Results 1  10
of
24
Elementary NonArchimedean Representations of Probability for Decision Theory and Games
 Suppes: Scientific Philosopher, Vol. I: Probability and Probabilistic Causality
, 1994
"... 1992 version is intended as a contribution to a two volume collection honouring Patrick Suppes, to be edited by Paul Humphreys and published by Kluwer Academic Publishers. ABSTRACT. In an extensive form game, whether a player has a better strategy than in a presumed equilibrium depends on the other ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
1992 version is intended as a contribution to a two volume collection honouring Patrick Suppes, to be edited by Paul Humphreys and published by Kluwer Academic Publishers. ABSTRACT. In an extensive form game, whether a player has a better strategy than in a presumed equilibrium depends on the other players ’ equilibrium reactions to a counterfactual deviation. To allowconditioning on counterfactual events with prior probability zero, extended probabilities are proposed and given the four equivalent characterizations: (i) complete conditional probability systems; (ii) lexicographic hierarchies of probabilities; (iii) extended logarithmic likelihood ratios; and (iv) certain ‘canonical rational probability functions ’ representing ‘trembles ’ directly as infinitesimal probabilities. However, having joint probability distributions be uniquely determined by independent marginal probability distributions requires general probabilities taking values in a space no smaller than the nonArchimedean ordered field whose members are rational functions of a particular infinitesimal. Elinor now found the difference between the expectation of an unpleasant event, however certain the mind may be told to consider it, and certainty itself. — Jane Austen, Sense and Sensibility, ch. 48.... a more attractive and manageable theory may result from a nonArchimedean representation.... One must keep in mind the fact that the refutability of axioms depends both on their mathematical form and their empirical interpretation. — Krantz, Luce, Suppes and Tversky (1971, p. 29).
Calculus and Numerics on LeviCivita Fields
, 1996
"... The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many cases, the ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many cases, the application of "infinitely small" numbers instead of "small but finite" numbers allows the use of the old numerical algorithm, but now with an error that in a rigorous way can be shown to become infinitely small (and hence irrelevant). While intuitive ideas in this direction have accompanied analysis from the early days of Newton and Leibniz, the first rigorous work goes back to LeviCivita, who introduced a number field that in the past few years turned out to be particularly suitable for numerical problems. While LeviCivita's field appears to be of fundamental importance and simplicity, efforts to introduce advanced concepts of calculus on it are only very new. In this paper, we address s...
Macneille completions and canonical extensions
 Transactions of the American Mathematical Society
"... Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical exten ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure. 1.
Neometric Spaces
, 1996
"... this paper is quite the opposite: we work within nonstandard analysis to formally prove, as promised in [9], that rich adapted spaces exist. Moreover, we explicitly show how nonstandard analysis provides the inspiration for the main notions and results presented in [9]. The paper [9] gives a large n ..."
Abstract

Cited by 12 (9 self)
 Add to MetaCart
this paper is quite the opposite: we work within nonstandard analysis to formally prove, as promised in [9], that rich adapted spaces exist. Moreover, we explicitly show how nonstandard analysis provides the inspiration for the main notions and results presented in [9]. The paper [9] gives a large number of applications of rich adapted spaces to probability theory. Nonstandard analysis gives us tools to dig deeper into the structure of subsets of metric spaces. The purpose of this paper is to refine these tools. The contents of this paper are as follows. In Section 2 we review the basic notions concerning neometric spaces from [9]. In Sections 3 and 4 we study these notions within the nonstandard setting. We give an explicit definition of basic and neocompact sets that captures the way internal sets are used in nonstandard probability practice, and then present a huge neometric family which contains all neometric spaces studied so far. In Section 5 we prove that the huge neometric family contains rich adapted spaces, and hence that rich adapted spaces exist. Section 6 gives a detailed study of the function spaces related to the existence theorems in [9]. In Section 7 we consider neometric spaces whose elements are functions from a probability space into another neometric space. We finish up in Section 8 with a stronger theory of neometric spaces which requires a saturated nonstandard universe where is a cardinal greater than ! 1 . In other publications we will present other aspects of our work. In [10] we develop the logical aspects that are behind our results. In [11] we discuss another nonstandard approach to neometric families which uses long sequences. The paper [17] gives a general quantifier elimination result showing that for many neometric families, every n...
