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: This article characterize the elements in the super summit set of e d in the dual Garside structure by studying the combinatorics of noncrossing partitions arising from periodic braids and provides a conjugating element γ.
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TL;DR: In this paper, it was shown that if H is a subgroup of a finite group G, the degree of H in the order divisor graph of the subgroups of G is greater orequal to 2.
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TL;DR: In this paper, a lower bound for the second Chern class of Fano manifold is established in terms of its index and degree, and it is shown that the rational map defined by this lower bound is birational.
Abstract: Let $X$ be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of $X$ in terms of its index and degree. As an application, if $Y$ is a $n$-dimensional Fano manifold with $-K_Y=(n-3)H$ for some ample divisor $H$, we prove that $h^0(Y,H)\geq n-2$. Moreover, we show that the rational map defined by $\vert mH\vert$ is birational for $m\geq 5$, and the linear system $\vert mH\vert$ is basepoint free for $m\geq 7$. As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most $4$ are also investigated.
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TL;DR: In this paper, positivity conditions for adjoint bundles of dimension k + tL with n-3 have been studied, where n is the number of non-degenerate quadratic singularities in the adjoint bundle.
Abstract: Let $(X,L)$ be a smooth polarized variety of dimension $n$. Let $A\in |L|$ be an effective irreducible divisor, and let $\Sigma$ be the singular locus of $A$. We assume that $\Sigma$ is a smooth subvariety of dimension $k\geq 2$, and codimension $c\geq 3$, consisting of non-degenerate quadratic singularities. We study positivity conditions for adjoint bundles $K_X+tL$ with $t\geq n-3$. Several explicit examples motivate the discussion.
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TL;DR: In this paper, the Dirichlet divisor problem was shown to be solvable in O(psi(n)n √ 1/4 ) time when n < 10−5.
Abstract: We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(\psi(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor - n\log{n} - (2\gamma-1)n \Big\rvert $$ and $ \psi(n) $ - any positive function that increases unboundedly as $ n \to \infty $. The result is achieved under the hypothesis: $$ \Big \{\frac{n}{x} \Big \} \sim w_x $$ where $ w_x $ is uniformly distributed over $ [0,1) $ random variable with a values set $ \{0, \frac {1} {x}, \ldots, \frac{x-1}{x} \} $ and the value accepting probability $ p = \frac{1}{x} $.
The paper concludes with a numerical argument in support of the hypothesis being true. It is shown that the expectation: $$\mu_{1} \Big[\sum_{x=1}^{n}\Big(\frac{n}{x} - \frac{x-1}{2x}\Big) \Big]= (2n+1)H_{\lfloor\sqrt{n}\rfloor} - \lfloor\sqrt{n}\rfloor^{2} - \lfloor\sqrt{n}\rfloor + C$$ has deviation from $D(n)$ is less than $R(n)$ in absolute value for all $n < 10^{5}$.