• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.147-159
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• 2013

Hamada ([8]) and Maruta ([17]) proved the minimum length $n_3(6,\;d)=g_3(6,\;d)+1$ for some ternary codes. In this paper we consider such minimum length problem for $q{\geq}4$, and we prove that $n_q(6,\;d)=g_q(6,\;d)+1$ for $d=q^5-q^3-q^2-2q+e$, $1{\leq}e{\leq}q$. Combining this result with Theorem A in [4], we have $n_q(6,\;d)=g-q(6,\;d)+1$ for $q^5-q^3-q^2-2q+1{\leq}d{\leq}q^5-q^3-q^2$ with $q{\geq}4$. Note that $n_q(6,\;d)=g_q(6,\;d)$ for $q^5-q^3-q^2+1{\leq}d{\leq}q^5$ by Theorem 1.2.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.419-425
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• 2008

For $q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2$, we prove the non-existence of a $[g_q(6,d),6,d]_q$ code and we give a $[g_q(6,d)+1,6,d]_q$ code by constructing appropriate 0-cycle in the projective space, where $g_q (k,d)={{\sum}^{k-1}_{i=0}}{\lceil}\frac{d}{q^i}{\rceil}$. Consequently, we have the minimum length $n_q(6,d)=g_q(6,d)+1\;for\;q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2\;and\;q{\geq}3$.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.623-640
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• 2005

본 논문에서는 양정수n의 q-유사를 이용한 q-이항정리, q-차례곱, q-차분 연산자, q-적분에 관한 최근의 기본개념을 정리하고, 이를 적용하여 q-베타함수와 q-감마함수의 새로운 q-유사에 대한 특징을 간략하게 논한다.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1179-1192
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• 2018

Several addition formulas for a general class of q-Appell sequences are proved. The q-addition formulas, which are derived, involved not only the generalized q-Bernoulli, the generalized q-Euler and the generalized q-Genocchi polynomials, but also the q-Stirling numbers of the second kind and several general families of hypergeometric polynomials. Some q-umbral calculus generalizations of the addition formulas are also investigated.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1181-1188
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• 2010

q-Volkenborn integrals ([8]) and fermionic invariant q-integrals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25]) studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by $${\varsigma}E,q,\varepsilon(s)=[2]q \sum\limits_{n=0}^\infty\frac{(-1)^n\epsilon^nq^{sn}}{[n]_q}$$ where 0 < q < 1, $\mathfrak{R}$(s) > 1, $\varepsilon{\in}T_p$, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.53-62
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• 2006

A Fock representation of q-commutation relation is studied by constructing a q-Fock space as the space of the representation, the q-creation and q-annihilation operators (-1 < q < 1). In the case of 0 < q < 1, the q-Fock space is interpolated between the Boson Fock space and the full Fock space. Also, a unitary decomposition of the q-Fock space $(q\;{\neq}\;0)$ is studied.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.193-200
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• 2017

Given a pair p, q of relative prime positive integers, we have uniquely determined positive integers x, y, u and v such that vx-uy = 1, p = x + y and q = u + v. Using this property, we show that$${\sum\limits_{1{\leq}i{\leq}x,1{\leq}j{\leq}v}}\;{t^{(i-1)q+(j-1)p}\;-\;{\sum\limits_{1{\leq}k{\leq}y,1{\leq}l{\leq}u}}\;t^{1+(k-1)q+(l-1)p}$$ is the Alexander polynomial ${\Delta}_{p,q}(t)$ of a torus knot t(p, q). Hence the number $N_{p,q}$ of non-zero terms of ${\Delta}_{p,q}(t)$ is equal to vx + uy = 2vx - 1. Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary 2.8); Let q be a positive integer> 1 and let k be a positive integer. Then we have $$\begin{array}{rccl}(1)&N_{kq}+1,q&=&2k(q-1)+1\\(2)&N_{kq}+q-1,q&=&2(k+1)(q-1)-1\\(3)&N_{kq}+2,q&=&{\frac{1}{2}}k(q^2-1)+q\\(4)&N_{kq}+q-2,q&=&{\frac{1}{2}}(k+1)(q^2-1)-q\end{array}$$ where we further assume q is odd in formula (3) and (4). Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type t(kq + 2, q) and t(kq + q - 2, q) in [5] agree with $N_{kq+2,q}$ and $N_{kq+q-2,q}$ respectively.

• Journal of Korean Society of Forest Science
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• v.87 no.2
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• pp.194-200
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• 1998

The difference in leaf bud flushing times of Quercus species and their hybrids was compared among the trees from natural forest, seedlings from natural pollination and artificial crossing. In the natural forest of the Hiruzen experimental forest of Tottori University, Japan, Quercus species flushed in the order of Q. crispula > Q. anguste-lepidota > Q. takatorensis ${\fallingdotseq}$ Q. serrate > Q. dentata. The flushing time of seedlings from natural pollination was in the order of Q. aliena ${\fallingdotseq}$ Q. crispula > Q. angustelepidota > Q. takatorensis ${\fallingdotseq}$ Q. serrate > Q. dentata. The flushing times of $F_1$ hybrids by artificial crossing were between those of parental species. From the result of interspecific hybridization, it is shown that the earliness of flushing in Quercus species is a genetic character considered hereditable to progenies.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.623-635
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• 2014

Classications of (${\alpha},{\beta}$)-fuzzy subalgebras of BCK/BCI-algebras are discussed. Relations between (${\in},{\in}{\vee}q$)-fuzzy subalgebras and ($q,{\in}{\vee}q$)-fuzzy subalgebras are established. Given special sets, so called t-q-set and t-${\in}{\vee}q$-set, conditions for the t-q-set and t-${\in}{\vee}q$-set to be subalgebras are considered. The notions of $({\in},q)^{max}$-fuzzy subalgebra, $(q,{\in})^{max}$-fuzzy subalgebra and $(q,{\in}{\vee}q)^{max}$-fuzzy subalgebra are introduced. Conditions for a fuzzy set to be an $({\in},q)^{max}$-fuzzy subalgebra, a $(q,{\in})^{max}$-fuzzy subalgebra and a $(q,{\in}{\vee}q)^{max}$-fuzzy subalgebra are considered.

• Honam Mathematical Journal
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• v.38 no.4
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• pp.795-807
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• 2016

More general form of an (${\in},{\in}{\vee}q_k$)-fuzzy subsemigroup is considered. The notions of (${\in},q^{\delta}_k$)-fuzzy subsemigroup, ($q^{\delta}_0,{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup are introduced, and related properties are investigated. Characterizations of an (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup are considered. Conditions for an (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup to be a fuzzy subsemigroup are provided. Relations between ($q^{\delta}_0,{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup, (${\in},q^{\delta}_k$)-fuzzy subsemigroup and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup are discussed.