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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 paper, the error term of the general $m$-th Riesz mean of the arithmetical function for any positive integer $m \ge 1$ was studied.
Abstract: Let $k\geq 1$ be an integer. Let $\delta_k(n)$ denote the maximum divisor of $n$ which is co-prime to $k$. We study the error term of the general $m$-th Riesz mean of the arithmetical function $\delta_k(n)$ for any positive integer $m \ge 1$, namely the error term $E_m(x)$ where
\[ \frac{1}{m!}\sum_{n \leq x}\delta_k(n) \left( 1-\frac{n}{x} \right)^m = M_{m, k}(x) + E_{m, k}(x). \] We establish a non-trivial upper bound for $\left | E_{m, k} (x) \right |$, for any integer $m\geq 1$.
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TL;DR: In this article, a generalized key equation with the majority coset scheme of Duursma was proposed for algebraic-geometric codes, which corrects up to half of the Goppa distance with complexity O(n**2.81).
Abstract: A new effective decoding algorithm is presented for arbitrary algebraic-geometric codes on the basis of solving a generalized key equation with the majority coset scheme of Duursma. It is an improvement of Ehrhard's algorithm, since the method corrects up to the half of the Goppa distance with complexity order O(n**2.81), and with no further assumption on the degree of the divisor G.
TL;DR: In this paper, the authors studied an asymptotic behavior of the sequence defined as T n (μ) = (τ(n))−1\({\max _{1 \leqslant t \LEQslant \left[ {{n^{1/\mu }}} \right]}}\), where τ(n + t) denotes the number of natural divisors of given positive integer n.
Abstract: In this paper for μ > 0 we study an asymptotic behavior of the sequence defined as T n (μ) = (τ(n))−1\({\max _{1 \leqslant t \leqslant \left[ {{n^{1/\mu }}} \right]}}\) {τ(n + t)}, where τ(n) denotes the number of natural divisors of given positive integer n. The motivation of this observation is to explore whether τ-function oscillates rapidly.
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TL;DR: In this article, the authors improved the lower bound for higher syzygies of a polarized abelian surface by using the conditions on the Newton-Okounkov body and extending the results of Lazarsfeld-Pareschi-Popa on abelians.
Abstract: Let $(X,L)$ be a polarized abelian surface and $f:Y \rightarrow X$ a double cover of $X$ branched over a smooth divisor $B=D^{\otimes 2}$. Based on the theory of infinitesimal Newton-Okounkov body, we show the $N_{p}$ property of $f^{*}L$ by using the conditions on $L-D$, and extend the results of Lazarsfeld-Pareschi-Popa on abelian surfaces. As an application, we improve the existing lower bound of $(L^{2})$ for higher syzygies of $L$.
TL;DR: In this article, the semigroup of all n × n upper triangular matrices over Fq under matrix multiplication is defined, and T ∗(n,q) is the group of all inverti...
Abstract: Let Fq be a finite field with q elements, n ≥ 2 a positive integer, T(n,q) the semigroup of all n × n upper triangular matrices over Fq under matrix multiplication, T∗(n,q) the group of all inverti...