Model Approach to Nonstandard Analysis in the Context of Axiomatic Set Theory

In this paper, the model approach to nonstandard analysis is developed on the basis of Zermelo—Fraenkel axiomatic set theory with atoms. The traditional consideration of the standard superstructure V as the primary object of nonstandard analysis is justified. Set-theoretic axioms for the nonstandard system *V are obtained.     

Intuitionistic nonstandard bounded modified realisability and functional interpretation

We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles.The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability.We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments.     

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.     

Global gradient estimates for divergence-type elliptic problems involving general nonlinear operators

We study nonlinear elliptic problems with nonstandard growth and ellipticity related to an N-function. We establish global Calderón–Zygmund estimates of the weak solutions in the framework of Orlicz spaces over bounded non-smooth domains. Moreover, we prove a global regularity result for asymptotically regular problems which are getting close to the regular problems considered, when the gradient variable goes to infinity.     

A nonstandard proof of the Eberlein-Smulian theorem

The Eberlein-Smulian theorem on the equivalence of weak compactness and the finite intersection property of bounded closed convex sets is given a short elementary proof by applying Abraham Robinson's nonstandard characterization of compactness.

     

Alpha-theory: An elementary axiomatics for nonstandard analysis

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.     

Relative arithmetic

In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y ‐standard’. Thus, objects are not (non)standard in an absolute sense, but (non)standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A trial-free-boundary method for computing Batchelor flows

The solutions of Batchelor flows in bounded domains are computed by a nonstandard trial-free-boundary method.     

Nonstandard integration theory in topological vector lattices

This paper develops a Daniell-Stone integration theory in topological vector lattices. Starting with an internal, vector valued, positive linear functionalI on an internal lattice of vector valued functions, we produce a nonstandard hull valued integralJ satisfying the monotone convergence theorem. Nonstandard hulls form a natural extension of infinite dimensional spaces and are equivalent to Banach space ultrapower constructions. The first application of our integral is a construction of Banach limits for bounded, vector valued sequences. The second example yields an integral representation for bounded and quasibounded harmonic functions similar to that of the Martin boundary. The third application uses our general integral to extend the Bochner integral.     

Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ$ \mathbb{F}\mathfrak{M} $ \mathbb{F}\mathfrak{M} over a model $ \mathfrak{M} $ \mathfrak{M} . We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.     

Hyperfinite stochastic integration for Lévy processes with finite-variation jump part

This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part.As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm.     

Controlling predator-prey discrete dynamics utilizing a threshold policy with hysteresis

This paper introduces a threshold policy with hysteresis (TPH) for the control of the logistic one-species model, the Lotka-Volterra and Rosenzweig-MacArthur two species density-dependent predator-prey models. A nonstandard scheme is used for the discretization of the models since it results in preservation of the qualitative characteristics of the continuous-time models. Two theorems that establish the global stability of the discrete logistic model subject to the threshold policy (TP) and the TPH are proved. The proposed policy (TPH) is more realistic than a pure threshold policy (TP) proposed earlier in the literature and changes the dynamics of the system in such a way that a low amplitude bounded oscillation, far from the extinction region, is achieved. Furthermore, it can be designed by a suitable choice of so called virtual equilibrium points in a simple and intuitive manner.     

Divergence L 2-Coercivity Inequalities

This paper derives sharp L ²-coercivity inequalities for the divergence operator on bounded Lipschitz regions in ? ⁿ . They hold for fields in H(div,Ω) that are orthogonal to N(div). The optimal constants in the inequality are defined by a variational principle and are identified as the least eigenvalue of a nonstandard boundary value problem for a linear biharmonic type operator. The dependence of the optimal constant under dilations of the region is described and a generalization that involves weighted surface integrals is also proved. When n = 2, this also yields a similar coercivity result for the curl operator.     

Transfer principles in nonstandard intuitionistic arithmetic

 Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. Received: 7 January 2000 / Revised version: 26 March 2001 / Published online: 12 July 2002     

Equilibria with nonstandard prices in vector lattice overlapping generations economies

The paper investigates overlapping generations (OLG) economies in vector lattices framework. Agents' preferences are assumed uniformly proper, though they may be nontransitive and incomplete. Existence is stated for the “equilibrium with nonstandard prices,” a notion that may be looked upon as a particular (or generalized, in other aspect) case of known “compensated equilibria” of OLG-economies. The difference is that compensated values are described via explicit formula given in nonstandard analysis terms. This approach enables more clear economic interpretation and shows some new properties of compensated values, such as their linearity over agents' endowments. Also it allows easy to prove the existence of equilibria under classical additional assumptions on agents' endowments.     

A Poincar�� Inequality for Orlicz�CSobolev Functions with Zero Boundary Values on Metric Spaces

We prove a Poincaré inequality for Orlicz–Sobolev functions with zero boundary values in bounded open subsets of a metric measure space. This result generalizes the (p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality for Orlicz–Sobolev functions on a Euclidean space, proved by Fuchs and Osmolovski (J Anal Appl (Z.A.A.) 17(2):393–415, 1998). Using the Poincaré inequality for Orlicz–Sobolev functions with zero boundary values we prove the existence and uniqueness of a solution to an obstacle problem for a variational integral with nonstandard growth.     

Existence and uniqueness results for a class of quasi-hemivariational inequalities

In this paper we are concerned with the study of a nonstandard quasi-hemivariational inequality. Using a fixed point theorem for set-valued mappings the existence of at least one solution in bounded closed and convex subsets is established. We also provide sufficient conditions for which our inequality possesses solutions in the case of unbounded sets. Finally, the uniqueness and the stability of the solution are analyzed in a particular case.     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.     

The Global Optimization of Variational Problems with Discontinuous Solutions

We consider variational problems in which the slope of the admissible curves is not necessarily bounded, so that they admit discontinuous solutions. A problem is first reformulated as one consisting of the minimization of an integral in a space of functions satisfying a set of integral equalities; this is then transfered to a nonstandard framework, in which Loeb measures take the place of the functions and a near-minimizer can always be found. This is mapped back to the standard world by means of the standard part map; its image is a minimizer, so that the optimization is global. The minimizer is shown to be the solution of an infinite dimensional linear program and by well-proven approximation procedures a finite dimensional linear program is found by means of which nearly-optimal curves can be constructed for the original problem. A numerical example is given.