Automatic Differentiation as Nonarchimedean Analysis
, 1992
"... It is shown how the techniques of automatic differentiation can be viewed in a broader context as an application of analysis on a nonarchimedean field. The rings used in automatic differentiation can be ordered in a natural way and form finite dimensional real algebras which contain infinitesimals. ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
It is shown how the techniques of automatic differentiation can be viewed in a broader context as an application of analysis on a nonarchimedean field. The rings used in automatic differentiation can be ordered in a natural way and form finite dimensional real algebras which contain infinitesimals. Some of these algebras can be extended to become a Cauchycomplete realclosed nonarchimedean field, which forms an infinite dimensional real vector space and is denoted by L. On this field, a calculus is developed. Rules of differentiation and certain fundamental theorems are discussed. A remarkable property of differentiation is that difference quotients with infinitely small differences yield the exact derivative up to an infinitely small error. This is of historical interest since it justifies the concept of derivatives as differential quotients. But it is also of practical relevance; it turns out that the algebraic operations used to compute derivatives in automatic differentiation are...
Nonstandard hulls of BanachLie groups and algebras
 Nova J. Algebra Geom
, 1992
"... Abstract. We propose a new construction of BanachLie groups and algebras via nonstandard analysis. A major “standard ” application is the Local Theorem which to certain extent reduces the problem of associating a Lie group to a given BanachLie algebra to a similar problem for finitely generated Li ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We propose a new construction of BanachLie groups and algebras via nonstandard analysis. A major “standard ” application is the Local Theorem which to certain extent reduces the problem of associating a Lie group to a given BanachLie algebra to a similar problem for finitely generated Lie subalgebras.
An operator equation and relativistic alterations in the time for radioactive decay
 Intern. J. Math. Math. Sci
"... ABSTRACT. In this paper, using concepts from the nonstandard physical world, the linear effect line element is derived. Previously, this line element was employed to obtain, with the exception of radioactive decay, various experimentally verified special theory relativistic alterations in physical m ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
ABSTRACT. In this paper, using concepts from the nonstandard physical world, the linear effect line element is derived. Previously, this line element was employed to obtain, with the exception of radioactive decay, various experimentally verified special theory relativistic alterations in physical measures. This line element is now used to derive, by means of separation of variables, an expression that predicts the same increase in the decay time for radioactive material as that predicted by the Einstein time dilation assumption. This indicates that such an increase in lifetime can be attributed to an interaction of the radioactive material with the subparticle field. Key Words and Phrases. Special relativity, separation of variables, radioactive decay, time dilation, nonstandard analysis, subparticle field. 1992 AMS Subject Classifications. 83A05, 03H10. 1. Introduction. In [7], a specific operator equation related to a partial differential equation, the Schwarzschild and linear effect line elements, and the concept of separation of
Analysis on the LeviCivita field, a brief overview
 CONTEMPORARY MATH., VOLUME 508,
, 2010
"... ..."
COMPACT LINEAR OPERATORS, A SURVEY
"... Abstract. In this paper we approach the Compact Linear Operators Theory by the methods of Nonstandard Analysis. We present new proofs of several known results and some new results as well. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. In this paper we approach the Compact Linear Operators Theory by the methods of Nonstandard Analysis. We present new proofs of several known results and some new results as well.
NONSTANDARD DISCRETE DERIVATIVES AND EXISTENCE THEOREMS FOR ODE
"... Abstract. We present nonstandard generalizations of Peano’s and Carathéodory’s Existence ..."
Abstract
 Add to MetaCart
Abstract. We present nonstandard generalizations of Peano’s and Carathéodory’s Existence