The British development of the theory of invariants (1841–1895)

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in 1859 Salmon, George. 1859. Lessons introductory to the modern higher algebra, Dublin: Hodges & Smith.  [Google Scholar] and later Edwin Bailey Elliott in 1895 Elliott, Edwin Bailey. 1895. An introduction to the algebra of quantics, Oxford: Oxford University Press.  [Google Scholar] codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals.     

Interplay between local and international journals: The case of Sicily, 1880–1920

In 1884, Giovan Battista Guccia founded first the Circolo Matematico di Palermo and then some years later its journal, the Rendiconti del Circolo Matematico di Palermo. Although historians of mathematics have published a number of works on the Circolo and the Rendiconti, there are very few systematic studies on mathematics in Sicilian periodicals. In our paper, we shall investigate the relationships between the “international” Rendiconti and the “local” proceedings published by the Sicilian academies located in Catania, Messina, and Palermo. What is the image of mathematics that emerges from these journals? What is the presence of Sicilian mathematicians among the authors in the different cases? May we recognize a Sicilian dynamics in mathematics and, if so, in what sense?     

The Tits–Kantor–Koecher Construction and Birepresentations of the Jordan Superpair SH(1, n)

Abstract

We introduce the concept of bimodule over a Jordan superpair and the Tits– Kantor–Koecher construction for bimodules. Using the construction we obtain the classification of irreducible bimodules over the Jordan superpair SH(1, n). We also prove semisimplicity for a class of finite dimensional SH(1, n)-bimodules for n ≥ 3.     

Mauro Picone,Sandro Faedo,and the numerical solution of partial differential equations in Italy (1928–1953)

In this paper we revisit the pioneering work on the numerical analysis of partial differential equations (PDEs) by two Italian mathematicians, Mauro Picone (1885–1977) and Sandro Faedo (1913–2001). We argue that while the development of constructive methods for the solution of PDEs was central to Picone’s vision of applied mathematics, his own work in this area had relatively little direct influence on the emerging field of modern numerical analysis. We contrast this with Picone’s influence through his students and collaborators, in particular on the work of Faedo which, while not the result of immediate applied concerns, turned out to be of lasting importance for the numerical analysis of time-dependentPDEs.     

The Mittag-Leffler Theorem: The origin,evolution, and reception of a mathematical result, 1876–1884

The Swedish mathematician Gösta Mittag-Leffler (1846–1927) is well-known for founding Acta Mathematica, often touted as the first international journal of mathematics. A “post-doctoral” student in Paris and Berlin between 1873 and 1876, Mittag-Leffler built on Karl Weierstrass? work by proving the Mittag-Leffler Theorem, which states that a function of rational character (i.e. a meromorphic function) is specified by its poles, their multiplicities, and the coefficients in the principal part of its Laurent expansion.     

The Thomas–Fermi Problem and Solutions of the Emden–Fowler Equation

Computational Mathematics and Mathematical Physics - A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential...     

The New Extensions of the Henstock–Kurzweil and the McShane Integrals of Vector-Valued Functions

In this paper, we define the Hake–Henstock–Kurzweil and the Hake–McShane integrals of Banach space valued functions defined on an open and bounded subset G of m-dimensional Euclidean space \(\mathbb {R}^{m}\). These are ”natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to bounded and open subsets of \(\mathbb {R}^{m}\). Our goal is not a generalization for the sake of generalization. Indeed, we will show theorems which reduce the study of our integrals to the study of McShane and Henstock–Kurzweil integrals. As applications, we will present Hake-type theorems for the Henstock–Kurzweil and the McShane integrals in terms of our integrals.     

The Hess–Appelrot case and quantization of the rotation number

This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.     

The Group Structure of the Normalizer of Γ0(N) After Atkin–Lehner

We determine the group structure of the normalizer of Γ₀(N) in SL ₂(?) modulo Γ₀(N). These results correct the Atkin–Lehner statement (Atkin and Lehner, 1970 Atkin , A. O. L. , Lehner , J. ( 1970 ). Hecke operator on Γ₀(N) . Math. Ann. 185 : 134 – 160 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 8).     

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

The Eigenstructure of Operators Linking the Bernstein and the Genuine Bernstein–Durrmeyer operators

We study the eigenstructure of a one-parameter class of operators ${U_{n}^{\varrho}}$ of Bernstein–Durrmeyer type that preserve linear functions and constitute a link between the so-called genuine Bernstein–Durrmeyer operators U _n and the classical Bernstein operators B _n . In particular, for ${\varrho\rightarrow\infty}$ (respectively, ${\varrho=1}$ ) we recapture results well-known in the literature, concerning the eigenstructure of B _n (respectively, U _n ). The last section is devoted to applications involving the iterates of ${U_{n}^{\varrho}}$ .     

The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes

We use Bochner’s subordination to give a representation of the genealogical structure associated with general continuous-state branching processes. We then apply this representation to connections between a branching process introduced by Neveu, and the coalescent process recently investigated by Bolthausen-Sznitman and others. Received: 25 March 1999 / Revised version: 13 September 1999 /?Published online: 11 April 2000     

The Goursat problem for the Einstein–Vlasov system: (II) The evolution of initial data

We solve, locally in time, the evolution problem associated with the Einstein–Vlasov (EV) system, the initial data being specified on two intersecting smooth null hypersurfaces. The proof of the obtained result relies heavily on a fixed point method deployed in appropriate weighted Sobolev spaces. The main tools of this method consist of adequate Sobolev inequalities and Moser estimates combined with energy inequalities for first-order and second-order linear hyperbolic partial differential equations.