a simplex based algorithm to solve separated continuous linear programs we consider the separated continuous linear programming problem with linear data we characterize the form of its optimal solution and present an algorithm which solves it in a finite number of steps using an analog of the simplex method in the space of bounded measurable functions

birkhoff integral for multi valued functions abstract the aim of this paper is to study birkhoff integrability for multi valued maps f cwk x where is a complete finite measure space x is a banach space and cwk x is the family of all non empty convex weakly compact subsets of x it is shown that the birkhoff integral of f can be computed as the limit for the hausdorff distance in cwk x of a net of riemann sums n an f tn we link birkhoff integrability with debreu integrability a notion introduced to replace sums associated to correspondences when studying certain models in mathematical economics we show that each debreu integrable multi valued function is birkhoff integrable and that each birkhoff integrable multi valued function is pettis integrable the three previous notions coincide for finite dimensional banach spaces and they are different even for bounded multi valued functions when x is infinite dimensional and x is assumed to be separable we show that when f takes values in the family of all non empty convex norm compact sets of a separable banach space x then f is pettis integrable if and only if f is birkhoff integrable in particular these pettis integrable f s can be seen as single valued pettis integrable functions with values in some other adequate banach space incidentally to handle some of the constructions needed we prove that if x is an asplund banach space then cwk x is separable for the hausdorff distance if and only if x is finite dimensional

parity of class numbers and witt equivalence of quartic fields we show that 27 out of the 29 witt equivalence classes of quartic number fields can be represented by fields of class number 1 it is known that the remaining two classes contain solely fields of even class numbers we show that these two classes can be represented by fields of class number 2

the mcshane integral in weakly compactly generated spaces abstract di piazza and preiss asked whether every pettis integrable function defined on 0 1 and taking values in a weakly compactly generated banach space is mcshane integrable in this paper we answer this question in the negative moreover we give a counterexample where the target banach space is reflexive

measurable selectors and set valued pettis integral in non separable banach spaces kuratowski and ryll nardzewski s theorem about the existence of measurable selectors for multi functions is one of the keystones for the study of set valued integration one of the drawbacks of this result is that separability is always required for the range space in this paper we study pettis integrability for multi functions and we obtain a kuratowski and ryll nardzewski s type selection theorem without the requirement of separability for the range space being more precise we show that any pettis integrable multi function f cwk x defined in a complete finite measure space with values in the family cwk x of all non empty convex weakly compact subsets of a general non necessarily separable banach space x always admits pettis integrable selectors and that moreover for each a the pettis integral afd coincides with the closure of the set of integrals over a of all pettis integrable selectors of f as a consequence we prove that if x is reflexive then every scalarly measurable multi function f cwk x admits scalarly measurable selectors the latter is also proved when x w is angelic and has density character at most 1 in each of these two situations the pettis integrability of a multi function f cwk x is equivalent to the uniform integrability of the family supx f x bx r results about norm borel measurable selectors for multi functions satisfying stronger measurability properties but without the classical requirement of the range banach space being separable are also obtained

convergence and approximation of optimization problems in this paper we consider a general optimization problem op and study the convergence and approximation of optimal values and optimal solutions to changes in the cost function and the set of feasible solutions by a general op we mean that the cost function and the constraints are defined on a hausdorff topological space first we obtain convergence results for a general op and then we present an application of these results on the approximation of the optimal value and the optimal solutions for the so called general capacity problem in metric spaces

regularity aspects of non additive set multifunctions abstract in this paper continuity properties of set multifunctions such as regularity and continuity from above and below as well as others are introduced by way of a wijsman topology some of the relationships between them are established different examples counterexamples and applications are provided

gauge integrals and selections of weakly compact valued multifunctions abstract in the paper henstock mcshane birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general banach space x in particular the existence of selections integrable in the same sense of the corresponding multifunctions has been considered

witt equivalence of global fields ii relative quadratic extensions this paper explores the consequences of the hasse principle for witt equivalence of global fields in the case of relative quadratic extensions we are primarily interested in generating the witt equivalence classes of quadratic extensions of a given number field and we study the structure of the class the number of classes and the structure of the set of classes along the way we reprove several results obtained earlier in the absolute case of the rational ground field giving unified and short proofs based on the hasse principle

