Topic
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 paper, it was shown that the anti-m$-canonical map of the Cartier divisor is birational on its image for all ε ≥ 0.
Abstract: For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $\varphi_{-m}$ is birational onto its image for all $m\geq 39$. By a weak $\mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $\mathbb{Q}$-Cartier, nef and big. For weak $\mathbb{Q}$-Fano 3-folds, we prove that $\varphi_{-m}$ is birational onto its image for all $m\geq 97$.
4 citations
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30 Apr 2020TL;DR: In this article, some concepts relating to Mersenne primes and perfect numbers were revisited and how to partition perfect numbers into odd cubes for odd prime, also the formula that partitions perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts.
Abstract: Mersenne primes are specific type of prime numbers that can be derived using the formula , where is a prime number. A perfect number is a positive integer of the form where is prime and is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime . Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.
4 citations
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TL;DR: In this article, it was shown that the moduli space of the Hitchin system admits a natural enhancement to a holomorphic symplectic manifold which is called ${\mathcal M}_H(r,d)$ by trivializing the restriction of the vector bundles underlying the stable Higgs bundles.
Abstract: Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${\mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.
4 citations
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TL;DR: In this article, the exact order of the cuspidal divisor of the rational Eisenstein prime of the Hecke ring was shown to be in the form of an odd prime.
Abstract: Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $\Gamma_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}]
eq 0$, where $\mathcal{C}(N)$ is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor $\mathcal{C}^D_{M, N}$, defined in Section 4, and the index of $\mathcal{I}^D_{M, N}$ in $\mathbb{T}(N)\otimes \mathbb{Z}_\ell$.
4 citations
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06 Nov 1991TL;DR: A floating-point division cell consisting of partial remainder data register for storing parallel-particle-remainder data or third-order divisor data is presented in this article.
Abstract: A floating-point division cell consisting of partial remainder data register for storing parallel-partial-remainder data or third partial remainder data, divisor data register for storing parallel-divisor data or third divisor data, low-order divisor data generator for receiving the low-order portion of the divisor data and generating low-order divisor data, low-order partial remainder calculator for obtaining low-order multi-divisor data by multiplying the low-order divisor data and a multiple of 2 together and calculating new low-order partial remainder data by subtracting or adding the low-order multi-divisor data from/to the low-order portion of the partial remainder data, high-order divisor data generator for receiving the high-order portion of the divisor data and generating high-order divisor data, and high-order partial remainder calculator for obtaining high-order multi-divisor data by multiplying the high-order divisor data and a multiple of 2 together and calculating new high-order partial remainder data by subtracting or adding the high-order multi-divisor data from/to the high-order portion of the partial remainder data.
4 citations