Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Reduced divisors and embeddings of tropical curves

Abstract: Given a divisor $D$ on a tropical curve $\Gamma$, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system $|D|$. This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the linear system, and in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points on tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve $C$ and a divisor $D$ of non-negative rank on $C$, reduced divisors equivalent to $D$ define a morphism from $C$ to the complete linear system $|D|$, which is described in terms of Wronskians.

Mechanism and method for integer divide involving pre-alignment of the divisor relative to the dividend

Abstract: A mechanism for dividing an integer dividend by an integer divisor to generate an integer quotient operates by aligning the divisor relative to the dividend such that a right-most bit of the divisor is aligned with a bit M of the dividend. The divisor is compared to an integer value whose right-most bits are equal to bits of the dividend which are aligned with bits of the divisor. As a result of this comparison, quotient bits which positionally correspond to the dividend bit M and to bits of the dividend which are located to the left of the dividend bit M are cleared to zero. Also as a result of the comparison, the dividend is divided by the divisor as aligned relative to the dividend to thereby generate values for any uncleared quotient bits.

Fast division using accurate quotient approximations to reduce the number of iterations

Abstract: A class of iterative integer division algorithms is presented based on lookup table Taylor-series approximations to the reciprocal. The algorithm iterates by using the reciprocal to find an approximate quotient and then subtracting the quotient multiplied by the divisor from the dividend to find a remaining dividend. Fast implementations can produce an average of either 14 or 27 b per iteration, depending on whether the basic or advanced version of this method is implemented. Detailed analyses are presented to support the claimed accuracy per iteration. Speed estimates using state-of-the-art ECL (emitted coupled logic) components show that this method is faster than the Newton-Raphson technique and can produce 53-b quotients of 53-b numbers in about 28 or 22 ns for the basic and advanced versions. >

The shifted convolution of divisor functions

Abstract: We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$, which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of $d_3(n)$ and Fourier coefficents of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.