partially finite convex programming part ii explicit lattice models in part i of this work we derived a duality theorem for partially finite convex programs problems for which the standard slater condition fails almost invariably our result depended on a constraint qualification involving the notion ofquasi relative interior the derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function the differentiability of various convex functions in lattices was considered at the end of part i in part ii we shall apply our results to a number of more concrete problems including variants of semi infinite linear programming l1 approximation constrained approximation and interpolation spectral estimation semi infinite transportation problems and the generalized market area problem of lowe and hurter 1976 as in part i we shall use lattice notation extensively but as we illustrated there in concrete examples lattice theoretic ideas can be avoided if preferred by direct calculation

the birkhoff integral and the property of bourgain in this paper we study the birkhoff integral of functions f x defined on a complete probability space with values in a banach space x we prove that if f is bounded then its birkhoff integrability is equivalent to the fact that the set of compositions of f with elements of the dual unit ball zf x f x bx has the bourgain property a non necessarily bounded function f is shown to be birkhoff integrable if and only if zf is uniformly integrable and has the bourgain property as a consequence it turns out that the range of the indefinite integral of a birkhoff integrable function is relatively norm compact we characterize the weak radon nikod m property in dual banach spaces via birkhoff integrable radon nikod m derivatives we also point out that a recently introduced notion of unconditional riemann lebesgue integrability coincides with the notion of birkhoff integrability some other applications are given

a general solution of the monge kantorovich mass transfer problem abstract the solvability and the absence of a duality gap for the primal and the dual monge kantorovich mass transference programs for arbitrary hausdorff topological spaces are established

duality and an algorithm for a class of continuous transportation problems in this paper we treat the problem of transferring mass at least cost from one line segment to another when there is a continuous cost function cx y giving the cost of transferring material from the point x on the first line segment to the point y on the second the mass has to be arranged with uniform density on the second line segment after the transfer this is a one dimensional form of the well known mass transfer problem it is an infinite dimensional linear program we discuss duality theory for this problem and give an algorithm which converges to an optimal solution

witt rings of hasse domains of global fields abstract we describe the structure of the witt group w o s k of any hasse domain o s k of a global field k by explicitly determining its cyclic direct summands and generators of each such summand using this description we find a sufficient condition for the existence of witt ring isomorphism w o s k w o s l we give also the necessary and sufficient condition for the existence of witt ring isomorphism wk wl of global fields k and l mapping the witt ring w o s k onto the witt ring w o s l

places of algebraic function fields in arbitrary characteristic we consider the zariski space of all places of an algebraic function field f k of arbitrary characteristic and investigate its structure by means of its patch topology we show that certain sets of places with nice properties e g prime divisors places of maximal rank zero dimensional discrete places lie dense in this topology further we give several equivalent characterizations of fields that are large in the sense of f pop s annals paper embedding problems over large fields we also study the question whether a field k is existentially closed in an extension field l if l admits a k rational place in the appendix we prove the fact that the zariski space with the zariski topology is quasi compact and that it is a spectral space

semi infinite transportation problems abstract a semi infinite transportation dual program pair is specified which involves general pairings of linear spaces stemming from an infinite number of destination requirements but where in the primal program least cost flows of goods are sought from only a finite number of origins to these destinations building on the work of m j todd solving the generalized market area problem management sci 24 1978 1549 1554 a finite dimensional dual unconstrained concave program is developed for the primal semi infinite program but without certain measure theoretic restrictions on the cost functions themselves optimality conditions for the dual program pair are specified involving generalized column number conditions which parellel but extend those of the classical finite dimensional transportation problem

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

optimal solution of a monge kantorovitch transportation problem abstract suppose that p and q are probabilities on a separable banach space e it is known that if p q satisfies certain regularity conditions and a random variable x has law p then there exists a function f e e such that the function f x has the law q and the random pair x f x is an optimal coupling for the monge kantorovitch problem in this paper we provide an approximation of the function f when the law q is discrete thenwe extend this main result to any law q the proofs are based on a relationship between optimal couplings and nonlinear equations

on the witt equivalence of fields of characteristic 2 let e f be two fields of characteristic 2 and let w e w f be the witt rings of non singular symmetric bilinear forms over e and f in this note it is proved that if dime2 e dimf2 f 2 then e f is equivalent with w e w f

