Une nouvelle preuve du théorème de point fixe de Handel
M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 348 (1996) 2637--2662] S Matsumoto [Topol. Appl. 104 (2000) 191--214]) to study homeomorphisms of surfaces.
AN EXTENSION OF THE JORDAN CURVE THEOREM
$S$ . Then $f$ is extensible to a homeomorphism of $S^{2}$ onto itself. In this paper we shall give an extension of this theorem, namely, a necessary and sufficient condition for the extensibility of a $h\mathfrak{c}me\mathfrak{c}mcr_{f}hi^{c}.m$ of a lccally connected continuum. Throughout this paper it is understocd that all sets and mappings are considered on $S^{a}$ . The sum of three arcs $ap,$ $bp,$ $cp$ having a single point $pjn$ common is called a Y-set. In a natural way we can define the positive and negative orientations on any Y-set. Deflnition. A $1^{\backslash }.omcomorphi\backslash mf$ of a locally ccnnected continuum $M$ is said to be sense-preserving, if $f$ does not change the orientaticn of each Y-set in $M$ . Likewise a sense-reversing hcmecmorphism will be cefired. The main result of this paper is the following: Theorem. Let $M$ be a locally connected continuum and $f$ a hcmecmorphism of M. A necessary ond suffcient condition for the extensibihty of $f$ to a hcmecmorphism of $S^{2}$ onto itself is that $f$ is either $sense-\phi reserving$ or sense-rezersirg. The necessity of the condition is clear. Hence vc will shcw that $tl:e$ conclition is sufficient. But it is sufficient to provc the sufficiency for tl.c case, vshere
A new method of near field analysis
The problem of the accurate determination of the total electromagnetic field produced by a current distribution on a circular disc type aperture is examined. For comparison of the near and Fresnel region solutions, an exact Maxwellian integral is used as the starting point. Aperture and space coordinates are separated by the use of Bessel function addition theorems, allowing the field to be expressed as a sum of terms containing radial and angular integrals. The radial integrals involve only the disc (aperture) distribution and coordinates while the angular integrals involve only the elevation angle. The distance from origin appears implicitly in the series coefficients. These integrals are suitable for high speed computations and have been evaluated for a number of cases. The results are compared with the small angle and general Fresnel formula values for various observation angles, distances from the aperture, and aperture distributions.