37 research outputs found
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Planar 2-homogeneous commutative rational vector fields
In this paper we prove the following result: if two 2-dimensional
2-homogeneous rational vector fields commute, then either both vector fields
can be explicitly integrated to produce rational flows with orbits being lines
through the origin, or both flows can be explicitly integrated in terms of
algebraic functions. In the latter case, orbits of each flow are given in terms
of -homogeneous rational functions as curves . An
exhaustive method to construct such commuting algebraic flows is presented. The
degree of the so-obtained algebraic functions in two variables can be
arbitrarily high.Comment: 23 page
Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales
El objetivo de esta tesis es dar un algoritmo para decidir si un sistema explicitable de ecuaciones diferenciales kJiferenciales de orden superior sobre las funciones racionales complejas, dado simbólicamente,admite
!Soluciones liouvillianas no nulas, calculando una (de laforma dada por un teorema de Singer) en caso
!afirmativo. mediante métodos numérico-simbólicos del tipo Introducido por van der Hoeven.donde el uso de álculo numérico no compromete la corrección simbólica. Para ello se Introduce untipo de grupos algebraicos lineales, los grupos euriméricos, y se calcula el cierre eurimérico del grupo de Galois diferencial,mediante una modificación del algoritmo de Derksen y van der Hoeven, dado por los generadores de Ramis.Departamento de Algebra, Análisis Matemático, Geometría y Topologí
Non-linear estimation is easy
Non-linear state estimation and some related topics, like parametric
estimation, fault diagnosis, and perturbation attenuation, are tackled here via
a new methodology in numerical differentiation. The corresponding basic system
theoretic definitions and properties are presented within the framework of
differential algebra, which permits to handle system variables and their
derivatives of any order. Several academic examples and their computer
simulations, with on-line estimations, are illustrating our viewpoint
Swinging Atwood's Machine: Experimental and Theoretical Studies
A Swinging Atwood Machine (SAM) is built and some experimental results
concerning its dynamic behaviour are presented. Experiments clearly show that
pulleys play a role in the motion of the pendulum, since they can rotate and
have non-negligible radii and masses. Equations of motion must therefore take
into account the inertial momentum of the pulleys, as well as the winding of
the rope around them. Their influence is compared to previous studies. A
preliminary discussion of the role of dissipation is included. The theoretical
behaviour of the system with pulleys is illustrated numerically, and the
relevance of different parameters is highlighted. Finally, the integrability of
the dynamic system is studied, the main result being that the Machine with
pulleys is non-integrable. The status of the results on integrability of the
pulley-less Machine is also recalled.Comment: 37 page
Dynamical Casimir Effect in a Leaky Cavity at Finite Temperature
The phenomenon of particle creation within an almost resonantly vibrating
cavity with losses is investigated for the example of a massless scalar field
at finite temperature. A leaky cavity is designed via the insertion of a
dispersive mirror into a larger ideal cavity (the reservoir). In the case of
parametric resonance the rotating wave approximation allows for the
construction of an effective Hamiltonian. The number of produced particles is
then calculated using response theory as well as a non-perturbative approach.
In addition we study the associated master equation and briefly discuss the
effects of detuning. The exponential growth of the particle numbers and the
strong enhancement at finite temperatures found earlier for ideal cavities turn
out to be essentially preserved. The relevance of the results for experimental
tests of quantum radiation via the dynamical Casimir effect is addressed.
Furthermore the generalization to the electromagnetic field is outlined.Comment: 48 pages, 8 figures typos corrected & references added and update
Description and control of decoherence in quantum bit systems
The description and control of decoherence of quantum bit systems have become a field of increasing interest during the last decade. We discuss different techniques to estimate and model decoherence sources of solid state quantum bit realizations.
At first, we derive a microscopic, perturbation theoretical approach for Lindblad master equations of a spin-Boson model at low temperatures.
A different sort of decoherence is investigate by means of the bistable fluctuator model.
For this particular but nevertheless for solid state qubits relevant noise source, we present a suitably designed dynamical decoupling method (so-called quantum bang-bang).
This works as a high-pass filter, suppressing low frequency parts of the noise most effectively and thus being a promising method to compensate the ubiquituous 1/f noise.
Furthermore, we investigate the behaviour of a two coupled spin system exposed to collective and localized bath.
For this dressed-spin system we receive by means of scaling-analysis in first order a quantum phase diagram.
On that we can identify the various quantum dynamical and entanglement phases
Torsion points, Pell's equation, and integration in elementary terms
The main results of this paper involve general algebraic differentials on a general pencil of algebraic curves. We show how to determine if is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of André and Hrushovski and with the Grothendieck-Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin-Mumford type allied to the Zilber-Pink conjectures: we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations over a polynomial ring. We determine whether the Pell equation (with squarefree ) is solvable for infinitely many members of the pencil