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  1.  34
    Topometric Spaces and Perturbations of Metric Structures.Itaï Ben Yaacov - 2008 - Logic and Analysis 1 (3-4):235-272.
    We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development in Ben Yaacov I and Usvyatsov A, Continuous first order logic and local stability. Trans Am Math Soc, (...)
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  2.  8
    Noetherian Varieties in Definably Complete Structures.Tamara Servi - 2008 - Logic and Analysis 1 (3-4):187-204.
    We prove that the zero-set of a C ∞ function belonging to a noetherian differential ring M can be written as a finite union of C ∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ∞ functions over the reals, but more generally for definable C (...)
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  3.  22
    Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis.Todor D. Todorov & Hans Vernaeve - 2008 - Logic and Analysis 1 (3-4):205-234.
    We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between (...)
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  4.  8
    Asymptotics of Families of Solutions of Nonlinear Difference Equations.Imme P. Berg - 2008 - Logic and Analysis 1 (2):153-185.
    One method to determine the asymptotics of particular solutions of a difference equation is by solving an associated asymptotic functional equation. Here we study the behaviour of the solutions in an asymptotic neighbourhood of such individual solutions. We identify several types of attraction and repulsion, which range from almost orthogonality to almost parallelness. Necessary and sufficient conditions for these types of behaviour are given.
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  5.  20
    Nonlinear Stochastic Integrals for Hyperfinite Lévy Processes.Tom Lindstrøm - 2008 - Logic and Analysis 1 (2):91-129.
    I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form $\sum_{s=0}^t\phi(\omega,dl_{s},s)$ and $\prod_{s=0}^t\psi(\omega,dl_{s},s)$ , where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps (...)
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  6.  18
    On the Constructive Dedekind Reals.Robert S. Lubarsky & Michael Rathjen - 2008 - Logic and Analysis 1 (2):131-152.
    In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy (...)
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  7.  5
    Asymptotics of Families of Solutions of Nonlinear Difference Equations.Imme P. van den Berg - 2008 - Logic and Analysis 1 (2):153-185.
  8.  6
    Introducing.Nigel Cutland - 2008 - Logic and Analysis 1 (1).
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