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64
On the Boltzmann equation
 Arch. Rational Mech. Anal
, 1972
"... 1. Introduction Let \Omega ae Rn be a strictly convex domain with C1 boundary and inward normal ~n(x). Consider in \Omega the stationary, nonlinear Boltzmann equation for hard and soft forces with Grad's angular cutoff, ..."
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Cited by 42 (0 self)
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1. Introduction Let \Omega ae Rn be a strictly convex domain with C1 boundary and inward normal ~n(x). Consider in \Omega the stationary, nonlinear Boltzmann equation for hard and soft forces with Grad's angular cutoff,
A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN
, 1998
"... The aim of this paper is to provide a viable measuretheoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors ..."
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Cited by 26 (12 self)
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The aim of this paper is to provide a viable measuretheoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the `right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the KarhunenLoeve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abun...
Distributional properties of correspondences on Loeb spaces
 YENENG SUN Journal of Functional Analysis
, 1996
"... We present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a Loeb space, which include closedness, convexity, compactness, purification, and semicontinuity. We also note that all the properties reported in the main theorems are no ..."
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Cited by 10 (9 self)
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We present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a Loeb space, which include closedness, convexity, compactness, purification, and semicontinuity. We also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit Lebesgue interval. 1996 Academic Press, Inc. 1.
Noncooperative games on hyperfinite Loeb spaces
, 1999
"... We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large nonanonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of ..."
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Cited by 9 (5 self)
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We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large nonanonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such gametheoretic situations asymptotic implementability, homogeneity and measurabilityand argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so rep...
Integration Of Correspondences On Loeb Spaces
 TRANS. AM. MATH. SOC
, 1997
"... We study the Bochner and Gel # fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semic ..."
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Cited by 8 (6 self)
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We study the Bochner and Gel # fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space, we show that exact versions of these two results hold in the case we study. We also note that our results on a hyperfinite Loeb space capture the nature of the corresponding asymptotic results for the large finite case; but the unit Lebesgue interval fails to provide such a framework.
Strong nonstandard completeness for fuzzy logics
 Soft Comput
, 2008
"... In this paper we are going to introduce the notion of strong nonstandard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic C is strong nonstandard complete means that, for any countable theory ..."
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Cited by 8 (2 self)
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In this paper we are going to introduce the notion of strong nonstandard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic C is strong nonstandard complete means that, for any countable theory Γ over C and any formula ϕ such that Γ 6`C ϕ, there exists an evaluation e of Cformulas into a Calgebra A such that the universe of A is a nonArchimedean extension [0, 1]? of the real unit interval [0, 1], e is a model for Γ, but e(ϕ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinitevalued events are complete with respect to classes of models defined starting from nonstandard measures, that is measures taking value in [0, 1]?.
Ultraproducts in Analysis
 IN ANALYSIS AND LOGIC, VOLUME 262 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 2002
"... ..."
Numerosities of labelled sets: a new way of counting
 Advances in Math., 173
, 2003
"... Abstract. The notions of “labelled set ” and “numerosity ” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ..."
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Cited by 7 (1 self)
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Abstract. The notions of “labelled set ” and “numerosity ” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC. Introduction. Similarly as cardinals and ordinals, the “numerosities ” we present in this paper originate as an attempt to extending the notion of finite cardinality. By considering suitable “labellings”, we show that a notion of numerosity for (countable) infinite sets can be defined in such a way that the usual properties of finite cardinalities
The Complete Removal of Individual Uncertainty: Multiple Optimal Choices and Random Exchange Economies
, 1999
"... this paper is to develop some measuretheoretic methods for the study of large economic systems with individualspecific randomness and multiple optimal actions. In particular, for a suitably formulated continuum of correspondences, an exact version of the law of large numbers in distribution is cha ..."
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Cited by 6 (2 self)
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this paper is to develop some measuretheoretic methods for the study of large economic systems with individualspecific randomness and multiple optimal actions. In particular, for a suitably formulated continuum of correspondences, an exact version of the law of large numbers in distribution is characterized in terms of almost independence, which leads to several other versions of the law of large numbers in terms of integration of correspondences. Widespread correlation due to multiple optimal actions is also shown to be removable via a redistribution. These results allow the complete removal of individual risks or uncertainty in economic models where nonunique best choices are inevitable. Applications are illustrated through establishing stochastic consistency in general equilibrium models with idiosyncratic shocks in endowments and preferences. In particular, the existence of "global" solutions preserving microscopic independence structure is shown in terms of competitive equilibria for the cases of divisible and indivisible goods as well as in terms of core for a case with indivisible goods where a competitive equilibrium may not exist. An important feature of the idealized equilibrium models considered here is that standard results on measuretheoretic economies are now directly applicable to the case of random economies. Some asymptotic interpretation of the results are also discussed. It is also pointed out that the usual unit interval [0, 1] can be used as an index set in our setting, provided that it is endowed together with some sample space a suitable larger measure structure.
Finite Models of Elementary Recursive Nonstandard Analysis
, 1996
"... This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in [2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Recursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERN ..."
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This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in [2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Recursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERNA proves the main axioms of the Chuaqui and Suppes system. Then a finitary consistency proof of ERNA will be given; in particular, we will show that PRA, the system of primitive recursive arithmetic, which is generally recognized as capturing Hilbert's notion of finitary, proves the consistency of ERNA. From the consistency proof we can extract a constructive method for obtaining finite approximations of models of nonstandard analysis. We present an isomorphism theorem for models that are finite substructures of infinite models. 1 Introduction This paper continues and extends the development of a constructive system of nonstandard analysis begun by Chuaqui and Suppes in [2, 9]. The ...