James R. Clay

Ambiguity of the Transfer Function with Partially Coherent Illumination

The one-dimensional case of the image of a sinusoidal transmittance distribution in partially coherent illumination (with the quasimonochromatic approximation) is described analytically, and shown generally to bear a nonlinear relation to the object. It is shown that the significant parameter is the ratio of coherence interval to the diameter of the Airy disk (or diffraction spot) of the imaging lens. It is further shown that since the spatial frequency of the object is related to coherence interval, typical nonlinear effects can take place at low frequencies. Since the transfer function is defined only for the incoherent limit without ambiguity, an apparent transfer function, dealing only with the image component which existed in the object, is used for comparison. The harmonics generated by the nonlinear behavior are ignored, and the variation of transfer function is observed to be a function of coherence and input modulation. It becomes apparent that the transfer function, as currently defined and measured, is inadequate to describe optical-system performance under all conditions of illumination.

Generating balanced incomplete block designs from planar near rings

Abstract In [5], Ferrero demonstrates a connection between a restricted class of planar near rings and balanced incomplete block designs (BIBD). In particular, if the order v of an abelian group ( A , +) is relatively prime to 6, then Ferrero constructs a planar near ring ( A , +, ·) whose blocks Ag + h , g ≠ 0, are the blocks of a BIBD with parameters (v, β, k, r, λ) = (v, v(v − 1) 2 , 3, 3(v − 1) 2 , 3) . In this paper we show: (1) that the restrictions imposed by Ferrero on planar near rings can be relaxed, (2) the construction of two large family of BIBDs, and (3) that the additive group of a planar near ring need not be abelian in contrast to that of a planar near field [10].

BLOCK DESIGNS FROM FROBENIUS GROUPS AND PLANAR NEAR-RINGS

Publisher Summary Ferrero showed, for the first time, that balanced incomplete block designs can be obtained from certain planar near-rings, that each finite planar near-ring is derived from a Frobenius group, and that each finite Frobenius group gives many planar near-rings, all of which yield the same tactical configuration. Later, Ferrero uses a powerful result of Hall to show that each of these finite planar near-rings, or finite Frobenius groups, give several different partially balanced incomplete block designs. Some of Ferreros results were extended, but most of the work in this direction had been done or directed by Ferrero. The research has developed to a point where the important ideas have surfaced. This chapter presents a unified, fairly self-contained, and streamlined development of these ideas.

Circular Block Designs from Planar Near-Rings

Publisher Summary This chapter discusses circular block designs from planar near-rings. Planar near-rings whose additive group is ( C , +), the additive group of complex numbers, have provided direction and motivation for constructing finite planar near-rings with interesting geometric interpretations. In particular, balanced incomplete block designs (BIBD) have resulted in two distinct, but related ways. Inspiration is taken from a planar near-ring on ( C , +) to obtain a third way of constructing BIBDs, and in this case, the blocks of some of the designs have properties analogous to those of circles in the euclidean plane. Further, the chapter describes computer program. The computer program used to generate the data in the table was written in BASIC, and was compiled for the Zenith Z-100 Desktop Computer. BASIC is usually an “interpreter” language. The program, left in “interpreter” form, would generate the same data, but the time required would be greatly increased.

More balanced incomplete block designs from Frobenius groups

Abstract From a Frobenius group G = N ×σ Φ with kernel N and complement Φ, one can sometimes construct new balanced incomplete block designs by taking as blocks the translations in N of blocks made from the union of the nontrivial orbits of Φ of a and −a in N together with the trivial orbit {0}.

The punctured plane is isomorphic to the unit circle

Abstract This note shows that the multiplicative group of non-zero complex numbers and the multiplicative group of complex numbers of absolute value 1 are isomorphic as groups.

Research in near-ring theory using a digital computer

AbstractIn this paper we wish to show how the computer has played a valuable role in research in the theory of near-rings. Basically, the author has used the computer to generate examples of near-rings to be applied for meaningful conjectures and counter-examples.All the near-rings of order less than eight are listed in [2]. Since there is only one non-abelian group of order less than eight, it is natural to still be curious what happens when one tries to construct a near-ring from a non-abelian group. The methods used by the author to construct near-rings from groups will be illustrated on the two non-abelian groups of order 8. Specifically, for each non-abelian group of order 8, it was decided to construct all near-rings enjoying one of the following four properties: P1.near-ring with identity:P2.near-rings without two-sided zero;P3.near-rings with no zero divisors;P4.idempotent near-rings; i.e. near-rings for whichx2=x for allx.

The structure of dilation groups of generalized affine planes

The concept of a generalized affine plane was introduced in [2]. To each such generalized affine plane there is a group of (bijective) dilations and a subgroup of translations. The subgroup of translations gives a nearring of trace preserving quasi-endomorphisms and there is a subgroup fo the translations, called the semi-identities, that give an ideal in this near-ring. The quotient nearring is a skew field.This paper is concerned with the structure of the various subgroups of dilations that arise from the geometry of the generalized affine planes. In particular, it shall be seen that the translation group, together with the cosets of a family of subgroups, will in turn be a generalized affine plane.

Compound Closed Chains in Circular Planar Nearrings

Publisher Summary This chapter discusses the concept of compound closed chains in circular planar nearrings. In 1986, the concept of a circular planar nearring was introduced at Combinatorics ’86 at Passo della Mendola. This idea is motivated by taking the field of complex numbers and constructing the planar nearring. Also introduced in Combinatorics ’86 at Passo della Mendola was a table of all the nontrivial circular planar nearrings, where nontrivial circular planar nearrings is the additive group of a prime field. By studying various examples from this table, one can find properties of these circles analogous to those in the usual euclidean plane. The chapter also discusses the torus phenomenon, the concept of simple closed 2-link chains, and the simple and compound closed s- link chains.

The Near-Ring of Some One Dimensional Noncommutative Formal Group Laws

Publisher Summary This chapter discusses the near-ring of some one-dimensional non-commutative formal group laws. In the chapter, A will denote either the ring of integers modulo p 2 , p a prime, or K[t]/(t 2 ), where K is a field of characteristic p, and t is an indeterminant.