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Register a new userOn the history of ring geometry (with a thematical overview of literature)
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Dirk Keppens
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In this survey paper we give an historical and at the same time thematical overview of the development of ring geometry from its origin to the current state of the art. A comprehensive up-to-date list of literature is added with articles that treat ring geometry within the scope of incidence geometry.
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We present some episodes from the history of interactions between geometry and physics over the past century.
Appearing in 1921 as an equation for small-amplitude waves on the surface of an infinitely deep liquid, the Nekrasov equation quickly became a source of new results. This manifested itself both in the field of mathematics (theory of nonlinear integral equations of A.I. Nekrasov; 1922, later - of N.N. Nazarov; 1941), and in the field of mechanics (transition to a fluid of finite depth - A.I. Nekrasov; 1927 and refusal on the smallness of the wave amplitude - Yu.P. Krasovskii; 1960).The main task of the author is to find out the prehistory of the Nekrasov equation and to trace the change in approaches to its solution in the context of the nonlinear functional analysis development in the 1940s - 1960s. Close attention will be paid to the contribution of European and Russian mathematicians and mechanics: A.M. Lyapunov, E. Schmidt, T. Levi-Civita, A. Villat, L. Lichtenstein, M.A. Krasnoselskii, N.N. Moiseev, V.V. Pokornyi, etc. In the context of the development of qualitative methods for the Nekrasov equation investigating, the question of the interaction between Voronezh school of nonlinear functional analysis under the guidance of Professor M.A. Krasnoselskii and Rostov school of nonlinear mechanics under the guidance of Professor I.I. Vorovich.
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals. Some expressions which are (amazingly) related to pi have been evaluated. In addition, a continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits. The development of powerful computers has played a fundamental role in these achievements of calculus.
Felix Kleins so-called Erlangen Program was published in 1872 as professoral dissertation. It proposed a new solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory. The given translation was made in 1892 by Dr. M. W. Haskell and transcribed by N. C. Rughoonauth. We replaced bibliographical data in text and footnotes with pointers to a complete bibliography section.
Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchys infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchys work challenges received views on Cauchys role in the history of analysis and geometry. We demonstrate the viability of Cauchys infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson
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