Showing papers on "Divisor published in 1975"

The Estimation of Arma Models

Abstract: In estimating a vector model, $\Sigma B(j)x(n-j)=\Sigma A(j)\epsilon(n-j), A(0)=I_r, E(\epsilon(m)\epsilon(n)')=\delta_{mn}K$ it is suggested that attention be confined to cases where $g(z) =\Sigma A(j)z^j, h(z)=\Sigma B(j)z^j$ have determinants with no zeroes inside the unit circle and have $I_r$ as greatest common left divisor and where $\1brack A(p)\vdots B(q) \rbrack$ is of rank r, where p, q are the degrees of g, h, respectively. It is shown that these conditions ensure that a certain estimation procedure gives strongly consistent estimates and the last of the conditions is probably necessary for this to be so, when the first two are satisfied. The strongly consistent estimation procedure may serve to initiate an iterative maximisation of a likelihood.

On the largest prime divisor of an odd perfect number. II

Abstract: It is proved here that every odd perfect number has a prime factor greater

Enhanced apparatus for binary quotient, binary product, binary sum and binary difference generation

Abstract: Three modular arrays structured from a common module are connected together a first way to form a binary quotient by successive approximations, or a second way to form a binary product. Any one of the three modular arrays may be used to add or subtract two binary numbers. To divide, one array is utilized to effectively form the reciprocal of the binary divisor, most significant bit first, by successive approximation. Control circuitry, including a carry detector, dictates the formation of the shift and add sequence that effectively represents the reciprocal of the divisor by controlling the positioning of the divisor before each addition step so that the product is a series of binary ones. The add and shift sequence utilized to generate the series of binary ones, as it is evolving, is also being utilized to manipulate the dividend, thereby forming the quotient, most significant bit first. In effect, the dividend is being multiplied by the reciprocal of the divisor so as to form an approximate product of the dividend and reciprocal of the divisor, most significant bit first. This product is actually an increasingly precise approximation of the quotient of the dividend and divisor. The binary product of two numbers is formed, most significant bit first, by manipulating the multiplicand according to an add and shift sequence determined by use of the multiplier.

Modular apparatus for accelerated generation of a quotient of two binary numbers

Abstract: A plurality of modular arrays, each structured from a common module, are connected together so as to form a binary quotient by successive approximations. For divisors that fall into that group of numbers that have reciprocals with a reasonably short period, the forming of a quotient with such a divisor and any dividend can be greatly accelerated after the add and shift sequence for the first period of the divisor reciprocal is obtained. A unity array, divisor array, dividend array, and quotient array may all be of equal length, but must be longer than the length of the periods of the reciprocals of the divisors utilized. The reciprocal of the divisor is effectively formed in the divisor array by generating a shift and add sequence that will produce a product that is a series of binary ones. After the first period of the divisor reciprocal is formed, the binary bits of the reciprocal start to repeat for the second period, and so on. By using the formed shift and add sequence that effectively represents the reciprocal of the divisor for a single period, to manipulate the dividend, the dividend is effectively multiplied by the reciprocal of the divisor, producing a product, most significant bit first, that is the quotient of the dividend and divisor. After obtaining the shift and add sequence representative of the first period of the divisor reciprocal, the quotient has been formed, most significant bit first, to a precision equal to the number of bits in the first period of the divisor reciprocal. The precision of the quotient can now be doubled by adding the formed quotient with itself after the quotient addend is shifted to the right the number of bit positions to which the quotient is precise. At the next step, the quotient precision can be quadrupled, and so on.