Non-Euclidean Geometry in Russian Art History: On a Little-Known Application of a Scientific Theory

The author has previously proposed that there are at least six different definitions of “reverse” or “inverse” perspective, i.e. the principle of organizing pictorial space in the icon. Reverse perspective is still a largely unresolved art historical problem. The author focuses on one of the six defi nitions, the one least familiar to Western scholars—namely, the view, common in Russian art-historical writing at the beginning of the twentieth century, that space in the icon is a visual analogue of non-Euclidean geometry. Russian mathematician-turned-theologian and priest Pavel Florensky claimed that the space of the icon is that of non-Euclidean geometry and truer to the way human vision functions. The author considers the scientifi c validity of Florensky's claim.

Sofia Vasílyevna Kovalévskaya (1850–1891)

The fascinating life of Russian mathematician Sofia Kovalévskaya is narrated in this chapter, along with her problems in being accepted as an equal in the academic world.

Love and Mathemetics. Sofia Kovalevska – Science and Struggle for Happiness

The author follows the professional successes and the personal experience of the one of the first women in the modern science – the Russian mathematician Sofia Kovalevska. The dialogue between her strong will, energy and intellect in the male world of science and her delicate sensibility and deep emotionality in her literary works has been analysed and the conflict points has been outlined. The author tries to answer to the question why despite of her professional successes Sofia Kovalevska felt unhappy, why the question of happiness is key question in her literary works and memoirs. Due to the tension between the profession-al and personal discourses a sense of polyphony and lack of satisfaction is marking her narratives. An attempt has been made to place Kovalevska in the trends of the Russian feminism in the second half of the 19th century.

On Insolvability of the 4-th Hilbert Problem for Hyperbolic Geometries

The article proves the insolvability of the 4-th Hilbert Problem for hyperbolic geometries. It has been hypothesized that this fundamental mathematical result (the insolvability of the 4-th Hilbert Problem) holds for other types of non-Euclidean geometry (geometry of Riemann (elliptic geometry), non-Archimedean geometry, and Minkowski geometry). The ancient Golden Section, described in Euclid’s Elements (Proposition II.11) and the following from it Mathematics of Harmony, as a new direction in geometry, are the main mathematical apparatus for this fundamental result. By the way, this solution is reminiscent of the insolvability of the 10-th Hilbert Problem for Diophantine equations in integers. This outstanding mathematical result was obtained by the talented Russian mathematician Yuri Matiyasevich in 1970, by using Fibonacci numbers, introduced in 1202 by the famous Italian mathematician Leonardo from Pisa (by the nickname Fibonacci), and the new theorems in Fibonacci numbers theory, proved by the outstanding Russian mathematician Nikolay Vorobyev and described by him in the third edition of his book “Fibonacci numbers”.

From Butterflies to Hurricanes

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the Kolmo–Arnol–Mos (KAM) theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.

THE FIRST "LOST" INTERNATIONAL CONFERENCE ON NONLINEAR OSCILLATIONS (I.C.N.O.)

From January 28 to 30, 1933, was held at the Institut Henri Poincaré (Paris) the first International Conference of Nonlinear Oscillations organized at the initiative of the Dutch physicist Balthazar Van der Pol and of the Russian mathematician Nikolaï Dmitrievich Papaleksi. The discovery of this forgotten event, whose virtually no trace remains, was made possible thanks to the report written by Papaleksi on his return to USSR. This document has revealed, both the list of participants who included French mathematicians — Alfred Liénard, Élie and Henri Cartan, Henri Abraham, Eugène Bloch, Léon Brillouin, Yves Rocard — and, the content of presentations and discussions. The analysis of the minutes of this conference presented here for the first time highlights the role and involvement of the French scientific community in the development of the theory of nonlinear oscillations.

Works of A.A. Samarskii on Computational Mathematics

This is a review of the main results in computational mathematics that were obtained by the eminent Russian mathematician Alexander Andreevich Samarskii (February 19, 1919 – February 11, 2008). His outstanding research output addresses all the main questions that arise in the construction and justification of algorithms for the numerical solution of problems from mathematical physics. The remarkable works of A.A. Samarskii include statements of the main principles re- quired in the construction of difference schemes, rigorous mathematical proofs of the stability and convergence of these schemes, and also investigations of their algorith- mic implementation. A.A. Samarskii and his collaborators constructed and applied in practical calculations a large number of algorithms for solving various problems from mathematical physics, including thermal physics, gas dynamics, magnetic gas dynam- ics, plasma physics, ecology and other important models from the natural sciences.