mathematicae application sinica english series 23 2007 141 148 by the result we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain respectively moreover we give some equivalent results to the variational principle

behavior of solutions near manifolds of periodic solutions abstract consider an autonomous differential system x f x of dimension n that admits a k dimensional invariant manifold in x space represented by a k parameter family of periodic solutions then k characteristic multipliers of the corresponding variational equation are equal to 1 under the hypothesis that none of the remaining n k characteristic multipliers has modulus 1 the behavior of the solutions of x f x near is investigated the main results are the description of a neighborhood of in terms of stable and unstable manifolds and the statement that whenever a solution of x f x approaches the invariant manifold as t it tends to with asymptotic amplitude and phase

differentiability properties of convex operators abstract in this paper we show that a convex operator between a weak asplund space and a suitable ordered locally convex space is gateaux differentiable on a dense g set of its domain

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

existence theorems for variational inclusion problems and the set valued vector ekeland variational principle in a complete metric space abstract in this paper we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in ekeland s sense and the existence results for some variants of set valued vector ekeland variational principles in a complete metric space our results contain ekeland s variational principle as a special case and our approaches are different to those for any existence theorems for such problems

conjugate convex operators convex mappings from a locally convex space x into f f are considered where f is an ordered topological vector space and an arbitrary greatest element adjoined to f in view of applications to the polarity theory of convex operators the possibility is investigated of representing a convex mapping taking values in f as a supremum of continuous affine mappings

on the size distribution of firms additional evidence from the g7 countries abstract we analyze the average size distribution of a pool of the g7 group s firms over the period 1987 2000 in particular firm sizes are measured employing different proxies and after conditioning on business cycle phases we find that i the empirical distributions are all consistent with a power law ii point estimates suggest that only in limited cases the exponent is equal to 1 i e the resulting size distribution generally is not zipf iii regardless of the variable employed to measure firm sizes firms are distributed more equally during recessions than during expansions

a generalized ekeland vector variational principle and its applications in optimization abstract in this paper we give a generalized ekeland vector variational principle by using the principle we extend and improve the related results in sharp efficiency in the framework of locally convex spaces we introduce two kinds of generalized sharp efficiencies and prove that they are equivalent in particular we show that a sharp efficient solution with respect to an interior point of the ordering cone is also one with respect to every interior point moreover we introduce the generalized takahashi s condition and the generalized hamel s condition for vector valued functions from the generalized ekeland principle we deduce that the two conditions are equivalent from this we discuss the relationship between the distance of f x from e f x and the distance of x from e f where e f x denotes the efficient point set of f x and e f denotes the efficient solution set

asymptotic phase revisited abstract we discuss the classical problem on asymptotic phase of periodic orbits of planar systems the existence of asymptotic phase for non hyperbolic periodic orbits is completely determined with hypotheses on the derivatives of a poincare map and a return time map smoothness of the vector field turns out to be crucial for existence of asymptotic phase for hyperbolic periodic orbits a new proof for the existence of invariant foliations is provided

on maximal element theorems variants of ekeland s variational principle and their applications abstract in this paper we establish several different versions of generalized ekeland s variational principle and maximal element theorem for functions in complete metric spaces the equivalence relations between maximal element theorems generalized ekeland s variational principle generalized caristi s common fixed point theorems and nonconvex maximal element theorems for maps are also proved moreover we obtain some applications to a nonconvex minimax theorem nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces

invariant manifolds with asymptotic phase for nonautonomous difference equations abstract for autonomous difference equations with an invariant manifold conditions are known which guarantee that a solution approaching this manifold eventually behaves like a solution on this manifold in this paper we extend the fundamental result in this context to difference equations which are nonautonomous and whose solutions are guaranteed only in forward time

vector valued variational principles in the context of vector valued extensions of variational principles we are dealing with functions taking values in a banach space partially ordered by a closed convex pointed cone we introduce and study a new notion of semi continuity connected with the order and we improve the vector valued extensions of deville godefroy zizler perturbed minimization principle

invariant manifolds with asymptotic phase this paper is concerned with invariant manifolds allowing a generic approach to solutions of the underlying autonomous differential system only this means that any solution approaching such an invariant manifold m ultimately behaves like a particular solution on j4 in the sense that the difference n t i l of the two solutions tends to 0 as t cq in other words only solutions on the stable manifold of m can approach m as f cc invariant manifolds with that property are said to have an asymptotic phase the simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose floquet multipliers lie all but one in the open unit disc of the complex plane malkin 12 proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold more generally a k dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k floquet multipliers have modulus less than 1 see hale stokes 9 conditions for a torus to be asymptotically stable with asymptotic phase are given by samoilenko 13 and coppel 5 invariant manifolds without any specified structure are considered by fenichel 6 7 and in kirchgraber s note 111 moreover each center manifold has an asymptotic phase if the corresponding equilibrium is stable see carr al amood 4 and henry lo in any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold in each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability a negative answer to this question is given by the author in l 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase in l the manifold is supposed to be compact while the noncompact case is treated in 2 under the restriction that the generating family of periodic solutions has an amplitude independent period recently hale massatt 8 extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations in this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result a compact invariant manifold has an asymptotic phase if the manifold is hyperbolic assumption hl and carries a flow parallel in the sense of our assumption h2 this includes most of the aforementioned results

