extension theorems of continuous random linear operators on random domains abstract the central purpose of this paper is to prove the following theorem let u be a complete probability space b a normed linear space over the scalar field k e 2 b a separable random domain with linear subspace values and gr e k a continuous random linear operator where gr e x b x e denotes the graph of e then there exists a continuous random linear operator b k such that x x x e and sup x x b x 1 sup x x e x 1 for each in for the case where e is not separable a result similar to the above stated theorem is also given which generalizes and improves many previous results on random generalizations of the hahn banach theorem

random strict convexity and random uniform convexity in random normed modules abstract based on the analysis of stratification structure on random normed modules we first present random strict convexity and random uniform convexity in random normed modules then we establish their respective relations to classical strict and uniform convexity in the process some known important results concerning strict convexity and uniform convexity of lebesgue bochner function spaces can be obtained as a special case of our results further we also give their important applications to the theory of random conjugate spaces as well as best approximation finally we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces

two fixed point theorems in complete random normed modules and their applications to backward stochastic equations abstract let f f p be a filtered probability space with a filtration f f t t 0 t satisfying the usual conditions and t a finite time l 0 f 0 the algebra of equivalence classes of f 0 measurable real valued random variables on l p f t r d the usual function space and l f 0 p f t r d the l 0 f 0 module generated by l p f t r d the usual backward stochastic equations b s e s are studied for their terminal conditions in l p f t r d motivated by the study of continuous time conditional mean conditional convex risk measure portfolio selection this paper for the first time formulates and studies a more general class of b s e s with their terminal conditions in l f 0 p f t r d the main results of this paper are to prove two fixed point theorems in complete random normed modules which are respectively the random generalizations of banach contraction mapping principle and browder kirk fixed point theorem and give their applications to the general class of b s e s

on almost surely bounded semigroups of random linear operators menger proposed transferring the probabilistic notions of quantum mechanics to the underlying geometry following menger s idea the notion of random metric spaces is a random generalization of that of metric spaces and also plays an important role in the study of random operator equations the main difficulty of this article is to work out a skill to give a property peculiar to a special semigroup of random operators which is not involved in the classical case subsequently some random operators equations are studied in particular we discuss the schrodinger type random equation which considerably generalizes the corresponding result in the sense of skorohod

conditionally evenly convex sets and evenly quasi convex maps abstract evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces we introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem this notion is then applied to obtain the dual representation of conditionally evenly quasi convex maps which turns out to be a fundamental ingredient in the study of quasi convex dynamic risk measures

weak topologies for modules over rings of bounded random variables abstract in order to establish a functional analytic basis for representation theorems for conditional and multi period risk measures we study locally convex modules over the ring l g their topology is determined by seminorms as expected central mathematical tools of the analysis are hahn banach type and separation theorems which however have to be treated more carefully in the module case once a dual module is introduced one can establish a module version of the bipolar theorem we also prove the krein smulian as well as the alaoglu bourbaki theorem for modules for banach modules their reflexivity is characterized by a compactness criterium in a very weak topology

separation and duality in locally l 0 convex modules motivated by financial applications we study convex analysis for modules over the ordered ring l0 of random variables we establish a module analogue of locally convex vector spaces namely locally l0 convex modules in this context we prove hyperplane separation theorems we investigate continuity subdifferentiability and dual representations of fenchel moreau type for l0 convex functions from l0 modules into l0 several examples and applications are given

farkas lemma in random locally convex modules and minkowski weyl type results in l0 f rn abstract in this paper we investigate the structure of a finitely generated l 0 positively homogeneous and l 0 convex cone in a random locally convex module first we establish a farkas type characterization for such a kind of cone then we can further give some minkowski weyl type results for the specific case when the random locally convex module is the space l 0 f r n of equivalence classes of r n valued random variables

