Showing papers on "Division (mathematics) published in 1975"

On-line algorithms for division and multiplication

Abstract: In this paper we are considering problems of division and multiplication in a computational environment in which all basic arithmetic algorithms satisfy "on-line" property: to generate jth digit of the result it is necessary and sufficient to have argument(s) available up to the (j+δ)th digit, where the index difference 6 is a small positive constant. Such an environment, due to its potential to perform a sequence of operations in an overlapped fashion, could conveniently speed up an arithmetic multiprocessor structure or it could be useful in certain real-time applications, with inherent on-line properties. The on-line property implies a left-to-right digit-by-digit type of algorithm and consequently, a redundant representation, at least, of the results. For addition and subtraction such algorithms, satisfying on-line property, can be easily specified. Multiplication requires a somewhat more elaborate approach and there are several possible ways of defining an on-line algorithm. However, the existence of an on-line division algorithm is not obvious and its analysis appears interesting.

Digitally Controlled Absolute Voltage Division

Abstract: In this paper, a principle for absolute voltage division is presented. The division ratio of a voltage divider on this principle does not depend on the values of its elements but depends exclusively on the configuration of the divider network. Hence, calibration is not necessary and not even possible. Absolute voltage division is obtained by cyclically shifting the network elements along all positions in the divider network. Each position is maintained for an equally long time interval. The average output voltage of such a dynamic divider is identical to that of a static divider with the same network configuration but composed of elements which all have the same value. To verify the principle in practice, a digitally controlled resistive voltage divider has been built. It has been realized with easily available electronic components, such as carbon resistors with ± 5-percent tolerance, junction field-effect transistors as electronic switches and digital integrated circuits for the generation of the switch drive signals. The inaccuracy of this divider is less than 5 ×10-6 and the temperature coefficient of the division ratio is less than 5 × 10-8/°C from 0°C to 40°C. It is expected that the performance, of this prototype can be improved.

Pipeline Iterative Arithmetic Arrays

Abstract: Observations are made on the effect of pipelining iterative arrays for multiplication. It is suggested that the best type of latched array for multiplication is the save-carry iterative array. The effect of pipelining on other iterative arrays, including the general multiply/divide array is discussed.

Educational gaming apparatus

Abstract: An educational gaming apparatus consisting of a set of numbered blocks, at least six in number and dodecahedron in shape, and a single operator block, also of dodecahedron configuration, each of the number blocks bearing one of the digits zero to nine on each of its faces with an unbiased distribution of the digits zero to nine on the set of numbered blocks, the operator block bearing one or more of the basic mathematical symbols of addition, subtraction, multiplication or division or their equivalent on each of its faces such that at least one of the basic mathematical symbols or their equivalent is present on an uppermost face section of the operator block for any random toss by an individual who can then arrange the blocks in such an order that the uppermost faces of the blocks indicate a mathematical problem and its solution.

Space division network for time-division switching systems

Abstract: A high-capacity, low-blocking time-division switching system including a three-stage space-division network is disclosed. In order to achieve high capacity and low blocking the space-division network is physically large resulting in delays of the Pulse Code Modulated (PCM) data words transmitted therethrough which are large in comparison to the individual time slots. In order to effectively increase the time available for the transmission of data words through the space-division network, communication paths through the individual network stages are completed in sequence and overlapped in time. The result of this operation is that a path through each network stage is completed for a time less than or equal to a time slot but the time between establishing the first stage path and removing the last stage path is sufficient to pass an entire data word.

Definition einer Kahan-Arithmetik und ihre Implementierung

Abstract: To get an Interval Arithmetic, which allows the division by any interval containing zero, the known arithmetic of the finite closed intervals is enlarged, under application of the rules for calculation with "±∞", to an arithmetic of infinite, closed intervals. With the help of this arithmetic a so-called "Kahan-Arithmetic" is constructed for intervals which contain "±∞" in their interieur.

Semi static time division multiplex slot assignment

Abstract: SEMI STATIC TIME DIVISION MULTIPLEX SLOT ASSIGNMENT Abstract of the Disclosure A plurality of time division multiplexed time slots circulate on a loop communication link, some of said time slots being permanently as-signed to terminals connected to the loop and others being temporarily assigned to terminals such as on an every third or every fifth slot basis by an extra slot count command. One use of a write echo command allows even permanently assigned time slots to be temporarily utilized by a terminal having a very high data rate.

Applications of Replacement Theory

Abstract: This paper describes a practical procedure for applying existing theory to obtain “approximate optimum” replacement policies for homogeneous pieces of equipment. In addition, the problem of detecting changes in time-to-failure distributions, the relation of these distributions to the life cycles of equipment, and common mistakes that can result in improper policies are discussed. Application of the procedure is illustrated by two examples encountered by Union Carbide Corporation—Nuclear Division personnel.

A Note on Base –2 Arithmetic Logic

Abstract: Circuits for performing arithmetic operations using base –2 representations are considered. Study of the counting process leads to a negative binary up-down counter and new simple methods for positive-negative base conversions. The advantage of employing carry-borrow rather than carry-only during additions is pointed out. Certain special features of negation, arithmetic shift, multiplication, and division in base –2 are described.

