Showing papers on "Division (mathematics) published in 1983"

The Fair Division of a Fixed Supply Among a Growing Population

Abstract: We reconsider the traditional problem of fair division. Division principles should be general enough to accommodate changes in what is to be divided as well as variations in the number of agents among whom the division is to take place. In the usual treatment of the question, this number is assumed to be fixed. Here, we explicitly allow it to vary. Agents are assumed to have von Neumann–Morgenstern utility functions and division problems are defined as subsets of the utility space. Our approach is axiomatic. Apart from the familiar axioms of Pareto-optimality, scale invariance, continuity and anonymity, we formulate and impose an axiom of monotonicity with respect to changes in the number of agents, stating that if sacrifices have to be made to support an additional agent, then everybody should contribute. There is a unique division principle that satisfies them all: it is the natural extension to the n-person case of the two-person solution proposed by Kalai and Smorodinsky.

Method and apparatus for quantitatively dividing zigzag folded sheet of paper

Abstract: A method for quantitatively dividing a zigzag folded sheet of paper comprises continuously folding a sheet of paper to obtain a zigzag folded sheet of paper having a plurality of divisions connected to one another, continuously accumulating the divisions one on top of another, allowing a separator with a cutting edge to be inserted, when the continuous accumulation constitutes a first unit composed of a prescribed number of divisions, between the uppermost division of the first unit and the division connected to the uppermost division and becoming the lowermost division of a second unit to be subsequently obtained, thereby partitioning the first unit, allowing the first unit and the separator to descend to accumulate the divisions which will constitute the subsequent unit on the separator and allow the separator to advance immediately before the fold between the aforementioned uppermost and lowermost divisions, and allowing the cutting edge to advance away from the separator to cut off the fold.

Characterization of Division Algebras of Minimal Rank and the Structure of their Algorithm Varieties

Abstract: The purpose of this paper is to show that every finite dimensional division algebra of minimal rank, i.e., of minimal complexity with respect to the noncommutative model of computation, is a finite simple field extension. Moreover, we investigate the structure of the variety of optimal algorithms for the computation of the multiplication in such fields modulo the isotropy group of the problem.

Digital signal processing system employing logarithms to multiply and divide

Abstract: A digital signal processing system applies numbers to the multiplied or divided to address a memory storing a logarithm table to produce the logarithms of the numbers. These logarithms are added for multiplication or substracted for division to produce the logarithm of the result number which is applied to address a memory storing an antilogarithm table to produce the result number. The base of the logarithms is selected in accordance with the magnitudes of the numbers to be multiplied or divided so as to utilize substantially the entire range of magnitudes of the digital representation of the logarithms.

Finite Precision Rational Arithmetic: An Arithmetic Unit

Abstract: The foundations of an arithmetic unit performing the add, subtract, multiply, and divide operations on rational operands are developed. The unit uses the classical Euclidean algorithm as one unified algorithm for all the arithmetic operations, including rounding. Binary implementations are discussed, based on techniques known from SRT division, and utilizing ripple-free borrow-save and carry-save addition. Average time behavior is investigated.

Omission and subsequent estimation of zero sequency coefficients of transformed digitized images to facilitate data compression

Abstract: A method of transmission or storage of a television picture involves dividing it into a plurality of blocks, subjecting them individually to a two-dimensional unitary transformation, transmitting or storing the transform coefficients and reconstructing the original blocks by the use of the inverse transformation. To reduce the data rate, the zero sequency coefficient is omitted for most or all the blocks, and the mean levels of the reconstructed blocks are adjusted to reduce visible brightness changes between them. The adjustment may be made to reduce the means square differences between the elements in two blocks along the common boundary. The division into blocks may give common elements in two blocks along each inter-block boundary. If a picture transition coincides with a block division an indication of this may be sent so that differences between elements along the division are ignored.

Magnetic length or angle measuring system having improved magnetic sensor arrangement

Abstract: A magnetic measuring instrument for measuring the relative position of two objects includes a scanning unit which incorporates a magnet positioned to scan a measuring division. The measuring division is provided with alternately magnetically conducting and magnetically non-conducting zones. Measuring signals of high signal quality are generated because the induction generated by the magnet in the measuring division runs only in planes perpendicular to the longitudinal extent of the measuring division (the measuring direction x). Magnetic field sensitive sensor elements in the air gap between the magnet and the measuring division scan the air gap induction to generate position dependent measuring signals.

The External Evidence for the Division of Propertius, Book 2

Abstract: hominum risu cavillatus. The diplomatic situation then prevailing in the two Arab kingdoms lent itself to the intentionally barbed question: Would the Senate which refused the toga to one mini-monarch allow another the pretension of wearing it? There is as yet no evidence that Abgar II was refused this honor, but the known outline of his reign does not exclude such a possibility.22 The variant readings or emendations noted above are much less satisfactory or convincing precisely because they do not address themselves to the multiple factors involved. We believe that the solution proposed above satisfies every consideration. The word to be restored must compare and contrast geographically, historically, and politically with Commagenum in the same sentence. Such a word must accommodate itself to a passage composed for delivery with intentional rhetorical flourishes. This same word must demonstrate that it would have been instantly comprehensible to Cicero's Senate colleagues. It must also plausibly demonstrate that it lends itself easily to errors of manuscript transmission. Lastly, it must satisfy the joking ambiance which pervades all portions of the letter to which it relates. Our examination of this crux benefits especially from a review of the historical and geographical background of Cicero's letter, important aspects overlooked by other commentators. We therefore submit that there is a solid basis, textual and contextual, for believing that the emendation we propose is preferable to anything so far suggested in the discussion of this notorious crux.

