on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

density results for proper efficiencies several density results are established for different notions of proper efficiency in vector optimization without the requirement of ordering cones being boundedly based for a compact set the set of henig proper efficient points is shown to be dense in the set of all efficient points density theorems for borwein s proper efficiency and for positive scalarizable efficiency follow immediately with appropriate assumptions

on the uniqueness of solutions to continuity equations abstract we obtain sufficient conditions for the uniqueness of solutions to the cauchy problem for the continuity equation in classes of measures that need not be absolutely continuous

neumann problems with time singularities in this paper we study the existence and uniqueness of solutions to a nonlinear neumann problem for a scalar second order ordinary differential equation u atu f t u u where a 0 and f t x y satisfies the local caratheodory conditions on 0 t xrxr

theoretical and numerical studies of nonlinear shell equations abstract we study the solution field m of a parameter dependent nonlinear two point boundary value problem presented by troger and steindl h troger a steindl nonlinear stability and bifurcation theory springer wien new york 1991 this problem models the buckling of a thin walled spherical shell under a uniform external static pressure the boundary value problem is formulated as an abstract operator equation t x 0 in appropriate banach spaces by exploiting the equivariance of t we obtain detailed informations about the structure of m these theoretical results are used to compute efficiently interesting parts of m with numerical standard techniques bifurcation diagrams a stability diagram and pictures of deformed shells are presented

collocation methods for index 1 daes with a singularity of the first kind we study the convergence behavior of collocation schemes applied to approximate solutions of bvps in linear index 1 daes which exhibit a critical point at the left boundary such a critical point of the dae causes a singularity within the inherent ode system we focus our attention on the case when the inherent ode system is singular with a singularity of the first kind apply polynomial collocation to the original dae system and consider different choices of the collocation points such as equidistant gaussian or radau points we show that for a well posed boundary value problem for daes having a sufficiently smooth solution the global error of the collocation scheme converges with the order o h s where s is the number of collocation points superconvergence cannot be expected in general due to the singularity not even for the differential components of the solution the theoretical results are illustrated by numerical experiments

positive proper efficient points and related cone results in vector optimization theory positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems a geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points it is shown that these results are applicable in the normed vector lattices c a b l p and l p for 1 leqq p leqq infty and that previous related results which required the ordering cone to have a compact or weak compact base are not applicable in many normed vector lattices including c a b l p and l p for 1 leqq p leqq infty

proper efficiency and tradeoffs in multiple criteria and stochastic optimization the mathematical equivalence between linear scalarizations in multiobjective programming and expected value functions in stochastic optimization suggests to investigate and establish further conceptual analogies between these two areas in this paper we focus on the notion of proper efficiency that allows us to provide a first comprehensive analysis of solution and scenario tradeoffs in stochastic optimization in generalization of two standard characterizations of properly efficient solutions using weighted sums and augmented weighted tchebycheff norms for finitely many criteria we show that these results are generally false for infinitely many criteria in particular these observations motivate a slightly modified definition to prove that expected value optimization over continuous random variables still yields bounded tradeoffs almost everywhere in general further consequences and practical implications of these results for decision making under uncertainty and its related theory and methodology of

the 2d euler equations and the statistical transport equations we prove the existence of weak solutions for the forward and backward statistical transport equations associated with the 2d euler equations such solutions can be interpreted respectively as a statistical lagrangian and a statistical eulerian description of the motion of the fluid

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

analysis of a new error estimate for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind the estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle we prove that for collocation methods which are not superconvergent the error estimate is asymptotically correct as an essential prerequisite we derive convergence results for collocation methods applied to nonlinear singular problems

the convergence of shooting methods for singular boundary value problems we investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems particular attention is paid to the well posedness of the process it is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available moreover convergence results for a perturbed newton iteration are discussed

anticipating flows on the wiener space generated by vector fields of low regularity abstract sobolev spaces for banach space valued wiener functionals are constructed using the operator norm for the h derivatives the corresponding sobolev norms are shown to be strictly weaker than the usual malliavin sobolev norms existing results on flows on an abstract wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew adjoint operator and a non linear part whose regularity is measured by the new sobolev norms this is equivalent to a non linear perturbation of a semi group of rotations

