Abstract:
A self-contained study of monopole configurations of pure Yang-Mills theories and a discussion of their charges is carried out in the language of principal bundles. A n-dimensional monopole over the sphere S^n is a particular type of principal connection on a principal bundle over a symmetric space K/H which is K-invariant, where K=SO(n+1) and H=SO(n). It is shown that principal bundles over symmetric spaces admit a unique K-invariant principal connection called canonical, which also satisfy Yang-Mills equations. The geometrical framework enables us to describe their associated field strengths in purely algebraic terms and compute the charge of relevant (Yang-type) monopoles avoiding the use of coordinates. Besides, two corrections on known results are performed in this paper. First, it is proven that the Yang monopole should be considered a connection invariant by Spin(5) instead of by SO(5), as Yang did in his original article J. Math. Phys. 19(1), pp. 320-328 (1978). Second, unlike the way suggested in Class. Quantum Grav. 23, pp. 4873-4885 (2006), we give the correct characteristic class to be used to calculate the charge of the monopoles studied by Gibbons and Townsend.

Abstract:
We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. Additionally, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.

Abstract:
We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system.

Abstract:
We introduce the concept of multidimensional antithetic as the absolute minimum of the covariance defined on the orthogonal group by $A\mapsto Cov(f(\xi),f(A\xi))$ where $\xi$ is a standard $N$-dimensional normal random variable and $f:\mathbb{R}^{N}\to\mathbb{R}$ is an almost everywhere differentiable function. The antithetic matrix is designed to optimise the calculation of $E[f(\xi)]$ in a Monte Carlo simulation. We present an iterative annealing algorithm that dynamically incorporates the estimation of the antithetic matrix within the Monte Carlo calculation.

Abstract:
This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples.

Abstract:
We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, are characterized by a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

Abstract:
We prove Knudsen's law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles. Dynamics are randomly perturbed and the corresponding random map studied under a skew-type deterministic representation which is shown to be ergodic and exact.

Abstract:
We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to the process of stabilization of constraints, both in Lagrangian and Hamiltonian formalisms. The connections between both formalisms are established by means of the time-evolution operator.

Abstract:
This article highlights the benefits of incorporating the qualitative perspective, and in particular the use of life stories, in studies based on the life-course perspective. These benefits will be illustrated using as an example a study of the effects of social protection measures on career paths based on the theoretical underpinnings of SEN's (1987, 1992, 1993) capability approach. The life-course perspective is a very good methodological option for evaluating the extent to which social protection systems are adapted to the present situation in labor markets, where many trajectories are non-linear and unstable. Studies adopting such a perspective use research designs in which secondary statistical data are central, whereas qualitative data are used little, if at all. This article will explain how these quantitative approaches can be improved by using semi-structured interviews and a formalized qualitative analysis to identify turning points, critical events, transitions and stages, in which social protection measures and resources play a significant role in redressing or channeling personal and career paths. Paying attention to the person and his or her agency—not only at a given time but in a broader perspective that embraces past episodes and projections into the future—is essential in order to consider how individuals use the resources at their disposal and to effectively evaluate the effects of social protection measures. URN: http://nbn-resolving.de/urn:nbn:de:0114-fqs1103152 Este artículo muestra los beneficios de la incorporación de la perspectiva cualitativa, y en particular del uso de los relatos de vida, en estudios basados en la perspectiva del curso de vida. Estos beneficios se ilustran mostrando un estudio sobre los efectos de las medidas de protección social en trayectorias profesionales basado en los fundamentos teóricos del enfoque de las capacidades de SEN (1987, 1992, 1993). La perspectiva del curso de vida es una buena opción metodológica para evaluar si los sistemas de protección social se adaptan a la situación actual de los mercados laborales pues numerosas trayectorias son no-lineales e inestables. Los estudios que adoptan esta perspectiva usan dise os de investigación en que los datos estadísticos secundarios ocupan un papel central, mientras que los datos cualitativos apenas tienen relevancia o directamente no se emplean. Este artículo muestra cómo estos estudios pueden ser mejorados mediante el uso de entrevistas semiestructuradas y un análisis cualitativo formal que permitan identificar puntos de inflexión, acontecimientos cr