• Journal of the Chungcheong Mathematical Society
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• v.27 no.1
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• pp.1-8
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• 2014

Recently, q-Dedekind-type sums related to q-Euler polynomials was studied by Kim in [T. Kim, Note on q-Dedekind-type sums related to q-Euler polynomials, Glasgow Math. J. 54 (2012), 121-125]. It is aim of this paper to consider a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight related to modified q-Euler polynomials with weight by using Kim's p-adic q-integral.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1345-1356
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• 2017

For a given number field K, we show that the ranks of elliptic curves over K are uniformly finitely bounded if and only if the weak Mordell-Weil property holds in all (some) ultrapowers $^*K$ of K. We introduce the nonstandard weak Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*{\mathbb{Z}}$-module, where $^*{\mathbb{Z}}$ is an ultrapower of ${\mathbb{Z}}$, and we show that the nonstandard weak Mordell-Weil property is equivalent to the weak Mordell-Weil property in $^*K$. In a saturated nonstandard number field, there is a nonstandard ring of integers $^*{\mathbb{Z}}$, which is definable. We can consider definable abelian groups as $^*{\mathbb{Z}}$-modules so that the nonstandard weak Mordell-Weil property is well-defined, and we conclude that the nonstandard weak Mordell-Weil property and the weak Mordell-Weil property are equivalent. We have valuations induced from prime numbers in nonstandard rational number fields, and using these valuations, we identify two nonstandard rational numbers.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.651-656
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• 2016

Let G be a finite group and nse(G) be the set of numbers of elements of G of the same order. In this paper, we prove that the simple group $Sz(2^{2m+1})$, where $2^{2m+1}-1$ is a prime number, is uniquely determined by $nse(Sz(2^{2m+1}))$ and ${\mid}Sz(2^{2m+1}){\mid}$.

• Proceedings of the KSME Conference
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• 2002.08a
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• pp.655-658
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• 2002

The present paper investigated the effect of ultrasonic vibrations on the melting process of a phase-change material (PCM). Furthermore, the present study considered constant heat-flux boundary conditions unlike many of the previous researches, which had adopted constant wall-temperature conditions. Therefore in the study, modified dimensionless numbers such as Stefan and Rayleigh were adopted to represent heat transfer results. The experimental results revealed that ultrasonic vibrations accompanied the effects like agitation, acoustic streaming, cavitation, and oscillating fluid motion, accelerating the melting process as much as 2.5 times, compared with the result of natural melting (i. e., the case without ultrasonic vibration). Such effects are believed to be a prime mechanism in the overall melting process when ultrasonic vibrations were applied. Subsequently, energy could be saved by applying the ultrasonic vibrations to the natural melting In addition, various time-wise dimensionless numbers provided a conclusive evidence of the important role of the ultrasonic vibrations on the melting phenomena of the PCM.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.115-128
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• 2015

Plumbing surfaces of links were introduced to study the geometry of the complement of the links. A basket surface is one of these plumbing surfaces and it can be presented by two sequential presentations, the first sequence is the flat plumbing basket code found by Furihata, Hirasawa and Kobayashi and the second sequence presents the number of the full twists for each of annuli. The minimum number of plumbings to obtain a basket surface of a knot is defined to be the basket number of the given knot. In present article, we first find a classification theorem about the basket number of knots. We use these sequential presentations and the classification theorem to find the basket number of all prime knots whose crossing number is 7 or less except two knots $7_1$ and $7_5$.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.365-374
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• 2013

Let $k=\mathbb{F}_q(T)$ be a rational function field over the finite field $\mathbb{F}_q$, where q is a power of an odd prime number, and $\mathbb{A}=\mathbb{F}_q[T]$. Let ${\gamma}$ be a generator of $\mathbb{F}^*_q$. Let $\mathcal{H}_n$ be the subset of $\mathbb{A}$ consisting of monic square-free polynomials of degree n. In this paper we obtain an asymptotic formula for the mean value of $L(1,{\chi}_{\gamma}{\small{D}})$ and calculate the average value of the ideal class number $h_{\gamma}\small{D}$ when the average is taken over $D{\in}\mathcal{H}_{2g+2}$.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.549-559
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• 2018

In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.

