the undecidability of some equivalence problems concerning ngsm s and finite substitutions abstract it is shown to be undecidable whether an e free finite substitution and a two state simple e free ngsm are equivalent on b consequently the state minimization of such ngsm s and the nonterminal minimization of linear grammars having two nonterminals are undecidable applications to some other decision problems concerning e free finite substitutions and linear grammars are also given finally it is shown that the time equivalence problem is undecidable for propagating ol systems i e for iterated e free finite substitutions this solves an open problem presented in 4

test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

caristi s fixed point theorem and selections of set valued contractions we show without the axiom of choice that the zermelo theorem implies directly a restriction of the caristi fixed point theorem to continuous functions under the axiom of choice this restriction is proved to be equivalent to caristi s theorem we also discuss kirk s problem on an extension of the caristi theorem and we establish two selection theorems for set valued contractions

polyhedral sweeping processes with unbounded nonconvex valued perturbation abstract a polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable hilbert space polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes we consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time the widely used assumption of lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of h lipschitzness the existence of solutions is proved for this sweeping process

on language equations with invertible operations the paper studies language equations of the type x l r and l y r where l and r are given languages and is an invertible binary word language operation for most of the considered insertion and deletion operations the existence of both a solution and a singleton solution to these equations proves to be decidable for given regular l and r in case l is a context free language and r is a regular one the existence of a solution is generally undecidable the results can be extended to more complex linear equations systems of linear equations as well as for equations of higher degree

pareto optimizing and scalarly stationary sequences the paper deals with vector optimization problems where the solution set the weakly efficient set may be empty we generalize the results known in scalar optimization concerning minimizing and stationary sequences the relations between these two concepts are given

some new results on grubbs s estimators abstract consider a two way classification with n rows and r columns and the usual model of analysis of variance except that the error components of the model may have heterogeneous variances 2 j j 1 r by columns grubbs 1948 provided unbiased estimators qj of 2 j that depend on column sums of squared residuals sj when r 3 the joint distributions of the sj and the qj are given for the first time in closed form two tests proposed by russell and bradley 1958 are examined when r 3 one for variance homogeneity and the second for one possible disparate variance a simple distribution is found for the test statistic of the first test and its non null distribution is derived also the distribution of the second test statistic was known to be the central variance ratio distribution in the null case and now its ratio to a parameter of noncentrality is shown to have that same distribution in the non null case when r 4 n 4 the joint distribution of the sj is also given in closed form

a brezis browder principle on partially ordered spaces and related ordering theorems abstract through a simple extension of brezis browder principle to partially ordered spaces a very general strong minimal point existence theorem on quasi ordered spaces is proved this theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems and three general ekeland variational principles in optimization problems where the objective space is quasi ordered then they are applied to prove strong minimal point existence results generalizations of bishop phelps lemma in linear spaces and ekeland variational principles in set valued optimization problems through a set solution criterion

the equivalence problem of multitape finite automata abstract using a result of b h neumann we extend eilenberg s equality theorem to a general result which implies that the multiplicity equivalence problem of two nondeterministic multitape finite automata is decidable as a corollary we solve a long standing open problem in automata theory namely the equivalence problem for multitape deterministic finite automata the main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group

nonlinear error bounds for lower semicontinuous functions on metric spaces as a development of the theory of linear error bounds for lower semicontinuous functions defined on complete metric spaces introduced in aze et al nonlinear anal 49 643 670 2002 and refined in aze and corvellec esaim control optim calc var 10 409 425 2004 we propose a similar approach to nonlinear error bounds based on the notion of strong slope the variational principle and the change of metric principle the latter allowing to obtain sharp estimates for such error bounds through a reduction to the linear case

