6,060 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
Variations for Some Painlev\'e Equations
This paper first discusses irreducibility of a Painlev\'e equation . We
explain how the Painlev\'e property is helpful for the computation of special
classical and algebraic solutions. As in a paper of Morales-Ruiz we associate
an autonomous Hamiltonian to a Painlev\'e equation . Complete
integrability of is shown to imply that all solutions to are
classical (which includes algebraic), so in particular is solvable by
''quadratures''. Next, we show that the variational equation of at a given
algebraic solution coincides with the normal variational equation of
at the corresponding solution. Finally, we test the Morales-Ramis
theorem in all cases to where algebraic solutions are present,
by showing how our results lead to a quick computation of the component of the
identity of the differential Galois group for the first two variational
equations. As expected there are no cases where this group is commutative
Non-Integrability of Some Higher-Order Painlev\'e Equations in the Sense of Liouville
In this paper we study the equation which is one of the higher-order
Painlev\'e equations (i.e., equations in the polynomial class having the
Painlev\'e property). Like the classical Painlev\'e equations, this equation
admits a Hamiltonian formulation, B\"acklund transformations and families of
rational and special functions. We prove that this equation considered as a
Hamiltonian system with parameters , , , is not integrable in Liouville sense by means of
rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis
approach. Then we study the integrability of the second and third members of
the -hierarchy. Again as in the previous case it
turns out that the normal variational equations are particular cases of the
generalized confluent hypergeometric equations whose differential Galois groups
are non-commutative and hence, they are obstructions to integrability
A Characterization of Reduced Forms of Linear Differential Systems
A differential system , with
is said to be in reduced form if where
is the Lie algebra of the differential Galois group of
. In this article, we give a constructive criterion for a system to be in
reduced form. When is reductive and unimodular, the system is in
reduced form if and only if all of its invariants (rational solutions of
appropriate symmetric powers) have constant coefficients (instead of rational
functions). When is non-reductive, we give a similar characterization via
the semi-invariants of . In the reductive case, we propose a decision
procedure for putting the system into reduced form which, in turn, gives a
constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr
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
Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields
In this paper we analyse the integrability of a dynamical system describing
the rotational motion of a rigid satellite under the influence of gravitational
and magnetic fields. In our investigations we apply an extension of the Ziglin
theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric
satellite the system does not admit an additional real meromorphic first
integral except for one case when the value of the induced magnetic moment
along the symmetry axis is related to the principal moments of inertia in a
special way.Comment: 39 pages, 4 figures, missing bibliography was adde
A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
Let with be a differential linear
system. We say that a matrix is a {\em reduced
form} of if and there exists such that . Such a form is
often the sparsest possible attainable through gauge transformations without
introducing new transcendants. In this article, we discuss how to compute
reduced forms of some symplectic differential systems, arising as variational
equations of hamiltonian systems. We use this to give an effective form of the
Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page
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