a banach lattice approach to convergent integrably bounded set valued martingales and their positive parts denote by cf x the set of all nonempty convex closed subsets of a separable banach space x let be a complete probability space and denote by l1 cf x the complete metric space of equivalence classes of a e equal integrably bounded cf x valued functions for any preassigned filtration i we describe the space of convergent integrably bounded cf x valued martingales in terms of the closure of i 1 l1 i cf x in l1 cf x in particular we provide a formula to calculate the join of two such martingales and the positive part of such a martingale our object is achieved by considering the more general setting of a near vector lattice s d endowed with a riesz metric d by means of radstrom s embedding theorem for such spaces a link is established between the space of convergent martingales in s and the space of convergent martingales in the radstrom completion r s of s this link provides information about the former space of martingales via known properties of measure free martingales in riesz normed vector lattices applicable to r s we also apply our general results to the spaces of convergent ck x valued martingales where ck x denotes the set of all nonempty convex compact subsets of x

the general pettis sugeno integral of vector multifunctions relative to a vector fuzzy multimeasure abstract in this paper we define and study the general pettis sugeno integral of a vector multifunction relative to a vector fuzzy multimeasure both taking values in a locally convex space x ordered by a closed convex pointed cone x with nonempty interior several classic properties of this integral and some estimation and comparative results with the pettis lebesgue integral are established

witt equivalence of algebraic function fields over real closed fields we examine the conditions for two algebraic function fields over real closed fields to be witt equivalent we show that there are only two witt classes of algebraic function fields with a fixed real closed field of constants real and non real ones the first of them splits further into subclasses corresponding to the tame equivalence this condition has a natural interpretation in terms of both orderings the associated harrison isomorphism maps 1 pt fans onto 1 pt fans and geometry and topology of associated real curves the bijection of points is a homeomorphism and these two curves have the same number of semi algebraically connected components finally we derive some immediate consequences of those theorems in particular we describe all the witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field

on the equivalence of mcshane and pettis integrability in non separable banach spaces abstract we show that mcshane and pettis integrability coincide for functions f 0 1 l 1 where is any finite measure on the other hand assuming the continuum hypothesis we prove that there exist a weakly lindelof determined banach space x a scalarly null hence pettis integrable function h 0 1 x and an absolutely summing operator u from x to another banach space y such that the composition u h 0 1 y is not bochner integrable in particular h is not mcshane integrable

witt equivalence classes of quartic number fields it has recently been established that there are exactly seven witt equivalence classes of quadratic number fields and then all quadratic and cubic number fields have been classified with respect to witt equivalence in this paper we have classified number fields of degree four using this classification we have proved the conjecture of szymiczek about the representability of witt equivalence classes by quadratic extensions of quadratic fields

exact solutions to the transportation problem on the line given distributions w of production and v of consumption on the line the monge kantorovich problem is to decide which producer should supply each consumer in order to minimize the total transportation costs here cost will be assumed to be a strictly concave function of the distance which translates into an economy of scale for longer trips and may encourage cross hauling the resulting solutions display a hierarchical structure that reflects a striking separation into local and global scales also found in the real world moreover this structure can be exploited to reduce the infinite dimensional linear problem to a convex minimization in m variables where 2 m 2 counts the number of times that w v changes sign a combinatorial algorithm is then derived which yields exact solutions by optimizing a certain finite sequence of convex separable network flows

enumeration of quartic fields of small discriminant with the mixed type case now completed all algebraic number fields of degree 4 with absolute discriminant 106 have been enumerated methods from the totally real and totally complex cases were used without major mod ification isomorphism of fields was determined by a method similar to one of lenstra the t2 criterion of pohst was applied to reduce the number of redundant examples

duality gap function in infinite dimensional linear programming abstract the concept of duality gap function in infinite dimensional linear programming is considered in this paper basic properties of the function and two theorems on its behavior are obtained by using duality theorems with interior conditions as illustrations for the results we investigate the parametric versions of an example due to d gale and parametric linear programs on spaces of continuous functions the notions of riemann stieltjes integral and function of bounded variation have been shown to be very useful for our investigations