nonadditivity in a two way classification is it interaction or nonhomogeneous variance abstract nonadditivity in a two way table with one observation per cell may be due either to row or to column related nonhomogeneous variance or to interaction between the row and column factors a general model that can detect both of these types of nonadditivity is discussed the data alone are not sufficient to distinguish between interaction and row or column related nonhomogeneous variance when there is one observation per cell a study of the fitted coefficients in the proposed model together with the scientific background of the problem can however provide the data analyst with the information necessary to develop a useful interpretation for the observed nonadditivity examples are included to illustrate the proposed methodology in one example involving the effects of four varieties of wheat in 13 locations it is shown that the variety related nonhomogeneous variance detected by several authors is more usefully interpreted as a level dependent interaction due to the differences among the yie

local completeness drop theorem and ekeland s variational principle by using a very general drop theorem in locally convex spaces we obtain some extended versions of ekeland s variational principle which only need assume local completeness of some related sets and improve hamel s recent results from this we derive some new versions of caristi s fixed points theorems in the framework of locally convex spaces we prove that danes drop theorem ekeland s variational principle caristi s fixed points theorem and phelps lemma are equivalent to each other

some equivalent formulations of the generalized ekeland s variational principle and their applications abstract in this paper we first give a generalized takahashi s existence theorem from the existence theorem we establish some equivalence relations between generalized caristi s fixed point theorems and generalized ekeland s variational principles some applications to the existence theorems of weak sharp minima and global error bounds for lower semicontinuous functions are also given

some new results on stability of ekeland s ϵ variational principle abstract in this paper we give an example and point out that solutions of ekeland s variational principle are not always lower semicontinuous in infinite dimensional banach spaces even with respect to the uniform metric further the example shows that the solutions need not be almost lower semicontinuous when the convergence of sequence of functions is weakened to painleve kuratowski epigraphical convergence to provide some results of stability we prove the almost lower semicontinuity of solutions in a general framework

mapping theorems for gateaux differentiable and accretive operators let p be a nonlinear operator mapping the banach space x into the banach space y there are many approaches to studying solvability of the equation px y for y e y a considerable number of which involve local or infinitesimal assumptions on the operator p this is due in part to the fact that p frequently arises from a differential or integral operator which has monotonicity or smoothness properties for which local apriori estimates are readily obtainable in this paper we derive mapping theorems for operators p satisfying two different kinds of local assumptions differentiability assumptions and monotonicity assumptions our approach in the former case relies on an abstract newton kantorovich scheme which was pioneered by m altman see for example l and further developed in 19 in the latter case we extend some results of f e browder using a combination of both new and standard techniques in each case our basic technique involves an application of a new version of ekeland s theorem which we derive from a fundamental maximal principle of h brezis and f e browder in section 2 of this paper we derive our new version of ekeland s theorem and in section 3 apply this result to nonlinear operators using different and by now standard techniques we prove a domain invariance result in section 4 needed for our result an accretive operators

ekeland s variational principle for vectorial multivalued mappings in a uniform space abstract in this paper we establish ekeland s variational principle and an equilibrium version of ekeland s variational principle for vectorial multivalued mappings in the setting of separated sequentially complete uniform spaces our approaches and results are different from those in chen et al 2008 hamel 2005 and lin and chuang 2010 13 14 15 as applications of our results we study vectorial caristi s fixed point theorems and takahashi s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated sequentially complete uniform space we also apply our results to study maximal element theorems which are unified methods of several variational inclusion problems our results contain many known results in the literature fang 1996 21 and will have many applications in nonlinear analysis

ekeland s variational principle minimax theorems and existence of nonconvex equilibria in complete metric spaces abstract in this paper we introduce the concept of function which generalizes the concept of w distance studied in the literature we establish a generalized ekeland s variational principle in the setting of lower semicontinuous from above and functions as applications of our ekeland s variational principle we derive generalized caristi s common fixed point theorems a generalized takahashi s nonconvex minimization theorem a nonconvex minimax theorem a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces we also prove that these theorems also imply our ekeland s variational principle

