Order preserving and order reversing operators on the class of convex functions in Banach spaces

A remarkable result by S. Artstein-Avidan and V. Milman states that, up to pre-composition with affine operators, addition of affine functionals, and multiplication by positive scalars, the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on $\mathbb{R}^n$ is the identity operator, and the only fully order reversing one acting on the same set is the Fenchel conjugation. Here fully order preserving (reversing) mappings are understood to be those which preserve (reverse) the pointwise order among convex functions, are invertible, and such that their inverses also preserve (reverse) such order. In this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinous proper convex functions defined on arbitrary infinite dimensional Banach spaces.Comment: 19 pages; Journal of Functional Analysis, accepted for publication; a better presentation of certain parts; minor corrections and modifications; references and thanks were adde

The cardinal of various monoids of transformations that preserve a uniform partition

In this paper we give formulas for the number of elements of the monoids ORm x n of all full transformations on it finite chain with tun elements that preserve it uniform m-partition and preserve or reverse the orientation and for its submonoids ODm x n of all order-preserving or order-reversing elements, OPm x n of all orientation-preserving elements, O-m x n of all order-preserving elements, O-m x n(+) of all extensive order-preserving elements and O-m x n(-) of all co-extensive order-preserving elements

(Strongly) zero-dimensional ordered spaces

Includes bibliographical references.The relationship between transitive uniform spaces and zero-dimensional topological spaces was first established by Banaschewski [1957], and was later investigated by Levine [1969]. The theory of transitive quasi-uniform spaces is treated in [Fletcher and Lindgren 1972], [Brummer 1984] and [Kiinzi 1990, 1992a, 1992b,1993]; a convenient presentation for our purpose is to be found in [Fletcher and Lindgren 1982]. After Reilly [1972] introduced the notion of zero-dimensionality in bitopological spaces, Birsan [1974] and Halpin [1974] studied the relationship between transitive quasi-uniform spaces and zero-dimensional bitopological spaces. In this thesis we define a notion of zero-dimensionality in ordered topological spaces and examine the relationship between transitive quasi-uniform spaces and zero-dimensional ordered topological spaces. To a large extent, our presentation is influenced by the situation in bitopological spaces (cf. [Halpin 1974] and [Birsan 1974]), and uses the commutative diagrams which occur in [Schauerte 1988] and [Brummer 1977, 1982]. We also study strongly zero-dimensional ordered topological spaces and their relation with functorial quasi-uniformities. In this respect, our results are influenced by those of [Fora 1984], [Banaschewski and Brummer 1990] and [Kiinzi 1990] for strongly zero-dimensional bitopological spaces

Topological Algebraic Structure in the Density Topology and on Souslin Lines

This research investigates which topological algebraic structures can exist on two types of topological spaces: the real line R with the density topology; and any linearly ordered topological space (LOTS) satisfying the countable chain condition (CCC) that is not separable (i.e. any Souslin Line). Some surprising results are established in the density topology when considering the common group operations on R. Indeed, this research shows that addition and multiplication are not topological group operations in this space. These theorems are then generalized to show that there are no topological group operations on R with the density topology. The case of cancellative topological semigroups, however, is left as an open question.On the other hand, the conditions of existence of topological algebraic structures on Souslin lines is rather completely determined by this work. The main results in this space are that paratopological groups do not exist on any Souslin line, but cancellative topological semigroups do exist. The research on this space culminates with the construction of a cancellative topological semigroup on a Souslin line

The cardinal of various monoids of transformations that preserve a uniform partition

Bulletin of the Malaysian Mathematical Sciences SocietyIn this paper we give formulas for the number of elements of the monoids OR mxn of all full transformations on a nite chain with mn elements that preserve a uniform m-partition and preserve or reverse the orientation and for its submonoids OD mxn of all order-preserving or order-reversing elements, OP mxn of all orientation- preserving elements, O mxn of all order-preserving elements, O+ mxn of all extensive order-preserving elements and O- mxn of all co-extensive order-preserving elements

Perron-Frobenius theorem for multi-homogeneous mappings with applications to nonnegative tensors

The Perron-Frobenius theorem for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative tensors. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings defined on a product of cones and their associated eigenvectors. By considering a vector valued version of the Hilbert metric, we prove several Perron-Frobenius type results for these mappings. We discuss the existence, the uniqueness and the maximality of nonnegative and positive eigenvectors of multi-homogeneous mappings. We prove a Collatz-Wielandt formula and a multi-linear Birkhoff-Hopf theorem. We study the convergence of the normalized iterates of multi-homogeneous mappings and prove convergence rates. Applications of our main results include the study of the (p,q)-singular vectors of nonnegative matrices, the p-eigenvectors, rectangular (p,q)-singular vectors and (p_1,...,p_d)-singular vectors of nonnegative tensors, the generalized DAD problem and the discrete generalized Schrödinger equation arising in multi-marginal optimal transport. We recast these problems in the multi-homogeneous framework and explain how our theorems can be used to refine, improve and offer a new point of view on previous results of the literature.Das Perron-Frobenius Theorem für nichtnegative Matrizen wurde auf homogene, ordnungserhaltende Abbildungen auf einem Kegel erweitert und, in letzter Zeit, auf nichtnegative Tensoren. Wir vereinheitlichen beide Ansätze, indem wir das Konzept der ordnungserhaltenden, multi-homogenen Abbildungen, die auf einem Produkt von Kegeln definiert sind, sowie deren zugehörige Eigenvektoren einführen. Indem wir eine vektorisierte Version der Hilbert-Metrik in Betracht ziehen, beweisen wir für diese Abbildungen mehrere Perron-Frobenius-Typ Ergebnisse. Wir diskutieren die Existenz, die Einzigartigkeit und die Maximalität nichtnegativer und positiver Eigenvektoren multihomogener Abbildungen. Wir beweisen eine Collatz-Wielandt-Formel und einen multi-linearen Birkhoff-Hopf Satz. Wir untersuchen die Konvergenz der normierten Iterationen von multi-homogenen Abbildungen und beweisen Konvergenzraten. Anwendungen unserer Hauptergebnisse umfassen die Untersuchung der (p,q)-singulären Vektoren nichtnegativer Matrizen, der p-Eigenvektoren, rechteckiger (p,q)-singulärer Vektoren und (p_1,...,p_d)-singulärer Vektoren nichtnegativer Tensoren, das generalisierte DAD-Problem und die diskrete generalisierte Schrödinger Gleichung, die im Zusammenhang mit multi-marginalem optimalen Transport auftritt. Wir übertragen diese Probleme in den multi-homogenen Rahmen und erklären, wie unsere Theoreme verwendet werden können, um frühere Ergebnisse der Literatur zu verfeinern, zu verbessern und eine neue Sichtweise auf diese zu bieten