• 전기의세계
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• v.28 no.6
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• pp.61-77
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• 1979

This paper is concerned with minimization process of the binary logic function. This paper describes an algorithm called the SIMPLE TABLE METHOD that well suited to minimization a switching function of any number of variables. For the Simple Table construction, a theorem based upon the numerical properties of the logic function is derived from the relationships governing minterms of the given function. Finally the minimal sum of products can be obtained in terms of the Direct Method or the Indirect Method from the table and table characteristics derived from the Simple Table. The properties and table characteristics used in this paper are described. All the minterms of a switching function are manipulated only by decimal numbers, not binary numbers. Some examples are used as a vehicle to guide the readers who are familiar with the Karnaugh map and Quine-McCluskey tabular method to this New method. These examples not only treat how to handle Don't Care miterms but also show the multiple output functions.

• Honam Mathematical Journal
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• v.44 no.4
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• pp.521-534
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• 2022

In this paper, we introduce a new class 𝓡^k_λ(λ ≥ 1, k is any positive real number) of univalent complex functions, which consists of functions f of the form f(z) = z + Σ^∞_n=2 a_nzⁿ (|z| < 1) satisfying the subordination condition $$(1-{\lambda}){\frac{f(z)}{z}}+{\lambda}f^{\prime}(z){\prec}{\frac{1+r^2_kz^2}{1-k{\tau}_kz-{\tau}^2_kz^2}},\;{\tau}_k={\frac{k-{\sqrt{k^2+4}}}{2}$$, and investigate the Fekete-Szegö problem for the coefficients of f ∈ 𝓡^k_λ which are connected with k-Fibonacci numbers $F_{k,n}={\frac{(k-{\tau}_k)^n-{\tau}^n_k}{\sqrt{k^2+4}}}$ (n ∈ ℕ ∪ {0}). We obtain sharp upper bound for the Fekete-Szegö functional |a₃-𝜇a²₂| when 𝜇 ∈ ℝ. We also generalize our result for 𝜇 ∈ ℂ.

• East Asian mathematical journal
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• v.28 no.2
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• pp.159-172
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• 2012

In all Mathematics I Textbooks(Kim, S. H., 2010; Kim, H. K., 2010; Yang, S. K., 2010; Woo, M. H., 2010; Woo, J. H., 2010; You, H. C., 2010; Youn, J. H., 2010; Lee, K. S., 2010; Lee, D. W., 2010; Lee, M. K., 2010; Lee, J. Y., 2010; Jung, S. K., 2010; Choi, Y. J., 2010; Huang, S. K., 2010; Huang, S. W., 2010) in high schools in Korea these days, it is written and taught that for a positive real number $a$, $a^{\frac{m}{n}}$ is defined as $a^{\frac{m}{n}}={^n}\sqrt{a^m}$, where $m{\in}\mathbb{Z}$ and $n{\in}\mathbb{N}$ have common prime factors. For that situation, the author shows his opinion that the definition is not well-defined and $a^{\frac{m}{n}}$ must be defined as $a^{\frac{m}{n}}=({^n}\sqrt{a})^m$, whenever $^n\sqrt{a}$ is defined, based on the field axiom of the real number system including rational number system and natural number system. And he shows that the following laws of exponents for reals: $$\{a^{r+s}=a^r{\cdot}a^s\\(a^r)^s=a^{rs}\\(ab)^r=a^rb^r$$ for $a$, $b$>0 and $s{\in}\mathbb{R}$ hold by the completeness axiom of the real number system and the laws of exponents for natural numbers, integers, rational numbers and real numbers are logically equivalent.