Showing papers on "Division (mathematics) published in 1976"

New Algorithms and Lower Bounds for the Parallel Evaluation of Certain Rational Expressions and Recurrences

Abstract: The parallel evaluation of rational expressions is considered. New algorithms which minimize the number of multiplication or division steps are given. They are faster than the usual algorithms when multiplication or division takes more time than addition or subtraction. It is shown, for example, that xn can be evaluated in two steps of parallel division and ⌈log2n⌉ steps of parallel addition, while the usual algorithm takes ⌈log2n⌉ steps of parallel multiplication.Lower bounds on the time required are obtained in terms of the degree of the expressions to be evaluated. From these bounds, the algorithms presented in the paper are shown to be asymptotically optimal. Moreover, it is shown that by using parallelism the evaluation of any first-order rational recurrence of degree greater than 1, e.g. yi+1 = 1/2;(yi + a/yi), and any nonlinear polynomial recurrence can be sped up at most by a constant factor, no matter how many processors are used and how large the size of the problem is.

Digital double differential phase-locked loop

Abstract: A first generator provides a reference pulse train having a predetermined reference frequency. A second generator including a voltage controlled oscillator provides a bit clock having a repetition frequency locked to the repetition frequency of the bits of digital data. A first divider coupled to the first generator divides the reference frequency by a selected one of a first division factor and a second division factor different than the first division factor. A fourth divider coupled to the second generator divides the repetition frequency of the bit clock by a selected one of a third division factor and a fourth division factor different than the first, second and third division factors. A phase comparator coupled to the first and second dividers compare the phase of the output signals of the first and second dividers and produces a control signal proportional to the phase difference between the output signals of the first and second dividers. The control signal is coupled through a loop filter to the voltage controlled oscillator to lock the phase of the bit clock to the bits of the digital data. Logic circuitry coupled to the digital data source, the second generator and the first and second dividers produce a phase slip rate signal to select the appropriate one of the first and second division factors for the first divider and to select the appropriate one of the third and fourth division factors for the second divider.

On the Number of Multiplications Required for Matrix Multiplication

Abstract: In this paper we give a new algorithm for matrix multiplication which for n large uses $n^2 + o(n^2 )$ multiplications to multiply $n \times p$ matrices by $p \times n$ matrices provided $p \leqq \log _2 n$. Multiplication and division by 2 is necessary in this algorithm. This is to be compared with $pn^2 $ for the standard algorithm and $ \simeq p^{.58} n^2 + o(n^2 )$ for an algorithm of Hopcroft and Kerr [1] which, however, requires no multiplication and division by 2.

A fast division technique for constant divisors

Abstract: A fast algorithm for division by constant divisors is presented. The method has proved very useful implemented as microcode on a binary machine, and can be adapted directly into hardware. The mathematical foundations of the algorithm are presented as well as some performance measures.

Time division metering system

Abstract: An improved system for metering a plurality of fluids is described. The instant system utilizes time division switching in flow valves to provide accurate metering of a plurality of fluids from individual reservoirs.

Existentially closed structures and Jensen’s principle

Abstract: AssumeV=L, or even ◊ M 1, there is no uncountable locally finite group which can be embedded in every uncountable universal locally finite group. Similar results hold for existentially closed groups and division rings.

Refrigeration system with magnetic linkage

Abstract: A vuilleumier-cycle refrigeration system of the type including first and second communicating, sealed, cylindrical, vessels having first and second displacers respectively mounted therein in sliding, sealing contact with the inner walls thereof, includes magnetic linkage between the two displacers for driving the displacers within the sealed vessels without contaminating the sealed vessels. With regard to the magnetic linkage, there are smaller, auxiliary cylindrical vessels coupled respectively to the first and second vessels, having smaller diameters, and thinner walls than the first and second vessels. Rods attached to the displacers and extending into the auxiliary vessels have magnetic material followers attached thereto. Magnetic-material drivers are moveable along the auxiliary vessels, opposite the followers, to thereby attract and move the magnetic-material followers and their attached displacers. This is a division of application Ser. No. 665,848, filed Mar. 17, 1976, now U.S. Pat. No. 4,118,943.

A recursive radix conversion formula and its application to multiplication and division

Abstract: A recursive formula for number conversion from one radix representation to another radix representation is presented. This formula differs from the existing ones in two major aspects. First, it utilizes a digit shift technique which provides faster accumulation of higher significant digits in the final result. Second, it is suitable for parallel computation, so that the length of time necessary for number conversion can be shortened. Thus the longer the digit number, the more appreciation in conversion time-saving will result. Applications of the recursive formula are studied in multiplication and division for negative radix numbers as well as for positive radix numbers. The multiplication and division presented here are especially useful for computations of η word precision, since successive bit-carrying propagation to the most significant digit hardly ever occurs.

Angle measuring device with a telescope

Abstract: An angle measuring device comprises a base, a telescope seated thereupon for rotation about an axis, a graduated circle having a line division, a micrometer for subdividing said line division, and photoelectric scanners for detecting angular values from said line division and from said micrometer. A graduated plate provided with a reticle is located in said telescope. Said reticle has a zone limited by two line marks which are arranged in parallel and symmetrically to and on either side of one of the reticle lines. Said two lines are spaced from said reticle line by a width which is at least double the size of one division interval of the graduated circle line division.

