As known, the problem of choosing ‘‘good’’ nodes is a central one in polynomial interpolation. While the problem is essentially solved in one dimension (all good nodal sequences are asymptotically equidistributed with respect to the arc-cosine metric), in several variables it still represents a substantially open question. In this work we consider new nodal sets for bivariate polynomial interpo...

A straightforward application of the variational principle to null strings meets difficulties since string’s world-sheets are degenerate. It is known that in this case can be formulated with help two-vector density on string world-sheet which plays a role Lagrange multipliers. shown recently suggested stress-energy tensor derived by variation over background metric action used describe tensionl...

We develop the formulation of turbulence in terms of the functional integral over the phase space configurations of the vortex cells. The phase space consists of Clebsch coordinates at the surface of the vortex cells plus the Lagrange coordinates of this surface plus the conformal metric. Using the Hamiltonian dynamics we find an invariant probability distribution which satisfies the Liouville ...

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians L : R × R × Ω → R, where Ω is a compact metric space on which R acts through an action which leaves L invariant. This setting allow us to generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians. Our main result is the existence of stationary Mather measures invariant und...

Abstract We continue our study of the mixed Einstein–Hilbert action as a functional pseudo-Riemannian metric and linear connection. Its geometrical part is total scalar curvature on smooth manifold endowed with distribution or foliation. develop variational formulas for quantities extrinsic geometry metric-affine space use them to derive Euler–Lagrange equations (which in case space-time are an...

We present two methods for generating trajectories and motion interpolants that minimize energy functionals on SE(3), the group of rigid body displacements. We treat SE(3) as a Lie Group and endow it with a left-invariant Riemannian metric derived from energy considerations. The first method is based on a relaxation formulation where the Euler-Lagrange equations for the minimization problem are...

The aim of this paper is to provide a formal link between an oscillatory neural model, whose phase is represented by a difference equation, and the Mumford and Shah functional. A Riemannian metric is induced by the pattern of neural connections, and in this setting the difference equation is studied. Its Euler–Lagrange operator Γ-converges as the dimension of the grid tends to 0 to the Mumford ...

Setting. Let (X, g) be a given Riemannian manifold and let ∇ be the corresponding LeviCivita connection. Let G be a compact Lie group with a bi-invariant Riemannian metric κ. Let π : Q → X be a principal bundle with structure group G acting on the left, let A be a principal connection on Q, and let B be the curvature of A. 46 4. Wong’s Equations and Coordinate Formulas Now define the Lagrangian...

An introduction into the theory of locally anisotropic spaces (modelled as vector bundles provided with compatible nonlinear and distinguished linear connections and metric structures and containing as particular cases different types of Kaluza–Klein and/or extensions of Lagrange and Finsler spaces ) is presented. The conditions for consistent propagation of closed strings in locally anisotropi...

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians L : Rn ×Rn ×Ω → R, where Ω is a compact metric space on which Rn acts through an action which leaves L invariant. This setting allow us to generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians. Our main result is the existence of stationary Mather measures invariant un...