unconditional schauder decompositions and stopping times in the lebesgue bochner spaces we extend troitsky s study of martingales in banach lattices to include stopping times results from the theory of unconditional schauder decompositions and multipliers are used to derive an optional stopping theorem for unbounded stopping times we also apply these techniques to convergent nets of stopped processes as well as to unconditional schauder decompositions in vector valued l p spaces 1 p

structure of some noetherian si rings abstract we describe the structure of rings over which every cyclic or finitely generated right module is a direct sum of a projective module and an injective module

the domination property for efficiency and bishop phelps theorem in locally convex spaces abstract we introduce the notion of closedness for any bounded convex set and investigate the relationship between closedness and local closedness moreover we study some properties of convex sets by using closedness local closedness and convexity we give two main theorems on the domination property and the existence of efficient points in locally convex spaces from these two main theorems we deduce a number of corollaries which improve the related known results as an application of the main results we obtain several support point theorems in locally convex spaces which generalize the famous bishop phelps theorem

rings whose modules have maximal or minimal injectivity domains in a recent paper alahmadi alkan and lopez permouth defined a module m to be poor if m is injective relative only to semisimple modules and a ring to have no right middle class if every right module is poor or injective we prove that every ring has a poor module and characterize rings with semisimple poor modules next a ring with no right middle class is proved to be the ring direct sum of a semisimple artinian ring and a ring t which is either zero or of one of the following types i morita equivalent to a right pci domain ii an indecomposable right si ring which is either right artinian or a right v ring and such that soc tt is homogeneous and essential in tt and t has a unique simple singular right module or iii an indecomposable right artinian ring with homogeneous right socle coinciding with the jacobson radical and the right singular ideal and with unique non injective simple right module in case iii either tt is poor or t is a qf ring with j t 2 0 converses of these cases are discussed it is shown in particular that a qf ring r with j r 2 0 and homogeneous right socle has no middle class

geometry of cones and an application in the theory of pareto efficient points in this article we give a new criterion for the existence of a bounded base for a cone p of a normed space x also if p is closed we give a partial answer to the problem is 0 a point of continuity of p if and only if 0 is a denting point of p the above problems have applications in the theory of pareto efficient points

rings and modules characterized by opposites of injectivity abstract in a recent paper aydo du and lopez permouth have defined a module m to be n subinjective if every homomorphism n m extends to some e n m where e n is the injective hull of n clearly every module is subinjective relative to any injective module their work raises the following question what is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules namely injectives we show using a dual opposite injectivity condition that such a ring r is isomorphic to the direct product of a semisimple artinian ring and an indecomposable ring which is i a hereditary artinian serial ring with j 2 0 or ii a qf ring isomorphic to a matrix ring over a local ring each case is viable and conversely i is sufficient for the said property and a partial converse is proved for a ring satisfying ii using the above mentioned classification it is also shown that such rings coincide with the fully saturated rings of trlifaj except possibly when von neumann regularity is assumed furthermore rings and abelian groups which satisfy these opposite injectivity conditions are characterized

on artinian rings with restricted class of injectivity domains abstract in a recent paper of alahmadi alkan and lopez permouth a ring r is defined to have no simple middle class if the injectivity domain of any simple r module is the smallest or largest possible er lopez permouth and sokmez use this idea of restricting the class of injectivity domains to classify rings and give a partial characterization of rings with no middle class in this work we continue the study of the property of having no simple middle class we give a structural description of right artinian right nonsingular rings with no right middle class we also give a characterization of right artinian rings that are not si to have no middle class which gives rise to a full characterization of rings with no middle class furthermore we show that commutative rings with no middle class are those artinian rings which decompose into a sum of a semisimple ring and a ring of composition length two also artinian rings with no simple middle class are characterized we demonstrate our results with several examples

a description of norm convergent martingales on vector valued lp spaces abstract norm convergent martingales on tensor products of banach spaces are considered in a measure free setting as a consequence we obtain the following characterization for convergent martingales on vector valued l p spaces let be a probability space x a banach space and n an increasing sequence of sub algebras of in order for f n n n 1 to be a convergent martingale in l p x 1 p it is necessary and sufficient that for each i n there exists a convergent martingale x i n n n 1 in l p and y i x such that for each n n we have f n s i 1 x i n s y i for all s where i 1 lim n x i n l p and lim i y i 0

