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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 ArticleDOI
TL;DR: In this paper, the authors studied the variation of the Newton-Okounkov bodies with respect to the field of rational functions of a smooth projective algebraic surface and a big divisor.
Abstract: Given a smooth projective algebraic surface X, a point $$O\in X$$ and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely near O, in the sense that there is a sequence of blowups $$X' \rightarrow X$$ , mapping the smooth, irreducible rational curve $$E\subset X'$$ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case $$X=\mathbb {P}^2$$ .

24 citations

Journal ArticleDOI
01 Feb 1960

24 citations

Journal ArticleDOI
TL;DR: In this paper, the authors established convolution sums of functions for the divisor sums for certain values of the Glaisher constant, such as the sum of the number of representations of a function as a sum of triangular numbers.
Abstract: One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.

23 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a comparative analysis of the most representative methods for parameterization of rotation matrices in three dimensions, including Cayley's factorization, and conclude that Cayley factorization is the most robust method when particularized to three dimensions.
Abstract: The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, 3D image processing, computer graphics, etc. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in $\R^3$, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to $3{\times}3$ rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from $4{\times}4$ rotation matrices, is the most robust method when particularized to three dimensions.

23 citations

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed F-theory and Type IIB orientifold compactifications to study α-corrections to the four-dimensional, $$ \mathcal{N} $$ = 1 effective actions.
Abstract: In this work we analyze F-theory and Type IIB orientifold compactifications to study α′-corrections to the four-dimensional, $$ \mathcal{N} $$ = 1 effective actions. In particular, we obtain corrections to the Kahlermoduli space metric and its complex structure for generic dimension originating from eight-derivative corrections to eleven-dimensional supergravity. We propose a completion of the G2R3 and (∇G)2R2-sector in eleven-dimensions relevant in Calabi-Yau fourfold reductions. We suggest that the three-dimensional, $$ \mathcal{N} $$ = 2 Kahler coordinates may be expressed as topological integrals depending on the first, second, and third Chern-forms of the divisors of the internal Calabi-Yau fourfold. The divisor integral Ansatz for the Kahler potential and Kahler coordinates may be lifted to four-dimensional, $$ \mathcal{N} $$ = 1 F-theory vacua. We identify a novel correction to the Kahler potential and coordinates at order α′2, which is leading compared to other known corrections in the literature. At weak string coupling the correction arises from the intersection of D7-branes and O7-planes with base divisors and the volume of self-intersection curves of divisors in the base. In the presence of the conjectured novel α′-correction resulting from the divisor interpretation the no-scale structure may be broken. Furthermore, we propose a model independent scenario to achieve non-supersymmetric AdS vacua for Calabi-Yau orientifold backgrounds with negative Euler-characteristic.

23 citations


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