Two methods for approximating minimal surfaces in parametric form are considered. One minimizes the area of the surface, and the other the energy of the surface. The convergence of the algorithm of the first method is proved. The application of the second method to the approximation of conformai maps is examined. Several examples of computations are given.

Let k be a positive integer. In this paper, we study a parametric family of the sets of integers {k, Ak + 4A, (A+ 1)k + 4(A+ 1), d}. We prove that if d is a positive integer such that the product of any two distinct elements of that set increased by 4 is a perfect square, then d = (A4 + 2A3 +A2)k3 + (8A3 + 12A2 + 4A)k2 +(20A2 + 20A + 4)k + (16A+ 8) for 1 ≤ A ≤ 22 and A ≥ 51767.

We introduce the concept of two-color multistep cascading for vectorial parametric wave mixing in optical media with quadratic (second-order or chi(2)) nonlinear response. We demonstrate that the multistep cascading allows light-guiding-light effects with quadratic spatial solitons. With the help of the so-called "almost exact" analytical solutions, we describe the properties of parametric wave...

Let n ≥ 3, v1(x), . . . , vn−1(x) ∈ Z[x] and u ∈ {−1, 1}, then x(x− v1(a)y) · · · (x− vn−1(a)y) + uy = ±1 is called a parametrized familiy of Thue equations, if a ∈ Z and the solutions x, y are searched in Z; cf. Thomas (1993). There are several results concerning parametrized families of cubic and quartic families of Thue equations, see Mignotte et al. (1996) and the references therein. Thomas...

We demonstrate spatial solitons via twin-beam second-harmonic generation in hexagonal lattices realized by poling lithium niobate planar waveguides. These simultons can be steered by acting on power, direction, and wavelength of the fundamental frequency input.