• East Asian mathematical journal
• /
• v.23 no.1
• /
• pp.59-73
• /
• 2007

A new reverse of the generalised triangle inequality that complements the classical results of Diaz and Metcalf is obtained. Applications for inner product spaces and for complex numbers are provided.

• Journal for History of Mathematics
• /
• v.25 no.4
• /
• pp.69-82
• /
• 2012

The objective of this paper is to study De Morgan's perspective on teaching and learning complex numbers. De Morgan's didactical approaches reflect the process of development of his thoughts about algebra from universal arithmetic, symbolic algebra to meaning algebra. De Morgan develop his perspective on algebra by justifying and explaining complex numbers. This implies that teaching and learning complex numbers is a catalyst for mathematical development of De Morgan.

• Korean Journal of Plant Taxonomy
• /
• v.47 no.4
• /
• pp.304-307
• /
• 2017

We report somatic chromosome numbers for three species belonging to the Euphorbia pekinensis complex distributed in Far East Asia. In E. pekinensis populations distributed in Korea, 2n = 28 and 56 were found, while the Japanese native E. lasiocaula was also found at 2n = 28 and 56 and the Japanese endemic E. sinanensis was found at 2n = 20. Based on the number of chromosomes, E. lasiocaula distributed in Japan supports treatment as a variety of E. pekinensis rather than as a different species, while E. sinanensis should be recognized as a distinct species rather than as a variety of E. pekinensis. In the same populations of E. pekinensis and E. lasiocaula, diploid and tetraploid individuals were found, and the diversity of these chromosome numbers was consistent with the morphological diversity of these populations, suggesting the future evolutionary potential of this species.

• Journal of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.391-410
• /
• 2000

In the complex case, we construct a q-analogue of the Riemann zeta function q(s) and a q-analogue of the Dirichlet L-function L(s,X), which interpolate the 1-analogue Bernoulli numbers. Using the properties of p-adic integrals and measures, we show that Kummer type congruences for the q-analogue Bernoulli numbers are the generalizations of the usual Kummer congruences for the ordinary Bernoulli numbers. We also construct a q0analogue of the p-adic L-function Lp(s, X;q) which interpolates the q-analogue Bernoulli numbers at non positive integers.

• Journal of the Korean Mathematical Society
• /
• v.47 no.5
• /
• pp.1055-1075
• /
• 2010

In presented paper there were studied some properties of Benford's law. The existence of this law in not necessary large sets of numbers is a very interesting example that can show how the complex phenomena can appear in the positional number systems. Such systems seem to be very simple and intuitive and help us proceed with numbers. However, their simplicity in the case of usage in our lifetime is not necessary connected with the simplicity in the case of laws that govern them. Even if this laws indicate the existence of self-similar properties.

• School Mathematics
• /
• v.10 no.3
• /
• pp.357-374
• /
• 2008

The complex number system is supposed to introduce first chapter in the first grade of high school. When number system is expanded to complex numbers, the main aim is to understand preservation of algebraic structure with regard to the flow of curriculum and textbook. This research reviewed overall alternative articulation and treatment of textbooks from a logical viewpoint. Two research questions are developed below. First, in the structure of the current curriculum, when we consider student's 'level', how are the alternative articulation and treatment of textbooks in complex unit on a logical point of view? Second, What are more logical alternative articulation and treatment? What alternative articulation and treatment are suitable for a running goal? and what are the improvement which is definitive?

• Honam Mathematical Journal
• /
• v.32 no.4
• /
• pp.563-579
• /
• 2010

In this paper we introduce the new analogs of Genocchi numbers and polynomials. Finally, we investigate the zeros of the twisted (h,q)-extension of Genocchi polynomials ${G_{n,q,w}^{(h)}}$(x).

• The Pure and Applied Mathematics
• /
• v.20 no.2
• /
• pp.129-136
• /
• 2013

We research properties of ternary numbers with values in ${\Lambda}(2)$. Also, we represent dual ternary numbers in the sense of Clifford algebras of real six dimensional spaces. We give generation theorems in dual ternary number systems in view of Clifford analysis, and obtain Cauchy theorems with respect to dual ternary numbers.

• Journal of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.485-505
• /
• 2019

In 1935, D. H. Lehmer introduced and investigated generalized Euler numbers $W_n$, defined by $${\frac{3}{e^t+e^{wt}e^{w^2t}}}={\sum\limits_{n=0}^{\infty}}W_n{\frac{t^n}{n!}}$$, where ${\omega}$ is a complex root of $x^2+x+1=0$. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli and Euler numbers. These concepts can be generalized to the hypergeometric Bernoulli and Euler numbers by several authors, including Ohno and the second author. In this paper, we study more general numbers in terms of determinants, which involve Bernoulli, Euler and Lehmer's generalized Euler numbers. The motivations and backgrounds of the definition are in an operator related to Graph theory. We also give several expressions and identities by Trudi's and inversion formulae.

• Journal of the Korean Society of Combustion
• /
• v.10 no.3
• /
• pp.25-33
• /
• 2005

Fast-time instability is investigated for diffusion flames with Lewis numbers greater than unity by employing the numerical technique called the Evans function method. Since the time and length scales are those of the inner reactive-diffusive layer, the problem is equivalent to the instability problem for the $Li\tilde{n}\acute{a}n#s$ diffusion flame regime. The instability is primarily oscillatory, as seen from complex solution branches and can emerge prior to reaching the upper turning point of the S-curve, known as the $Li\tilde{n}\acute{a}n#s$ extinction condition. Depending on the Lewis number, the instability characteristics is found to be somewhat different. Below the critical Lewis number, $L_C$, the instability possesses primarily a pulsating nature in that the two real solution branches, existing for small wave numbers, merges at a finite wave number, at which a pair of complex conjugate solution branches bifurcate. For Lewis numbers greater than $L_C$, the solution branch for small reactant leakage is found to be purely complex with the maximum growth rate found at a finite wave number, thereby exhibiting a traveling nature. As the reactant leakage parameter is further increased, the instability characteristics turns into a pulsating type, similar to that for L < $L_C$.