Showing papers on "Remainder published in 1981"

Hybrid ray-mode formulation of parallel plane waveguide Green's functions

Abstract: A line-source excited parallel plane waveguide with perfectly conducting or surface impedance walls is investigated in the high frequency range where alternative field representations involve many rays or many modes. It is shown that a hybrid ray-mode field representation can be developed whereby the truncation remainder of a mode series can be expressed in terms of ray fields or, conversely, the truncation remainder of a ray series can be expressed in terms of guided modes. A great variety of appropriate ray-mode combinations, each determined from a physically well-founded criterion, is possible: with respect to propagation angles measured at the source, the retained modes are those whose characteristic plane wave propagation angles fill the void left by the truncated ray series and vice versa. The analysis thus clarifies and quantifies the intimate relation between a bundle of rays and a corresponding bundle of modes. The accuracy of various ray-mode mixtures, and computational simplifications gained thereby, are illustrated in several numerical examples covering different parameter ranges, with the conventional guided mode series serving as a reference solution.

Compatible hardware for division and square root

Abstract: Hardware for radix four division and radix two square root is shared in a processor designed to implement the proposed IEEE floating-point standard. The division hardware looks ahead to find the next quotient digit in parallel with the next partial remainder. An 8-bit ALU estimates the next remainder's leading bits. The quotient digit look-up table is addressed with a truncation of the estimate rather than a truncation of the full partial remainder. The estimation ALU and the look-up table are asymmetric for positive and negative remainders. This asymmetry reduces the width of the ALU and the number of minterms in the logic equations for thy look-up table. The square root algorithm obtains the correctly rounded result in about two division times using small extensions to the division hardware.

Error estimates for interpolatory quadrature formulae

Abstract: In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Polya type formulae. Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainderR[f] of a formula of orderm can be represented asc·f (m)(?). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases. Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.

A Remainder Term Estimate for the Normal Approximation in Classical Occupancy

Abstract: Let balls be thrown successively at random into $N$ boxes, such that each ball falls into any box with the same probability $1/N$. Let $Z_n$ be the number of occupied boxes (i.e., boxes containing at least one ball) after $n$ throws. It is well known that $Z_n$ is approximately normally distributed under general conditions. We give a remainder term estimate, which is of the correct order of magnitude. In fact we prove that $0.087/\max(3, DZ_n) \leqq \sup_x |P(Z_n < x) - \Phi((x - EZ_n)/DZ_n)| \leqq 10.4/DZ_n.$

Hybrid ray-mode formulation of SH motion in a two-layer half-space

Abstract: A hybrid ray-mode formulation is presented as an attractive alternative to the generalized ray formulation or normal mode formulation for the construction of synthetic seismograms By the hybrid method, the motion is expressed as a combination of ray fields, modal fields, and a remainder The number of retained ray fields can be chosen at will, and the required number of modes is thereby determined Thus, the hybrid method quantifies the truncation effect of a ray series in terms of a certain number of modes, plus a remainder, and vice versa The remainder field provides the fine tuning that ensures continuity of the total motion; its effect is greatest near the arrival time of the last retained ray and diminishes soon thereafter The hybrid scheme incorporates within a single concise framework the advantages of a ray description for early observation times and of a modal description for later observation times Therefore, over a prolonged range of observation times, it is numerically more efficient and physically more appealing than either the pure ray or pure mode formulations These aspects are demonstrated on the simple example of a two-layer (sediment atop bedrock) half-space excited by a line source of SH waves in the semi-infinite bedrock Numerical results are presented for parameters employed in a similar calculation by Heaton and Helmberger and by Swanger and Boore

