The book    

 

The book presents a new type of arithmetic that allows us to represent infinite and infinitesimal numbers by a finite number of symbols and to execute arithmetical operations with them in the same manner as we are used to do with finite ones. The new approach is not related to the non-standard analysis and the problem of infinity is considered in the book in a coherent way different from those proposed by Georg Cantor, Abraham Robinson, and John Conway. However, the new approach does not contradict Cantor, it evolves his theory and supplies new more powerful tools to deal with infinite and infinitesimal quantities.   From the methodological point of view, the principle of Ancient Greeks ‘The part is less than the whole’ is adopted and applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new positional system with the infinite radix introduced in the book allows one to consider infinite, finite, and infinitesimal numbers as particular cases of a unique framework. The new viewpoint gives detailed answers to many questions and paradoxes regarding infinite and infinitesimal quantities. Particularly, applications of the new approach to limit theory, measure theory, and set theory are given.   The book is mainly addressed to mathematicians, computer scientists, philosophers, physicists, and students. However, it is written in a popular way in order to allow any person having a high school education and interests in the foundations of these sciences to understand it easily.

Book Reviews published in International Scientific Journals ·    Computational Management Science, 2007, vol. 4(1), pages 85-86, written by Prof. D. Trigiante ·    Cybernetics & Systems Analysis, 2006, vol. 42(6), page 183, written by Prof. V.P. Shylo ·    Optimization Methods and Software, 2006, vol. 21(6), pages 995–996, written by Prof. O. Prokopyev ·    International Journal of Unconventional Computing, 2006, vol. 2(2), pages 193-194, written by the Editor-in-Chief Prof. A. Adamatzky ·    Journal of Global Optimization, 2006, vol. 34, pages 157–158, written by the Editor-in-Chief Prof. P.M. Pardalos

The patent 

 

The main difference of the new approach with respect to non-standard analysis theories is its strong computational character opening new exiting area in the theory and practice of computations – Infinity Computing. In 2004 the author has filed the following international patent “Computer system  for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them” describing the Infinity Computerable to execute computations with infinite, finite, and infinitesimal numbers.

The Infinity Computer technology has been used to simulate the first prototype of the Infinity Calculator able to execute arithmetical operations with numbers having finite parts and/or infinitesimal and infinite parts of different orders. Here you can see a screenshot of the Infinity Calculator with an example of division and a brief comment.

About the author                                                                            

 

Yaroslav D. Sergeyev holds a Full Professorship reserved for distinguished scientists at the University of Calabria, Rende, Italy. He is also a part-time Professor at the N.I. Lobatchevsky State University, Nizhni Novgorod, Russia and Affiliated Researcher at the Institute of High Performance Computing and Networking. He was awarded his Ph.D. (1990) from the N.I. Lobatchevsky State University and his D.Sc. degree from the M.V. Lomonosov State University of Moscow (1996). His research interests include numerical analysis, parallel computations, and number theory. His list of publications contains more than 150 items including three books (the first has been published by Nizhni Novgorod University Press, the second by Kluwer Academic Publishers) and numerous papers in prestigious international journals. Additional information about the author can be found here.

Presentations

 

Results described in the book and in the patent have been presented at the following congresses:

 

·  tutorial, the 48th international workshop on Nonsmooth Analysis, Optimization and Applications, Erice, Italy, under the auspices of the International School of Mathematics “G. Stampacchia”,  May, 9–17, 2008;

·  invited plenary lecture, International Conference of Numerical Analysis and Applied Mathematics ICNAAM 2007, September 16-20, 2007;

·  invited lecture, the 8th Mediterranean Workshop and Topical Meeting "Novel Optical Materials and Applications", Cetraro, Italy, June 3-9, 2007;

·  tutorial, the International Workshop on Learning and Intelligent Optimization, Andalo, Italy, February 12-18, 2007;

·  invited one hour lecture, the 15th Euromicro Conference on Parallel, Distributed and Network-based Processing, Naples, February 7-9, 2007;

·  invited plenary lecture opening the MATHESIS Congress, Trento, Italy, November 2-4, 2006;

·  invited plenary lecture, the Third International Conference of Applied Mathematics, Plovdiv, Bulgaria, August 12-18, 2006;

·  invited plenary lecture, International Conference “Applied Optimization and Metaheuristic Innovations”, Yalta, Ukraine, July 17-23, 2006;

·  invited lecture, the second International Workshop on Variational Analysis and Partial Differential Equations, Erice, Italy, under the auspices of the International School of Mathematics “G. Stampacchia”, July 5-14, 2006;

·  round table “Infinity Computer and Calculus”, the 8th Congress of SIMAI (La Societa' Italiana di Matematica Applicata e Industriale), Ragusa (Sicily), May 22-26, 2006;

·  invited plenary lecture, the XXI Autumn Meeting of the Polish Information Processing Society, Katowice, Poland, December 5-9, 2005;

·  invited plenary lecture, the International Workshop on Global Optimization, Almeria, Spain, September 18-22, 2005;

·  tutorial, the International Conference on Complementarity, Duality, and Global Optimization in Science and Engineering, Blacksburg, Virginia, USA, August 15-17, 2005;

·  invited plenary lecture, International Conference on Difference Equations, Special Functions and Applications, Munich, July 25-30, 2005;

·  invited plenary lecture, International Conference “Numerical Analysis: the State of the Art”, Rende (CS), Italy, May, 19-21, 2005;

·  invited semi-plenary lecture at the International conference on Selected Problems of Modern Mathematics, dedicated to the 200th anniversary of K.G. Jacobi and the 750th anniversary of the Koenigsberg foundation, Kaliningrad, April, 4-8, 2005;

·  tutorial, the Workshop "Numerical Methods and Mathematical Software", Montecatini Terme (PT), January 31 - February 1, 2005;

·  invited plenary lecture, the VI-th International Congress on Mathematical Modelling, Nizhni Novgorod, Russia, September, 21-26, 2004;

·  opening invited lecture, the congress “Infinity in Mathematics, Physics, and Philosophy”, Pisa, Italy, 26.03.2004.

 

Results described in the book and in the patent have been also presented at the following universities, research institutes, and organizations:

 

Universities

Research Institutes, Academies, and Organizations

Aberdeen, Amsterdam, Bari, Bologna, Cagliari, Calabria, Cardiff, Catania, Ferrara, Florence, Kaliningrad, Milano, Modena, Moscow “M.V. Lomonosov”, Nizhni Novgorod “N.I. Lobatchevsky”, Padua, Palermo, Pisa, Reggio Calabria, Rome “La Sapienza”, Salerno, Siena, St. Petersburg, Trento, and Trieste

Russian Academy of Education, Moscow; Catania Astrophysical Observatory of the Italian National Institute of Astrophysics; Bologna Academy of Sciences; Italian Institute of Philosophical Studies, Naples; Moscow philosophical seminar “Philosophy of mathematics”; Institute of Control Sciences of the Russian Academy of Sciences, Moscow; Computing Centre of the Russian Academy of Sciences, Moscow; Institute of Computer Science of the Academy of Sciences of the Czech Republic, Prague; Institute of Cybernetics and Institute of Mathematics, both of the National Ukrainian Academy of Sciences, Kiev; Nizhni Novgorod Mathematical Society; Guarasci Foundation, Cosenza; Institute for Mathematical Modelling and Institute of Numerical Mathematics, both of the Russian Academy of Sciences, Moscow; Institute for High Performance Computing and Networking, Rende, and Institute of Applied Mathematics and Information Technology, Pavia, both of the Italian National Research Council