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The purpose of this paper is to identify all of the Cayley-Dickson doubling products. A Cayley-Dickson algebra $\mathbb{A}_{N+1}$ of dimension $2^{N+1}$ consists of all ordered pairs of elements of a Cayley-Dickson algebra $\mathbb{A}_{N}$ of dimension $2^N$ where the product $(a,b)(c,d)$ of elements of $\mathbb{A}_{N+1}$ is defined in terms of a pair of second degree binomials $\left(f(a,b,c,d),g(a,b,c,d)\right)$ satisfying certain properties. The polynomial pair$(f,g)$ is called a `doubling product.' While $\mathbb{A}_{0}$ may denote any ring, here it is taken to be the set $\mathbb{R}$ of real numbers. The binomials $f$ and $g$ should be devised such that $\mathbb{A}_{1}=\mathbb{C}$ the complex numbers, $\mathbb{A}_{2}=\mathbb{H}$ the quaternions, and $\mathbb{A}_{3}=\mathbb{O}$ the octonions . Historically, various researchers have used some but not all of these doubling products., Comment: 32 candidates for alternate Cayley-Dickson doubling products are winnowed down to 8 products. Four of these products produce, at the third doubling, the octonions and four produce non-octonion but alternative algebras

Regarding the Cayley-Dickson algebras as twisted group algebras, this paper reveals some basic periodic properties of these twists., Comment: 8 pages

Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form formula for the product of any two basis vectors of a Cayley-Dickson algebra., Comment: This doubling product does not produce the octonions from the quaternions, but produces an eight dimensional algebra with zero divisors

Gives an elementary exposition of the twisted group algebra rep- resentation of simple Clifford algebras

The effect of some properties of twisted groups on the associated algebras, particularly Cayley-Dickson and Clifford algebras. It is conjectured that the Hilbert space of square-summable sequences is a Cayley-Dickson algebra.

A twist property is developed which imparts certain properties on the twisted group algebra. These include an involution * satisfying (xy)*=y*x* and an inner product satisfying = and =. Examples of twisted group algebras having this property are the Cayley-Dickson algebras and Clifford algebras.

A catalog of all 32 Cayley-Dickson-like doubling products on ordered pairs (a,b),(c,d) for which (1,0) is the left and right identity and for which xx* and x*x equal the square of the norm of x given the conjugate (a,b)*=(a*,-b). Only eight of these are true Cayley-Dickson doubling products, since 24 of them do not satisfy the quaternion properties. Each of the eight Cayley-Dickson products has a distinctive representation in the Fano Plane.

The F64 frame grabber is a high performance video image acquisition and processing board utilizing the TMS320C40 and TMS34020 processors. The hardware is designed for the ISA 16 bit bus and supports multiple digital or analog cameras. It has an acquisition rate of 40 million pixels per second, with a variable sampling frequency of 510 kHz to MO MHz. The board has a 4MB frame buffer memory expandable to 32 MB, and has a simultaneous acquisition and processing capability. It supports both VGA and RGB displays, and accepts all analog and digital video input standards.

An automated docking system must have a reliable method for determining range and orientation of the passive (target) vehicle with respect to the active vehicle. This method must also provide accurate information on the rates of change of range to and orientation of the passive vehicle. The method must be accurate within required tolerances and capable of operating in real time. The method being developed at Marshall Space Flight Center employs a single TV camera, a laser illumination system and a target consisting, in its minimal configuration, of three retro-reflectors. Two of the retro-reflectors are mounted flush to the same surface, with the third retro-reflector mounted to a post fixed midway between the other two and jutting at a right angle from the surface. For redundancy, two additional retroreflectors are mounted on the surface on a line at right angles to the line containing the first two retro-reflectors, and equally spaced on either side of the post. The target vehicle will contain a large target for initial acquisition and several smaller targets for close range.