Showing papers on "Remainder published in 1980"
TL;DR: In this paper, a polynomial plus a remainder is represented as a Taylor series and the remainder can be manipulated in many ways to give different types of bounds, including integer order and nonstandard Sobolev-like spaces.
Abstract: Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
447 citations
TL;DR: In this article, it was shown that under some regularity conditions on the kernel, a one-sample $U$-statistic with kernel of degree two admits an asymptotic expansion with remainder term $o(N^{-1})$.
Abstract: It is shown that, under some regularity conditions on the kernel, a one-sample $U$-statistic with kernel of degree two admits an asymptotic expansion with remainder term $o(N^{-1})$.
77 citations
01 Jan 1980
TL;DR: In this article, the indefinite integrals of elementary functions are discussed, where the integral part must be separated out, if there is an integral part, and then the integral parts and the remainder must be integrated separately.
Abstract: This chapter discusses the indefinite integrals of elementary functions. To integrate an arbitrary rational function, the integral part must be separated out, if there is an integral part, and then the integral part and the remainder must be integrated separately. Integration of the remainder, which is then a proper rational function, is based on the decomposition of the fraction into elementary fractions, which are the so-called partial fractions. If some of the roots of the equation are imaginary, the fractions that represent conjugate roots of the equation are then grouped together. The chapter further presents equation forms containing certain binomials.
28 citations
22 citations
TL;DR: The existence of the light-cone expansion in renormalized perturbation theory is proved in this paper, which relies on the method of Anikin and Zavialov which applies new subtraction operators and allows the elimination of a remainder which is small for x 2 → 0.
20 citations
Patent•
IBM1
TL;DR: In this paper, a system for first justifying a text line according to a system minimum escapement unit, and then rejustifying the line by dividing each justified word space size by the printer minimum esca-tement unit to obtain a new quotient and remainder.
Abstract: A 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.
19 citations
17 citations
TL;DR: In this article, the authors apply Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.
Abstract: The purpose of this paper is the application of Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.
14 citations
Patent•
08 Jul 1980
TL;DR: In this article, a first summing unit sums up the coefficients a i's to provide a sum of a carry multiplied by 2 n and a sum portion less than 2 n.
Abstract: For division by a Mersenne number (2 n -1), a divided P is expressed as: ##EQU1## A first summing unit sums up the coefficients a i 's to provide a sum of a carry multiplied by 2 n and a sum portion less than 2 n . The carry is added to the sum portion to provide a similar sum. When the sum portion of such a sum becomes less than 2 n , the sum portion provides the remainder if it is less than the Mersenne number. If not, the remainder is rendered zero with an additional carry produced. A second summing unit calculates ##STR1## for i=1 to (k-2). Each summation is given by a carry multiplied by 2 n plus a sum portion with the carry from the first summing unit and the additional carry added to the summation for i=1 and with the carry from the summation i=i-1 added to the summation for i=i. The sum portions calculated by the second summing unit with successive addition of the carries provide zeroth through (k-2)-th coefficients to (2 n ) O through (2 n ) k-2 of a polynomial representative of the quotient, in which the carry from the summation for i=k-2, if not zero, provides the coefficient for (2 n ) k-1 . For a parallel processor system, a Mersenne prime is used as the number of memory modules. The system preferably comprises a divider of the type described.
11 citations
TL;DR: In this article, the number of renewals in the time interval [0, t] for an ordinary renewal process is approximately normally distributed under general conditions, and a remainder term estimate for this normal distribution approximation is given.
Abstract: It is well known that the number of renewals in the time interval [0, t] for an ordinary renewal process is approximately normally distributed under general conditions. We give a remainder term estimate for this normal distribution approximation. RENEWAL PROCESS; NORMAL APPROXIMATION; REMAINDER TERM
10 citations
TL;DR: In this article, the idea of the FrCchet derivative can be extended to both topological and non-normed pseudotopological linear vector spaces, where it is called a functional derivative.
Patent•
27 Oct 1980TL;DR: In this paper, a signal to be quantized is translated to a charge Q and the latter is multiplied by a fraction f to produce a fractional charge packet fQ. This last step is repeated for succeeding remainder charge packets a sufficient number of times until a total of n-1 fractional charges have been produced, where n is the number of quantization levels desired.
