• Communications of the Korean Mathematical Society
• /
• v.13 no.2
• /
• pp.233-242
• /
• 1998

Let $k = Q(\sqrt{m})$ be a real quadratic field. In this paper, the following theorems on p-divisibility of the class number h of k are studied for each prime pp. Theorem 1. If the discriminant of k has at least three distinct prime divisors, then 2 divides h. Theorem 2. If an odd prime p divides h, then p divides $B_{a,\chi\omega^{-1}}$, where $\chi$ is the nontrivial character of k, and $\omega$ is the Teichmuller character for pp. Theorem 3. Let $h_n$ be the class number of $k_n$, the nth layer of the $Z_p$-extension $k_\infty$ of k. If p does not divide $B_{a,\chi\omega^{-1}}$, then $p \notmid h_n$ for all $n \geq 0$.

• Communications of the Korean Mathematical Society
• /
• v.38 no.3
• /
• pp.715-723
• /
• 2023

In this paper, we construct explicitly an infinite family of primes P with h^±_P ≡ 0 (mod q^{deg P}), where h^±_P are the plus and minus parts of the divisor class number of the P-th cyclotomic function field over 𝔽_q(T). By using this result and Dirichlet's theorem, we give a condition of A, M ∈ 𝔽_q[T] such that there are infinitely many primes P satisfying with h^±_P ≡ 0 (mod p^e) and P ≡ A (mod M).

• Korean Journal of Mathematics
• /
• v.7 no.2
• /
• pp.227-233
• /
• 1999

Let $k$ be a real abelian field and $k_{\infty}={\bigcup}_{n{\geq}0}k_n$ be its $\mathbb{Z}_p$-extension for an odd prime $p$. For each $n{\geq}0$, we denote the class number of $k_n$ by $h_n$. The following is a well known theorem: Theorem. Suppose $p$ remains inert in $k$ and the prime ideal of $k$ above $p$ totally ramifies in $k_{\infty}$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$ for all $n{\geq}0$. The aim of this paper is to generalize above theorem: Theorem 1. Suppose $H^1(G_n,E_n){\simeq}(\mathbb{Z}/p^n\mathbb{Z})^l$, where $l$ is the number of prime ideals of $k$ above $p$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$. Theorem 2. Let $k$ be a real quadratic field. Suppose that $H^1(G_1,E_1){\simeq}(\mathbb{Z}/p\mathbb{Z})^l$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$ for all $n{\geq}0$.

• Journal of Forest and Environmental Science
• /
• v.33 no.3
• /
• pp.233-238
• /
• 2017

This study was carried out to figure out the number of logs for sawtimber by DBH and height class and to compare merchantable volume ratio by categorizing into sawtimber, lagging board and pulpwood, and others for Pinus densiflora, Pinus koraiensis, and Larix kaempferi. Logs for sawtimber were hardly produced in small DBH class of three species, but produced evidently from medium DBH class. In large DBH class, the number of logs for sawtimber were noticeably different among species: 4.3 logs for L. kaempferi, 2.6 logs for P. densiflora, and 1.0 logs for P. koraiensis on average. Similarly, merchantable volume ratio for sawtimber were largely different among species in large DBH class with higher than 15 m: 82% logs for L. kaempferi, 60% logs for P. densiflora, and 44% logs for P. koraiensis. When compared to the upper diameter and upper height by species with regard to the last log of a tree produced for sawtimber, upper diameter was smallest with 14.1 cm and upper height was highest with 12.2 m in L. kaempferi. Overall, L. kaempferi was considered as the more commercial species for sawtimber production than P. densiflora and P. koraiensis.

• The Pure and Applied Mathematics
• /
• v.7 no.1
• /
• pp.27-32
• /
• 2000

The quadratic fields generated by $x^2$=ax+1($\alpha\geq$1) are studied. The regulators are relatively small and are known at one. The class numbers are relatively large and easy to compute. We shall find all the values of p, where p=$\alpha^2$+4 is a prime in $\mathbb{Z}$, such that $\mathbb{Q}(\sprt{p})$ has class numbers 1, 3 and 5.

