We consider perturbations of Dirac type operators on complete, connected metric spaces equipped with a doubling measure. Under a suitable set of assumptions, we prove quadratic estimates for such operators and hence deduce that these operators have a bounded functional calculus. In particular, we deduce a Kato square root type estimate.

A new metric, the Multi-Dimensional aperiodic Merit Factor, is presented, and various recursive quadratic sequence constructions are given for which both the one and multi-dimensional aperiodic Merit Factors can be computed exactly. In some cases these constructions lead to Merit Factors with non-vanishing asymptotes.

We exploit the fact that, in Minkowski space-time, γ-matrices are possibly more fundamental than the metric to describe how gauge invariance at perturbative level enforces a Lagrangian for spinor electrodynamics with massless photons. The term quadratic in the potential arises naturally in the gauge-fixed Lagrangian but has vanishing coefficient.

We propose a hamiltonian formulation of the N = 2 supersym-metric KP type hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic hamiltonian structure which allows for several reductions of the KP type hierarchy. In particular, the third family of N = 2 KdV hierarchies is recovered. We also give an easy constrution of Wronskian solutions of the KP and KdV type equations.

We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier’s theorem of existence of optimal maps.

We present an explicit formula relating volumes of strata of meromorphic quadratic differentials with at most simple poles on Riemann surfaces and counting functions of the number of flat cylinders filled by closed geodesics in associated flat metric with singularities. This generalizes the result of Athreya, Eskin and Zorich in genus 0 to higher genera.

In this experimentation project, we introduce the Neighbourhood Component Analysis (NCA), a classification method combining k-nearest neighbour (KNN) and learned distance metric, originally proposed by Goldberger et al. in 2004. With an implementation in the R environment, we perform experiments on several datasets. Results of our experiments suggest that with the learned distance metric, KNN c...

This paper extends the RRT* algorithm, a recentlydeveloped but widely-used sampling-based optimal motion planner, in order to effectively handle nonlinear kinodynamic constraints. Nonlinearity in kinodynamic differential constraints often leads to difficulties in choosing appropriate distance metric and in computing optimized trajectory segments in tree construction. To tackle these two difficu...

Kinodynamic planning algorithms like RapidlyExploring Randomized Trees (RRTs) hold the promise of finding feasible trajectories for rich dynamical systems with complex, non-convex constraints. In practice, these algorithms perform very well on configuration space planning, but struggle to grow efficiently in systems with dynamics or differential constraints when using conventional proximity met...

We consider Covariance Matrix Adaptation schemes (CMA-ES [3], Gaussian Adaptation (GaA) [4]) and Randomized Hessian (RH) schemes from Leventhal and Lewis [5]. We provide a new, numerically stable implementation for RH and, in addition, combine the update with an adaptive step size strategy. We design a class of quadratic functions with parametrizable spectra to study the influence of the spectr...