Horospheres in hyperbolic geometry

In this paper we investigate the role of horospheres in Integral Geometry and Differential Geometry. In particular we study envelopes of families of horocycles by means of “support maps”. We define invariant “linear combinations” of support maps or curves. Finally we obtain Gauss-Bonnet type formulas and Chern-Lashof type inequalities.

Unified formalism for non-autonomous mechanical systems

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

On a class of exact solutions to the Fokker-Planck equations

In this paper we study under which circumstances there exists a general change of gross variables that transforms any FokkerPlanck equation into another of the OrnsteinUhlenbeck class that, therefore, has an exact solution. We find that any FokkerPlanck equation will be exactly solvable by means of a change of gross variables if and only if the curvature tensor and the torsion tensor associated with the diffusion is zero and the transformed drift is linear. We apply our criteria to the Kubo and Gompertz models.

The problem of physical coordinates in predictive Hamiltonian systems

In the Hamiltonian formulation of predictive relativistic systems, the canonical coordinates cannot be the physical positions. The relation between them is given by the individuality differential equations. However, due to the arbitrariness in the choice of Cauchy data, there is a wide family of solutions for these equations. In general, those solutions do not satisfy the condition of constancy of velocities moduli, and therefore we have to reparametrize the world lines into the proper time. We derive here a condition on the Cauchy data for the individuality equations which ensures the constancy of the velocities moduli and makes the reparametrization unnecessary.

On the relationship between connections and the asymptotic properties of predictive distributions

In a recent paper, Komaki studied the second-order asymptotic properties of predictive distributions, using the Kullback-Leibler divergence as a loss function. He showed that estimative distributions with asymptotically efficient estimators can be improved by predictive distributions that do not belong to the model. The model is assumed to be a multidimensional curved exponential family. In this paper we generalize the result assuming as a loss function any f divergence. A relationship arises between alpha connections and optimal predictive distributions. In particular, using an alpha divergence to measure the goodness of a predictive distribution, the optimal shift of the estimate distribution is related to alpha-covariant derivatives. The expression that we obtain for the asymptotic risk is also useful to study the higher-order asymptotic properties of an estimator, in the mentioned class of loss functions.

Geometria de superficies: una aproximació a la figura de Gausss

Gauss va publicar l’any 1827 Disquisitiones generales circa superficies curvas, obra que ha resultat fonamental en el desenvolupament de la geometria diferencial a partir del segle XIX. La documentació de la qual es disposa sobre la gènesi i el desenvolupament de les idees d’aquesta obra, ens permet, a més de presentar els principals resultats que hi apareixen, fer una aproximació a la figura de Gauss, al seu estil matemàtic

Differential epipolar constraint in mobile robot egomotion estimation

The estimation of camera egomotion is a well established problem in computer vision. Many approaches have been proposed based on both the discrete and the differential epipolar constraint. The discrete case is mainly used in self-calibrated stereoscopic systems, whereas the differential case deals with a unique moving camera. The article surveys several methods for mobile robot egomotion estimation covering more than 0.5 million samples using synthetic data. Results from real data are also given

Periodic solutions of second-order differential inclusions systems with p-Laplacian

Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p-Laplacian

Linear stochastic differential algebraic equations with constant coefficients

We consider linear stochastic differential-algebraic equations with constant coefficients and additive white noise. Due to the nature of this class of equations, the solution must be defined as a generalised process (in the sense of Dawson and Fernique). We provide sufficient conditions for the law of the variables of the solution process to be absolutely continuous with respect to Lebesgue measure.

Periodic solutions for nonautonomous second order differential inclusions systems with p- Laplacian

Using the nonsmooth variant of minimax point theorems, some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p-Laplacian.

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper, a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.

Limit Cycles for a class of three dimensional polynomial differential systems

"Vegeu el resum a l'inici del document del fitxer adjunt."

Reflected Backward Stochastic Differential Equation with Jumps and RCLL Obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left-hand limited obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.