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

the lagrange multiplier rule for multifunctions in banach spaces we study general constrained multiobjective optimization problems with objectives being closed multifunctions in banach spaces in terms of the coderivatives and normal cones we provide generalized lagrange multiplier rules as necessary optimality conditions of the above problems in an asplund space setting sharper results are presented

an alternative perspective on injectivity of modules abstract given modules m and n m is said to be n subinjective if for every extension k of n and every homomorphism n m there exists a homomorphism k m such that n for a module m the subinjectivity domain of m is defined to be the collection of all modules n such that m is n subinjective as an opposite to injectivity a module m is said to be indigent if its subinjectivity domain is smallest possible namely consisting of exactly the injective modules properties of subinjectivity domains and of indigent modules are studied in particular the existence of indigent modules is determined for some families of rings including the ring of integers and artinian serial rings it is also shown that some rings e g artinian chain rings have no middle class in the sense that all modules are either injective or indigent for various classes of modules such as semisimple singular and projective necessary and sufficient conditions for the existence of indigent modules of those types are studied indigent modules are analog to the so called poor modules an opposite of injectivity in terms of injectivity domains recently studied in papers by alahmadi alkan and lopez permouth and by er lopez permouth and sokmez relations between poor and indigent modules are also investigated here

unconditional martingale difference sequences in banach spaces abstract for a banach space y the question of whether l p y has an unconditional basis if 1 p and y has unconditional basis stood unsolved for a long time and was answered in the negative by aldous in this work we prove a weaker positive result related to this question we show that if y j is a basis of y and d i is a martingale difference sequence spanning l p then the sequence d i y j is a basis of l p y for 1 p moreover if 1 p and y j is unconditional then d i y j is strictly dominated by an unconditional tensor product basis in addition for 1 p we show that if d i l p is a martingale difference sequence then there exists a constant k 0 so that i j n i j y j d i l p y k i j n i j y j d i l p holds for every sequence y j y and every choice of finitely supported scalars i j

characterizing rings in terms of the extent of the injectivity and projectivity of their modules abstract given a ring r we define its right i profile resp right p profile to be the collection of injectivity domains resp projectivity domains of its right r modules we study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and vice versa we show that the i profile is isomorphic to an interval of the lattice of linear filters of right ideals of r and is therefore modular and coatomic in particular we give a practical characterization of the profile of a right artinian ring and offer an example of a ring without injective left middle class for with the same is not true on the right hand side we characterize the p profile of a right perfect ring and show through an example that the right p profile of a ring is not necessarily a set in addition we use our results to provide a characterization of a special class of qf rings in which the injectivity and projectivity domains of all modules coincide the study of rings in terms of their i or p profile was inspired by the study of rings with no right i or p middle class initiated in recent papers by er lopez permouth and sokmez and by holston lopez permouth and orhan ertas

rings whose cyclic modules have restricted injectivity domains abstract a poor module is one that is injective relative only to semisimple modules and a module is maximally injective if its domain of injectivity is a coatom in the lattice of domains of injectivity the so called injective profile of a ring a ring is said to have no right middle class if every right module is injective or poor we consider two families of rings characterized by their cyclic right modules those whose nonzero cyclic modules are poor property p and those whose nonzero cyclic modules are maximally injective property q we show that rings satisfying property q are precisely those rings that have no right middle class and property p structural properties of both classes of rings are obtained and it is shown in particular that a ring with property q is isomorphic to a matrix ring over a nonsemisimple local right artinian ring and is such that its right socle right singular ideal and jacobson radical coincide

some more density results for proper efficiencies in this paper we give sufficient conditions for a set and or a cone ensuring the density of henig s proper efficient points and positive proper efficient points in the set of efficient points in hausdorff locally convex spaces these conditions are based on the concept of the point of continuity an d generalize many existing results

poor modules with no proper poor direct summands abstract as a mean to provide intrinsic characterizations of poor modules the notion of a pauper module is introduced a module is a pauper if it is poor and has no proper poor direct summand we show that not all rings have pauper modules and explore conditions for their existence in addition we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule the second condition holds for the ring of integers and is just as significant as the first one for noetherian rings since in that context modules having poor pure submodules must themselves be poor it is shown that the existence of paupers is equivalent to the noetherian condition for rings with no middle class as indecomposable poor modules are pauper we study rings with no indecomposable right middle class i e the ring whose indecomposable right modules are pauper or injective we show that semiartinian v rings satisfy this property and also that a commutative noetherian ring r has no indecomposable middle class if and only if r is the direct product of finitely many fields and at most one ring of composition length 2 structure theorems are also provided for rings without indecomposable middle class when the rings are artinian serial or right artinian rings for which not having an indecomposable middle class suffices not to have a middle class include commutative noetherian and artinian serial rings the structure of poor modules is completely determined over commutative hereditary noetherian rings pauper abelian groups with torsion free rank one are fully characterized

choquet boundaries and efficiency this research paper is devoted to establish the coincidence between choquet boundaries and a new type of approximate efficient points sets in ordered hausdorff locally convex spaces being based on the first result established by us concerning such a property as this for pareto type efficient points sets and the corresponding choquet boundaries of non empty compact sets with respect to appropriate convex cones of real increasing and continuous functions thus the main result represents a strong connection between two great fields of mathematics the axiomatic theory of potential and vector optimization the present study contains also important relationships concerning strong optimization and approximate efficiency interesting examples pertinent remarks and some open problems

local completeness and pareto optimization in this paper we study the pareto optimization problem in the framework of locally convex spaces with the restricted assumption that only some related sets are locally complete