A Programming Example

Abstract: It very often happens that from the structure of the data we are able to predict the division of a problem into subproblems and therefore the program structure. The data may be the input, the output or an intermediate data structure. In some applications — as in our first example — there does not even exist a data structure as a whole but one element after the other is generated and immediately processed.

URR: Universal representation of real numbers

Abstract: A new internal representation is proposed for real numbers that has been named URR for Universal Representation of Real Numbers. This approach is based on a bisection method which is applied to real number intervals. With this method, the point of division increases or decreases double-exponentially in global area.

Binary coded decimal number division apparatus

Abstract: In preparation of addresses of a quotient prediction table used in a binary coded decimal number division scheme with predetermined bits of a dividend and a divisor in binary coded decimal representation, the addresses are modified with the redundant bits. The absolute bit number for the addresses is thus decreased, whereby data quantity and hence capacity of RAM required for implementing the quotient prediction table can be significantly reduced, while satisfactory function of the quotient prediction table being assured. The apparatus for the binary coded decimal number division is implemented inexpensively in a small size.

Apparatus and method for performing division with an extended range of variables

Abstract: A method and apparatus are disclosed which, for obtaining the quotient of division operations in a data processing apparatus, generate as an auxiliary value the inverse square root g of the divisor w. The auxiliary value g is then multiplied by the divident v, and the intermediate result again multiplied by the auxiliary value g. An improvement in operation of the data processing apparatus is obtained despite introduction of the auxiliary value because the range covered by the auxiliary value is significantly smaller than that of the direct inverse of the divisor. A preferred application is the area of signal processing in communications.

Programmable counter system

Abstract: A programmable counter system of the swallow operation type using binary counters is disclosed The counter comprises a prescaler (101) for frequency dividing an input signal by a frequency division factor "2 n -1" or "2 n ", upper and lower order bit counters (A and B) for counting down an output signal from the prescaler, a flip-flop (2) for selecting either the frequency division factor "2" - 1" or "2"" according to the logical level of the output signal of the counter A or B, and inverters (51 0 to 51 3 ) for level inverting programming data (PDO to PD3) and applying them to the lower order bit counter A, thereby setting a division number of the counter A to a complement of the binary code of the programming data

Communication arrangements for distributed control systems

Abstract: Time division switching system having distributed control processors (17, 18). The distributed processors exchange control messages and cooperate in the completion of talking paths between subscribers (23-26). Each control message includes an address portion defining the destination for the control message and is transmitted to a time-shared space division switch (10) on the same time multiplex lines (13-16) that speech representations are transmitted. Control messages are routed by the time-shared space division switch to a control distribution unit (31) where the address portion is interpreted. When the address portion defines a distributed processor which controls the time-shared space division switch, the control distribution untit transmits the associated control message directly to that processor. Alternatively, when the address portion defines one of the other distributed processors the control distribution unit transmits the associated control message to the defined processor via the time-shared space division switch. Additionally, each control message can comprise an intervention portion and an arrangement is associated with each distributed processor which responds to the intervention portions by altering the functions performed by an associated distributed processor.

A hidden line elimination method for curved surfaces

Abstract: An algorithm for displaying curved surfaces with hidden lines eliminated is described. The surfaces are displayed as rectangular grids of straight segments which approximate those surfaces. The algorith is essentially one of brute force. However, it divides the screen space into small 3D boxes to reduce the number of subpatches to be checked to determine the visibility of each grid point. It turned out that this division process made this algorithm about 10 times faster than the algorithm which had no use of the division. Expanding the algorithm for making contour drawings is very easy. Some methods for drawing silhouette lines are considered.

Fast Iterative Division of p-adic Numbers

Abstract: A fast iterative scheme based on the Newton method is described for finding the reciprocal of a finite segment p-adic numbers (Hensel code). The rate of generation of the reciprocal digits per step can be made quadratic or higher order by a proper choice of the starting value and the iterating function. The extension of this method to find the inverse transform of the Hensel code of a rational polynomial over a finite field is also indicated.

Mathematical game apparatus

Abstract: An apparatus for generating arithmetic expressions is disclosed. It comprises sets of number dice and of operator dice. The number dice are provided in two different colors, in order to remove ambiguity arising from the non-cumulative operations of subtraction and division, and to provide a means of designating negative numbers if so desired.

Device for the uniform or regular distribution of flowable media in a predetermined width

Abstract: In a device for the uniform or regular distribution of flowable media in a predetermined width with flow channels branched from a main feed by continuous division, in particular for treating web goods, the flow channels are produced in sheets or profile strips with cross-sections of any shape.