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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.


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Journal IssueDOI
23 Sep 2020
TL;DR: In this article, Erdős showed that if a primitive set is a 1-primitive set, then 1-1/p is the asymptotic density of the set of integers with a positive integer as an initial divisor.
Abstract: A generalization of primitive sets and a conjecture of Erdős, Discrete Analysis 2020:16, 13 pp. Call a set $A$ of integers greater than 1 _primitive_ if no element of $A$ divides any other. How dense can a primitive set be? An obvious example of a primitive set is the set ${\mathbb P}$ of prime numbers, which has the property that $\sum_{p\in {\mathbb P}}\frac 1{p\log p}<\infty$. (This follows easily from the prime number theorem, or even just Chebyshev's theorem.) In 1935 Erdős proved that the same is true of all primitive sets. That is, if $A$ is a primitive set, then $\sum_{a\in A}\frac 1{a\log a}<\infty$. We briefly sketch his very short argument. He starts with the observation that if $a$ is a positive integer and $P(a)$ is the largest prime factor of $a$, then $$\prod_{p< P(a)}(1-1/p)\geq \frac c{\log P(a)}\geq\frac c{\log a}$$ for some absolute constant $c>0$. Thus, it is sufficient to prove that $$\sum_{a\in A}\frac 1a\prod_{p

2 citations

Posted Content
TL;DR: In this article, the authors introduced a superpotential for partial flag varieties of type A. This is a map $W: Y √ √ to \mathbb{C}, where $Y √ is the complement of an anticanonical divisor on a product of Grassmannians.
Abstract: We introduce a superpotential for partial flag varieties of type $A$. This is a map $W: Y^\circ \to \mathbb{C}$, where $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map $W$ is expressed in terms of Plucker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Plucker coordinate mirror for Grassmannians. We show that in a distinguished cluster chart for $Y$, our superpotential agrees with earlier mirrors constructed by Eguchi--Hori--Xiong and Batyrev--Ciocan-Fontanine--Kim--van Straten. Our main tool is quantum Schubert calculus on the flag variety.

2 citations

Journal ArticleDOI
TL;DR: It is proved that the only integer m for which m(q2 + 1)∈A(q) is m-= 2 for q ≡ 3 (mod 4), m = 1 forq ≡ 1 ( mod 4) and the cap C(q^{2}+1) is complete.
Abstract: For any divisor k of q 4?1, the elements of a group of k th-roots of unity can be viewed as a cyclic point set C k in PG(4,q). An interesting problem, connected to the theory of BCH codes, is to determine the spectrum A(q) of maximal divisors k of q 4?1 for which C k is a cap. Recently, Bierbrauer and Edel [Edel and Bierbrauer (2004) Finite Fields Appl 10:168---182] have proved that 3(q 2 + 1)?A(q) provided that q is an even non-square. In this paper, the odd order case is investigated. It is proved that the only integer m for which m(q 2 + 1)?A(q) is m = 2 for q ? 3 (mod 4), m = 1 for q ? 1 (mod 4). It is also shown that when q ? 3 (mod 4), the cap $$C_{2(q^{2}+1)}$$ is complete.

2 citations

Posted Content
TL;DR: In this paper, it was shown that the presentation of affine variables of complexity one in terms of polyhedral divisor of Altmann-Hausen holds over an arbitrary field.
Abstract: We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisor of Altmann-Hausen holds over an arbitrary field. We describe also a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{G}$-varieties of complexity one over a field, where $\mathbf{G}$ is a (non-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties are described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.

2 citations

Patent
Martin Langhammer1
02 Oct 2008
TL;DR: In this paper, a specialized processing block is constructed to support asymmetric multiplication by providing programmable shifting of partial products, so that the partial products can be shifted one number of bits for symmetric multiplication and a different number of bit for asymmetric multiplications.
Abstract: Division can be performed in a programmable integrated circuit device by computing a relatively small number of bits of the inverse of the divisor, and then programming multipliers in a specialized processing block of the device to perform multiplication of the dividend and the inverted divisor. The specialized processing block is constructed to be able to be programmed to support such asymmetric multiplication by providing programmable shifting of partial products, so that the partial products can be shifted one number of bits for symmetric multiplication and a different number of bits for asymmetric multiplication. The process is performed recursively, by chaining a plurality of the specialized processing blocks, so that the result converges notwithstanding the relatively low precision of the inverted divisor.

2 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021157
2020172
2019127
2018120
2017140