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Register a new userTranslation of the C.J. Malmstens paper De integralibus quibusdam definitis, seriebusque infinitis
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Alexander Aycock
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The legacy of Jordans canonical form on Poincares algebraic practices. This paper proposes a transversal overview on Henri Poincares early works (1878-1885). Our investigations start with a case study of a short note published by Poincare on 1884 : Sur les nombres complexes. In the perspective of todays mathematical disciplines - especially linear algebra -, this note seems completely isolated in Poincares works. This short paper actually exemplifies that the categories used today for describing some collective organizations of knowledge fail to grasp both the collective dimensions and individual specificity of Poincares work. It also highlights the crucial and transversal role played in Poincares works by a specific algebraic practice of classification of linear groups by reducing the analytical representation of linear substitution to their Jordans canonical forms. We then analyze in detail this algebraic practice as well as the roles it plays in Poincares works. We first provide a micro-historical analysis of Poincares appropriation of Jordans approach to linear groups through the prism of the legacy of Hermites works on algebraic forms between 1879 and 1881. This mixed legacy illuminates the interrelations between all the papers published by Poincare between 1878 and 1885 ; especially between some researches on algebraic forms and the development of the theory of Fuchsian functions. Moreover, our investigation sheds new light on how the notion of group came to play a key role in Poincares approach. The present paper also offers a historical account of the statement by Jordan of his canonical form theorem. Further, we analyze how Poincare transformed this theorem by appealing to Hermites
Euler gives a long introduction, giving all the arguments for and against the use of divergent series in calculus and then gives his own definition of the sum of a diverging series. Then in the second half of this paper he evaluates the the 1-1+2-6+24-120+720-... on several ways and gets the sum 0.5963473621372. The paper is translated from Eulers Latin original into German.
This is a typeset version of Alan Turings declassified Second World War paper textit{Paper on Statistics of Repetitions}. See the companion paper, textit{The Applications of Probability to Cryptography}, also available from arXiv at arXiv:1505.04714, for Editors Notes.
This paper aims at shedding a new light on the novelty of Poincares Methodes nouvelles de la mecanique celeste. The latters approach to the three-body-problem has often been celebrated as a starting point of chaos theory in relation to the investigation of dynamical systems. Yet, the novelty of Poincares strategy can also be analyzed as having been cast out some specific algebraic practices for manipulating systems of linear equations. As the structure of a cast-iron building may be less noticeable than its creative fac{c}ade, the algebraic cast of Poincares strategy was broken out of the mold in generating the new methods of celestial mechanics. But as the various components that are mixed in some casting process can still be detected in the resulting alloy, this algebraic cast points to some collective dimensions of the Methodes nouvelles. It thus allow to analyze Poincares individual creativity in regard with the collective dimensions of some algebraic cultures. At a global scale, Poincares strategy is a testimony of the pervading influence of what used to play the role of a shared algebraic culture in the 19th century, i.e., much before the development of linear algebra as a specific discipline. This shared culture was usually identified by references to the equation to the secular inequalities in planetary theory. This form of identification highlights the long shadow of the great treatises of mechanics published at the end of the 18th century. At a more local scale, Poincares approach can be analyzed in regard with the specific evolution that Hermites algebraic theory of forms impulsed to the culture of the secular equation. Moreover, this papers shows that some specific aspects of Poincares own creativity result from a process of acculturation of the latter to Jordans practices of reductions of linear substitutions within the local algebraic culture anchored in Hermites legacy .
During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled vigorously over the organisation of the theory of bilinear forms. That theory promised a general and homogeneous treatment of numerous questions arising in various 19th-century theoretical contexts, and it hinged on two theorems, stated independently by Jordan and Weierstrass, that would today be considered equivalent. It was, however, the perceived difference between those two theorems that sparked the 1874 controversy. Focusing on this quarrel allows us to explore the algebraic identity of the polynomial practices of the manipulations of forms in use before the advent of structural approaches to linear algebra. The latter approaches identified these practices with methods for the classification of similar matrices. We show that the prac- tices -- Jordans canonical reduction and Kroneckers invariant computation -- reflect identities inseparable from the social context of the time. Moreover, these practices reveal not only tacit knowledge, local ways of thinking, but also -- in light of a long history tracing back to the work of Lagrange, Laplace, Cau- chy, and Hermite -- two internal philosophies regarding the significance of generality which are inseparable from two disciplinary ideals opposing algebra and arithmetic. By interrogating the cultural identities of such practices, this study aims at a deeper understanding of the history of linear algebra without focusing on issues related to the origins of theories or structures.
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