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

flows associated to tangent processes on the wiener space we prove under certain regularity assumptions on the coefficients that tangent processes namely semimartingales d a dx b d where a is an antisymmetric matrix generate flows on the classical wiener space main applications of the result can be found in the study of the geometry of path spaces

inclusion theorems for non explosive and strongly exposed cones in normed spaces abstract this paper is composed of five sections in the first one we classify cones into pseudonuclear intermediate and explosive the notion of strongly exposed cone and the equivalence between pseudonuclearity and strong exposition for cones in a normed space are recalled in the second one it is shown that several properties for a cone in a normed space exist each one being equivalent to strong exposition of the cone furthermore 2 5 an estimate by chung is improved in the third one through projections of rank one some cones that are an immediate extension of elliptic cones in elementary geometry are defined and it is established 3 3 that a cone is strongly exposed if and only if it can be contained in one of that type in the fourth one it is shown 4 2 that also the non explosivity of a cone is expressible in terms of metric properties more precisely a cone is non explosive if and only if it can be contained in a balanced cone of elliptic kind but in which the axis becomes a finite dimensional subspace it may be interesting to observe that the result does not depend on particular hypotheses e g reflexivity on the space in the fifth one it is put in evidence a property of a cone in a reflexive normed space which excludes that the cone is intermediate an orientative essential bibliography concludes the paper

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

numerical solution of singular boundary value problems with logarithmic singularities by pade approximation and collocation methods in this paper we present a method using quintic b spline and pade approximation for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficient we use the modified quintic b spline to approximate the first coefficients in the taylor expansion on which all the other coefficients depend moreover we introduce the superposed pade approximation the resulting perturbed initial value problem is then efficiently treated

integrated harnack inequalities on lie groups we show that the logarithmic derivatives of the convolution heat kernels on a uni modular lie group are exponentially integrable this result is then used to prove an integrated harnack inequality for these heat kernels it is shown that this integrated harnack inequality is equivalent to a version of wang s harnack inequality a key feature of all of these inequalities is that they are dimension independent finally we show these inequalities imply quasi invariance properties of heat kernel measures for two classes of infinite dimensional lie groups

super efficiency of vector optimization in banach spaces using variational analysis in terms of the clarke normal cone we consider super efficiency of vector optimization in banach spaces we establish some characterizations for super efficiency in particular dropping the assumption that the ordering cone has a bounded base we extend a result in borwein and zhuang j m borwein d zhuang super efficiency in vector optimization trans amer math soc 338 1993 105 122 to the nonconvex setting 2006 elsevier inc all rights reserved

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

the two dimensional euler equation a statistical study we construct surface type measures on the level sets for the renormalized energy which is an invariant quantity for the two dimensional periodic euler flow and prove the existence of weak solutions living on such level sets other classes of invariant measures for the motion are also introduced

on geometry of cones and some applications we thank the referee for suggestions that helped to improve the overall aspect of the manuscript this work has been partially supported by the generalitat valenciana project gv 2014 072 f garcia castano and by the ministerio de economia y competitividad and feder project mtm2014 57838 c2 2 p v montesinos

the application of explicit nystrom methods to singular second order differential equations abstract an important class of singular second order initial value problems is y 2 x y f x y 0 0 x x f y 0 a y 0 0 this class includes for example the well known singular equations of emden and liouville the purpos of this paper is to show the interesting result that explicit nystrom methods existing for the direct integration of special second order regular initial value problems can be used for the integration of this class of singular initial value problems and the methods show their proper respective orders of convergence this is justified mathematically and demonstrated computationally

a generalization of a theorem of arrow barankin and blackwell this paper presents a generalization of a known density theorem of arrow barankin and blackwell admissible points of convex sets in contributions to the theory of games h w kuhn and a w tucker eds princeton university press princeton nj 1953 this result is shown to hold even in a real normed space partially ordered by a bishop phelps cone

