Applications Of Nonstandard Analysis Research Articles

Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)

The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.

Applications of Nonstandard Analysis in Additive Number Theory

AbstractThis paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

Nonstandard extensions of uniform algebraic systems

Important applications of nonstandard analysis to the theory of Banach spaces are based on the construction of the nonstandard hull of a normed linear space [1,2]. By making use of the iterated nonstandard enlargements [3], in the present article we propose a universal construction for arbitrary uniform algebraic systems which allows one to study the nonstandard hulls and completions of such systems.

Infinitesimals, Microsimplexes and Elementary Homology Theory

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsRade T. ŽivaljevićR. T. Živaljević was born in 1954 in Sarajevo, Yugoslavia. He studied at the University of Belgrade, where he obtained both his Master's degree in combinatorial geometry and Ph.D. degree in nonstandard analysis under the supervision of Milosav Marjanović. His second (and last) Ph.D. degree (1985) was obtained in nonstandard analysis under the supervision of H. Jerome Keisler from the University of Wisconsin, Madison. Generally interested in applications of nonstandard analysis, he is a secret admirer of algebraic topology.

Applications of non-standard analysis to relativistic quantum mechanics. I

Some representations of the Dirac delta function are considered including a new representations. A new theory of Fourier transforms is developed which is better suited to use in physics than the standard theory. The work is of general interest as well as of relevance to subsequent articles. The authors then give a brief outline of the construction of field theory from quantum mechanics as facilitated by non-standard analysis and a theorem which enables the calculation of a cross section from plane wave states.

Application of nonstandard analysis to quantum mechanics

Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.