an ordered family of lorenz curves abstract a general method for building parametric functional families of lorenz curves generated from an initial lorenz curve which satisfies some regularity conditions is presented it is shown that these families can be ordered in a manner which leads to a hierarchy of lorenz curves the method starts from a generating lorenz curve l 0 p and builds the family by increasing the number of parameters which can be easily interpreted in terms of the elasticities of l 0 p the method is applied to a family we term the pareto family since they use the pareto lorenz curves as their generating curves the family is shown to contain an important number of classical lorenz curves used in the existing literature several properties of this family are analyzed these include the population function inequality measures and lorenz orderings a general method for the estimation of these family is given and applied to the pareto family finally an application is presented for data from various countries the results are very robust across data sources the pareto models fit very well in a number of applications

continuity and differentiability properties of convex operators this paper provides some theorems on the generic differentiability of convex operators between topological vector spaces these results are applied to give new conditions for subgradients of convex operators to be non empty and well behaved limiting examples are given

asymptotic stability regions via extensions of zubov s method i on considere un systeme differentiel x f t x satisfaisant les hypotheses suivantes f r n 1 r n est analytique f t x est presque periodique en t pour chaque x r n f t 0 0 t r la matrice principale fondamentale t s de y f x t 0 y satisfait t s exp t s t s avec et des constantes positives f t x f x t o x 0 x uniformement en t quand x 0

a generalized distance and enhanced ekeland s variational principle for vector functions we propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation this result is shown to contain probably a large part of existing versions of ekeland s variational principle evp we introduce the notion of a weak function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced evp for various settings under relaxed lower semicontinuity assumptions these principles conclude the existence not only of p strict minimizers of p perturbations of the considered vector function but also p sharp and p strong minimizers our results are proved to be stronger than the classical evp and many generalizations in the literature even in the usual one dimensional case by numerous corollaries and examples we include equivalent formulations of our enhanced evp as well

multiply broken power law densities as survival functions an alternative to pareto and lognormal fits abstract survival functions defined by multiply broken power law densities are introduced to model firm size data sets of four historical censuses reported in montebruno et al physica a 523 2019 858 the survival functions complementary cumulative distributions are obtained by least squares regression the probability distributions are inferred from the analytic survival functions the mean firm growth the growth volatility and entropy evolution are calculated over a 30 year period covered by the censuses a representation of the survival functions as single power law density with varying exponent depending on firm size is derived index functions i e rescaled logarithmic derivatives of survival functions are used to demonstrate that neither pareto power laws nor lognormal densities can accurately reproduce the tails of the firm size distributions like the survival functions the empirical rank size relations of the four censuses also admit representation by broken power law densities a generating mechanism for empirically obtained broken power law densities based on the fokker planck equation is proposed as well

closedness of the solution map for parametric weak vector equilibrium problems in this paper we introduced a new definition of vector topological pseudomonotonicity to study the parametric weak vector equilibrium problems the main result gives sufficient conditions for closedness of the solution map defined on the set of parameters

ekeland s principle for vector equilibrium problems abstract in this paper the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones the aim is to provide a vector version of ekeland s theorem related to equilibrium problems to prove this principle a weak notion of continuity of a vector valued function is considered and some of its properties are presented via the vector ekeland s principle existence results for vector equilibria are proved in both compact and noncompact domains

well posed equilibrium problems abstract in this paper we introduce some notions of well posedness for scalar equilibrium problems in complete metric spaces or in banach spaces as equilibrium problem is a common extension of optimization saddle point and variational inequality problems our definitions originates from the well posedness concepts already introduced for these problems we give sufficient conditions for two different kinds of well posedness and show by means of counterexamples that these have no relationship in the general case however together with some additional assumptions we show via ekeland s principle for bifunctions a link between them finally we discuss a parametric form of the equilibrium problem and introduce a well posedness concept for it which unifies the two different notions of well posedness introduced in the first part

adjoint process duality we study convex processes between topological vector spaces with particular emphasis on their adjoints this study is then applied to produce general duality results for classes of convex programs involving processes the use of processes allows one to exploit the symmetry of linear programming and to obtain significantly broader and stronger results

an alternative method for modeling the size distribution of top wealth abstract the pareto distribution has been widely applied in modeling the distribution of wealth as well as top incomes cities and firms however recent evidence has shown that the pareto distribution is not consistent with many situations in which it was previously considered applicable we propose an alternative method for estimating the upper tail distribution of wealth and suggest a new lorenz curve for building models to provide such estimates applying our new models to the forbes world s billionaire lists we show that they significantly outperform the pareto lorenz curve as well as some other popular alternatives

the existence of solutions for the system of vector quasi equilibrium problems in topological order spaces abstract this paper introduces two kinds of quasiconcave mappings which are different from the usual quasiconcave function we establish a result for the existence of solutions for the system of vector quasi equilibrium problems in the frame of topological order by providing a maximal elements version of the well known browder fixed points theorem