Riesz Spaces Valued Measures and Processes Bulletin De La S
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Order Convergence of Vector Measures on Topological Spaces
133 (2008) MATHEMATICA BOHEMICA No. 1, 19–27 ORDER CONVERGENCE OF VECTOR MEASURES ON TOPOLOGICAL SPACES Surjit Singh Khurana, Iowa City (Received May 24, 2006) Abstract. Let X be a completely regular Hausdorff space, E a boundedly complete vector lattice, Cb(X) the space of all, bounded, real-valued continuous functions on X, F the algebra generated by the zero-sets of X, and µ: Cb(X) → E a positive linear map. First we give a new proof that µ extends to a unique, finitely additive measure µ: F → E+ such that ν is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of E+-valued finitely additive measures on F are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov’s theorem about the convergent sequences of σ-additive measures is extended to the case of order convergence. Keywords: order convergence, tight and τ-smooth lattice-valued vector measures, mea- sure representation of positive linear operators, Alexandrov’s theorem MSC 2000 : 28A33, 28B15, 28C05, 28C15, 46G10, 46B42 1. Introduction and notation All vector spaces are taken over reals. E, in this paper, is always assumed to be a boundedly complete vector lattice (and so, necessarily Archimedean) ([??], [??], [??]). ′ ′ If E is a locally convex space and E its topological dual, then h·, ·i : E × E → Ê will stand for the bilinear mapping hx, fi = f(x). For a completely regular Hausdorff space X, B(X) and B1(X) are the classes of Borel and Baire subsets of X, C(X) and Cb(X) are the spaces of all real-valued, and real-valued and bounded, continuous functions on X and X is the Stone-Čech compactification of X respectively. -
Contents 1. Introduction 1 2. Cones in Vector Spaces 2 2.1. Ordered Vector Spaces 2 2.2
ORDERED VECTOR SPACES AND ELEMENTS OF CHOQUET THEORY (A COMPENDIUM) S. COBZAS¸ Contents 1. Introduction 1 2. Cones in vector spaces 2 2.1. Ordered vector spaces 2 2.2. Ordered topological vector spaces (TVS) 7 2.3. Normal cones in TVS and in LCS 7 2.4. Normal cones in normed spaces 9 2.5. Dual pairs 9 2.6. Bases for cones 10 3. Linear operators on ordered vector spaces 11 3.1. Classes of linear operators 11 3.2. Extensions of positive operators 13 3.3. The case of linear functionals 14 3.4. Order units and the continuity of linear functionals 15 3.5. Locally order bounded TVS 15 4. Extremal structure of convex sets and elements of Choquet theory 16 4.1. Faces and extremal vectors 16 4.2. Extreme points, extreme rays and Krein-Milman's Theorem 16 4.3. Regular Borel measures and Riesz' Representation Theorem 17 4.4. Radon measures 19 4.5. Elements of Choquet theory 19 4.6. Maximal measures 21 4.7. Simplexes and uniqueness of representing measures 23 References 24 1. Introduction The aim of these notes is to present a compilation of some basic results on ordered vector spaces and positive operators and functionals acting on them. A short presentation of Choquet theory is also included. They grew up from a talk I delivered at the Seminar on Analysis and Optimization. The presentation follows mainly the books [3], [9], [19], [22], [25], and [11], [23] for the Choquet theory. Note that the first two chapters of [9] contains a thorough introduction (with full proofs) to some basics results on ordered vector spaces. -
Arxiv:1812.04035V1 [Math.OA]
The order topology on duals of C∗-algebras and von Neumann algebras EMMANUEL CHETCUTI and JAN HAMHALTER Department of Mathematics Faculty of Science University of Malta Msida, Malta [email protected] Department of Mathematics Faculty of Electrical Engineering Czech Technical University in Prague Technicka 2, 166 27 Prague 6, Czech Republic [email protected] Abstract: For a von Neumann algebra M we study the order topology associated to s the hermitian part M∗ and to intervals of the predual M∗. It is shown that the order s topology on M∗ coincides with the topology induced by the norm. In contrast to this, it is proved that the condition of having the order topology associated to the interval [0, ϕ] equal to that induced by the norm for every ϕ M +, is necessary and sufficient for the ∈ ∗ commutativity of M . It is also proved that if ϕ is a positive bounded linear functional on a C∗-algebra A , then the norm-null sequences in [0, ϕ] coincide with the null sequences with ′ respect to the order topology on [0, ϕ] if and only if the von Neumann algebra πϕ(A ) is of finite type (where πϕ denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology on order-bounded parts of dual spaces are nonequivalent for all C∗-algebras that are not of Type I. 2010 MSC: 46L10, 4605, 46L30, 06F30 Key words: dual spaces of C∗-algebras, order topology arXiv:1812.04035v1 [math.OA] 10 Dec 2018 1. -
