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Showing papers on "Divisor published in 1986"


Journal ArticleDOI
TL;DR: The following conjecture of R. L. Graham is verified: ifn≧n0, wheren0 is an explicitly computable constant, then for anyn distinct positive integersa1,a2, ...,an the authors have ai/(ai,aj) ≧ ≧n, and equality holds only in two trivial cases.
Abstract: The following conjecture of R. L. Graham is verified: Ifn≧n 0, wheren 0 is an explicitly computable constant, then for anyn distinct positive integersa 1,a 2, ...,a n we have\(\mathop {\max }\limits_{i,j} \) a i /(a i ,a j ) ≧ ≧n, and equality holds only in two trivial cases. Here (a i ,a j ) stands for the greatest cnmmon divisor ofa i anda j .

17 citations


Patent
11 Sep 1986
TL;DR: In this article, the magnitudes of the most significant digits of the fractional parts of the dividend and divisor were compared to determine the leading zero quotient bits of a pair of binary coded, hexidecimal floating point numbers.
Abstract: In dividing a pair of binary coded, hexidecimal floating point numbers, leading zero quotient bits are eliminated by comparing the magnitudes of the most significant digits of the fractional parts of the dividend and divisor after the dividend and divisor have been normalized

16 citations


Journal ArticleDOI
TL;DR: The invariant factors of the least common left multiple and the greatest common right divisor of integral matrices are explored in this paper, where the invariant factor of the left multiple is also explored.
Abstract: The invariant factors of the least common left multiple and the greatest common right divisor of integral matrices are explored.

10 citations


Patent
11 Aug 1986
TL;DR: In this paper, a divider is used to reduce the number of cycles it takes to generate each quotient digit, and the predicted minimum possible quotient times the divisor is then subtracted from the partial remainder.
Abstract: A divider, which performs division in a base other than 2, that reduces in most cases the number of cycles it takes to generate each quotient digit. This involves predicting the minimum possible quotient digit in response to leading digits of the partial remainder and of the divisor. The predicted minimum possible quotient digit times the divisor is then subtracted from the partial remainder. If the result of the subtraction is less than the divisor, the predicted least possible quotient digit is the correct quotient digit. If the result of the subtraction is greater than the divisor, the divisor is subtracted iteratively from that result until the partial remainder falls below the value of the divisor. For each subtraction, the predicted quotient digit is incremented by one, so that a correct quotient digit results at the end of the iteration.

10 citations


Patent
29 Dec 1986
TL;DR: In this article, a method and apparatus for performing division which calculates a quotient from a dividend and a divisor by using recursive subtraction operations without using carry propagation for each subtraction operation is presented.
Abstract: A method and apparatus for performing division which calculates a quotient from a dividend and a divisor by using recursive subtraction operations without using carry propagation for each subtraction operation. The apparatus contains a circuit (16) for generating a plurality of quotient digits from the divisor and the dividend. The apparatus also contains a circuit (18, 20) for generating the quotient from the quotient digits.

8 citations


Patent
Tatsuya Sakai1, Sakou Ishikawa1
14 Jan 1986
TL;DR: In this paper, a high-speed dividing apparatus includes first and second carry-save adders and a half carry save adder and the outputs of the first carry-saving adder are connected to the inputs of the second carry saving adder, and a carry lookahead logic is connected to each adder.
Abstract: A high-speed dividing apparatus includes first and second carry-save adders and a half carry-save adder and the outputs of the first carry-save adder are connected to the inputs of the second carry-save adder and half carry-save adder. The first carry-save adder is capable of carrying out either the addition or the subtraction of the divisor. The second carry-save adder is adapted to carry out the subtraction of a divisor, and the half carry-save adder the addition thereof. The first and second carry-save adders generate half-sums and half-carries, and the half carry-save adder generates a half-carry. A half-sum of the divisor addition is obtained by inverting the half-sum of the second carry-save adder by an inverter. A pair of half-sum and half-carry is supplied to each of carry look-ahead logics. A carry look-ahead logic is connected to each adder. A quotient determining logic is adapted to determine quotient bits in response to outputs from carry-save adder and half carry-save adder and carry look-ahead logics. A selector control logic controls a selector in accordance with the quotient such that one of the pairs of the half-sums and half-carries of the divisor addition and divisor subtraction, and either the divisor or its inversion are supplied to the first carry-save adder. An arbitrary number of stages can be arranged in a binary tree configuration in the same manner.

7 citations



Patent
04 Apr 1986
TL;DR: In this article, a gear speed reducer 11 of constant rotation type coupled with a meshing clutch capable of transmitting a certain rotational component to a constant angle rotating output shaft which can convert the resolution ability with a replacement gear train is presented.
Abstract: PURPOSE:To eliminate that phenomenon which likely to occur in processing a gear etc., where dividing errors are concentrated to one groove by making integer equal division of the circumference with an integer multiple of the remainder angle, which has been left when the circumference was divided by an integer. CONSTITUTION:In this gear speed reducer 11 of constant rotation type coupled with a meshing clutch capable of transmitting a certain rotational component to a constant angle rotating output shaft which can convert the resolution ability with a replacement gear train Gn, first the given number of division is resolved into a compound fraction which does not have a common measure. Then the table is turned at steps of the angle {(360 deg.)X(denominator/divisor)}, and the clutch is disengaged to permit the replacement gear to idle for a certain angle. When the table is stopped, the work thereon is processed. This procedure is repeated, and processing with all divisors is completed when the table has turned for an angle corresponding to the denominator. By this arrangement, the surface of a disc, cylinder, etc. can be divisionally processed into the number of arbitrary integer.

