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We have applied the Laser‐Induced Fluorescence technique to thyroid tissues. A close correlation between spectral shapes and the degree of tissue abnormality have been found. In some cases fluorescence spectrum is formed by linear superposition of “healthy” and “pathological” basic spectral shapes. This allows to ca determine unambiguously the state of the tissue by means of any single spectral measurement. (© 2006 by Astro, Ltd. Published exclusively by WILEY‐VCH Verlag GmbH & Co. KGaA)

We show that the incenter of an isosceles triangle is a stable equilibrium point of the electrostatic potential of certain point charges placed at its vertices. To this end, explicit formulas for these charges are given and the hessian of their electrostatic potential is computed. The behavior of this hessian in a family of triangles with the given inscribed and circumscribed circles is investigated and its extremal values are computed. As an application, we prove that each point in the unit disc is a stable equilibrium point of a certain triple of point charges on its boundary, which yields an explicit scenario of robust electrostatic control in Euclidean discs. In conclusion, several examples are presented which show that analogous problems are interesting and feasible for convex quadrilaterals.

We consider the equilibrium points of the electrostatic potential of three mutually repelling point charges with Coulomb interaction placed at the vertices of a given triangle T. It is proven that for each point P inside the triangle T, there exists a unique collection of positive point charges, called stationary charges for P in T, such that P is a critical point of the electrostatic potential of these point charges placed at vertices of T in a fixed order. Explicit formulas for stationary charges are given, which are used to investigate the existence and geometry of stable equilibria arising in this setting. In particular, symbolic computations and computer experiments reveal that for an isosceles triangle T, the set S(T) of points P that are stable equilibria of their stationary charges is a non-empty open set containing the incenter of a triangle T. For a regular triangle, using symbolic computations, it appears possible to verify that the formulas for stationary charges define a stable mapping in the sense of Whitney having a deltoid caustic with three ordinary cusps. An interpretation of our results in terms of electrostatic ion traps is also given, and several plausible conjectures are presented.

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