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Let G be a set of elements on which there is defined a ternary operation (abc) satisfying the following postulates: (i) ((abc)de) = (ab(cde)), (ii) (abb) = a, (iii) (bba) = a. These postulates are shown to be independent and consistent. It is proved that any element u of G may be chosen as the identity of a group $G_{u}$ defined by ab=(aub) and that $(abc)=ab^{-1}c$. Further, the groups $G_{u}$ are isomorphic. Equivalent and weaker postulates are also considered. A geometrical interpretation of the ternary operation leads to an abstract definition of a set of free vectors. It is proved that any group may be converted into a group of free vectors and conversely.

I wish to express my gratitude to Garrett Birkhoff for his kind assistance and encouragement, without which this note would probably not have been written.

Lattice-ordered Groupoids and Some Related Problems. The idea of ideals in a commutative ring is a well-known one. We are able to multiply two ideals. For a given set of ideals we are able to find a unique smallest ideal containing any member of the set (greatest common divisor or g.c.d.) and a unique maximal ideal contained by any member of the set (least common multiple or l.c.m.). Under the operations of l.c.m. and g.c.d., the ideals form a lattice; and we find that multiplication distributes over g.c.d., i.e. A, B, C are three ideals, and (B, C) is the g.c.d. or join of B and C, then A · (B, C) = (A · B, A · C), This is merely a special example of a lattice-ordered groupoid, defined as follows. Let G be a lattice with join (a, b) and meet [a, b], with a multiplication a · b be defined over G such that a · (b, c) = (ab, ac), (b, c)a = (ba, ca). (This is called an operator lattice.) If multiplication is associative and an identity exists, then G is called a lattice-ordered groupoid.

Jeremiah Certaine was the fourteenth African American to earn a Ph.D. in Mathematics (University of Michigan). At this time half of all African American Ph.Ds in Mathematics were earned by students of the University of Michigan.

In 1947 Woodard was retired, Blackwell was chair, and two new regular faculty members had been added. One was Claytor, whose mathematical work, beginning with his 1933 doctoral thesis under Kline at Pennsylvania, had attracted considerable attention. It is generally agreed that his very promising career had been blunted by the racial restrictions he encountered .... Claytor was appointed as an associate professor. The new assistant professor was Jeremiah Certaine, who had just received his Ph.D. from Harvard, and who remained at Howard until 1951.

My teachers at Howard included Dr Elbert Cox, who was the first Black person in the United States to receive the PhD degree in mathematics. He received his doctorate from Cornell University in 1925. I also studied under Dudley Woodard and William Claytor who had PhD degrees from the University of Pennsylvania and David Blackwell, who was eminent enough to occupy later positions at Stanford University and the University of California at Berkeley. It was Jeremiah Certaine, the Black Harvard PhD, who helped me see the beauty of algebraic structure.

We present in this report the fundamental mathematical results upon which the NDA (Nuclear Development Associates) neutron calculational program has been based. All our results are derived from the neutron transport equation for a plane source, from which it is also possible to obtain certain results about point sources.

About 1955, J Certaine proposed a method for the numerical integration of the Boltzmann transport equation, which found its fruition in a code known as NIOBE [Numerical Integration of the Boltzmann Equation] for the lBM7090 series of computers. Basically, the transport part of the equation was evaluated in terms of the standard method of characteristic rays while a discrete energy treatment was used with the slowing clown integrals determined by Gauss- Legendre quadratures. The results of some NIOBE-calculated problems have been published and a good number more were performed in the early 1960s, but remain unpublished. After that the method sort of faded away, and is practically unknown today.

Dr Jeremiah Certaine, Howard University, has accepted the position of Manager, Department of Mathematics, Nuclear Development Corporation of America, White Plains, New York.

This chapter discusses the methods employed for the solution of the transport equation in slab geometry, transformation of the monoenergetic transport equation, a round-off free solution for the monoenergetic one-velocity single-region transport equation, the multi-region solution of the slab problem, and applications of the round-off free solution. The nuclear engineer or shield designer usually associates the Monte Carlo method with very long computing times. It turns out that at large distances from the source, very few particles can be expected to reach and to be recorded in some specific angular direction or energy range. For a reliable statistical answer, a sufficient number of particles are needed. This necessitates the tracing of the histories of an immensely large number of particles. Techniques like biased sampling and splitting and Russian roulette that increase the probability of an initial particle contributing to the final answer have been used to considerably decrease the time required for a solution. Monte Carlo calculations are resorted to only when the complications of geometry of the source and the region make it impossible to find an answer by other means. This method is particularly useful for the calculation of integral rather than differential quantities.