• Communications of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.401-414
• /
• 2019

Let ${\mathcal{F}}$ denote an algebraically closed field with characteristic not two. Fix an integer $d{\geq}3$, let $Mat_{d+1}({\mathcal{F}})$ denote the ${\mathcal{F}}$-algebra of $(d+1){\times}(d+1)$ matrices with entries in ${\mathcal{F}}$. An ordered pair of matrices A, $A^*$ in $Mat_{d+1}({\mathcal{F}})$ is said to be LB-TD form whenever A is lower bidiagonal with subdiagonal entries all 1 and $A^*$ is irreducible tridiagonal. Let A, $A^*$ be a Leonard pair in $Mat_{d+1}({\mathcal{F}})$ with fundamental parameter ${\beta}=2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.

• Communications of the Korean Mathematical Society
• /
• v.12 no.4
• /
• pp.935-950
• /
• 1997

When $\tau$ is a quasi-definite moment functional on P, the vector space of all real polynomials, we consider a symmetric bilinear form $\phi(\cdot,\cdot)$ on $P \times P$ defined by $$ \phi(p,q) = \lambad p(a)q(a) + \mu p(b)q(b) + <\tau,p'q'>, $$ where $\lambda,\mu,a$, and b are real numbers. We first find a necessary and sufficient condition for $\phi(\cdot,\cdot)$ and show that such orthogonal polynomials satisfy a fifth order differential equation with polynomial coefficients.

• Communications of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.417-445
• /
• 2020

The purpose of this article is to continue the study of q, 𝜔-special functions in the spirit of Wolfgang Hahn from the previous papers by Annaby et al. and Varma et al., with emphasis on q, 𝜔-Apostol Bernoulli and Euler polynomials, Ward-𝜔 numbers and multiple q, 𝜔power sums. Like before, the q, 𝜔-module for the alphabet of q, 𝜔-real numbers plays a crucial role, as well as the q, 𝜔-rational numbers and the Ward-𝜔 numbers. There are many more formulas of this type, and the deep symmetric structure of these formulas is described in detail.

• Proceedings of the KIEE Conference
• /
• 1998.07a
• /
• pp.241-243
• /
• 1998

This paper describes a new method for the faster shape optimization of the electromagnetic devices. In a conventional iterative method of shape design optimization using design sensitivity based on a finite element method, meshes for a new shape of the model are generated and a discretized system equation is solved using the meshes in each iteration. They cause much design time. To save this time, a polynomial approximation of the finite element solution with respect to the geometric design parameters using Taylor expansion is constructed. This approximate state variable expressed explicitly in terms of design parameters is employed in a gradient-based optimization method. The proposed method is applied to the shape design of quadrupole magnet.

• ETRI Journal
• /
• v.21 no.1
• /
• pp.28-39
• /
• 1999

Koblitz has suggested to use "anomalous" elliptic curves defined over ${\mathbb{F}}_2$, which are non-supersingular and allow or efficient multiplication of a point by and integer, For these curves, Meier and Staffelbach gave a method to find a polynomial of the Frobenius map corresponding to a given multiplier. Muller generalized their method to arbitrary non-supersingular elliptic curves defined over a small field of characteristic 2. in this paper, we propose an algorithm to speed up scalar multiplication on an elliptic curve defined over a small field. The proposed algorithm uses the same field. The proposed algorithm uses the same technique as Muller's to get an expansion by the Frobenius map, but its expansion length is half of Muller's due to the reduction step (Algorithm 1). Also, it uses a more efficient algorithm (Algorithm 3) to perform multiplication using the Frobenius expansion. Consequently, the proposed algorithm is two times faster than Muller's. Moreover, it can be applied to an elliptic curve defined over a finite field with odd characteristic and does not require any precomputation or additional memory.

• The Bulletin of The Korean Astronomical Society
• /
• v.38 no.2
• /
• pp.80-80
• /
• 2013

대규모 광도곡선 자료에서 다양한 주기변광성들의 정확한 주기를 효율적으로 검출하는 실험을 시도하였다. 실험을 위해 OGLE-III 맥동 변광성(RR Lyrae, Delta Scuti, Cepheid) 목록 중, I 필터로 관측된 총 31,324개의 광도 곡선을 사용하였다. 이 실험에 사용한 주기분석 알고리즘 MS_Period(Multi-Step period searching method)는 주기를 놓치지 않기 위해 두 가지 다른 방법(Multi Polynomial function, Phase Dispersion)으로 후보 주기를 구하고 정밀주기를 도출하기 위해 후보 주기 주변부를 Spline fitting을 통해 재탐색하는 방법이다. 기존의 MS_Period 방식은 주기 탐색 간격(dP/P)이 일정하였으나, 우리는 탐색 주기 구간을 나누고 짧은 주기에서는 작은 간격으로, 긴 주기에서는 보다 넓은 간격으로 주기를 탐색하는 과정을 추가하였다. 그 결과 98% 이상의 별에서 OGLE-III와 거의 일치하는 주기를 얻었으며, 긴 주기에서의 불필요한 정밀 탐색을 회피함으로써 분석시간도 단축되었다. 주기 결정이 어려운 경우들은 주로 1) periodogram에서 실제 주기가 아닌 1일 근처에서 noise보다 큰 peak가 보이는 경우, 2) 하나의 별에 대해 여러 주기가 비슷한 Phase diagram을 보이고, periodogram에서도 비슷한 peak를 갖는 경우, 3) OGLE-III의 주기와 전혀 다른 주기만 찾은 경우, 4) OGLE-III에서 제시하지 않은 혼합된 주기의 존재가 의심되는 경우인 것을 확인하였고, 각 사례들의 특징을 살펴보았다.