Numerical Analysis at the National Bureau of Standards

Abstract: In 1946/47, Dr. E. U. Condon (1902–1974), the fourth director of the National Bureau of Standards, arranged for the organization of a Division of Applied Mathematics, under the leadership of Dr. John H. Curtiss. A principal objective of this division was the development of the mathematics appropriate for the automatic digital computing machines then being constructed. An account is given here of some of the activities in numerical analysis carried out by that division in its first 25 years.

Exceptional Jordan division algebras over a field with a discrete valuation.

Abstract: Exceptional simple Jordan algebras play an important role in various branches of algebra, analysis, and geometry. Apart from being of central interest in the theory of Jordan algebras itself, they are, for example, closely connected with algebraic groups. In fact, äs Hijikata [2] has shown by means of Galois cohomology, classifying the simple algebraic groups of type F4 defined over a perfect field of characteristic not 2 or 3 amounts to the same äs classifying the central simple exceptional Jordan algebras over that field. In the present paper, we address ourselves to a particular aspect of the second one of these two problems: Over a field k which is complete under a discrete valuation we investigate the structure of central exceptional Jordan division algebras. It turns out that, pathologies of characteristic 2 and 3 being ignored, these are completely determined by the central exceptional Jordan division algebras and the central associative division algebras of degree 3 over the residue class field k. The proof requires a good deal of machinery, including, among other things, a lifting theorem for involutions of the second kind, which seems to have some interest of its own. We also determine the isotopy classes of exceptional Jordan division algebras over k. At the end of the paper, we sketch the implications of all this for the field K((Tl, . . . , Tn)*) of formal power series in n ̂ l indeterminates 7\\, . . ., Tn having coefficients in a given ground field K ). Our results show, somewhat surprisingly, that over ((7\\, . . . , TJ) every central simple exceptional Jordan algebra is reduced, whereas C((7\\, . . . , TJ) admits central exceptional Jordan division algebras for n > 3.

Construction of buildings by assembling prefabricated elements

Abstract: Construction of buildings by assembling prefabricated elements. The volumes constructed are the sum of elementary volumes resulting from the division of a right prism whose height is a dimension U which is taken as unit and whose base is an equilateral triangle whose sides are equal to 2 U, this division being made through a plane passing through one side of one of the bases and the apex of the other base in line with the apex opposite the said side.

Symmetrical odd-modulus frequency divider

Abstract: A frequency divider arrangement that can be used for division by an odd number and which provides a symmetrical output. It comprises an exclusive Or gate followed by a divide by 2N counter where (2N-1) equals the odd number. Input to the exclusive Or gate is from a pulse source and from the counter output.

Solid pickup equipment

Abstract: PURPOSE: In case the color element is irradiated with time division, the storage time of electric charge element is adjusted and the difference of light quantity between color elements as well as the difference between color elements of conversion characteristics of electric charge transfer element is compensated. In this way, color balance adjustment can be simplified. COPYRIGHT: (C)1977,JPO&Japio

Picture transmissin system

Abstract: PURPOSE: The encoded picture signal element of no correlation between frames is inserted in terms of time division multiplication into coded pictue signal element with correlation between frames. As a result, picture can be transmitted very effectively. COPYRIGHT: (C)1977,JPO&Japio

Time division communication system adapted to structural expansion

Abstract: A time division communication system is constructed with modular time division switches. The system includes 2n switch modules (n is a positive integer), and each switch module is composed of m line control units (m is a positive integer) for accomodating m lines and a time division switch control unit capable of controlling 2m line control units. The switch modules are assembled in pairs, and cross connections are provided between the switch modules of each pair to enable the respective time division switch control units to mutually send and receive time-division multiplexed information. In addition, the input and output terminals of the corresponding time division switch control units of the respective switch module pairs are interconnected so that respective ones of the module pairs may mutually send and receive time-division multiplexed information.

Oscillation type division count apparatus of scale division interval

Abstract: PURPOSE: Oscillating type division count apparatus of scale division interval, eliminating the error, caused by the velocity change of oscillator, by generating clock signal with oscillator. COPYRIGHT: (C)1977,JPO&Japio

Arithmetic board game

Abstract: A competitive board game in which two playing boards are used and a set of 117 tiles for placement upon the playing boards, the playing boards having spaces where the tiles are to be placed, some of the tiles having arithmetic addition, subtraction, multiplication and division symbols while other of the tiles have the numerical digits, the game using arithmetic so to be played.

Two simple windowing algorithms

Abstract: Two algorithms that solve the frequently occurring problem in computer graphics, windowing, are described. The primary distinction between the two algorithms is the need for multiplication and division in one of them. The algorithms are compared for running time and size on two commonly used minicomputer systems, the Digital Equipment Corporation PDP-9/15 and PDP-11. Not surprisingly the algorithm incorporating the extended arithmetic capability executes faster and the PDP-11 versions are substantially smaller.