On a class of polynomial Lagrangians

In the framework of finite order variational sequences a new class of Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are the horizontalization of forms on a jet space of lower order. We describe their properties together with properties of related objects, such as Poincar\'e--Cartan and Euler--Lagrange forms, momenta and momenta of generating forms, a new geometric object arising in variational sequences. Finally, we provide a simple but important example of special Lagrangian, namely the Hilbert--Einstein Lagrangian.Comment: LaTeX2e, amsmath, diagrams, hyperref; 15 page

On the bi-Hamiltonian Geometry of WDVV Equations

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).Comment: 21 pages, revised published version; exposition substantially improve

Systems of conservation laws with third-order Hamiltonian structures

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^{\alpha} \in \Lambda^2(W)$ such that $$\phi_{\beta \gamma}A^{\beta}\wedge A^{\gamma}=0,$$ for some non-degenerate symmetric $\phi$.Comment: 31 page

Tetrad gravity, electroweak geometry and conformal symmetry

A partly original description of gauge fields and electroweak geometry is proposed. A discussion of the breaking of conformal symmetry and the nature of the dilaton in the proposed setting indicates that such questions cannot be definitely answered in the context of electroweak geometry.Comment: 21 pages - accepted by International Journal of Geometric Methods in Modern Physics - v2: some minor changes, mostly corrections of misprint

Accessing extremes of mid-latitudinal wave activity: methodology and application

A statistical methodology is proposed and tested for the analysis of extreme values of atmospheric wave activity at mid-latitudes. The adopted methods are the classical block-maximum and peak over threshold, respectively based on the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD). Time-series of the ‘Wave Activity Index’ (WAI) and the ‘Baroclinic Activity Index’ (BAI) are computed from simulations of the General Circulation Model ECHAM4.6, which is run under perpetual January conditions. Both the GEV and the GPD analyses indicate that the extremes ofWAI and BAI areWeibull distributed, this corresponds to distributions with an upper bound. However, a remarkably large variability is found in the tails of such distributions; distinct simulations carried out under the same experimental setup provide sensibly different estimates of the 200-yr WAI return level. The consequences of this phenomenon in applications of the methodology to climate change studies are discussed. The atmospheric configurations characteristic of the maxima and minima of WAI and BAI are also examined

On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics

Using the theory of $1+1$ hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this class of solutions yields an infinite family of \emph{linear} solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.Comment: 19 page

Lagrangian reductive structures on gauge-natural bundles

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split structure on the relevant underlying principal bundle.Comment: 11 pages, remarks and comments added, this version published in ROM