Summary. We give a geometrical interpretation of the notion of µ-prolongations of vector fields and of the related concept of µ-symmetry for partial differential equations (extending to PDEs the notion of λ-symmetry for ODEs). We give in particular a result concerning the relationship between µ-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local µ-symmetries and nonlocal standard symmetries.

Summary. We give a geometrical characterization of λ-prolongations of vector fields, and hence of λ-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form µ, and we speak of µ-prolongations of vector fields and µ-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions.

Abstract. We introduce an extension of hamiltonian dynamics, defined on hyperkahler manifolds, which we call "hyperhamiltonian dynamics". We show that this has many of the attractive features of standard hamiltonian dynamics. We also discuss the prototypical integrable hyperhamiltonian systems, i.e. quaternionic oscillators. PACS: 45.90.+t , 45.20.Ji MSC: 53D99 , 37J99 , 70H99

Root reinforcement is a key factor when dealing with slope stability problems and is an important quantitative criterion for the evaluation of the protective function of forests against shallow landslides, as well as for the adoption of appropriate practices in protection forest management.Although many models have been developed to estimate root reinforcement, a reliable quantification that considers both its spatial and temporal variability still remains a challenge. This work aims to extend the understanding of the long term spatial and temporal dynamics of root reinforcement after forest harvest in subalpine spruce forests by supplying new experimental data and applying a state-of-the-art model.We estimated root reinforcement decay 5, 10 and 15 years after timber had been harvested in spruce stands in a small catchment in the Swiss Alps. We collected root distribution data at different distances from the trees and calibrated and validated a root distribution model (RootDis). To

We introduce, in the spirit of Witten's gauging of exterior differential, a deformed Lie derivative that allows a geometrical interpretation of λand µ-symmetries, in complete analogy with standard symmetries. The case of variational symmetries (both for ODEs and for PDEs) is also considered in this approach, leading to λand µ-conservation laws.