Abstract: A signal to be quantized is translated to a charge Q and the latter is multiplied by a fraction f to produce a fractional charge packet fQ. Then, another fractional charge packet is produced by multiplying the remainder Q(1-f) of the charge packet by f. This last step is repeated for succeeding remainder charge packets a sufficient number of times until a total of n-1 fractional charge packets have been produced, where n is the number of quantization levels desired. The successive fractional charge packets are compared with threshold levels of different amplitudes to determine the number of incremental charge packets, each of the same size, to be added to one another to form a quantized charge packet.
TL;DR: In this paper, the authors show that the resolvent of a linear Volterra integral or integrodifferential equation may be written as a finite sum of polynomials and exponentials, plus a remainder term which belongs to a weighted $L^1$-space.
Abstract: Conditions are found that guarantee that the resolvent of a linear Volterra integral or integrodifferential equation may be written as a finite sum of products of polynomials and exponentials, plus a remainder term which belongs to a weighted $L^1$-space. The kernel has the form $a(t) = c + b(t)$; here c is a constant and $b(t)$ belongs to the same weighted $L^1$-space. It is assumed that $b(t)$ satisfies a combination of moment and monotonicity hypotheses that is determined by the maximum of the orders of the zeros on $\operatorname{Re} z = 0$ of certain Laplace transform equations. The results extend to weighted $L^1$-spaces some recent $L^1$-remainder theorems due to K. B. Hannsgen (Indiana Univ. Math. J., 29 (1980), pp. 103–120). The results for resolvents are deduced from more general results for linear Volterra-Stieltjes equations. The proofs employ extensions of Banach algebra techniques used by the authors in an earlier related paper, where the hypotheses involve only moment conditions.
01 Nov 1980
TL;DR: In this article, the Routh-Hurwitz type algorithms or determinants are used to evaluate the quadratic factor of a polynomial in the last three terms or in the remainder of the remainder.
Abstract: Lin's method for finding a quadratic factor of a polynomial is too well known. Basically it is to perform long divisions on the polynomial in question by a quadratic factor shown in the last three terms of the polynomial or in the remainder. The new approach is to use Routh-Hurwitz type algorithms or determinants to evaluate the factor iteratively.
TL;DR: In this paper, the Parseval relation for the Mellin transform is used to obtain an explicit expression for the remainder in the asymptotic expansion of a class of integral transforms.
Abstract: In this paper we show how the Parseval relation for the Mellin transform can be used to obtain an explicit expression for the remainder in the asymptotic expansion of a class of integral transforms. The technique, with some modification, can be used to derive similar results for many other integral transforms which are not discussed here.
01 Jan 1980
TL;DR: In this paper, the problem of characterizing those spaces that have a compactification with countable remainder is dealt with, and the problem is solved by characterising those spaces with compactification and remainder.
Abstract: In this paper, we deal with the problem of characterizing those spaces that have a compactification with countable remainder.
01 Jan 1980
TL;DR: In this article, an asymptotic expansion for distribution functions (d.f) of Neyman's C(α) test statistics to order n−1/2 is obtained under weaker conditions than previously known.
Abstract: In the paper an asymptotic expansion (a.e.) for distribution functions (d.f’s) of Neyman’s C(α) test statistics to order n−1/2 (with a remainder 0(n−1/2) is obtained under weaker conditions than previously known (Theorem 2.2). The proof is based on a special theorem giving an a.e.for the d.f. of a statistic admitting a stochastic expansion (Theorem 2.1).
TL;DR: In this paper, a uniform asymptotic expansion of the Laplace integrals with explicit remainder terms is given, valid in the whole complex s−plane, for s = −ix, it provides the Fourier integral expansion.
Abstract: A uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.
Patent•
14 Feb 1980
TL;DR: In this paper, the upper-rank bits were limited to speed up the operation by performing the process only with the upper rank bits limited, in the unit in which division is made with the processing of multiplication for the approximate value Dt which is a reciprocal of the divisor D CONSTITUTION.
