Lemniscates and inequalities for the logarithmic capacities of continua

It is shown that if P(z) = z ⁿ + ? is a polynomial with connected lemniscate E(P) = {z: ¦P(z)¦ ≤ 1} and m critical points, then, for any n? m+1 points on the lemniscate E(P), there exists a continuum γ ? E(P) of logarithmic capacity cap γ ≤ 2^?1/n which contains these points and all zeros and critical points of the polynomial. As corollaries, estimates for continua of minimum capacity containing given points are obtained.     

On the Rate of Approximation of Closed Jordan Curves by Lemniscates

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve Γ in the complex plane {z} by lemniscates generated by polynomials P(z). In the present paper, we obtain quantitative upper bounds for the least deviations H _n (Γ) (in this metric) from the curve Γ of the lemniscates generated by polynomials of a given degree n in terms of the moduli of continuity of the conformal mapping of the exterior of Γ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of Γ. For the case in which the curve Γ is analytic, we prove that H _n (Γ) = O(q _n ), 0 ≤ q = q(Γ) < 1, n → ∞.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 861–876.Original Russian Text Copyright ©2005 by O. N. Kosukhin.     

Lemniscates 3D: a CAGD primitive?

A 3D lemniscate is an implicitly given surface which generalizes the well-known Bernoulli lemniscates curves and the Cassini ovals in 2D. It is characterized by placing a finite number of points in space (the foci) and choosing a constant (radius), its algebraic degree is twice the number of foci and it is always contained in the union of certain spheres centered at the foci. The distribution of the foci gives a rough idea of the 3D shapes that could be modeled with any of the connected components of the lemniscate. The position of the foci can be used to stretch and to produce knoblike features. Given a set of foci, for a small radius the lemniscate consists of a number of spherelike surfaces centered at the foci which do not touch each other. As the radius increases the disconnected pieces coalesce producing interesting surfaces. In order to make 3D lemniscates a potentially useful primitive for CAGD it is necessary to control the coalescing/splitting of the connected components of the lemniscate while we move the foci and change the radius, simultaneously. In this paper we offer tools towards this control. We look closely at the case of four noncoplanar foci. AMS subject classification 65D05, 65D17, 65D18This work was partially supported by grant G97 000651 of Fonacit, Venezuela.     

Natural harmonic oscillations of a heavy fluid in basins of complex shape

The natural harmonic oscillations of a heavy fluid in uniform-depth basins of complex shape, including those with three or more symmetry axes, are investigated within the shallow-water approximation. The waves are assumed to be gently sloping. The modes found are compared with the similar modes in an elliptic basin representing the class of basins with two symmetry axes. The mode characteristics associated with the number of basin symmetry axes and the basin shape are explored. Basins with two symmetry axes whose shape differs considerably from the elliptic, in particular, nonconvex basins, are considered. Both rotating and non-rotating basins are studied. The possibility of approximating the amplitudes of certain rotating-basin mode classes by Bessel functions is discussed.