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Register a new userA comparative review of recent researches in geometry
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Felix C. Klein
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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.
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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
A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete Mobius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.
The invariant mass method is used to identify the $^8$Be and $^9$B nuclei and Hoyle state formed in dissociation of relativistic nuclei in a nuclear track emulsion. It is shown that to identify these extremely short-lived states in the case of the isotopes $^9$Be, $^{10}$B, $^{10}$C, $^{11}$C, $^{12}$C, and $^{16}$O, it is sufficient to determine the invariant mass as a function of the angles in pairs and triples of He and H fragments in the approximation of the conservation of momentum per nucleon of the parent nucleus. According to the criteria established in this way, the contribution of these three unstable states was evaluated in the relativistic fragmentation of the $^{28}$Si and $^{197}$Au nuclei.
Due to the potential impact on the diagnosis and treatment of various cardiovascular diseases, work on the rheology of blood has significantly expanded in the last decade, both experimentally and theoretically. Experimentally, blood has been confirmed to demonstrate a variety of non-Newtonian rheological characteristics, including pseudoplasticity, viscoelasticity, and thixotropy. New rheological experiments and the development of more controlled experimental protocols on more extensive, broadly physiologically characterized, human blood samples demonstrate the sensitivity of aspects of hemorheology to several physiological factors. For example, at high shear rates to the red blood cells elastically deformation, imparting viscoelasticity, while and at low shear rates, they form rouleaux structures that impart additional, thixotropic behavior. In addition to these advances in experimental methods and validated data sets, significant advances have also been made in both microscopic simulations and macroscopic, continuum, modeling, as well as novel, multiscale approaches. We outline and evaluate the most promising of these recent advances. Although we primarily focus on human blood rheology, we also discuss recent observations on variations across some animal species that provide some indication on evolutionary effects.
The book is a unique phenomenon in Russian geometry education. It was first published in 1892; there have been more than 40 revised editions, and dozens of millions of copies (by these parameters it is trailing only Euclids Elements). Our edition is based on 41st edition (the stable edition of Nil Aleksandrovich Glagolev; its been in public domain since 2015). At a few places we reverted changes to the earlier editions; we also made more accurate historical remarks.
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