200 research outputs found
On Jordan's measurements
The Jordan measure, the Jordan curve theorem, as well as the other generic
references to Camille Jordan's (1838-1922) achievements highlight that the
latter can hardly be reduced to the "great algebraist" whose masterpiece, the
Trait\'e des substitutions et des equations alg\'ebriques, unfolded the
group-theoretical content of \'Evariste Galois's work. The present paper
appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte
der Mathematik (1868-1942) for providing an overview of Jordan's works. On the
one hand, we shall especially investigate the collective dimensions in which
Jordan himself inscribed his works (1860-1922). On the other hand, we shall
address the issue of the collectives in which Jordan's works have circulated
(1860-1940). Moreover, the time-period during which Jordan has been publishing
his works, i.e., 1860-1922, provides an opportunity to investigate some
collective organizations of knowledge that pre-existed the development of
object-oriented disciplines such as group theory (Jordan-H\"older theorem),
linear algebra (Jordan's canonical form), topology (Jordan's curve), integral
theory (Jordan's measure), etc. At the time when Jordan was defending his
thesis in 1860, it was common to appeal to transversal organizations of
knowledge, such as what the latter designated as the "theory of order." When
Jordan died in 1922, it was however more and more common to point to
object-oriented disciplines as identifying both a corpus of specialized
knowledge and the institutionalized practices of transmissions of a group of
professional specialists
Computing the differential Galois group of a parameterized second-order linear differential equation
We develop algorithms to compute the differential Galois group associated
to a parameterized second-order homogeneous linear differential equation of the
form where the coefficients are rational
functions in with coefficients in a partial differential field of
characteristic zero. Our work relies on the procedure developed by Dreyfus to
compute under the assumption that . We show how to complete this
procedure to cover the cases where , by reinterpreting a classical
change of variables procedure in Galois-theoretic terms.Comment: 14 page
Differential Equations and Finite Groups
AbstractThe classical solution of the Riemann–Hilbert problem attaches to a given representation of the fundamental group a regular singular linear differential equation. We present a method to compute this differential equation in the case of a representation with finite image. The approach uses Galois coverings of P1\{0,1,∞}, differential Galois theory, and a formula for the character of the Galois action of the space of holomorphic differentials. Examples are produced for the finite primitive unimodular groups of degree two and three
Two Examples of Toric Arrangements
We show that the integral cohomology algebra of the complement of a toric
arrangement is not determined by the poset of layers. Moreover, the rational
cohomology algebra is not determined by the arithmetic matroid (however it is
determined by the poset of layers).Comment: 12 pages, 1 figure, removed section
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