• Journal of the Korean Data and Information Science Society
• /
• v.3 no.1
• /
• pp.47-63
• /
• 1992

Bechhofer and Tamhane(1981) proposed Balanced Treatment Incomplete Block (BTIB) desings for comparing p test treatments with a control treatment in blocks of size ${\kappa}$. Notz and Tamhane(1983) solved the problem about determination of the minimal complete class for ${\kappa}=3$. However there are a number of design parameters for which BTIB designs do not exist. We suggest a new class of designs called Group Divisible Treatment Desings(GDTD's) that is a larger class including BTIB designs as a subclass. In this paper we give the minimal complete classes of generator designs for GDTD's with ${\kappa}=2,\;p{\geq}4(except\;prime\;number)\;and\;{\kappa}=3,\;p=4(2)6$.

• Korean Journal of Mathematics
• /
• v.26 no.2
• /
• pp.293-297
• /
• 2018

In the paper [4], we constructed 3-part of the Hilbert class field of imaginary quadratic fields whose class number is divisible exactly by 3. In this paper, we extend the result for any odd prime p.

• Journal of the Korean Mathematical Society
• /
• v.52 no.4
• /
• pp.751-763
• /
• 2015

Let ${\alpha}$ be a positive integer, and let $p_1$, $p_2$ be two distinct prime numbers with $p_1$ < $p_2$. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form $2^{\alpha}p_1p_2$ and $2^{\alpha}p_1^2p_2$, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form $p_1=2^{{\alpha}+1}-1$ and $p_2={\frac{p^2_1+p_1+1}{3}}$, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.

• Journal of Korean society of Dental Hygiene
• /
• v.20 no.1
• /
• pp.63-71
• /
• 2020

Objectives: This study provides basic data for developing practical teaching methods enabling efficient execution of the dental hygiene care process. Methods: A total of 197 dental hygiene students experienced in dental hygiene care process in Gwangju and Jeonnam were surveyed from June 1 to August 30, 2019 to study their class flow, professors-student relationships, and class satisfaction level. Post survey, statistical analysis was performed using frequency analysis, independent t-test, Pearson's correlation analysis, and multiple regression analysis. Results: 1. Class fl ow was high in three lecturers (3.56), four hours per week (3.39), and four hours per week (3.94). Class satisfaction was high in three lecturers (3.99) and four hours per week (3.90) (p<0.05). 2. There was a positive correlation between class flow and professor-student relationship (r=0.519), class fl ow and class satisfaction (r=0.566), and professor-student relationship and class satisfaction (r=0.838) (p<0.01). 3. The factors influencing class fl ow were the number of lecturers (β=0.442), class hours per week (β=-0.397), and class satisfaction (β=0.385). Conclusions: Apart from finding ways to improve class satisfaction for class flow in the dental hygiene care process, efforts are required to increase the number of lecturers and class hours per week for efficient class management. Further research is needed to develop practical teaching methods.

• Kyungpook Mathematical Journal
• /
• v.50 no.4
• /
• pp.545-556
• /
• 2010

A good amount of work has been done on degree of approximation of functions belonging to Lip${\alpha}$, Lip($\xi$(t),r) and W($L_r,\xi(t)$) and classes using Ces$\`{a}$ro, N$\"{o}$rlund and generalised N$\"{o}$rlund single summability methods by a number of researchers ([1], [10], [8], [6], [7], [2], [3], [4], [9]). But till now, nothing seems to have been done so far to obtain the degree of approximation of functions using (N,$p_n$)(C, 1) product summability method. Therefore the purpose of present paper is to establish two quite new theorems on degree of approximation of function $f\;\in\;Lip({\alpha},r)$ class and $f\;\in\;W(L_r,\;\xi(t))$ class by (N, $p_n$)(C, 1) product summability means of its Fourier series.