Lemniscate growth
نویسندگان
چکیده
It was recently noticed that lemniscates do not survive Laplacian growth [12] (2010). This raises the question: “Is there a growth process for which polynomial lemniscates are solutions?” The answer is “yes”, and the law governing the boundary velocity is reciprocal to that of Laplacian growth. Viewing lemniscates as solutions to a moving-boundary problem gives a new perspective on results from classical potential theory, and interesting properties emerge while comparing lemniscate growth to Laplacian growth.
منابع مشابه
Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem
Thirteen simple closed geodesics are found in the lemniscate. Among these are nine “mirrors”—geodesics of reflection symmetry—which generate the full octahedral group and determine a triangulation of the lemniscate as a disdyakis dodecahedron. New visualizations of the lemniscate are presented.
متن کاملInequalities for Jacobian elliptic functions and Gauss lemniscate functions
A new proof of inequalities involving Jacobian elliptic functions and their inverse functions are obtained. Similar results for the Gauss lemniscate functions are also established. Upper bounds for the inverse Jacobian elliptic functions and for the Gauss arc lemniscate functions are derived. 2012 Elsevier Inc. All rights reserved.
متن کاملShape control of 3D lemniscates
A 3D lemniscate is the set of points whose product of squared distance to a given finite family of fixed points is constant. 3D lemniscates are the space analogs of the classical lemniscates in the plane studied in [5]. They are bounded algebraic surfaces whose degree is twice the number of foci. Within the field of computer aided geometric design (CAGD), 3D lemniscates have been considered in ...
متن کاملCharacteristic polynomials and pseudospectra
In this paper, we study the ε-lemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as generalizations of eigenvalues. The question of numerical determination of the ε-lemniscate raises the problem of computing the characteristic polynomial p. We can express the coeffi...
متن کاملA Note on the Potential Flow past a Lemniscate and a General Method of Milne-thomson*
Basset's (1885) solution for potential flow past a lemniscate is (a) disproved and (b) reproduced by a naive application of "a general method" of MilneThomson. The correct solution is also given. The result is of interest for Rayleigh scattering, as well as for potential theory. 1. Lemniscate in uniform flow. We seek the complex potential associated with uniform, plane, potential flow past a le...
متن کامل