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24
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
Monotone Comparative Statics in Ordered Vector Spaces
, 2001
"... This paper considers ordered vector spaces with arbitrary closed cones and establishes a number of characterization results with applications to monotone comparative statics (Topkis (1978), Topkis (1998), Milgrom and Shannon (1994)). By appealing to the fundamental theorem of calculus for the Hensto ..."
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Cited by 3 (2 self)
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This paper considers ordered vector spaces with arbitrary closed cones and establishes a number of characterization results with applications to monotone comparative statics (Topkis (1978), Topkis (1998), Milgrom and Shannon (1994)). By appealing to the fundamental theorem of calculus for the HenstockKurzweil integral, we generalize existing results on increasing differences and supermodularity for C 1 or C 2 functions. None of the results are based on the assumption that the order is Euclidean. As applications we consider a teamwork game and a monopoly union model.
Aggregative Games and BestReply Potentials
"... This paper introduces quasiaggregative games and establishes conditions under which such games admit a bestreply potential. This implies existence of a pure strategy Nash equilibrium without any convexity or quasiconcavity assumptions. It also implies convergence of bestreply dynamics under so ..."
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Cited by 3 (0 self)
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This paper introduces quasiaggregative games and establishes conditions under which such games admit a bestreply potential. This implies existence of a pure strategy Nash equilibrium without any convexity or quasiconcavity assumptions. It also implies convergence of bestreply dynamics under some additional assumptions. Most of the existing literature’s aggregation concepts are special cases of quasiaggregative games, and many new situations are allowed for. An example is payoff functions that depend on own strategies as well as a linear combination of the mean and the variance of players’ strategies.
The distributional Denjoy integral
, 2006
"... Abstract. Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F ′ = f (distributional derivative) then the distributional integral of f is defined as ∫ ∞ f = F(∞)−F(−∞). It is shown that this simple definition g ..."
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Cited by 2 (2 self)
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Abstract. Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F ′ = f (distributional derivative) then the distributional integral of f is defined as ∫ ∞ f = F(∞)−F(−∞). It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock–Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, Hölder inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy’s descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed. 2000 subject classification: 26A39, 46E15, 46F05, 46G12 1
Beppo Levi's Theorem for the VectorValued McShane Integral and Applications
"... Using only elementary properties of the McShane integral for vectorvalued functions, we establish a convergence theorem which for the scalar case of the integral yields the classical Beppo Levi (monotone) convergence theorem as an immediate corollary. As an application, the convergence theorem is u ..."
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Using only elementary properties of the McShane integral for vectorvalued functions, we establish a convergence theorem which for the scalar case of the integral yields the classical Beppo Levi (monotone) convergence theorem as an immediate corollary. As an application, the convergence theorem is used to prove that the space of McShane integrable functions, although not usually complete, is ultrabornological. 1
The Integral: An Easy Approach after Kurzweil and Henstock.
"... A simple definition. Riemann’s integral of 1867 can be summarized as f(t)dt = lim � f(τi)(ti − ti−1). This summary conceals some of the complexity—for example, the limit is of a net, not a sequence—but it displays what we wish to emphasize: The integral is formed by combining the values f(τi) in a v ..."
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A simple definition. Riemann’s integral of 1867 can be summarized as f(t)dt = lim � f(τi)(ti − ti−1). This summary conceals some of the complexity—for example, the limit is of a net, not a sequence—but it displays what we wish to emphasize: The integral is formed by combining the values f(τi) in a very direct fashion. The values of f are used less directly in Lebesgue’s integral (1902), which � b can be described as limn→ ∞ a gn(t)dt. The approximating functions gn must be chosen carefully, using deep, abstract notions of measure theory. Simpler definitions are possible—for example, functional analysts might consider the metric completion of C[0, 1] using the L1 norm—but such a definition does not give us easy access to the Lebesgue integral’s simple and powerful properties such as the Monotone Convergence Theorem. We generally think in terms of those simple properties, rather than the various complicated definitions, when we actually use the Lebesgue integral.
Topologizing the Denjoy Space
"... Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces X ∆ whose bounded sets are precisely the subsets of KH[0, 1] that are equiintegrable and pointwise bounded. The resulting inductive limit topology on � ∆ X ∆ = KH[0, 1] is barreled, bornologi ..."
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Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces X ∆ whose bounded sets are precisely the subsets of KH[0, 1] that are equiintegrable and pointwise bounded. The resulting inductive limit topology on � ∆ X ∆ = KH[0, 1] is barreled, bornological, and stronger than both pointwise convergence and the topology given by the Alexiewicz seminorm, but it lacks the countability and compatibility conditions that are often associated with inductive limits. 1 Introduction. This paper is concerned with KH[0, 1], the space of all functions f: [0, 1] → R that are KH integrable (also known as Kurzweil, Henstock, DenjoyPerron, gauge, nonabsolute, or generalized Riemann integrable). We emphasize that we are considering individual functions, whereas most of the related literature
Gerald B. Folland
"... spirit is always mathematical. This is very possibly the best "mathematics for the nonmathematician" book that I have seenand that includes popular (nontextbook) books that one would find in a general bookstore. This one doesn't lapse into the pitfalls often found in such books. It preaches th ..."
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spirit is always mathematical. This is very possibly the best "mathematics for the nonmathematician" book that I have seenand that includes popular (nontextbook) books that one would find in a general bookstore. This one doesn't lapse into the pitfalls often found in such books. It preaches the beauty and fascination of mathematics in the introduction and then follows through. It does not cut mathematical corners: Its explanations are complete and accurate, its theorems are stated precisely, and its proofs are intuitive and often colloquially presented without being sloppy. The book could also be of benefit to mathematicians. Teachers often say that they learn something new every time they teach a course, but they usually mean something subtle. I have to confess that I learned some nonsubtle things from reading this book. For example, I hadn't heard of Newcomb's Paradox or Conway tiles, and I hadn't realized that no one knows whether the builders of the Parthenon were consciou
Vector Spaces ∗
"... This paper considers ordered vector spaces with arbitrary closed cones and establishes a number of characterization results with applications to monotone comparative statics (Topkis (1978), Topkis (1998), Milgrom and Shannon (1994)). By appealing to the fundamental theorem of calculus for the Hensto ..."
Abstract
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This paper considers ordered vector spaces with arbitrary closed cones and establishes a number of characterization results with applications to monotone comparative statics (Topkis (1978), Topkis (1998), Milgrom and Shannon (1994)). By appealing to the fundamental theorem of calculus for the HenstockKurzweil integral, we generalize existing results on increasing differences and supermodularity for C 1 or C 2 functions. None of the results are based on the assumption that the order is Euclidean. As applications we consider a teamwork game and a monopoly union model.