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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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TL;DR: In this article, the problem of determining whether a normal variety has a rationalizing divisor is addressed, and a criterion for cones to have such a rationalising divisors is given.
Abstract: Rational pairs generalize the notion of rational singularities to reduced pairs $(X,D)$. In this paper we deal with the problem of determining whether a normal variety $X$ has a rationalizing divisor, i.e. a reduced divisor $D$ such that $(X, D)$ is a rational pair. We give a criterion for cones to have a rationalizing divisor, and relate the existence of such a divisor to the locus of rational singularities of a variety.

1 citations

Journal ArticleDOI
John H. Jaroma1
TL;DR: In this article, the rank of apparition of an odd prime p in a specific Lehmer sequence is the index of the first term that contains p as a divisor, where p is the odd prime.
Abstract: We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, , U0 = 0, U1 = 1, and , , where R and Q are relatively prime integers and n ∈ {0,1,...}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We add that the rank of apparition of an odd prime p in a specific Lehmer sequence is the index of the first term that contains p as a divisor. In this paper, we obtain results that pertain to the rank of apparition of primes of the form 2 n p ± 1. Upon doing so, we will also establish rank of apparition results under more explicit hypotheses for some notable special cases of the Lehmer sequences. Presently, there does not exist a closed formula that will produce the rank of apparition of an arbitrary prime in any of the aforementioned sequences.

1 citations

Journal ArticleDOI
TL;DR: It is proven that for any triple ( n,q,\lambda ) with (n,q) = 1 the set of primitive idempotents gives rise to an orthogonal matrix.
Abstract: For any $$ \lambda \in GF(q)^{\ast} $$ a $$ \lambda $$ -constacyclic code $$ C^{n,q,\lambda } : = \,\langle g(x) \rangle $$ , of length $$ n $$ is a set of polynomials in the ring $$ GF(q)[x]/x^{n} - \lambda $$ , which is generated by some polynomial divisor $$ g(x) $$ of $$ x^{n} - \lambda $$ . In this paper a general expression is presented for the uniquely determined idempotent generator of such a code. In particular, if $$ g(x): = (x^{n} - \lambda) / P_{t}^{n,q,\lambda } (x) $$ , where $$ P_{t}^{n,q,\lambda } (x) $$ is an irreducible factor polynomial of $$ x^{n} - \lambda $$ , one obtains a so-called minimal or irreducible constacyclic code. The idempotent generator of a minimal code is called a primitive idempotent generating polynomial or, shortly, a primitive idempotent. It is proven that for any triple $$ (n,q,\lambda ) $$ with $$ (n,q) = 1 $$ the set of primitive idempotents gives rise to an orthogonal matrix. This matrix is closely related to a table which shows some resemblance with irreducible character tables of finite groups. The cases $$ \lambda = 1 $$ (cyclic codes) and $$ \lambda = - 1 $$ (negacyclic codes), which show this resemblance most clearly, are studied in more detail. All results in this paper are extensions and generalizations of those in van Zanten (Des Codes Cryptogr 75:315–334, 2015).

1 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that if and only if d is even, the linear system embeds S in a smooth rational normal scroll of dimension 3, where S is linearly equivalent to Q, where Q is a quadric on T.
Abstract: Let \((S,{\mathcal {L}})\) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle \({\mathcal {L}}\) of degree \(d > 35\). In this paper we prove that \(K^2_S\ge -d(d-6)\). The bound is sharp, and \(K^2_S=-d(d-6)\) if and only if d is even, the linear system \(|H^0(S,{\mathcal {L}})|\) embeds S in a smooth rational normal scroll \(T\subset {\mathbb {P}}^5\) of dimension 3, and here, as a divisor, S is linearly equivalent to \(\frac{d}{2}Q\), where Q is a quadric on T.

1 citations

Patent
03 Jan 2019
TL;DR: In this paper, a gear-shifting serializer-deserializer (SerDes) is provided that uses a first divisor value to form a divided clock while de-serializing a serial data stream prior to a lock detection.
Abstract: A gear-shifting serializer-deserializer (SerDes) is provided that uses a first divisor value to form a divided clock while de-serializing a serial data stream prior to a lock detection and that uses a second divisor value to form the divided clock value after the lock detection, wherein the second divisor value is greater than the first divisor value.

1 citations


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