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

the equivalence problem for single valued two way transducers on npdtol languages is decidable the equivalence problem for deterministic two way transducers was a longstanding problem finally shown decidable by e m gurari we show a generalization of this result namely that it is decidable whether two single valued two way transducers are equivalent on an npdtol language we obtain this result by showing that it is even decidable whether a two way transducer is single valued on an npdtol language our solution is based on the compactness property of the systems of equations over a finitely generated free monoid shown recently by j lawrence and m albert and also v s guba this was a longstanding open problem known as the ehrenfeucht conjecture our approach also shows that for one way transducers their single valuedness on a given hdtol language can be tested and thus the equivalence of two single valued one way transducers on an hdtol language is decidable too our results imply the decidability of the hdol and hdtol sequence equivalence problems also longstanding open problems solved rece

stability of efficient sets continuity of mobile polarities whenever for every x e d such that a x afi xo for i 1 2 n one hash x fi x for each i pareto optimization is also referred to as uecror multi objecfiue or multi criteria optimization multi objective maximization problems 0 1 have been nowadays extended to the situations where f is a mapping from x to a set y equipped with a transitive relation 2 this relation is frequently called a preference consider first the image a f d of the constraint d an element y of a is called 3 efficient up to indifference or in the broad sense if for every y e a y z y implies y 5 yo it is said to be strongly z eficient or efficient in the narrow sense if for each y e a y 2 y implies y y the two notions coincide for antisymmetric transitive relations we shall be primarily concerned with efficient points in the broad sense the set of efficient points of a with respect to 7 will be denoted by max a max z a and that of strongly efficient points by max a an element of y is called an efficient value of 0 1 if it is an efficient point of f d an element x0 of d is a solution of 0 1 if f xo is an efficient point of f d we shall use the latter s to denote the set of solutions of 0 1

a unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization in this paper we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now we establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations with the help of quasi efficient solutions a generalized subdifferential of a vector mapping is introduced which generates a number of approximate subdifferentials frequently used in optimization in a unifying way the generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization an application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given

generalized e quasi solutions in multiobjective optimization problems existence results and optimality conditions abstract this paper is concerned with approximate solutions of multiobjective optimization problems whose order cone is not necessarily the nonnegative orthant we introduce the concept of generalized e quasi solution that extends other well known approximate efficiency notions of the literature and we study the limit behavior of these solutions as approximations to the efficient and weak efficient sets moreover we prove several existence results and a bound for the generalized e quasi solution set under convexity assumptions and by using asymptotic analysis tools finally we develop optimality conditions for a particular case of these kinds of e quasi solutions in nonsmooth convex and nonconvex problems

equivalents to ekeland s variational principle in uniform spaces abstract a minimal element theorem on sequentially complete uniform spaces is presented that generalizes earlier results of pacific j math 55 2 1974 335 341 proc amer math soc 108 3 1990 707 714 two more equivalent formulations of the minimal element theorem are given it is applied to derive new versions of ekeland type theorems which improve results of j math anal appl 202 1996 398 412 finally the minimal element theorem is used to obtain an ekeland type theorem for functions with values in a linear space without topological structure

a constructive proof of the existence of a utility in revealed preference theory within the context of the standard model of rationality within economic modelling we show the existence of a utility function that rationalises a demand correspondence hence completely characterizes the associated preference structure by taking a dense demand sample this resolves the problem of revealed preferences under some very mild assumptions on the demand correspondence which are closely related to a number of established axioms in preference theory the proof establishes the existence of a limit of a sequences of indirect utilities that rationalise finite data sets where the sample size increases to infinity this limiting utility provides a rationalisation of the demand relation up to a rescaling this limiting indirect utility is unique and continuous on a set of full measure

equivalent formulations of ekeland s variational principle abstract we prove that for some 0 inf x f inf x f e there exists y x such that 0

a drop theorem without vector topology abstract danes drop theorem is extended to bornological vector spaces an immediate application is to establish ekeland type variational principle and its equivalence caristi fixed point theorem in bornological vector spaces meanwhile since every locally convex space becomes a convex bornological vector space when equipped with the canonical von neumann bornology qiu s generalization of danes work to locally convex spaces is recovered

equations over finite sets of words and equivalence problems in automata theory abstract several observations concerning equations over monoids of finite sets of words are made monoids considered are the monoid of finite prefix codes the monoid of all finite sets and the monoid of all finite multisets as well as some others as an application some known decidability results in automata theory are extended finally several open questions are raised

