The stability problem of functional equations was originated from a question of Ulam [66] concerning the stability of group homomorphisms: Let (G1, .) be a group and let (G2, ∗) be a metric group with the metric d(., .). Given ε > 0, does there exist a δ > 0, such that if a mapping h : G1 → G2 satisfies the inequality d(h(x1.x2), h(x1) ∗ h(x2)) < δ for all x1, x2 ∈ G1, then there exists a homom...

We determine the number of ordered factorisations of an arbitrary permutation on n symbols into transpositions such that the factorisations have minimal length and such that the factors generate the full symmetric group on n symbols. Such factorisations of the identity permutation have been considered by Crescimanno and Taylor in connection with a class of topologically distinct holomorphic map...

Given a multidimensional data set and a model of its density, we consider how to define the optimal interpolation between two points. This is done by assigning a cost to each path through space, based on two competing goals-one to interpolate through regions of high density, the other to minimize arc length. From this path functional, we derive the Euler-Lagrange equations for extremal motionj ...

Consider a frictionless surface S in a gravitational field that need not be uniform. Given two points A and B on S, what curve is traced out by a particle that starts at A and reaches B in the shortest time? This paper discusses this problem for simple surfaces such as surfaces of revolution. We first solve this more general problem using the Euler-Lagrange equation and conservation of mechanic...

This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of conjugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems a...

We develop in this paper a new framework for discrete calculus of variations when the actions have densities involving an arbitrary discretization operator. We deduce the discrete Euler-Lagrange equations for piecewise continuous critical points of sampled actions. Then we characterize the discretization operators such that, for all quadratic lagrangian, the discrete Euler-Lagrange equations co...

In this paper we give the definition of 1/2-harmonic maps u : D → N , where D is an open set of IR and N is a k dimensional submanifold N of IRm as critical points of the functional ∫ Ω |∆ 1/4u|2dx . We write the Euler-Lagrange equation satisfied in a weak sense by the 1/2-harmonic maps, which is a nonlocal partial differential equation involving the 1/2-Laplacian operator . We prove the local ...

We study the regularity properties of generalized solutions of the EulerLagrange equation of a functional involving capacity and perimeter, related to a conjecture of Pólya and Szegö.

A surface of codimension higher than one embedded in an ambient space possesses a connection associated with the rotational freedom of its normal vector fields. We examine the Yang-Mills functional associated with this connection. The theory it defines differs from Yang-Mills theory in that it is a theory of surfaces. We focus, in particular, on the Euler-Lagrange equations describing this surf...

Calculus of variations on time scales (we refer the reader to Section 2 for a brief introduction to time scales) has been introduced in 2004 in the papers by Bohner [2] and Hilscher and Zeidan [4], and seems to have many opportunities for application in economics [1]. In both works of Bohner and Hilscher&Zeidan, the Euler-Lagrange equation for the fundamental problem of the calculus of variatio...