• 호남수학학술지
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• 제36권2호
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• pp.467-472
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• 2014

With the properties of Riemann zeta function, we find an asymptotic formula of the sum for the absolute value of M$\ddot{o}$bius function, $\sum_{n{\leq}x}{\mid}{\mu}(n){\mid}$, using Dirichlet inversion formula.

• Korean Journal of Mathematics
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• 제28권2호
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• pp.361-378
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• 2020

The most apparent aspect of the present study is to introduce a new sequence space 𝚽^⋆(𝓛_p) derived by double Euler-Totient matrix operator. We examine its topological and algebraic properties and give an inclusion relation. In addition to those, the α-, β(bp)- and γ-duals of the space 𝚽^⋆(𝓛_p) are determined and finally, some 4-dimensional matrix mapping classes related to this space are characterized.

• 대한수학회논문집
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• 제31권4호
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• pp.667-682
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• 2016

In this article we show a recursive method to compute the coefficients of the minimal polynomial of cos($2{\pi}/n$) explicitly for $n{\geq}3$. The recursion is not on n but on the coefficient index. Namely, for a given n, we show how to compute ei of the minimal polynomial ${\sum_{i=0}^{d}}(-1)^ie_ix^{d-i}$ for $i{\geq}2$ with initial data $e_0=1$, $e_1={\mu}(n)/2$, where ${\mu}(n)$ is the $M{\ddot{o}}bius$ function.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.657-668
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• 2016

Given a partition ${\lambda}=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_l)$ of a positive integer n, let Tab(${\lambda}$, k) be the set of all tabloids of shape ${\lambda}$ whose weights range over the set of all k-compositions of n and ${\mathcal{OP}}^k_{\lambda}_{rev}$ the set of all ordered partitions into k blocks of the multiset $\{1^{{\lambda}_l}2^{{\lambda}_{l-1}}{\cdots}l^{{\lambda}_1}\}$. In [2], Butler introduced an inversion-like statistic on Tab(${\lambda}$, k) to show that the rank-selected $M{\ddot{o}}bius$ invariant arising from the subgroup lattice of a finite abelian p-group of type ${\lambda}$ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(${\lambda}$, k) and ${\mathcal{OP}}^k_{\hat{\lambda}}$. When k = 2, we also introduce a major-like statistic on Tab(${\lambda}$, 2) and study its connection to the inversion statistic due to Butler.

• Clinical and Experimental Pediatrics
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• 제60권9호
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• pp.282-289
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• 2017

Purpose: Recent advancements in molecular techniques have greatly contributed to the discovery of genetic causes of unexplained developmental delay. Here, we describe the results of array comparative genomic hybridization (CGH) and the clinical features of 27 patients with global developmental delay. Methods: We included 27 children who fulfilled the following criteria: Korean children under 6 years with global developmental delay; children who had at least one or more physical or neurological problem other than global developmental delay; and patients in whom both array CGH and G-banded karyotyping tests were performed. Results: Fifteen male and 12 female patients with a mean age of $29.3{\pm}17.6months$ were included. The most common physical and neurological abnormalities were facial dysmorphism (n=16), epilepsy (n=7), and hypotonia (n=7). Pathogenic copy number variation results were observed in 4 patients (14.8%): 18.73 Mb dup(2)(p24.2p25.3) and 1.62 Mb del(20p13) (patient 1); 22.31 Mb dup(2) (p22.3p25.1) and 4.01 Mb dup(2)(p21p22.1) (patient 2); 12.08 Mb del(4)(q22.1q24) (patient 3); and 1.19 Mb del(1)(q21.1) (patient 4). One patient (3.7%) displayed a variant of uncertain significance. Four patients (14.8%) displayed discordance between G-banded karyotyping and array CGH results. Among patients with normal array CGH results, 4 (16%) revealed brain anomalies such as schizencephaly and hydranencephaly. One patient was diagnosed with Rett syndrome and one with $M{\ddot{o}}bius$ syndrome. Conclusion: As chromosomal microarray can elucidate the cause of previously unexplained developmental delay, it should be considered as a first-tier cytogenetic diagnostic test for children with unexplained developmental delay.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제26권4호
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• pp.351-362
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• 2012

행렬 및 벡터공간을 다루는 선형대수학은 사회의 복잡한 현상을 선형화 과정을 거쳐 선형연립방정식이라는 단순한 형태의 수학 문제로 바꾼 후 실제로 해결하는 데 결정적으로 기여한다. 이와 같은 이유로 20세기 중반까지 추상적인 고등수학 과목으로만 여겨지던 선형대수학이 현재는 자연-공학-사회계열 분야 학생의 대부분이 배우는 기본 교과목이 되었다. 본 연구에서는 초기 선형대수학의 발전에 기여한 중국, 일본, 그리고 서양의 수학자들에 대하여 다룬다. 선형대수학은 <산수서>, <구장산술>, 세키 고와, 뫼비우스, 그라스만 실베스터, 케일리 등을 거치면서 비선형적으로 발전해왔다. 우리는 새로 발굴한 내용을 중심으로 초기 선형대수학의 발전과정을 소개한다.