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

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

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

the first order necessary conditions for nonsmooth variational and control problems the first order necessary conditions for nonsmooth problems of lagrange bolza multiple integral and optimal control are given the necessary conditions are expressed in terms of differential inclusions we deal with some new differentials that generalize the differential of frechet in particular the pontryagin maximum principle is proved without the assumption of continuous differentiability

a near subtraction result in metric spaces the locally closed graph case abstract this paper concerns a near subtraction result for multifunctions with locally closed graph in metric spaces

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

stability results for ekeland s e variational principle and cone extremal solutions given x a banach space and f x a a proper lower semicontinuous function which is bounded from below the ekeland s e variational principle asserts the existence of a point xi in x which we call e extremal with respect to f which satisfies fu fxi e u xi for all u x u xi by using set convergence notions kuratowski painleve mosco bounded hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e ext f the set of such e extremal points the key for the geometrical understanding of such properties is to consider the equivalent phelps extremization principle which given a closed set d in x and a partial ordering with respect to a pointed cone associates the set of elements of d maximal with respect to this order direct or potential applications are given in various fields multicriteria optimization numerical algorithmic calculus of variations

testing the homogeneity of variances in a two way classification summary approximations have been obtained to the null distribution of bartlett s type of statistic for testing the homogeneity of one way variances in a two way classification simulation studies show that the approximations give satisfactory results even for a very small number of rows and columns

quantitative stability of variational systems ii a framework for nonlinear conditioning stability results of lipschitz and holder type are obtained for the solutions and optimal values of optimization problems when perturbations are measured in terms of the rho epi distance

multiplicities a deterministic view of nondeterminism abstract we discuss four important problems in automata theory each of which has three natural variants a deterministic one a nondeterministic one and as an in between case a nondeterministic one with multiplicities our goal is to support a general view that the third variant is much closer to the first one than to the second one

isometries for the legendre fenchel transform it is shown that on the space of lower semicontinuous convex functions defined on r the conjugation map the legendre fenchel transform is an isometry with respect to some metrics consistent with the epi topology we also obtain isometries for the infinite dimensional case hilbert space and reflexive banach space but this time they correspond to topologies finer than the mosco epi topology

distribution free tests for one way homoscedasticity in a two way classification homoscedasticity is an important assumption for the analysis of variance of data from a two way table a preliminary test for equality of row column variances is often in order moreover the row column variance may be a parameter characterizing physical or biological properties of the treatment associated with a row column e g in the analysis of measurement errors or in the analysis of phenotypic stability in this case a test for homoscedasticity is central to the statistical analysis unfortunately parametric procedures available for this problem are quite sensitive to the underlying assumption of normality this paper examines two rank tests given by nassar and hiihn 1987 biometrics 43 45 53 and hiihn and nassar 1989 biometrics 45 977 1000 which are applicable for detecting homoscedasticity in a two way classification these tests are approximately valid for normally distributed errors but they may be anticonservative for nonnormal distributions a modification is suggested that leads to a distribution free test for homoscedasticity in a two way layout simulations indicate that under normality the modified test has low power compared to parametric tests

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

existence of minimizers on drops for a boundedly generated drop a e a property which holds for instance whenever e is bounded where a belongs to a real banach space x and e subset x is a nonempty convex set we show that for every lower semicontinuous function h x longrightarrow mathbb r cup infty that satisfies sup delta 0 inf x in e delta b x h x h a b x being the unitary open ball in x there exists bar x in a e such that h a ge h bar x and bar x is a strict minimizer of h on the drop bar x e

on the existence of prime decompositions we investigate factorizations of regular languages in terms of prime languages a language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps we give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition we show that there exist non regular unary languages that do not have prime decompositions we also consider infinite factorizations of unary languages