efficient mesh selection for collocation methods applied to singular bvps we describe a mesh selection strategy for the numerical solution of boundary value problems for singular ordinary differential equations this mesh adaptation procedure is implemented in our matlab code sbvp which is based on polynomial collocation we prove that under realistic assumptions our mesh selection strategy serves to approximately equidistribute the global error of the collocation solution thus enabling to reach prescribed tolerances efficiently moreover we demonstrate that this strategy yields a favorable performance of the code and compare its computational effort with other implementations of polynomial collocation

scalarization in vector optimization in this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem this will be done for weakly minimal and minimal solutions respectively finally the results will be applied to a certain class of approximation problems

the application of runge kutta schemes to singular initial value problems a theory for explicit runge kutta schemes applied to the initial value problem for a first order system of differential equations with a singularity of the first kind is developed it is shown that in general the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems 1 introduction in this paper we examine the application of explicit runge kutta schemes to the numerical solution of the problem

nystrom methods and singular second order differential equations chawla jain and subramanian studied the application of nystrom methods to a class of singular initial value problems following their approach we generalize this class by applying the nystrom method to the initial value problem for an equation of the form y p t y q t y t 0 email protected 0 1 where p has a certain specified type of singularity and q is sufficiently differentiable and then we determine the order of convergence this is followed by computational evidence

degrees of efficiency and degrees of minimality in this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure usually different scalarizing functions are used in order to obtain the various solutions of the vector problem here we consider different kinds of solutions of the same scalarized problem our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency besides the usual notions of weakly efficient and efficient points which are characterized as global and strict global solutions of the scalarized problem we also consider some restricted notions of efficiency such as strict and proper efficiency which are characterized as tikhonov well posed minima and sharp minima for the scalarized problem

flows associated to adapted vector fields on the wiener space the existence and uniqueness of a flow associated to an adapted vector field on the wiener space with d a d b d are proved by a modified picard s iteration method mainly under the conditions of exponential integrability concerning b as well as the first order malliavin gradients of a and b a newton leibnitz type inequality for a kind of malliavin differentiable functionals is also proved which is the key point to prove the above result and has an independent interest

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

higher order optimality conditions for henig efficient solutions in set valued optimization in this paper higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set valued maps are introduced necessary and sufficient conditions for henig efficient solutions to a constrained set valued optimization problem are given by employing the higher order generalized epiderivatives

iterated defect correction for the solution of singular initial value problems we investigate the convergence properties of the iterated defect correction idec method based on the implicit euler rule for the solution of singular initial value problems with a singularity of the first kind we show that the method retains its classical order of convergence which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure

properly maximal points in product spaces we study maximal points in a locally convex space partially ordered by a convex cone with a bounded base properly maximal points are defined and compared with other concepts of efficiency existence and density theorems are given which unify and generalize several results known in recent literature particular attention is paid on properly maximal points in a product space which has an interesting application in obtaining a multiplier rule for convex set valued problems in a general setting

stability and sensivity analysis for conical regularization of linearly constrained least squares problems in hilbert spaces abstract in this paper we follow the conical regularization approach given in khan and sama 2013 14 for a linearly constrained least square problem in hilbert spaces this regularization can be seen as a family of linearly constrained least problem that is parametrized by a positive parameter e we perform a stability and sensitivity analysis by using set valued analysis and duality tools as a consequence we prove that the stability of the optimal value function the regularity of the unperturbed problem and the norm boundeness of the regularized multipliers are equivalent properties moreover under an additional regularity condition we prove the stability of the regularized solutions and we find a computation formula for the contingent derivative of the optimal value function in terms of any multiplier of the unperturbed problem and the w contingent derivative of the trajectory of regularized solutions finally we provide two examples to illustrate our theoretical results

two generalizations of a theorem of arrow barankin and blackwell in 1953 arrow barankin and blackwell proved that if r n is equipped with its natural ordering and if a is a closed convex subset of r n then the set of points in a that can be supported by strictly positive linear functionals is dense in the set of all efficient maximal points of a in this note two generalizations of this result are given the first of these is in the setting of a dual system and requires relatively weak assumptions on the ordering cone but a rather strong compactness assumption on the set a the second generalization which is in the setting of a locally convex space relaxes the compactness assumption on the set a but demands more stringent assumptions on the ordering cone this second result was recently obtained by petschke for normed spaces m petschke on a theorem of arrow barankin and blackwell siam j control optim 28 1990 pp 395 401 the proof given here is substantially different from that given by petschke