Hybrid High-Order Methods. a Primer with Application to Solid Mechanics. Matteo Cicuttin, Alexandre Ern, Nicolas Pignet
Hybrid high-order methods. A primer with application to solid mechanics. Matteo Cicuttin, Alexandre Ern, Nicolas Pignet To cite this version: Matteo Cicuttin, Alexandre Ern, Nicolas Pignet. Hybrid high-order methods. A primer with applica- tion to solid mechanics.. 2021. hal-03262027 HAL Id: hal-03262027 https://hal.archives-ouvertes.fr/hal-03262027 Preprint submitted on 16 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Hybrid high-order methods. A primer with applications to solid mechanics∗ Matteo Cicuttin†, Alexandre Ern‡, Nicolas Pignet§ June 16, 2021 ∗This is a preprint of the following work: M. Cicuttin, A. Ern, N. Pignet, Hybrid high-order methods. A primer with applications to solid mechanics, Springer, (in press) 2021, reproduced with the permission of the publisher. †University of Liège (Montefiore Institute), Allée de la découverte 10, B-4000 Liège, Belgium ‡CERMICS, École des Ponts, 6 & 8 avenue Blaise Pascal, F-77455 Marne-la-vallée cedex 2, France and INRIA Paris, F-75589 France §EDF R&D, 7 Boulevard Gaspard Monge, F-91120 Palaiseau, France Preface Hybrid high-order (HHO) methods attach discrete unknowns to the cells and to the faces of the mesh. -
Fast, Adaptive Algorithms for Flow Problems
FAST, ADAPTIVE ALGORITHMS FOR FLOW PROBLEMS by Michael Edward McLaughlin B.S. in Mathematics, Ohio University, 2014 M.A. in Mathematics, University of Pittsburgh, 2018 Submitted to the Graduate Faculty of the Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2020 UNIVERSITY OF PITTSBURGH DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Michael Edward McLaughlin It was defended on March 30th, 2020 and approved by William Layton, Professor of Mathematics, University of Pittsburgh Michael Neilan, Associate Professor of Mathematics, University of Pittsburgh Catalin Trenchea, Associate Professor of Mathematics, University of Pittsburgh Peyman Givi, Distinguished Professor of Mechanical Engineering, University of Pittsburgh Dissertation Director: William Layton, Professor of Mathematics, University of Pittsburgh ii FAST, ADAPTIVE ALGORITHMS FOR FLOW PROBLEMS Michael Edward McLaughlin, PhD University of Pittsburgh, 2020 Time-accurate simulations of physical phenomena (e.g., ocean dynamics, weather, and combustion) are essential to economic development and the well-being of humanity. For example, the economic toll hurricanes wrought on the United States in 2017 exceeded $200 billon dollars. To mitigate the damage, the accurate and timely forecasting of hurricane paths are essential. Ensemble simulations, used to calculate mean paths via multiple real- izations, are an invaluable tool in estimating uncertainty, understanding rare events, and improving forecasting. The main challenge in the simulation of fluid flow is the complexity (runtime, memory requirements, and efficiency) of each realization. This work confronts each of these challenges with several novel ensemble algorithms that allow for the fast, effi- cient computation of flow problems, all while reducing memory requirements. -
The Order Convergence Structure
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Indagationes Mathematicae 21 (2011) 138–155 www.elsevier.com/locate/indag The order convergence structure Jan Harm van der Walt∗ Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa Received 20 September 2010; received in revised form 24 January 2011; accepted 9 February 2011 Communicated by Dr. B. de Pagter Abstract In this paper, we study order convergence and the order convergence structure in the context of σ- distributive lattices. Particular emphasis is placed on spaces with additional algebraic structure: we show that on a Riesz algebra with σ-order continuous multiplication, the order convergence structure is an algebra convergence structure, and construct the convergence vector space completion of an Archimedean Riesz space with respect to the order convergence structure. ⃝c 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved. MSC: 06A19; 06F25; 54C30 Keywords: Order convergence; Convergence structure; Riesz space 1. Introduction A useful notion of convergence of sequences on a poset L is that of order convergence; see for instance [2,7]. Recall that a sequence .un/ on L order converges to u 2 L whenever there is an increasing sequence (λn/ and a decreasing sequence (µn/ on L such that sup λn D u D inf µn and λn ≤ un ≤ µn; n 2 N: (1) n2N n2N In case the poset L is a Riesz space, the relation (1) is equivalent to the following: there exists a sequence (λn/ that decreases to 0 such that ju − unj ≤ λn; n 2 N: (2) ∗ Tel.: +27 12 420 2819; fax: +27 12 420 3893. -
