Abstract: (starting from an arbitrary x1 and r1 = b Ax1, Pt = r1) converges to the solution of(1) in n steps or less. Theoretically, the process (2) can be carried out in such a manner that only one multiplication of A times a vector is required at each step, although in practice two such multiplications are often employed. In many applications we are led to equations of the form (1) where A is nonsingular and symmetric but not positive definite. For example the problem of minimizing the quadratic form

Abstract: SUMMARY The response of lymphocytes from neonatally thymectomized rats to various mitogens was studied in vitro. Lymph node lymphocytes were stimulated with phytohcmagglutinin, poleweed mitogen, or allogeneic cells treated with mitomycin-C. A significant depression in the uptake of tritiated thymidine occured in response to all mitogens, although the cll survival in unstimulated cultures from thymectomized animals was normal. In vitro lymphocyte response to mitogens correlates well with tests of cell-mediated immunity in rats and, like these, appears to depend upon a thymic influence.

Abstract: A reciprocal conversion technique for obtaining the quotient of two numbers and the reciprocal of a number. A predetermined number of leading bits of the mantissa of the denominator is used as an entry into a table used for locating the required number of shifts and adds or shifts and subtracts to form a standard from of a denominator. Significant precision control and the semireciprocal of the normalized fraction is formed in successive multiplication steps. The reciprocal of the normalized fraction is formed and the quotient can thereafter be determined with a final multiplication step.

Abstract: Winograd has considered the time necessary to perform numerical addition and multiplication and to perform group multiplication by means of logical circuits consisting of elements each having a limited number of input lines and unit delay in computing their outputs. In this paper the same model as he employed is adopted, but a new lower bound is derived for group multiplication—the same as Winograd's for an Abelian group but in general stronger. Also a circuit is given to compute the multiplication which, in contrast to Winograd's, can be used for non-Abelian groups. When the group of interest is Abelian the circuit is at least as fast as his. By paralleling his method of application of his Abelian group circuit, it is possible also to lower the time necessary for numerical addition and multiplication.

Abstract: Apparatus and method for obtaining the reciprocal of a number and the quotient of two numbers is disclosed. The dividend and divisor, after left justification of the most significant ''''one'''' of each, are supplied to an array of combinatorial logic, the output of which is a group of polynomials having positive and negative terms. Arithmetic means are provided for subtracting the negative terms of the polynomials from the positive terms thereof to obtain the reciprocal of the divisor. This reciprocal may thereafter be multiplied by the dividend by well-known multiplication means to form the desired quotient. The apparatus and method perform the described arithmetic functions according to a flow-through scheme, where a flow-through scheme is defined as a scheme not requiring iterative techniques.

Abstract: I. Systems of exponential polynomials. Tarski has observed in [2] that there is no decision procedure for the elementary theory of complex numbers with addition, multiplication, and exponentiation. In this section a stronger result is proved: there is no uniform procedure for deciding whether a system of equations built up using addition, multiplication, and exponentiation has a solution in complex numbers. We write Rat(x) if x is rational, Int(x) if x is an integer, and Nat(x) if x is a natural number. In what follows, unless otherwise specified, variables range over the complex numbers. It is easy to verify that

Abstract: This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication by i. In a previous paper (8), we had considered curves with complex multiplication by √ − 2. Here we shall look at the case when the ring of complex multiplications is isomorphic to the ring Z[ω], where ω 3 = 1, ω ≠ 1.

Abstract: The representation of integers by a place value sequence of digits taken from a radix polynomial is one of the profound mathematical developments of all times. It provides an elegant foundation for mathematical questions where algorithmic procedures are needed, such as how to computationally effect the operations of addition, subtraction, multiplication, division and comparison. Nevertheless, as invaluable as the place value representation system is to computational mathematics, most questions regarding the mathematical structure of the integers are usually answered with proofs which make no reference to any representational form of the integers, but are based solely on an abstract characterization of the integers.

Abstract: A general method for evaluation of transition matrices is presented. The technique involves determination of the inverse of the modified Vandermonde matrix followed by its post multiplication with the initial vector.

Abstract: This paper presents an algebraic solution to the problem: given a diagrammatic representation of a woven design (weave), determine (1) the size and type of loom patterning mechanism necessary to produce it; (2) the initial conditions of this loom, i.e., the way the threads are connected to the patterning mechanism; and (3) the dynamic control information to this mechanism. First, the problem is solved for weaves of one layer (two dimensions). Then the method is extended to two-layer (three dimensional) weaves. It is then generalized to n layers (three dimensions).The method of solution represents the given weave as a binary matrix W and the weaving process as the multiplication of two binary matrices L and H. The product of this multiplication is the design W. The method shows how to compute L and H given W. A technique is developed for reducing multilayer woven structures to a single layer W in order to compute the L and H matrices more easily.

Abstract: The above material contains twelve headings that indicate a collection of families of models. Many of these collections consist of several families. For example, the description of the collection of number-line models refers to at least seven families. There are listed here more than 20 families of models for addition and subtraction on the whole numbers which are easily distinguished from one another. Research has revealed that there are also more than 20 well-known models for the other operations of multiplication and division of whole numbers.

Abstract: berg system, a method for addition, multiplication, division, and square root extraction which depends primarily upon the memorization of a great many rules and secondarily upon the use of mental arithmetic. Trachtenberg's approach to computation suggests some interesting experimentation in the teaching of whole-number arithmetic. We would like to point out some of the directions experimentation might take, in the hope that readers of this article will be stimulated to try them. One feature of the Trachtenberg system is a collection of rules for multiplication by 3, 4, 5, 6, 7, 8, and 9. In most cases this is unnecessary, since the majority of students memorize the multiplication tables with little serious difficulty. Once these tables have been memorized the usual mul-

Abstract: Let A be an H-space and K a space. It is well known that [K, A] is a loop. Suppose A has a comultiplication as well, that is, cat A < 2. Then we shall prove that [K, A] is a Moufang loop. This generalises a result of C. W. Norman who proved this for the case where A is the circle, the 3-sphere or the 7-sphere. It also improves the known result that [K, A] is a diassociative loop if A has a comultiplication as well, since Moufang loops are diassociative.

Abstract: A he number line is a powerful tool that can be used to illustrate the arithmetical operations and their relationships. Unfortunately, when the topic of multiplication and division involving two negative numbers occurs, there seems to be little or no information regarding how to represent these problems on the number line. With the use of two interpretations for the + and signs, both multiplication and division involving negative numbers can be easily represented and their inverse nature illustrated.

Abstract: With a suitable adder organization it is possible to overlap the adder operation during a binary multiplication and significantly decrease the overall multiplication time. The method is explained and a prototype multiplier described. The new technique provides a very economical method of obtaining a reasonably fast multiplier.

Abstract: The time and incremental complexity required to perform two-operand addition using logical circuitry are compared for nonredundant and minimally redundant encodings of the operands. The comparison is extended to multi-operand addition and two-operand multiplication.