In this talk I will discuss a method to prove the absence of critical points for the Helmholtz equation in 3D. The key element of the approach is the use of multiple frequencies in a fixed range, and the proof is based on the spectral analysis of the associated problem. This question is strictly connected with the Radó-Kneser-Choquet theorem, whose direct extension to the Helmholtz equation or ...

At every point, a smooth plane curve can be approximated, to second order, by a circle; this circle is called osculating. One may think of the osculating circle as passing through three infinitesimally close points of the curve. A vertex of the curve is a point at which the osculating circle hyper-osculates: it approximates the curve to third order. Equivalently, a vertex is a critical point of...

for the polynomial planar vector fields with a hyperbolic or nilpotent critical point at the origin, the monodromy problem has been solved, but for the strongly degenerate critical points this problem is still open. when the critical point is monodromic, the stability problem or the center- focus problem is an open problem too. in this paper we will consider the polynomial planar vector fields ...

In this paper, using the best proximity theorems for an extensionof Brosowski's theorem. We obtain other results on farthest points. Finally, wedene the concept of e- farthest points. We shall prove interesting relationshipbetween the -best approximation and the e-farthest points in normed linearspaces (X; ||.||). If z in W is a e-farthest point from an x in X, then z is also a-best approximati...

We prove a central limit theorem concerning the number of critical points in large cubes of an isotropic Gaussian random function on a Euclidean space.

Given a complex cubic polynomial p(z) = (z − 1)(z − r1)(z − r2) with |r1| = 1 = |r2|, where are the critical points? Marden’s Theorem tells us that the critical points are the foci of the Steiner ellipse of 41r1r2. In this paper we further explore the structure of these critical points. If we let Tγ be the circle of diameter γ passing through 1 and 1 − γ, then there are α, β ∈ [0, 2] such that ...

we provide fuzzy quasi-metric versions of a fixed point theorem ofgregori and sapena for fuzzy contractive mappings in g-complete fuzzy metricspaces and apply the results to obtain fixed points for contractive mappingsin the domain of words.