A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction

Starting with Maxwell’s equations, we derive the fundamental results of the Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories of scalar diffraction and scattering. These results are then extended to cover the case of vector electromagnetic fields. The famous Sommerfeld solution to the problem of diffraction from a perfectly conducting half-plane is elaborated. Far-field scattering of plane waves from obstacles is treated in some detail, and the well-known optical cross-section theorem, which relates the scattering cross-section of an obstacle to its forward scattering amplitude, is derived. Also examined is the case of scattering from mild inhomogeneities within an otherwise homogeneous medium, where, in the first Born approximation, a fairly simple formula is found to relate the far-field scattering amplitude to the host medium’s optical properties. The related problem of neutron scattering from ferromagnetic materials is treated in the final section of the paper.

ON HILBERT FUNCTIONS OF GENERAL INTERSECTIONS OF IDEALS

Let $I$ and $J$ be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of $I$ and $g(J)$, where $g$ is a general change of coordinates. Our main result gives a generalization of Green’s hyperplane section theorem.

An RS-KKM Theorem in FC-Metric Spaces with the Application to Fixed Points

In this paper, RS-KKM mappings are introduced and an RS-KKM theorem is established in noncompact complete FC-metric spaces. As applications, a Ky Fan section theorem, a maximal element theorem and a Fan-Browder type fixed point theorem are obtained in noncompact complete FC-metric spaces. These results unify, improve and generalize some known results in recent literature.

GENERALIZED ZARISKI–VAN KAMPEN THEOREM AND ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES

We formulate and prove a generalization of Zariski–van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz–Zariski–van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.