Abstract:
We will introduce linear operators and obtain their exact norms defined on the function spaces X and Z 5. These operators are constructed from the Euler-Lagrange type cubic functional equations and their Pexider versions.

Abstract:
The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators.

Abstract:
The equations of Euler-Lagrange elasticity describe elastic deformations
without reference to stress or strain. These equations as previously published
are applicable only to quasi-static deformations. This paper extends these
equations to include time dependent deformations. To accomplish this, an
appropriate Lagrangian is defined and an extrema of the integral of this
Lagrangian over the original material volume and time is found. The result is a
set of Euler equations for the dynamics of elastic materials without stress or
strain, which are appropriate for both finite and infinitesimal deformations of
both isotropic and anisotropic materials. Finally, the resulting equations are
shown to be no more than Newton's Laws applied to each infinitesimal volume of
the material.

Abstract:
We study the Euler-Lagrange cohomology and explore the symplectic or multisymplectic geometry and their preserving properties in classical mechanism and classical field theory in Lagrangian and Hamiltonian formalism in each case respectively. By virtue of the Euler-Lagrange cohomology that is nontrivial in the configuration space, we show that the symplectic or multisymplectic geometry and related preserving property can be established not only in the solution space but also in the function space if and only if the relevant closed Euler-Lagrange cohomological condition is satisfied in each case. We also apply the cohomological approach directly to Hamiltonian-like ODEs and Hamiltonian-like PDEs no matter whether there exist known Lagrangian and/or Hamiltonian associated with them.

Abstract:
The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.

Abstract:
We derive a new symmetric hyperbolic formulation of the Einstein-Euler equations in Lagrange coordinates that are adapted to the Frauendiener-Walton formulation of the Euler equations. As an application, we use this system to show that the densitized lapse and zero shift coordinate systems for the vacuum Einstein equations are equivalent to Lagrange coordinates for a fictitious fluid with a specific equation of state.

Abstract:
We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings.

Abstract:
We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings.

Abstract:
We establish lower bounds for norms and CB-norms of elementary operators on B(H). Our main result concerns the operator Tx = axb + bxa and we show its norm is at least the product of the norms of a and b, proving a conjecture of M. Mathieu. We also establish some other results and formulae for the CB norm and norm of such T for special cases.