the algebraic structure of finitely generated l0 f k modules and the helly theorem in random normed modules abstract let k be the scalar field of real numbers or complex numbers and l 0 f k the algebra of equivalence classes of k valued random variables defined on a probability space f p in this paper we first characterize the algebraic structure of finitely generated l 0 f k modules and then combining the recently developed separation theorem in random locally convex modules we prove the helly theorem in random normed modules with the countable concatenation property under the framework of random conjugate spaces at the same time a simple counterexample shows that it is necessary to require the countable concatenation property

a characterization for an l μ k topological module to admit enough canonical module homomorphisms let k be the scalar field of all real or complex numbers let a be a finite measure space and let l k be the algebra of the equivalence classes of all k valued measurable functions defined on a l k is a topological algebra over k when endowed with the topology of convergence locally in measure topological modules over this topological algebra l k briefly l k topological modules are an extensive class of topological modules which arise naturally in the course of the study of the theory of probabilistic normed spaces the purpose of this paper is to show that an arbitrary regular l k topological module admits enough canonical module homomorphisms if and only if all of its quasi free submodules of finite rank are complemented in the sense of topological modules

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

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 bishop phelps theorem in complete random normed modules endowed with the e λ topology abstract in this paper we adopt a new approach so that we can prove that the bishop phelps theorem in complete random normed modules still holds under the e topology which solves an open problem posed in t x guo y j yang ekeland s variational principle for an l 0 valued function on a complete random metric space j math anal appl 389 2012 1 14

stability in locally l0 convex modules and a conditional version of james compactness theorem abstract locally l 0 convex modules were introduced in filipovic et al 2009 10 as the analytic basis for the study of conditional risk measures later the algebra of conditional sets was introduced in drapeau et al 2016 8 in this paper we study locally l 0 convex modules and find exactly which subclass of locally l 0 convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory second we provide a version of the classical james theorem of characterization of weak compactness for conditional banach spaces finally we state a conditional version of the fatou and lebesgue properties for conditional convex risk measures and as application of the developed theory we establish a version of the so called jouini schachermayer touzi theorem for robust representation of conditional convex risk measures defined on a l type module

a basic strict separation theorem in random locally convex modules abstract the central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module a series of interesting corollaries of the basic theorem are also included

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

relations between some basic results derived from two kinds of topologies for a random locally convex module the purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies namely the e topology and the stronger locally l0 convex topology for a random locally convex module first we give an extremely simple proof of the known hahn banach extension theorem for l0 linear functions as well as its continuous variant then we give the relations between the hyperplane separation theorems in d filipovic m kupper n vogelpoth separation and duality in locally l0 convex modules j funct anal 256 2009 3996 4029 and a basic strict separation theorem in t x guo h x xiao x x chen a basic strict separation theorem in random locally convex modules nonlinear anal 71 2009 3794 3804 in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies as applications of the fact we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the e topology are still valid under the locally l0 convex topology which considerably enriches financial applications of random normed modules

the james theorem in complete random normed modules the purpose of this paper is to generalize the classical james theorem characterizing the reflexivity of banach spaces to one characterizing the random reflexivity of complete random normed modules

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

the mean ergodic theorem on random reflexive random normed modules in this paper we prove a mean ergodic theorem for an almost surely power bounded random linear operator on a random reflexive random normed module which general izes and improves several known important results

the algebra of conditional sets and the concepts of conditional topology and compactness abstract the concepts of a conditional set a conditional inclusion relation and a conditional cartesian product are introduced the resulting conditional set theory is sufficiently rich in order to construct a conditional topology a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets it is proved that the conditional power set is a complete boolean algebra and a conditional version of the axiom of choice the ultrafilter lemma tychonoff s theorem the borel lebesgue theorem the hahn banach theorem the banach alaoglu theorem and the krein smulian theorem are shown

hilbert a modules abstract we consider real pre hilbert modules h on archimedean f algebras a with unit e we provide conditions on a and h such that a riesz representation theorem for bounded continuous a linear operators holds