• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.963-975
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• 2003

The aim of this work is to construct twisted q-L-series which interpolate twisted q-generalized Bernoulli numbers. By using generating function of q-Bernoulli numbers, twisted q-Bernoulli numbers and polynomials are defined. Some properties of this polynomials and numbers are described. The numbers $L_{q}(1-n,\;X,\;{\xi})$ is also given explicitly.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.107-110
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• 1996

Usng the Pochhammer symbol $(\lambda)_n$ given by $$ (1.1) (\lambda)_n = {1, if n = 0 {\lambda(\lambda + 1) \cdots (\lambda + n - 1), if n \in N = {1, 2, 3, \ldots}, $$ we define a general double hypergeometric series by [3, pp.27] $$ (1.2) F_{q:s;\upsilon}^{p:r;u} [\alpha_1, \ldots, \alpha_p : \gamma_1, \ldots, \gamma_r; \lambda_1, \ldots, \lambda_u;_{x,y}][\beta_1, \ldots, \beta_q : \delta_1, \ldots, \delta_s; \mu_1, \ldots, \mu_v; ] = \sum_{l,m = 0}^{\infty} \frac {\prod_{j=1}^{q} (\beta_j)_{l+m} \prod_{j=1}^{s} (\delta_j)_l \prod_{j=1}^{v} (\mu_j)_m)}{\prod_{j=1}^{p} (\alpha_j)_{l+m} \prod_{j=1}^{r} (\gamma_j)_l \prod_{j=1}^{u} (\lambda_j)_m} \frac{l!}{x^l} \frac{m!}{y^m} $$ provided that the double series converges.

• Journal of Radiation Protection and Research
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• v.23 no.3
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• pp.197-210
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• 1998

Regulations for the safe transport of radioactive material published by IAEA Safety Standard Series ST-l is reviewed and summarized. Safety Series No.115(International standard of radiation protection and safety for ionizing radiation and radiation sources), which reflected the new recommendation of ICRP60 published in 1991, has been a important encouragement for IAEA to revise their safety series related to the transportation of radioactive materials. IAEA Safety, Standard Series No. ST-l is summarized by comparing IAEA Safety Series No.6 regarding radiation protection system and its implementation, technical standards of packages, concept of Q system and exemption of regulation. The IAEA regulations of transportation of radioactive materials is summarized from the viewpoint of radiation protection and safety assessment. Research on transportation system of radioactive waste is suggested as a further study.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.693-704
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• 2012

We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n{\geq}1$, we have $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1;3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.

• Journal of the Korean Statistical Society
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• v.21 no.1
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• pp.47-58
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• 1992

The autocorrelation function structures of bilinear time series model BL(p, q, r, s), $r \geq s$ are obtained and shown to be analogous to those of ARMA(p, l), l=max(q, s). Simulation studies are performed to investigate the adequacy of Akaike information criteria for identification between ARMA(p, l) and BL(p, q, r, s) models and for determination of orders of BL(p, q, r, s) models. It is suggested that the model of having minimum Akaike information criteria is selected for a suitable model.

• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.913-926
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• 2011

The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.271-289
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• 2018

Let 0 < q < ${\infty}$. In this study, we introduce the spaces ${\mathcal{BV}}_q$ and ${\mathcal{LS}}_q$ of q-bounded variation double sequences and q-summable double series as the domain of four-dimensional backward difference matrix ${\Delta}$ and summation matrix S in the space ${\mathcal{L}}_q$ of absolutely q-summable double sequences, respectively. Also, we determine their ${\alpha}$- and ${\beta}-duals$ and give the characterizations of some classes of four-dimensional matrix transformations in the case 0 < q ${\leq}$ 1.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.419-431
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• 2017

In this paper, we define a new kind of formal power series rings by using Gaussian binomial coefficients and investigate some properties. More precisely, we call such a ring a Gaussian series ring and study McCoy's theorem, Hermite properties and Noetherian properties.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-284
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• 2019

The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

• Journal of the Mineralogical Society of Korea
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• v.10 no.2
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• pp.97-104
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• 1997

Based on the method of determination for relative stability of each phase from the difference among the interaction parameters of the phases consisting the mixed layer, the types of interactions between layers were specified and interaction parameter between layers in ordered domain was analytically derived as a function parameter between layers in ordered domain was analytically derived as a function of not only temperature and mole fraction of layers but also ordering parameter. Interaction parameter between the different layers in ordered phase, L is as follows:{{{{ {L }_{1 } (X,Q,T)= { C} over { Q} -4(1-2Q) { L}^{2 } - { RT} over {2} ln { 1} over {2 } - { 2RT} over { { X}_{ s} } ln { { 4QX}`_{s } ^{2 } } over {(1- { X}_{s }- { QX}_{s })( { X}_{s }- {QX }_{s } ) } }}}}L2 is the interaction parameter between ordered and disordered phase in domain and is the mole fraction of the domain which represent the infinite length of mixed layer mineral and Q and C are the reaction progress parameter and arbitrary constant, respectively. This equation was used for the I/S mixed layer clay minerals to infer the relative stability of R1 type I/S mixed layer in the temperature range from 373K to 450K. The result of calculation suggest that, owing to the decrease in interaction parameter with increasing temperature. The interaction parameter decreases more rapidly with decreasing mole fraction of smectite in domain, which is consistent with the fact that the probability of finding the series smectite layer is lo in the domain with small mole fraction of smectite layers in natural system.