Geometrical Optics from the Ground up

We review, with proper derivation and proofs, the common undergraduate formulas for building images of objects using a system of lenses with spherical surfaces. This is done using the first-order approximation which assumes that light rays deviate from the symmetry axis by only small angles. Yet, even this most basic approximation results in surprisingly complex theory, which is then applied to explain workings of everyday optical instruments.

Chaos in Planar, Circular, Restricted Three-Body Problem

In this article we analyze the motion of a test particle of a planar, circular, restricted three-body problem in resonance, using the Kustaanheimo-Stiefel formalism. We show that a good qualitative description of the motion can be reduced to three simple equations for semi-major axis, eccentricity and resonance angle. Studying these equations reveals the onset of chaos, and sheds a new light on its weak nature. The 7:4 resonance is used as an example.

Understanding Lens Aberrations

In most textbooks, lens aberrations are usually described in the briefest possible manner, without any attempt for their proper derivation. At the same time, monographs which do go into more detail are often inaccessible to most students and non-specialists interested in deeper understanding of this topic. This article tries to fill this gap and provide an introduction to what happens when basic formulas of Geometrical Optics are extended by third-order terms in Taylor’s expansion of sin (α). The presentation is accessible to most undergraduate students as it requires only some knowledge of basic calculus and planar geometry. The resulting five aberrations are then described in detail, including a novel derivation of the exact shape of coma. A simple Mathematica program is included to facilitate numerical exploration of the magnitude of the resulting aberrations for various optical systems.

Solving Electrical Circuits via Graph Theory

Solving for currents of an electrical circuit with resistances and batteries has always been the ultimate test of proper understanding of Kirchoff’s rules. Yet, it is hardly ever emphasized that a systematic solution of more complex cases requires good understanding of the relevant part of Graph theory. Even though this is usually not covered by Physics’ curriculum, it may still be of interest to some teachers and their mathematically inclined students, who may want to learn details of the rigorous approach. The purpose of this article is to provide a concise derivation of a linear set of equations leading to a unique solution of the problem at hand. We also present a simple computer program which builds such a solution for circuits of any textbook size.

Some Penrose transforms in complex differential geometry Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this article, we review a construction in the complex geometry often known as the Penrose transform. We then present two new applications of this transform. One concerns the construction of symmetries of the massless field equations from mathematical physics. The otherconcerns obstructions to the embedding of CR structures on the three-sphere.

Zero divisors and prime elements of bounded semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.

Coleman Automorphisms of Generalized Dihedral Groups

Let D be a generalized dihedral group and Autcol（D） its Coleman automorphism group. Denote by Outcol（D） the quotient group of Autcol（D） by Inn（D）, where Inn（D） is the inner automorphism group of D. It is proved that either Outcol（D） = i or Outcol（D） is an elementary abelian 2-group whose order is completely determined by the cardinality of π（D）. Furthermore, a necessary and sufficient condition for Outcol（D） = 1 is obtained. In addition, whenever Outcol（D） ≠ 1, it is proved that Autcol（D） is a split extension of Inn（D） by an elementary abelian 2-group for which an explicit description is given.

Some ＊-Clean Group Ringst

A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.