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

the clark ocone formula for vector valued wiener functionals abstract the classical representation of random variables as the ito integral of nonanticipative integrands is extended to include banach space valued random variables on an abstract wiener space equipped with a filtration induced by a resolution of the identity on the cameron martin space the ito integral is replaced in this case by an extension of the divergence to random operators and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense a complete characterization of measure preserving transformations in wiener space is presented as an application of this generalized clark ocone formula

pareto optimization in topological vector spaces to identify the efficient point of a set is an important problem in vector optimization the existence of efficient points in infinite dimensional space has been studied by many authors such as 2 10 12 15 17 19 21 23 25 etc to study pareto efficiency in locally convex topological vector space in 1983 the second author isac 14 has introduced an important notion of nuclear cone which has also other applications as for example in the fixed point theory in the best approximation theory in the study of vector optimization for multivalued functions in the study of conically bounded sets and in the study of nuclearity of topological vector space about these one can refer to hyers isac rassias book 11 and references therein in 1990 pontini 20 defined a more general notion of pseudonuclear in general topological vector space may not be locally convex pontini remarked that in a normed vector space a cone is pseudo nuclear if and only if it is nuclear but in a general topological vector space this result is not true in 11 13 isac hyers and rassias discussed the existence of efficient points in locally convex topological space ordered by pointed nuclear cone and got some general results in book 11 and paper 13 they also posed the following open problem

computation formulas and multiplier rules for graphical derivatives in separable banach spaces abstract in this paper we present new computation formulas for the contingent epiderivative and hypoderivative of a set valued map taking values in a banach space with a shrinking schauder basis these formulas are established in terms of the fourier coefficients and in particular in terms of the derivatives of the component maps associated with the schauder basis as an application we obtain multiplier rules for vector optimization problems in terms of the derivatives of the component maps extending classical results from smooth multiobjective optimization problems

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

density theorems for ideal points in vector optimization abstract several results are established concerning the density of the set of ideal points in the set of minimal solutions of positive support functionals of sets in normed linear spaces the above ideal points are defined and several characterizations and sufficient conditions for their existence are also stated

asymptotically correct error estimation for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind as an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts this requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class

the divergence of banach space valued random variables on wiener space the domain of definition of the divergence operator on an abstract wiener space w h is extended to include w valued and open image in new window valued integrands the main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in s extended domain have divergence zero these results are then applied to the analysis of quasiinvariant flows induced by w valued vector fields and among other results it turns out that these divergence free vector fields are responsible for generating measure preserving flows

analytical and numerical treatment of a singular initial value problem in avalanche modeling we discuss a leading edge model used in the computation of the run out length of dry flowing avalanches the model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics existence uniqueness and smoothness properties of the analytical solution are shown we also prove the existence of a unique root of the solution moreover we present a fortran 90 code for the numerical computation of the run out length the code is based on a solver for singular initial value problems which is an implementation of the acceleration technique known as iterated defect correction based on the implicit euler method

the lagrange multiplier rule for super efficiency in vector optimization in this paper we study constrained multiobjective optimization problems with objectives being closed graph multifunctions in banach spaces in terms of the coderivatives and clarke s normal cones we establish lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems

on a singular boundary value problem arising in the theory of shallow membrane caps abstract we investigate the following singular boundary value problem which originates from the theory of shallow membrane caps t 3 u t t 3 1 8 u 2 t a 0 u t b 0 t 2 4 0 lim t 0 t 3 u t 0 u 1 0 where a 0 b 0 and are given constants we show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes we illustrate the theory by numerical experiments in which we used the new version of the matlab code sbvp based on polynomial collocation to approximate the solution of the membrane problem

on a theorem of arrow barankin and blackwell a well known theorem of arrow barankin and blackwell states that if bf r n is equipped with the natural ordering then for every compact convex subset s of bf r n the set of properly minimal elements of s is dense in the set of minimal elements of s in this note a result of jahn is used to show a generalization of the density theorem of arrow barankin and blackwell it will be shown that this theorem holds in a real normed space that is partially ordered by a convex cone with a closed bounded base

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