A theorem concerning homologies in a compact space

A New Theorem Concerning Scattering of Electromagnetic Waves in Multilayered Metamaterial Spherical Structures

The proposed theorem in this paper is indicative of a kind of duality in the propagation of waves in the dual media of and in the spherical structures. Independent of wave frequency, the number of layers, their thickness, and the type of polarization, this theorem holds true in case of any change in any of these conditions. Theorem: Consider a plane wave incident on a multilayered spheric...

A Theorem concerning Six Points

1. Introduction and statement of results. "Let three points be specified on a line. Then one of the points is in the interior of the line segment joining the other two, and one of the points is exterior to the line segment joining the other two." This elementary statement concerning the ordering of three points on a line is capable of various extensions. Thus, e.g., it is easy to prove the foll...

A Theorem concerning Closed and Compact Point Sets Which Lie in Connected Domains

PROOF. For each point P of D, there exists a region gp of Gi containing P such that gp is a subset of D. By Axiom 2, there exists a connected domain dp containing P which is a subset of gp. Let Ui denote the collection of all domains dp for each point P of D. The point set M is closed and compact, and hence, by Theorem 22 of Chapter I, it is covered by a finite subcollection W\ of U\. By Theore...

A Theorem concerning Systems of Residue Classes

By system we mean a multi-set whose elements are unordered but may occur repeatedly. FollowingŠ. Znám [8] we use a(n) to denote the residue class { x ∈ Z : x ≡ a (mod n) }. For a system (1) A = {a i (n i)} k i=1 of residue classes, the n i are called its moduli. Definition. An integer T is said to be a covering period of (1) if it is a period of the characteristic function of the set k i=1 a i ...

On images of continuous functions from a compact manifold to Euclidean space