Singular sets and remainders

Abstract: This paper characterizes the singular sets of several traditional classes of continuous mappings associated with compactifications. By relating remainders of compactifications to singular sets of mappings with compact range, new results are obtained about each. All spaces to be considered here are locally compact and Hausdorff and all functions are continuous. The compact subsets of a space Z will be denoted by %z. For/? G Z, 9l(/?) (9tK(/?)) denotes the family of open neighborhoods (open neighborhoods with compact closure) of /?. For a mapping /: X -* Y, Cain [2] defined the singular set off to be S(/) = {/? G r|V£/ G 9L(/»), 3F G %Y,p G F c U and f~l(F) <2 %x}. In a later paper [3], he further explored the nature of S (/). Whyburn [14] gave an alternate definition. Various authors have seen methods of constructing a compactification of X given a map /: X —> Y where Y is compact. Perhaps the earliest (in the context of this paper) was Loeb [9]. Others who should be mentioned are Steiner and Steiner [11], [12], Magill [10], Blakley, Gerlits and Magill [1], and Choo [7]. Chandler and Tzung [6] defined the remainder induced by f to be £(/)= n{clYf(X\\F)\\FG%x} and proved that (whenever Y is compact) there is a compactification aX with aX \\ X homeomorphic to £(/). We show here that §(/) and £(/) are the same set. We can then use what was previously known about singular sets to obtain results about remainders and vice versa. We will also explore this set for three types of mappings commonly encountered in studying compactifications: (i) the composition of a mapping of X into Y and the inclusion of Y into a compactification of Y; (ii) the composition of a mapping of X into Y and a quotient mapping of Y; and (iii) the evaluation mapping into a product generated by the inclusion of X into a compactification and by a mapping of X into a compact space K. Received by the editors April 9, 1980 and, in revised form, October 1, 1980. AMS (MOS) subject classifications (1970). Primary 54D35, 54D40, 54C10.

Fluid filter cartridge and method of its construction

Abstract: An insert for placement between concentric cylindrical filter elements comprises in a one-piece construction a tubular cylindrical body and pairs of parallel slits in the body each forming therebetween a strip of material connected to the remainder of the body at the ends of the strip. The strips are bowed inwardly away from the remainder of the body between their ends to form protrusions extending transversely from the remainder of the body.

On a method of asymptotic evaluation of multiple integrals

Abstract: In this paper, some of the formal arguments given by Jones and Kline [J. Math. Phys., v. 37, 1958, pp. 1-28] are made rigorous. In particular, the reduction procedure of a multiple oscillatory integral to a one-dimensional Fourier transform is justified, and a Taylor-type theorem with remainder is proved for the Dirac 8-function. The analyticity condition of Jones and Kline is now replaced by infinite differentiability. Connections with the asymptotic expansions of Jeanquartier and Malgrange are also discussed.

The Universal Domination of Geometry.

Abstract: Contrary to popular belief, the idea of "pure" geometry, divorced from algebra, is quite recent (early nineteenth century) and was completely foreign to the Greeks. They had no algebra in the modern sense, but their geometry was really an "algebro-geometry," a complex mixture of purely geometric arguments, and computations with ratios of segments. For instance, most properties of conics are proved by Apollonius by using what we now would call the cartesian equation of the curve with respect to two axes consisting of a tangent to the conic and the corresponding diameter. On the other hand, the solution of quadratic equations is presented as a geometric construction of areas. Finally the method of solving an algebraic problem by finding the intersection of two curves is one of the oldest in Greek mathematics: the solution given by Maenechmus to the problem of "duplicating the cube" (i.e., finding a ratio x/y such that (x/y)3 = a/b, a given ratio) was to intersect the two curves xy = ab, ay = x2, and this introduced the conics in mathematics. As soon as the invention of coordinates enormously enlarged the scope of geometry, both Descartes and Newton emphasized the possibility of using this new tool to solve far more complicated equations by intersection of curves. Thus, again contrary to what most people think, the method of coordinates worked both ways, linking algebra and geometry rather than replacing one by the other. This possibility of interpreting algebraic problems in geometric terms had a great appeal for mathematicians ever since the end of the seventeenth century. Unfortunately it was limited to problems dealing with 2 or 3 independent variables, but with the development of Mechanics and Astronomy, problems with an arbitrary number of variables (the "degrees of freedom") became more and more common; furthermore, the passage from n variables to n + 1 variables usually did not modify the algebraic treatment in an appreciable way. The temptation to use in such problems a language inspired by geometry thus became irresistible by the middle of the nineteenth century; after 1870 it was generally agreed that one could use in mathematics a conventional language, derived from ordinary geometry, without of course claiming any more that it corresponded to an underlying physical reality. Instead of speaking of a system of n numbers, one would say it was a "point in n-dimensional space;" the set of such systems satisfying a linear equation would be called a "hyperplane," etc. The success of this idea has been amazing, and has proliferated during the last century in an unexpected variety of ways. In the remainder of this paper I would like to emphasize some of the highlights of this evolution.