Abstract: PURPOSE:To speed up the operation by performing the process only with the upper rank bits limited, in the unit in which division is made with the processing of multiplication for the approximate value Dt which is a reciprocal of the divisor D CONSTITUTION:The X input corresponding to the value Q(j), Y input corresponding to the value dt and the remainder R(j) are inputted to the multiplication processing section 1, and the carry output C of the multiplication corresponding to the left side of equation (1) and the sum output S are respectively set to the registers 2 and 3 The upper rank bit of the outputs C and S is fed to the addition processing section 4 to obtain the step quotient Q (j+1) On the other hand, the remainder R(j+1) at that time is returned to the processing section 1 from the registers 2 and 3 The remainder processing section 6 repeats specified processing to obtain the final remainder at the remainder generator 9 The quotient generator 5 integrates the step quotients Q(O) obtained every step The processing speed can be increased by limiting the number of processing bits at the processing section 4
TL;DR: The following questions are investigated: 1.) How large may Y be? 2.) When is there necessarily an edge contained in all the members of Y?
TL;DR: In this paper, an asymptotic expansion of the remainder term is established containing the more valid terms the smaller the exponent a is, and it appears that the exact behavior of remainder term can be determined.
Abstract: As a generalization of the sphere problem (cf. VINOGRADOV [7]) the present paper deals with estimates for the number of lattice points (points of the space with integral coordinates) in regions defined by xa+ya+za≦Ra, x≧0, y≧0, z≧0. In particuliar, the case 0 2 has been treated by KRATZEL [2]), and it appears that the exact asymptotic behaviour of the remainder term can be determined. Furthermore an asymptotic expansion of the remainder term is established containing the more valid terms the smaller the exponent a is.
Patent•
12 Jun 1980
TL;DR: In this article, it was shown that it is possible to generate various random numbers of a high random number property by dividing a counted value by an indicated divisor at a time indicated by a controller and taking out the remainder to generate random numbers smaller than the remainder.
Abstract: PURPOSE:To make it possible to generate various random numbers of a high random number property by dividing a counted value by an indicated divisor at a time indicated by a divisor controller and taking out the remainder to generate random numbers smaller than the divisor CONSTITUTION:Noise from noise generation source 1 is converted to a square wave by amplitude limiter 2, and the number of square waveforms of a random period is always counted by counter 3 Progress of this counting is random in respect to time, and the counter is reset to start counting from 0 again when the counted value reaches a maximum counted value The counted value is divided by an indicated divisor at a time indicated by divisor controller 5 in divider 4, and the remainder is taken out from remainder output terminal 6 of divider 4, thus generating random numbers smaller than the remainder Thus, various random numbers of a high random number property can be generated
Patent•
14 Jul 1980
TL;DR: In this article, it was shown that it is possible to determine any strict error within a fixed range by finding value R 0 approximate to the reciprocal number of divisor D for a division process between dividend N and divisors D.
Abstract: PURPOSE:To make it possible to determine any strict error within a fixed range by finding value R0 approximate to the reciprocal number of divisor D for a division process between dividend N and divisor D. CONSTITUTION:Approximate value R0 of the reciprocal of divisor D is found and approximate value R1 of the reciprocal of the product of divisor D and approximate value R0 is also found; and approximate value Ri+1 of the reciprocal of the product of D and approximate values R0-Ri is similarly obtained. Then, approximate values R0-Rn obtained in the above-mentioned way are processed according to (formula-A) and after the process is repeated fixed times until fixed error precision is obtained, expedient quotient Qn is obtained. Next, quotient Qn is rounded and under the condition the obtained remainder satisfies, corrections are made. This arithmetic is attained by a unit composed of normalization circuit parts 1 and 7, size-comparison and fine-adjustment processing part 2, R0 index table, multiplier 4, digit-matching processing part 5, adder 6, buffer or bus 8, and registers 9-16.
TL;DR: In this article, some monotonicity results for the remainder terms in the asymptotic expansion for x ∞ of the function Jv(x)Jv+n(x), + Yv (xYv +n+n+x), v e R, n e Z are given.
Abstract: Some monotonicity results are given for the remainder terms in the asymptotic expansion for x ∞ of the function Jv(x)Jv+n(x) + Yv(xYv+n(x), v e R, n e Z.