Fixed Point Theorems on Ordered Vector Spaces Jin Lu Li1†,Congjunzhang2† and Qi Qiong Chen3†
Li et al. Fixed Point Theory and Applications 2014, 2014:109 http://www.fixedpointtheoryandapplications.com/content/2014/1/109 RESEARCH Open Access Fixed point theorems on ordered vector spaces Jin Lu Li1†,CongJunZhang2† and Qi Qiong Chen3† *Correspondence: [email protected] Abstract 3Department of Applied Mathematics, Nanjing University of In this paper, we introduce the concept of ordered metric spaces with respect to Science and Technology, some ordered vector spaces, which is an extension of the normal metric spaces. Then Xiaolingwei Street, Nanjing, Jiangsu we investigate some properties of ordered metric spaces and provide several fixed 210094, China †Equal contributors point theorems. As applications we prove several existence theorems for best ordered Full list of author information is approximation. available at the end of the article MSC: 46B42; 47H10; 58J20; 91A06; 91A10 Keywords: ordered vector spaces; order-continuity; ordered Lipschitz condition; order-preserving mapping; best ordered approximation; fixed point 1 Introduction Let S be a nonempty set and let X be a Banach space. Let K be a closed convex cone in X. TheconemetricdefinedonS, with respect to the cone K in X, is a bifunction d : S×S → X, for which the following properties hold: (a) d(u, v) ∈ K and d(u, v)=if and only if u = v; (b) d(u, v)=d(v, u); (c) d(u, s)+d(s, v)–d(u, v) ∈ K, for all s, u, v ∈ S. Note that if we take X to be the set of real numbers and take K =[,∞), then any cone metric space with respect to K turns out to be a normal metric space. -
Semantics of Higher-Order Probabilistic Programs with Conditioning
Semantics of higher-order probabilistic programs with conditioning Fredrik Dahlqvist Dexter Kozen University College London Cornell University [email protected] [email protected] Abstract—We present a denotational semantics for higher- exponentials. To accommodate conditioning and Bayesian order probabilistic programs in terms of linear operators between inference, we introduce ‘Bayesian types’, in which values are Banach spaces. Our semantics is rooted in the classical theory of decorated with a prior distribution. Based on this foundation, Banach spaces and their tensor products, but bears similarities with the well-known semantics of higher-order programs a` we give a type system and denotational semantics for an la Scott through the use ordered Banach spaces which allow idealized higher-order probabilistic language with sampling, definitions in terms of fixed points. Being based on a monoidal conditioning, and Bayesian inference. rather than cartesian closed structure, our semantics effectively We believe our approach should appeal to computer sci- treats randomness as a resource. entists, as it is true to traditional Scott-style denotational se- mantics (see IV-D). It should also appeal to mathematicians, I. INTRODUCTION statisticians and§ machine learning theorists, as it uses very Probabilistic programming has enjoyed a recent resurgence familiar mathematical objects from those fields. For example, of interest driven by new applications in machine learning a traditional perspective is that a Markov process is just a and statistical analysis of large datasets. The emergence of positive linear operator of norm 1 between certain Banach probabilistic programming languages such as Church and lattices [7, Ch. 19]. These are precisely the morphisms of our Anglican, which allow statisticians to construct and sample semantics. -
Math.FA] 28 May 2017 [10]
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institutional Repository of the Islamic University of Gaza ORDER CONVERGENCE IN INFINITE-DIMENSIONAL VECTOR LATTICES IS NOT TOPOLOGICAL Y. A. DABBOORASAD1, E. Y. EMELYANOV1, M. A. A. MARABEH1 Abstract. In this note, we show that the order convergence in a vector lattice X is not topological unless dim X < ∞. Furthermore, we show that, in atomic order continuous Banach lattices, the order convergence is topological on order intervals. 1. Introduction A net (xα)α∈A in a vector lattice X is order convergent to a vector x ∈ X if there exists a net (yβ)β∈B in X such that yβ ↓ 0 and, for each β ∈ B, there is an αβ ∈ A satisfying |xα − x|≤ yβ for all α ≥ αβ. In this case, we o write xα −→ x. It should be clear that an order convergent net has an order bounded tail. A net xα in X is said to be unbounded order convergent to a o vector x if, for any u ∈ X+, |xα −x|∧u −→ 0. In this case, we say that the net uo xα uo-converges to x, and write xα −→ x. Clearly, order convergence implies uo-convergence, and they coincide for order bounded nets. For a measure uo space (Ω, Σ,µ) and a sequence fn in Lp(µ) (0 ≤ p ≤ ∞), we have fn −→ 0 o iff fn → 0 almost everywhere; see, e.g., [8, Remark 3.4]. Hence, fn −→ 0 in Lp(µ) iff fn → 0 almost everywhere and fn is order bounded in Lp(µ). -