Algorithms for solution updating due to large changes in system parameters

Abstract: A set of algorithms for solution updating due to large changes in system parameters is derived. The algorithms are derived by partitioning the changes in the values of the elements from the remainder of the system and then applying tearing procedures to solve tthe partitioned system of equations. The algorithms obtained include new ones as well as most of the previously proposed ones for solving this problem. Sparse matrix solution techniques are used whenever possible, and the computational requirements of the algorithms are assessed and compared.

Detecting redundant digital codewords using a variable criterion

Abstract: Fast response detection of repeatedly transmitted redundant digital codewords is provided by checking the same bit-position in consecutive n bit codewords. The number m of codewords checked is increased until the number of equal bit values reaches a varying threshold j m dependent upon m, or until m reaches a maximum M. This sequence is repeated for each of the remaining bit-positions of the codeword. Faster response is provided when an early threshold is reached, eliminating the need to continue checking the same bit-position in the remainder of the M codewords.

Remainder term in the Weyl-Selberg asymptotic formula

Abstract: One derives a refinement of the Wey1-Selberg asymptotic formula for an arbitrary Fuchsian group of the first kind with a noncompact fundamental domain.

Division control system

Abstract: PURPOSE:To shorten an operation cycle time, by performing the operation for generation of the partial remainder and the generation of the control signal in parallel. CONSTITUTION:A dividend, a divisor, and the 3/2 of the divisor are set to A register 1, D register 2, and register 3, respectively, and the A register output, the D register output, and so on are selected by selecting circuits 7 and 8, and outputs of selecting circuits 7 and 8 are added or subtracted by adder and subtractor 11, and the result is set to R register 15. The prescribed upper priority digits of outputs of selecting circuits 7 and 8 are sent to control signal forecasting circuits 9 and 10 and are added. The output of control signal forecasting circuit 10 is selected if carry is generated from prescribed digits of adder and subtractor 11, and the output of circuit 9 is selected if carry is not generated. This output forecasts and determines a multiple divisor, a shift quantity of the partial remainder, and addition or subtraction of adder and subtractor 11 in the next operation cycle. Thus, since the operation for generation of the partial remainder and the operation for forecasting of the control signal are performed in parallel, the operation cycle time is shortened.

Method of decoding phrases and obtaining a readout of events in a text processing system

Abstract: Method of decoding stored phrases and obtaining a readout of events in a text processing system controlled by a processor (2) with program (50, 51, 52) and tables (53) stored in a memory (4). The method includes comparing a phrase made up of a number of words encoded on a byte valuelfrequency of use basis and included in a phrase table (57) with the words of a decode table (58) arranged on a byte value/frequency of use basis, a pointer associated with each word pointing to a word stored in a word table (59). Upon a compare, the number of bits information associated with the word resulting in a compare is used to determine any phrase remainder. If there is a remainder, the remainder is compared with the decode table to obtain a match. The compare/remainder determination sequencing continues until all words in the phrase have been decoded.