the drop theorem the petal theorem and ekeland s variational principle on montre que le principe variationnel d ekeland ou plus precisement une forme legerement modifiee est une consequence d un resultat geometrique connu sous le terme de theoreme de la goutte

convergence of minimal sets in convex vector optimization we study the behavior of the minimal sets of a sequence of convex sets left a n right converging to a given set a the main feature of the present work is the use of convexity properties of the sets a n and a to obtain upper and lower convergence of the minimal frontiers we emphasize that we study both kuratowski painleve convergence and attouch wets convergence of minimal sets moreover we prove stability results that hold in a normed linear space ordered by a general cone in order to deal with the most common spaces ordered by their natural nonnegative orthants e g c left left a b right right l p and l p left mathbb r right for 1 leq p leq infty we also make a comparison with the existing results related to the topics considered in our work

another isometry for the legendre fenchel transform abstract a new distance is introduced on the space of extended real valued lower semicontinuous convex functions defined on reflexive banach spaces it is shown that with this notion of distance the legendre fenchel is an isometry

persistence and stability of solutions of hamilton jacobi equations abstract given a convergent sequence of hamiltonians h n and a convergent sequence of initial data g n for the first order evolutionary hamilton jacobi equation we look for conditions ensuring that the sequences u n and v n of lax solutions and hopf solutions respectively converge the convergences we deal with are variational convergences we take advantage of several recent results giving criteria for the continuity of usual operations

new stability results for ekeland s ϵ variational principles for vector valued and set valued maps in this paper we discuss the stability of ekeland s variational principles for vector valued and set valued maps when the dominating cone is a closed pointed convex cone whose interior may be empty we provide a new approach to the study of the stability of ekeland s variational principles for vector valued and set valued maps without scalarization

the density of extremal points in ekeland s variational principle in this paper we investigate the density of extremal points appeared in ekeland s variational principle by introducing radial intersections of sets we give a very general result on the density of extremal points in the framework of locally convex spaces this solves a problem proposed by g isac in 1997 from the general result we deduce several convenient criterions for judging the density of extremal points which extend and improve a result of f cammaroto and a chinni using the equivalence between ekeland s variational principle and caristi s fixed point theorem we obtain some density results on caristi s fixed points

some generalizations of ekeland type variational principle with applications to equilibrium problems and fixed point theory abstract in this paper we introduce the concept of a q function defined on a quasi metric space which generalizes the notion of a function and a w distance we establish ekeland type variational principles in the setting of quasi metric spaces with a q function we also present an equilibrium version of the ekeland type variational principle in the setting of quasi metric spaces with a q function we prove some equivalences of our variational principles with caristi kirk type fixed point theorems for multivalued maps the takahashi minimization theorem and some other related results as applications of our results we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps we also extend the nadler s fixed point theorem for multivalued maps to a q function and in the setting of complete quasi metric spaces as a consequence we prove the banach contraction theorem for a q function and in the setting of complete quasi metric spaces the results of this paper extend and generalize many results appearing recently in the literature

parametrized multicriteria optimization continuity and closedness of optimal multifunctions this paper studies continuity properties of the marginal multifunction and solution multifunction for optimization problems with respect to a preorder relation among others we present a number of results which are generalizations to multicriteria problems of theorems due to c berge w hogan t tanino and y sawaragi

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

the strong ekeland variational principle the strong drop theorem and applications on donne une nouvelle demonstration du theoreme de la goutte generalisee en utilisant des formes renforcees du principe variationnel d ekeland et du theoreme de la goutte et donne de nouveaux resultats et de nouvelles demonstrations de resultats connus sur les proprietes generiques des problemes de minimisation convexes et non convexes

ekeland s variational principle in locally convex spaces and the density of extremal points in this paper we prove a general version of ekeland s variational principle in locally convex spaces where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function from this we obtain a number of special versions of ekeland s principle which include all the known extensions of the principle in locally convex spaces moreover we give a general criterion for judging the density of extremal points in the general ekeland s principle which extends and improves the related known results