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In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and nonstandard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...) 

In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The part is less than the whole’. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a compute  the Infinity Computer – introduced recently in [18]. The new approach does not (...) 

There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer  the Infinity Computer  able to work numerically with finite, infinite, and infinitesimal number. It is proved that the Infinity Computer is able (...) 

The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multitape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the instrument (...) 