Dividing circuit device

Abstract: PURPOSE:To increase a processing speed by foreseeing a shift value from a dividend and divisor and by making a shift right after deciding the shift value at the timing when an intermediate remainder is obtained. CONSTITUTION:On the basis of the contents of registers 1 and 2, arithmetic part 6 finds (A-B)>=B... CO0, (A-B)X2 >=B... CO2, and ...(A-B)X2 >=B... COn simultaneously with the subtraction processing of adding circuit 3. Then, shift-value foreseeing circuit 7 determines shift value (i) satisfying 2B>(A-B)X2>=B on the basis of the above-mentioned CO0, CO1, CO2.... Next, shifter 4 shifts intermediate remainder (A-B) by the above-mentioned shift value (i) immediately at the timing when intermediate remainder (A-B) is obtained and the result is set as a new intermediate remainder in register 1.

Division error detecting system

Abstract: PURPOSE:To output the error signal with a simple constitution, by adding a comparatively simple hardware to detect the error of the control system of the dividing device. CONSTITUTION:In the repeated operation cycle, the dividend from devidend A register 1 is applied to control signal generating circuits 8 and 9, and is applied to adder and subtractor 10 for operation between the partial remainder and a proper multiple of the divisor. One of outputs of circuits 8 and 9 is selected according to the signal of adder and subtractor 10 by selecting circuit 11, and the output is stored in register 12. The multiple of the divisor determined by contents of register 12 and the divisor from D register 2 where the multiplier is stored are added or subtracted by adder and subtractor 10, and the partial remainder of the result is stored in register 14. The quotient is forecasted on a basis of the output of register 12 by forecasting circuit 13, and the quotient is forecasted on a basis of the output of register 14 by forecasting circuit 15, and outputs of forecasting circuits 13 and 15 are compared with each other by comparing circuit 16; and if they do not agree with each other, the error is output from comparing circuit 16.

Display system for remainder time of cassette tape

Abstract: PURPOSE:To display the remainder time accurately without reference to the kind, thickness, etc., of a tape by using the rotational speeds of both supply and take-up reels, the tape winding number of the take-up reel, and a fixed constant. CONSTITUTION:Rotational speeds of bodies 9 and 12 of revolution of supply and take-up reels 1 and 2 are detected by rotation detectors 8 and 11 respectively and supplied to rotational-period detection parts 14 and 15, so that rotational frequencies of both the supply and take-up reels will be detected respectively. Counter part 16 at the take-up side serves as a tape counter and arithmetic part 17 receives the outputs of those detection part and counter part and also calculates tape remainder time t1 by using a previously stored constant according to expression (1). Rotational speeds Vs of those reels are found by expression (2). Further, An1 is a constant determined by the thickness and speed of the tape and previously calculated by measurement.

Invariance Principles for Rank Statistics for Testing Independence

Abstract: In this paper we consider a general class of rank order statistics for testing independence in bivariate populations. Each statistic is represented as a sum of independent and identically distributed (i.i.d) random variables and a remainder term. Suitable order (a.s.) of the remainder term is found and then some invariance principles are obtained. The results obtained are extensions of the results of Chernoff and Savage (1958), Bhuchongkul (1964), Bahadur (1966), Ruymgaart et al. (1972), Sen and Ghosh (1974) and Lai (1975).

Automatic line justification process' in a text processing system comprising a plurality of printers

Abstract: not available for EP0042045Abstract of corresponding document: US4298290A system for first justifying a text line according to a system minimum escapement unit, and then rejustifying the line according to a minimum escapement unit for a printer which is to be utilized in printing the line. The first justification of the line is in a normal manner. That is, any residue is divided by the number of word spaces on the line to obtain a quotient and any remainder. The extent of word space expansion is then the value of the quotient plus the remainder until exhausted. Rejustification for the printer being utilized is accomplished by dividing each system justified word space size by the printer minimum escapement unit to obtain a new quotient and remainder. Each system justified word space is then converted to a value including the obtained quotient for each space, and the remainders for all spaces are accumulated for adding to the first word space.