MAXIMAL D-IDEALS in a RIESZ SPACE
Can. J. Math., Vol. XXXV, No. 6, 1983, pp. 1010-1029 MAXIMAL d-IDEALS IN A RIESZ SPACE CHARLES B. HUIJSMANS AND BEN DE PAGTER 1. Introduction. We recall that the ideal / in an Archimedean Riesz space L is called a d-ideal whenever it follows from/ e / that {/} c /. Several authors (see [4], [5], [6], [12], [13], [15] and [18] ) have considered the class of all d-ideals in L, but the set^ of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact that^, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L. The main purpose of the present paper is to investigate the topological properties of^ and to compare^ to other structure spaces of L, such as the space Jt of minimal prime ideals and the space Qe of all ^-maximal ideals in L (where e > 0 is a weak order unit). This study of^ makes it for instance possible to place some recent results of F. K. Dashiell, A. W. Hager and M. Henriksen [7], concerning the order completion of C(X) in the general setting of Riesz spaces. After establishing some preliminary results in Section 2, it is proved in Section 3 that for any uniformly complete Riesz space L with weak order unit the spaced is a compact quasi-F-space. This result makes it possible to use^ for the description of the order completion of L, which is done in Section 4. -
UNBOUNDED P-CONVERGENCE in LATTICE-NORMED VECTOR LATTICES
UNBOUNDED p-CONVERGENCE IN LATTICE-NORMED VECTOR LATTICES A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY MOHAMMAD A. A. MARABEH IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS MAY 2017 Approval of the thesis: UNBOUNDED p-CONVERGENCE IN LATTICE-NORMED VECTOR LATTICES submitted by MOHAMMAD A. A. MARABEH in partial fulfillment of the require- ments for the degree of Doctor of Philosophy in Mathematics Department, Middle East Technical University by, Prof. Dr. Gülbin Dural Ünver Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Mustafa Korkmaz Head of Department, Mathematics Prof. Dr. Eduard Emel’yanov Supervisor, Department of Mathematics, METU Examining Committee Members: Prof. Dr. Süleyman Önal Department of Mathematics, METU Prof. Dr. Eduard Emel’yanov Department of Mathematics, METU Prof. Dr. Bahri Turan Department of Mathematics, Gazi University Prof. Dr. Birol Altın Department of Mathematics, Gazi University Assist. Prof. Dr. Kostyantyn Zheltukhin Department of Mathematics, METU Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last Name: MOHAMMAD A. A. MARABEH Signature : iv ABSTRACT UNBOUNDED p-CONVERGENCE IN LATTICE-NORMED VECTOR LATTICES Marabeh, Mohammad A. A. Ph.D., Department of Mathematics Supervisor : Prof. Dr. Eduard Emel’yanov May 2017, 69 pages The main aim of this thesis is to generalize unbounded order convergence, unbounded norm convergence and unbounded absolute weak convergence to lattice-normed vec- tor lattices (LNVLs). -
Nonstandard Hulls of Ordered Vector Spaces
NONSTANDARD HULLS OF ORDERED VECTOR SPACES A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY HASAN GÜL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS DECEMBER 2015 Approval of the thesis: NONSTANDARD HULLS OF ORDERED VECTOR SPACES submitted by HASAN GÜL in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department, Middle East Technical University by, Prof. Dr. Gülbin Dural Ünver ___________________ Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Mustafa Korkmaz ___________________ Head of Department, Mathematics Prof. Dr. Eduard Emelyanov ___________________ Supervisor, Mathematics Dept., METU Examining Committee Members : Prof. Dr. Zafer Nurlu __________________ Mathematics Dept., METU Prof. Dr. Eduard Emelyanov ___________________ Mathematics Dept., METU Prof. Dr. Bahri Turan __________________ Mathematics Dept., Gazi University Prof. Dr. Birol Altın __________________ Mathematics Dept., Gazi University Assist. Prof. Dr. Konstyantyn Zheltukhin __________________ Mathematics Dept., METU Date: 25 December 2015 I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name: Hasan Gül Signature: iv ABSTRACT NONSTANDARD HULLS OF ORDERED VECTOR SPACES Gül, Hasan Ph. D., Department of Mathematics Supervisor : Prof. Dr. Eduard Emelyanov December 2015, 57 pages This thesis undertakes the investigation of ordered vector spaces by applying nonstandard analysis. We introduce and study two types of nonstandard hulls of ordered vector spaces.