54 research outputs found
Set-valued Brownian motion
No full textBrownian motions, martingales, and Wiener processes are introduced and
studied for set valued functions taking values in the subfamily of compact convex
subsets of arbitrary Banach spaces X. The present paper is an application of the paper
(Labuschagne et al. in Quaest Math 30(3):285–308, 2007) in which an embedding
result is obtained which considers also the ordered structure of the family of compact convex subsets of a Banach space X and of Grobler and Labuschagne (J Math Anal
Appl 423(1):797–819, 2015; J Math Anal Appl 423(1):820–833, 2015) in which these
processes are considered in f-algebras.Moreover, in the space of continuous functions
defined on a Stonian space, a direct Levy’s result follows
On set-valued cone absolutely summing maps
No full textSpaces of cone absolutely summing maps are generalizations of Bochner spaces L(p)(mu, Y), where (Omega, Sigma, mu) is some measure space, 1 <= p < infinity and Y is a Banach space. The Hiai-Umegaki space L(1)[Sigma, cbf(X)] of integrably bounded functions F : Omega -> cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L(1)(mu, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of L(1)[Sigma, cbf(X)], and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued function
Ergodic theory and the Strong Law of Large Numbers on Riesz spaces
No full textAbstractIn [W.-C. Kuo, C.C.A. Labuschagne, B.A. Watson, Discrete-time stochastic processes on Riesz spaces, Indag. Math. (N.S.) 15 (3) (2004) 435–451], we introduced the concepts of conditional expectations, martingales and stopping times on Riesz spaces. Here we formulate and prove order theoretic analogues of the Birkhoff, Hopf and Wiener ergodic theorems and the Strong Law of Large Numbers on Riesz spaces (vector lattices)
Rehabilitation for Cas Labuschagne : good enough, timely enough?
No full textDATA BESKIKBAARHEIDSVERKLARING : Datadeling is nie van toepassing nie, aangesien geen nuwe
data in die studie geskep of ontleed is nie.This research is part of the project, ‘Contextualized Reformed Theology in South Africa’, directed by Dr Andre Ungerer of the Reformed Theological College, Faculty of Theology, University of Pretoria.Special Collection: Wim Dreyer Dedication, sub-edited by Jaco Beyers (University of Pretoria, South Africa).During the General
Church Assembly of 2010, the Netherdutch Reformed Church of Africa (NRCA) confessed her
previous justification of apartheid as wrongdoing. It led to serious dissatisfaction in the more
conservative part of the Church, eventually ending in the painful church schism of 2011.
Afterwards, the leadership had to handle several related issues. And then, out of the blue, an
outstanding matter arose from the past’s nebulae – unfinished business between the Church
and prof. Cas Labuschagne. He was a dissident with prof. Albert Geyser who acted against
the church’s justification of apartheid during the sixties. After a fierce struggle and much
desperation, he left South Africa for the Netherlands in 1967. Almost 40 years later, in 2008,
correspondence occurred between the NRCA and Labuschagne to see if reconciliation was possible,
but without any positive results. Labuschagne distrusted the Church’s motives and argued that
only a public excuse would settle matters. Due to the run-up to the decision in 2010 and the
church schism in 2011, the case gathered some cyber dust. Professors A.G. van Aarde, J.
Buitendag and W.A. Dreyer played a prominent role in bringing the outstanding matter of Cas
Labuschagne to the attention of the author of this article. It led to a visit in 2017 and more
correspondence to clarify issues. Eventually, reconciliation had occurred only a few months
before he died in 2019.
CONTRIBUTION : This article contributes to revealing an untold story in which different leaders
of the NRCA played a prominent role. It is about a struggle for reconciliation, events behind
the scenes, and a story with a good end – events the church and the academic community
should know about.http://www.hts.org.zaam2024Reformed Theological CollegeNon
Volatility models applied to risk measurement after the global financial crisis
No full textAbstract: Please refer to full text to view abstract.M.Com. (Investment Management
Characterizing the one-sided tensor norms Δ<sub><i>P</i></sub>and<sup><i>T</i></sup>Δ<sub><i>P</i></sub>
No full text
Lax proper maps of locales
No full textAbstractWe give a proof of localic Priestley duality. Our approach is based on lax proper maps of locales, which provide a vehicle for presenting the Priestley version of full Stone duality constructively and preserve spatial intuitions
On the uniform density of C(X) ⊗ C(Y) in C(X × Y)
No full textAbstractWe prove that if X and Y are compact Hausdorff spaces, then every f ∈ C(X × Y)+, i.e. f(x, y) ≥ 0 for all (x, y) ∈ X × Y, can be approximated uniformly from below and above by elements of the form ∑i=1nfigi, where fi ∈ C(X)+ and gi ∈ C(Y)+ for i = 1, 2, …, n. The proof uses only elementary topology. We use this result, in conjuction with Kakutani's M-spaces representation theorem, to obtain an alternative proof for a known property of Fremlin's Riesz space tensor product of Archimedean Riesz spaces
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