• Honam Mathematical Journal
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• v.36 no.2
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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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• v.28 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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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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• v.56 no.3
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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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• v.60 no.9
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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.

• Communications of Mathematical Education
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• v.26 no.4
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• pp.351-362
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• 2012

Until the 1950s, linear algebra was considered only as one of abstract and advanced mathematics subject among in graduate mathematics courses, mainly dealing with module in algebra. Since the 1960s, it has been a main subject in undergraduate mathematics education because matrices has been used all over. In Korea, it was considered as a course only for mathematics major students until 1980s. However, now it is a subject for all undergraduate students including natural science, engineering, social science since 1990s. In this paper, we investigate the early history of linear algebra and its development from a historical perspective and mathematicians who made contributions. Secondly, we explain why linear algebra became so popular in college mathematics education in the late 20th century. Contributions of Chinese and H. Grassmann will be extensively examined with many newly discovered facts.