An L{structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in L are internal. A nonstandard universe is said to satisfy the {isomorphism property if for any two internally presented L{structures A and B, where L has less than many symbols, A is elementarily equivalent to B implies that A is isomorphic to B. In this paper we prove t...

An anomaly in circuit theory is the disappearance of some of the stored energy when two capacitors, one charged and the other uncharged, are connected together through resistanceless wires. Nonstandard analysis shows that, when the wires are taken to have infinitesimally small but nonzero resistance, the energy dissipated in the wires equals that substantial amount of energy that had disappeare...

In this paper a proof of the Jordan curve theorem will be presented. Some familiarity with the basic notions of nonstandard analysis is assumed. The rest of the paper is selfcontained except for some standard theorems about polygons. The theorem will be proved in what ought to be a natural way: by approximation by polygons. This method is not usually found in the standard proofs since the appro...

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Patrick Suppes and Richard Sommer, who also proved its consistency inside PRA. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes, of which Michal Rössler and Emil Jeřábek have recently proposed a weakened version. We add a Π1-transfer ...

Noises: a nonstandard analysis. Thanks to the nonstandard formalization of fast oscillating functions, due to P. Cartier and Y. Perrin, an appropriate mathematical framework is derived for new non-asymptotic estimation techniques, which do not necessitate any statistical analysis of the noises corrupting any sensor. Various applications are deduced for multiplicative noises, for the length of t...

In the past few decades, nonstandard methods, as a branch of mathematical logic, have been successfully applied to obtain new results in additive/combinatorial number theory (cf. [BJ, Ji1, Ji2, Ji3, Ji4, Ji5, Ji6, JK, Le1, Le2, Le3, Le4]). Although the nonstandard techniques in these applications are elementary from a nonstandard analyst’s point of view, it is extremely difficult for a reader w...

this paper concerns with the modeling and construction of a fifth order method for two dimensional acoustic wave equation in heterogenous media. the method is based on a standard discretization of the problem on smooth regions and a nonstandard method for nonsmooth regions.the construction of the nonstandard method is based on the special treatment of the interface using suitable jump condition...