Results 1 to 10 of about 133,292 (167)
Upper bound estimate of incomplete Cochrane sum
Open Mathematics, 2017 By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.
Tianping Zhang
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Reporting and methodological quality of systematic reviews underpinning clinical practice guidelines for low back pain: a meta-epidemiological study [PDF]
Frontiers in Pain ResearchBackgroundLow back pain (LBP) is the leading musculoskeletal disorder worldwide and a major cause of disability, health care utilization, and economic burden.
Adam Khan +10 more
doaj +2 more sources
On the generalized Cochrane sum with Dirichlet characters
AIMS Mathematics, 2023 <abstract><p>In this paper, we defined a new generalized Cochrane sum with Dirichlet characters, and gave the upper bound of the generalized Cochrane sum with Dirichlet characters. Moreover, we studied the asymptotic estimation problem of the mean value of the generalized Cochrane sum with Dirichlet characters and obtained a sharp ...
Zhefeng Xu
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A note on the hyper Cochrane sum
Indian Journal of Pure and Applied Mathematics, 2013 Let \(q\) and \(h\) be integers with \(q\geq 3\) and \((h,q)=1\). Let \(\mathbf{m}=\left(m_1,\cdots,m_k,m_{k+1}\right)\in \mathbb{Z}^{k+1}\). The hyper Cochrane sum is defined as following: \[ C\left(h,q;\mathbf{m},k\right)=\mathop{\sum_{a_1=1}^{q}}_{(a_1,q)=1}\cdots \mathop{\sum_{a_k=1}^{q}}_{(a_k,q)=1}\overline{B}_{m_1}\left(\frac{a_1}{q}\right ...
Tianping Zhang
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On the mean value of general Cochrane sum
Proceedings of the Japan Academy Series A: Mathematical Sciences, 2010 The general Cochrane sum is defined by \(C(h,\chi_{1},p)=\sum_{a\leq p-1}\chi_{1}(a)((\frac{\bar{a}}{p}))((\frac{ah}{p}))\). The authors obtain \[ \sum_{h\leq p-1}C^{2}(h,\chi_{1},p)=\frac{1}{180}\prod_{p_{1}\in\mathcal{A}}\left(\frac{p_{1}^{2}+1}{p_{1}^{2}-1}\right)^{2} +O(p^{1+\varepsilon}), \] where \(\mathcal{A}\) is the set of quadratic residues ...
Ren, Dongmei, Yi, Yuan
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On a Cochrane Sum and Its Hybrid Mean Value Formula
Journal of Mathematical Analysis and Applications, 2002 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Zhang Wenpeng
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On the order of the high-dimensional Cochrane sum and its mean value
Journal of Number Theory, 2006 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Zhang Wenpeng
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On Cochrane sums over short intervals
Journal of Mathematical Analysis and Applications, 2009 \textit{W. Zhang} [Int. J. Math. Math. Sci. 32, No. 1, 47--55 (2002; Zbl 1107.11310)] gave an asymptotic formula for the mean square value of Cochrane sums \(C(h,k)\). This paper provides a corresponding result for sums over short intervals when \(k\) is squarefree. In particular, when \(k\) is an odd prime \(p\), the formula takes the form \[ \sum_{h \
Yuan Yi
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On a Cochrane sum and its hybrid mean value formula (II)
Journal of Mathematical Analysis and Applications, 2002 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Wenpeng Zhang
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A note on the Cochrane sum and its hybrid mean value formula
Journal of Mathematical Analysis and Applications, 2003 T. Cochrane introduced a sum analogous to the Dedekind sum as follows: \[ c(h,k)=\sum_{a \text{mod} k, (a,k)=1}\left(\left(\frac{\bar{a}}{k}\right)\right)\left(\left(\frac{ah}{k}\right)\right) \] where \(h\) is any integer, \(k\) is a positive integer, \(a\bar{a}\equiv 1 \mod k\) and \(((x))=x-[x]-1/2\) if \(x\) is not an integer; or \(((x))=0\) if \(x\
Wenpeng Zhang, Huaning Liu
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