the pettis integral for multi valued functions via single valued ones abstract we study the pettis integral for multi functions f cwk x defined on a complete probability space with values into the family cwk x of all convex weakly compact non empty subsets of a separable banach space x from the notion of pettis integrability for such an f studied in the literature one readily infers that if we embed cwk x into l b x by means of the mapping j cwk x l b x defined by j c x sup x c then j f is integrable with respect to a norming subset of b l b x a natural question arises when is j f pettis integrable in this paper we answer this question by proving that the pettis integrability of any cwk x valued function f is equivalent to the pettis integrability of j f if and only if x has the schur property that is shown to be equivalent to the fact that cwk x is separable when endowed with the hausdorff distance we complete the paper with some sufficient conditions involving stability in talagrand s sense that ensure the pettis integrability of j f for a given pettis integrable cwk x valued function f

henstock kurzweil pettis integrability of compact valued multifunctions with values in an arbitrary banach space abstract the aim of this paper is to describe henstock kurzweil pettis hkp integrable compact valued multifunctions such characterizations are known in case of functions see di piazza and musial 2006 16 it is also known see di piazza and musial 2010 19 that each hkp integrable compact valued multifunction can be represented as a sum of a pettis integrable multifunction and of an hkp integrable function invoking to that decomposition we present a pure topological characterization of integrability having applied the above results we obtain two convergence theorems that generalize results known for hkp integrable functions we emphasize also the special role played in the theory by weakly sequentially complete banach spaces and by spaces possessing the schur property

a new result on impulsive differential equations involving non absolutely convergent integrals in this paper we obtain as an application of a darbo type theorem global solutions for differential equations with impulse effects under the assumption that the function on the right hand side is integrable in the henstock sense we thus generalize several previously given results in literature for ordinary or impulsive equations

on solutions to the mass transfer problem this paper studies the monge kantorovich mass transfer mt problem on metric spaces with possibly unbounded cost function conditions are given under which the mt problem is solvable and furthermore an optimal solution can be obtained as the weak limit of a sequence of optimal solutions to suitably approximating mt problems

an algorithm for semi infinite transportation problems in this paper we consider a class of semi infinite transportation problems we develop an algorithm for this class of semi infinite transportation problems the algorithm is a primal dual method which is a generalization of the classical algorithm for finite transportation problems the most important aspect of our paper is that we can prove the convergence result for the algorithm finally we implement some examples to illustrate our algorithm

generalized equilibrium results for games with incomplete information milgrom and weber milgrom p r weber r j 1985 distributional strategies for games with incomplete information math oper res 10 619 632 gave an existence result for a nash equilibrium in a game with incomplete information using their notion of a distributional strategy here we obtain a substantial improvement of their existence result in terms of the more traditional concept of a behavioral strategy this improvement is reached very naturally as an application of a theory of weak convergence for transition probabilities which is recapitulated extensively in this paper also a new result on the weak convergence of product transition probabilities is included

a lusin type measurability property for vector valued functions abstract we study the class of riemann measurable vector valued functions based on a lusin type property this class contains all riemann integrable functions and is closely related to the restricted versions of the mcshane and henstock integrals the m and h integrals defined by means of lebesgue measurable gauges our developments are in the spirit of the riemann type integral theory for real valued functions in particular we prove that a bounded riemann measurable vector valued function is m integrable

on the comparison of aumann and bochner integrals we introduce the aumann integral in the finitely additive setting and we compare it with the bochner integral

strong duality of the monge kantorovich mass transfer problem in metric spaces this paper studies the monge kantorovich mass transfer mt problem on metric spaces and with an unbounded cost function conditions are given under which the strong duality condition holds that is mt and its dual mt are both solvable and their optimal values coincide

extreme points of sets of randomized strategies in constrained optimization and control problems this paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems we first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment like constraints conditions are given under which those extreme points are randomizations of at most n 1 deterministic strategies this result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic allocation like optimization problems and their young relaxations similar results are obtained for constrained markov control processes in borel spaces

strong integral of multifunctions relative to a fuzzy measure abstract in this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of 0 with respect to a fuzzy measure unlike various ways of defining such an integral e g aumann way using different selections dunford way riemann way we use the sugeno method directly applied to multifunctions the originality of our definition is that the value of the integral is a real scalar and not usually a set some classical integral properties a convergence theorem and relationships with other integrals are presented