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.

Extreme positive ternary sextics

Abstract: We study nonnegative (psd) real sextic forms $q(x_0,x_1,x_2)$ that are not sums of squares (sos). Such a form has at most ten real zeros. We give a complete and explicit characterization of all sets $S\subset\mathbb{P}^2(\mathbb{R})$ with $|S|=9$ for which there is a psd non-sos sextic vanishing in $S$. Roughly, on every plane cubic $X$ with only real nodes there is a certain natural divisor class $\tau_X$ of degree~$9$, and $S$ is the real zero set of some psd non-sos sextic if, and only if, there is a unique cubic $X$ through $S$ and $S$ represents the class $\tau_X$ on $X$. If this is the case, there is a unique extreme ray $\mathbb{R}_+q_S$ of psd non-sos sextics through $S$, and we show how to find $q_S$ explicitly. The sextic $q_S$ has a tenth real zero which for generic $S$ does not lie in $S$, but which may degenerate into a higher singularity contained in $S$. We also show that for any eight points in $\mathbb{P}^2(\mathbb{R})$ in general position there exists a psd sextic that is not a sum of squares and vanishes in the given points.

Prescaled integer division

Abstract: We describe a high radix integer division algorithm where the divisor is prescaled and the quotient is postscaled without modifying the dividend to obtain an identity N=Q/sup *//spl times/D+R/sup */ with the quotient Q/sup */ differing from the desired integer quotient Q only in its lowest order high radix digit. Here the "oversized" partial remainder R/sup */ is bounded by the scaled divisor with at most one additional high radix digit selection needed to reduce the partial remainder and augment the quotient to obtain the desired integer division result N=Q/spl times/D+R with 0/spl les/R/spl les/D-1. We present a high radix multiplicative version of this algorithm where a k/spl times/p digit base /spl beta/ rectangular aspect ratio multiplier allows quotient digit selection in radix /spl beta//sup k-1/ with a cost of only one k/spl times/p digit multiply per high radix digit, plus the fixed pre- and post-scaling operation costs. We also present a Booth radix 4 additive version of this algorithm where appropriately compressed representation of the partial remainder with Booth digits {-2, -1, 0, 1, 2} allows successive quotient digit selection from the leading partial remainder digit without the iterative table lookups required in SRT division.

A note of perfect nonlinear functions

Abstract: Perfect nonlinear functions are of importance in cryptography. By using Galois rings and investigating the character values of corresponding relative difference sets, we construct a perfect nonlinear function from $\mathbb{Z}^{n}_{p_{2}}$ to $\mathbb{Z}^{m}_{p_{2}}$ where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from $\mathbb{Z}^{2}_{2_{p}}$ to $\mathbb{Z}_{2_{p}}$ if and only if p=2, and that there doesn't exist a perfect nonlinear function from $\mathbb{Z}^{2n}_{2k_{l}}$ to $\mathbb{Z}^{m}_{2k_{l}}$ if m>n and l(l is odd) is self-conjugate modulo 2k(k≥1) .

The probability of generating the symmetric group with a commutator condition

Abstract: Let B(n) be the set of pairs of permutations from the symmetric group of degree n with a 3-cycle commutator, and let A(n) be the set of those pairs which generate the symmetric or the alternating group of degree n. We find effective formulas for calculating the cardinalities of both sets. More precisely, we show that #B(n)/n! is a discrete convolution of the partition function and a linear combination of divisor functions, while #A(n)/n! is the product of a polynomial and Jordan's totient function. In particular, it follows that the probability that a pair of random permutations with a 3-cycle commutator generates the symmetric or the alternating group of degree n tends to zero as n tends to infinity, which makes a contrast with Dixon's classical result. Key elements of our proofs are Jordan's theorem from the 19th century, a formula by Ramanujan from the 20th century and a technique of square-tiled surfaces developed by French mathematicians Lelievre and Royer in the beginning of the 21st century. This paper uses and highlights elegant connections between algebra, geometry, and number theory.