2 citations


Patent
30 May 1986
TL;DR: In this article, a special calculation to square-law information obtained from temperature information from a temperature measuring means and on the one hand, controlling the action of the frequency dividing circuit and the crystal oscillating circuit respectively is presented.
Abstract: PURPOSE:To execute the temperature compensation with high accuracy by executing the special calculation to square-law information obtained from temperature information from a temperature measuring means and on the one hand, controlling the action of the frequency dividing circuit and on the other hand, the crystal oscillating circuit respectively. CONSTITUTION:The signal of a crystal oscillating circuit 101 is divided up to a timer unit signal by frequency dividing circuits 102 and 103, on the other hand, the contents of the frequency dividing circuit 103 is compared with the contents of the setting A123 by a comparing circuit 122, a gate is opened at the special time width only, and the output signal of a ring oscillator 121, in which an oscillating frequency is changed in proportion to temperature, is counted by a counter B. After the counting data are reflected so that the right and left can be the same to a peak temperature of a crystal vibrator, the time width in proportion to the square-law of temperature information is made by a square-law circuit 127. Based upon the quotient obtained by dividing the square-law information with the fixed divisor, the action of the frequency dividing circuit is controlled, and on the other hand, based upon the remainder obtained by dividing with the designated divisor while the value in proportion to the quotient from the square-law information is subtracted, the crystal oscillating circuit is controlled. Thus, the correction can be executed by changing the value of the divisor and the producing variance influence can not be received.

2 citations


Journal ArticleDOI
TL;DR: In this paper, the authors established a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp ( n, Z ) which satisfy X 2 + I = 0 (resp. X 2+ X + X + I) and the set for hermitian forms of rank n over Z [√−1] (resp.

1 citations


Journal ArticleDOI
01 Jan 1986
TL;DR: In this paper, the geometric genus of a nearly absolutely isolated hypersurface singularitiy of dimension 2 is found by using the canonical resolution. But it is not the case that the singularity can be embedded in a complex number field.
Abstract: A formula for the geometric genus of a nearly absolutely isolated hypersurface singularitiy of dimension 2 is found by using the canonical resolution. An upper bound for the fundamental cycle of such singularity is also given. Introduction. Let -r: M -V be a resolution of an isolated singularity of dimension 2. The number dim H?(V, R17r*Q(M)) is defined to be the geometric genus of the singularity. We study the case in which V can be embeded in C3. Our major result is a formula (Theorem 2) for the geometric genus when the singularity is nearly absolutely isolated (see ?2 for definition). Our proof is based on the canonical resolution of an m-tuple point, which is developed in the first section. The case m = 2 is well-known (cf. [2, pp. 47-48]). Finally we give a bound for the fundamental cycle of that kind of singularity as well as for its self-intersection number (Theorem 3). The base field is the complex number field C. A singular point p always means a hypersurface point of dimension 2. Sometimes we use the same notation for a line bundle and its corresponding divisor if it will not cause confusion. 1. m-tuple covering. Let m > 2. Let Y be a smooth surface covered by affine open sets {U i? , . Let C0, ... , Cm2 be effective divisors on Y locally defined in Ui by equations cs i = 0 (0 < s < m -2, i E I). Suppose there is a line bundle F over Y with transition function { fij } over { Ui n Uj ) such that (1) c5 = fiT7 Sc'j for all 0 < s < m 2. Let 0i be the fibre coordinates over Ui. Then the equations (2) 4im + C2_2 -2 + _ _ _ +Co,i 0 give rise to a surface X in F and the projection map from F to Y induces a finite morphism f: X -Yof degree m. DEFINITION. The surface X constructed as above is called the m-tuple cover of Y with branch locus data (CO, * * *, C. 2 ) Let Di be the discriminant of the equation (2) for i E I. Then { Di }i, I give rise to a divisor on Y, denoted by D. Obviously D is the branch locus of the map f. The map f is called totally ramified at a point p E Y if f '(p) consists of one point. Received by the editors September 12, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 14J17. ?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Book ChapterDOI
01 Jan 1986
TL;DR: In this paper, the authors consider only prime divisors of n and ask, for given order of magnitude of n, how many prime factors are there typically and how many different ones are there?
Abstract: Here we consider only prime divisors of n and ask, for given order of magnitude of n, “how many prime divisors are there typically?” and “how many different ones are there?” Some of the answers will be rather counterintuitive. Thus, a 50-digit number (1021 times the age of our universe measured in picoseconds) has only about 5 different prime factors on average and — even more surprisingly — 50-digit numbers have typically fewer than 6 prime factors in all, even counting repeated occurrences of the same prime factor as separate factors.