• Journal of the Chungcheong Mathematical Society
• /
• v.16 no.2
• /
• pp.75-78
• /
• 2003

The Tychonoff product theorem is one of the most fundamental theorems in general topology. As is well-known, the proof of the Tychonoff product theorem relies on the axiom of choice. The converse was also conjectured by S. Kakutani and Kelley [1] then resolved this conjecture in his historical short note on 1950. However, the original proof due to Kelley has a flaw. According to this observation, we provide a correction of the proof in this paper.

• Communications of the Korean Mathematical Society
• /
• v.9 no.4
• /
• pp.853-855
• /
• 1994

The object of present note is to give a very short proof of Stirling's formula which uses only a formula for the generalized zeta function. There are several proofs for this formula. For example, Dr. E. J. Routh gave an elementary proof using Wallis' theorem in lectures at Cambridge ([5, pp.66-68]). We can find another proof which used the Maclaurin summation formula ([5, pp.116-120]). In [1], they used the Central Limit Theorem or the inversion theorem for characteristic functions. In [2], pp. Diaconis and D. Freeman provided another proof similarly as in [1]. J. M. Patin [7] used the Lebesgue dominated convergence theorem.

• Communications of the Korean Mathematical Society
• /
• v.25 no.2
• /
• pp.185-191
• /
• 2010

Srivastava noticed the existence of three additional complete triple hypergeometric functions $H_A$, $H_B$ and $H_C$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series $H_A$ and $H_C$. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's $H_C$ with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a $_3F_2$ obtained earlier by Lavoie et al.. Some special cases of our results are also considered.

• Kyungpook Mathematical Journal
• /
• v.47 no.3
• /
• pp.449-454
• /
• 2007

InIn this paper we consider generalizations of the classical Dixon's theorem and the classical Whipple's theorem on the sum of a $_3F_2$. The results are derived with the help of generalized Watson's theorem obtained earlier by Mitra. A large number of results contiguous to Dixon's and Whipple's theorems obtained earlier by Lavoie, Grondin and Rathie, and Lavoie, Grondin, Rathie and Arora follow special cases of our main findings.

• Korean Journal of Mathematics
• /
• v.8 no.1
• /
• pp.73-86
• /
• 2000

We study the Sobolev spaces to the generalized Sobolev spaces $H^s_{\mathcal{G}}$ of exponential type based on the Silva space $\mathcal{G}$ and investigate its properties such as imbedding theorem and structure theorem. In fact, the imbedding theorem says that for $s$ > 0 $u{\in}H^s_{\mathcal{G}}$ can be analytically continued to the set {$z{\in}\mathbb{C}^n{\mid}{\mid}Im\;z{\mid}$ < $s$}. Also, the structure theorem means that for $s$ > 0 $u{\in}H^{-s}_{\mathcal{G}}$ is of the form $$u={\sum_{\alpha}\frac{s^{{|\alpha|}}}{{\alpha}!}D^{\alpha}g{\alpha}$$ where $g{\alpha}$'s are square integrable functions for ${\alpha}{\in}\mathbb{N}^n_0$. Moreover, we introduce a classes of symbols of exponential type and its associated pseudo-differential operators of exponential type, which naturally act on the generalized Sobolev spaces of exponential type. Finally, a generalized Bessel potential is defined and its properties are investigated.

• Communications of Mathematical Education
• /
• v.23 no.4
• /
• pp.1079-1092
• /
• 2009

The purpose of this study is to investigate Newton's binomial theorem that was on epistemological basis of the emergent background and developmental course of infinite series and power series. Through this investigation, it will be examined how finding the approximate of square root of given numbers, the method of the inverse method of fluxions by Newton, and Gregory and Mercator series were developed in the course of history of mathematics. As the result of this study pedagogical analysis and discussion of the history of mathematics on Newton's binomial theorem will be presented.

• Communications of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.841-854
• /
• 2019

In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $S^{\delta}[{\alpha}]$, $0{\leq}{\alpha}<1$, $-{\infty}<{\delta}<{\infty}$ which has been introduced and studied by Kumar [17] (see also [20], [21], [22], [23]). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bound for the Bloch's constant for all functions in that family.

• Korean Journal of Mathematics
• /
• v.21 no.3
• /
• pp.325-330
• /
• 2013

In this paper, we prove Trotter-Kato theorem in the weak topology if $X^*$ is a uniformly convex Banach space.

• Kyungpook Mathematical Journal
• /
• v.63 no.1
• /
• pp.29-36
• /
• 2023

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated R-module M is Noetherian if and only if for every prime ideal 𝔭 of R with Ann(M) ⊆ 𝔭, there exists a finitely generated submodule N^𝔭 of M such that 𝔭M ⊆ N^𝔭 ⊆ M(𝔭), where M(𝔭) = {x ∈ M | sx ∈ 𝔭M for some s ∈ R \ 𝔭}. In this paper, we generalize the Parkash and Kour version of Cohen's theorem for Noetherian modules to S-Noetherian modules and w-Noetherian modules.

• East Asian mathematical journal
• /
• v.17 no.1
• /
• pp.147-158
• /
• 2001

The Gauss summation theorem plays a key role in the theory of(generalized) hypergeometric series. The authors study several proofs of the theorem and consider some applications of it.