In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing  a few applications of this approach with the former methods.

All the existing computers are able to do arithmetical operations only with finite numerals. Operations with infinite and infinitesimal quantities could not be realized. The paper describes a new positional system with infinite radix allowing us to write down finite, infinite, and infinitesimal numbers as particular cases of a unique framework. The new approach both gives possibilities to do ca...

An infinitesimal theory for bending and earthquaking in onedimensional dynamics is developed. It is shown that any tangent vector to Teichmuller space is the initial data for a bending and for an earthquaking ordinary differential equation. The discussion involves an analysis of infinitesimal bendings and earthquakes, the Hilbert transform, natural bounded linear operators from a Banach space o...

By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type A may be transformed into modules over Lusztig’s infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we f...

In this paper we study the infinitesimal-time case of the so called multilinear constraints that exist for each subsequence in a sequence of images. These constraints link the infinitesimal motion of the image points with the infinitesimal viewer motion. The analysis is done both for calibrated and uncalibrated cameras. Two simplifications are also presented for the uncalibrated camera case. On...

Optimal robust M-estimates of a multidimensional parameter are described using Hampel’s infinitesimal approach. The optimal estimates are derived by minimizing a measure of efficiency under the model, subject to a bounded measure of infinitesimal robustness. To this purpose we define measures of efficiency and infinitesimal sensitivity based on the Hellinger distance. We show that these two mea...