Information transmitting device

Abstract: PURPOSE:To get time division function of a transmitting device improved for detection of condition change, by reading the real time in parallel with the content of condition change in each apparatus at all branch stations.

Controlled frequency division frequency divider circuit

Abstract: To control the division ratio of an input frequency applied to a loop circuit including an adder, the overflow output of which is fed back to one of its inputs, the division denominator is applied to a comparator to be compared with the overflow output of the adder to select, in accordance with the comparison in the comparator, with respect to a selected denominator number, the frequency division, so that the output of the adder will be representative of a converted frequency as selected with respect to the denominator number.

Frequency-signalling circuit for a telephone

Abstract: A frequency signaling circuit includes a plurality of channels, each channel having division means for frequency division of a clock signal, the division ratio being selected in accordance with data defining the required frequency. Stochastic conversion means are provided for converting the output of the frequency division means into pulse trains with the repetition rate of the required frequency. The stochastic conversion means includes a noise generating circuit, a value generating circuit for generating signals representing a predetermined sequence of values at a rate determined by the output of the division means and a comparison circuit for comparing the values produced by the value generating circuit with the values represented by the outputs of the noise generating circuit and for producing a pulse whenever a predetermined relation exists between the values.

British socialist theories of the division of power by area

Abstract: MOST works on British local government leave the impression that John Stuart Mill was the last political theorist to have thought seriously about the importance of local government in a democratic polity. The aim of this article is to show that both the Fabians and guild socialists also deserve attention, not just because their theoretical work on local and regional government has been largely neglected but also because it is of considerable relevance to many of today’s current problems. These problems have been recently investigated by both the RedcliffeMaud and Kilbrandon Royal Commissions. The tragedy of these Royal Commissions is that fundamentally they were both considering, in isolation from each other, the same problem-the division of power by area.’ They were both concerned with deciding how many levels of government there should be in England and in the United Kingdom and what boundaries and functions each level should have. If the problem is viewed in this light, questions about the values, structures, boundaries, and functions of traditional local government are subsumed in these broader concerns-which is as it should be if we want to view the problem clearly and comprehensively. Had there been just one royal commission dealing with this broad subject, we could have expected a more sophisticated theoretical underpinning to the justifications for its reform proposals than we received in the Redcliffe-Maud report and more meaningful consideration given to devolving power throughout the United Kingdom that we received in the majority Kilbrandon report.2 The Fabians and guild socialists had the advantage of not being fettered by official terms of reference. Neither did they face the complications of Celtic nationalism which were forced on the Kilbrandon commission. Although the prescriptive work of the Fabians and guild socialists is obviously flawed by a lack of political reality, their general approach contains numerous insights for the student of modern government. Because they were concerned with developing a collectivist value structure and programme to replace the individualist ideals of the previous century, much of their writing tended to negate traditional values relating to local autonomy and self government. Nevertheless, both groups of socialist theorists did pay considerable attention to the role of local and regional government in their ideal society. A discussion of their attempts to relate this aspect of their theory to their more general socialist beliefs forms the central part of this article. * This is a revised version of a paper presented at the meeting of the Canadian Political Science Association at St. Johns, Newfoundland in June 1971. This concept was first developed by Arthur Maass: see his ‘Division of Powers: An Areal Analysis’, in A. Maass, ed., Area and Power (Glencoe, 1959), pp. 9-26. A recent British book which implicitly uses this concept is J . P. Mackintosh, The Devolution ofpower (London, 1968). Lord Crowther-Hunt and Professor A. Peacock‘s Memorandum of Dissent, Report of Royal Commissionon the Constitution, Vol. 11) largely avoids this fault. They struggle with the relationship of their proposals to the recent local government re-organization in Chap. V.

Continental European Division of the International Association for Dental Research—Abstracts of Papers Presented at the Thirteenth Annual Meeting

Abstract: S D151

A method for plotting curves defined by implicit equations

Abstract: A square chain of points is found that approximates a plane curve of the form F (x,y) =0. In case F (x,y) =0 represents a conic section, the method requires no multiplication or division, and so is well adapted to hardware or small computer software implementation. The algorithm may also be used to find approximate solutions of differential equations of the form F1dx + F2dy = 0.

Folded space-time-space switching network

Abstract: A switching network provides interconnections for time division channels utilizing a folded space-time-space network. The network modifies the ratios of operation frequency between space division switches and time division switches to increase the efficiency of transmission while minimizing the total volume of equipment.

A dividing apparatus of ellipsoidal pulse for numerical control

Abstract: PURPOSE:An optional ellipsoid is divided to pulses by only adding a simple circuit to the pulse division of a circle.

Pedal device for electronic musical instrument

Abstract: PURPOSE: To simplify the circuit configuration by extracting the lowest sound of keyboard switches in time division multiplex signals after converting the information of keyboard switches into the time division multiplex signals. COPYRIGHT: (C)1977,JPO&Japio

Time division multiple control system

Abstract: PURPOSE:Time division multiple control system in which system can be composed, without consideration of delay compensation to synchronizing of main control part and controlled part by transmission delay

Apparatus for correcting dynamic unbalance of rotary body

Abstract: PURPOSE:To provide apparatus for automatically correcting dynamic unbalance of a rotary body by the radiation of a laser beam, wherein the number of unbalance collection working plane is made plural, and the laser beam is used in time division thereby doubling the cost-effect ratio of the apparatus