• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.2
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• pp.91-97
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• 2005

If ${\beta}\;>\;1$, then every non-negative number x has a ${\beta}-expansion$, i.e., $$x\;=\;{\epsilon}_0(x)\;+\;{\frac{\epsilon_1(x)}{\beta}}\;+\;{\frac{\epsilon_2(x)}{\beta}}\;+\;{\cdots}$$ where ${\epsilon}_0(x)\;=\;[x],\;{\epsilon}_1(x)\;=\;[\beta(x)],\;{\epsilon}_2(x)\;=\;[\beta(({\beta}x))]$, and so on ([x] denotes the integral part and (x) the fractional part of the real number x). Let T be a transformation on [0,1) defined by $x\;{\rightarrow}\;({\beta}x)$. It is well known that the relative frequency of $k\;{\in}\;\{0,\;1,\;{\cdots},\;[\beta]\}$ in ${\beta}-expansion$ of x is described by the T-invariant absolutely continuous measure ${\mu}_{\beta}$. In this paper, we show the mod M normality of the sequence $\{{\in}_n(x)\}$.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1175-1187
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• 2017

The spectral problem $$-y^{{\prime}{\prime}}+q(x)y={\lambda}y,\;0 < x < 1, \atop y(0)cos{\beta}=y^{\prime}(0)sin{\beta},\;0{\leq}{\beta}<{\pi};\;{\frac{y^{\prime}(1)}{y(1)}}=h({\lambda})$$ is considered, where ${\lambda}$ is a spectral parameter, q(x) is real-valued continuous function on [0, 1] and $$h({\lambda})=a{\lambda}+b-\sum\limits_{k=1}^{N}{\frac{b_k}{{\lambda}-c_k}},$$ with the real coefficients and $a{\geq}0$, $b_k$ > 0, $c_1$ < $c_2$ < ${\cdots}$ < $c_N$, $N{\geq}0$. The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented.

• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.127-133
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• 2012

It is well known that if the ${\beta}$-expansion of any nonnegative integer is finite, then ${\beta}$ is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the ${\beta}$-expansion of the polynomial part of ${\beta}$ is finite if and only if ${\beta}$ is a Pisot series. Consequently we give an other proof of Scheiche theorem about finiteness property in $\mathbb{F}_q((x^{-1}))$. Finally we show that if the base ${\beta}$ is a Pisot series, then there is a bound of the length of the fractional part of ${\beta}$-expansion of any polynomial P in $\mathbb{F}_q[x]$.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.591-599
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• 2009

For given ${\alpha},{\omega}\;{\in}\;{\mathbb{R}}$ and ${\beta}$ > 1, let $T_{{\beta},{\alpha},{\omega}}$ be the skew-product transformation on the torus, [0, 1) ${\times}$ [0, 1) defined by (x, y) ${\longmapsto}\;({\beta}x,y+{\alpha}x+{\omega})$ (mod 1). In this paper, we give a criterion of ergodicity and weakly mixing for the transformation $T_{{\beta},{\alpha},{\omega}}$ when the natural extension of the given ${\beta}$-transformation can be viewed as a generalized baker's transformation, i.e., they flatten and stretch and then cut and stack a two-dimensional domain. This is a generalization of theorems in [10].

• Honam Mathematical Journal
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• v.43 no.2
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• pp.259-267
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• 2021

Given 𝛽 > 1, we consider real numbers whose 𝛽-expansions are Sturmian words. When the slope of Sturmian words varies, their behaviors have been well studied from analytical point of view. The regularity enables us to find the Fourier series expansion, while the singularity at rational slopes yields a new kind of trigonometric series representing 𝜋.

• Journal of the Korean Society of Safety
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• v.12 no.4
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• pp.27-38
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• 1997

The effect of geometry factors on the combined mode stress intensity factor behaviors of a slant crack in a non-uniform thickness material was analysed by 2-dimensional theoretical analysis. The analysis is based on the Laurent's series expansions of complex potentials where the complex coefficients of the series are determined from the compatibility and the equilibrium conditions of the thickness interface and the stress free conditions of the crack surface. In numerical calculations the perturbation technique is employed. The expressions for the crack tip stress intensity factor are given in the form of power series of dimensionless crack length $\lamda$, and the function of crack slant angle $\alpha$ and thickness ratio $\beta$. The results of numerical calculations for each problems are represented as the correction factors F($\lamda$, $\alpha$, $\beta$). The results clearly show the following characteristics : The correction factors of the combined mode stress intensity factors for a non-uniform thickness material can be defined in the form of F($\lamda$, $\alpha$, $\beta$). The stress intensity factor values for a given crack length are decreased with increase of thickness ratio $\beta$.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.485-506
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• 2023

This paper investigates an extended form Hurwitz-Lerch zeta function, as well as related integral images, ordinary and fractional derivatives, and series expansions, using the term extended beta function. We establish a connection between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. Furthermore, we present a probability distribution application of the extended Hurwitz-Lerch zeta function ζ^𝛿,𝜇_𝜈,λ. Several results, both known and new, are shown to follow as special cases of our findings.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.637-647
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• 2015

For generalized continued fraction (GCF) with parameter ${\epsilon}(k)$, we consider the size of the set whose partial quotients increase rapidly, namely the set $$E_{\epsilon}({\alpha}):=\{x{\in}(0,1]:k_{n+1}(x){\geq}k_n(x)^{\alpha}\;for\;all\;n{\geq}1\}$$, where ${\alpha}$ > 1. We in [6] have obtained the Hausdorff dimension of $E_{\epsilon}({\alpha})$ when ${\epsilon}(k)$ is constant or ${\epsilon}(k){\sim}k^{\beta}$ for any ${\beta}{\geq}1$. As its supplement, now we show that: $$dim_H\;E_{\epsilon}({\alpha})=\{\frac{1}{\alpha},\;when\;-k^{\delta}{\leq}{\epsilon}(k){\leq}k\;with\;0{\leq}{\delta}&lt;1;\\\;\frac{1}{{\alpha}+1},\;when\;-k-{\rho}&lt;{\epsilon}(k){\leq}-k\;with\;0&lt;{\rho}&lt;1;\\\;\frac{1}{{\alpha}+2},\;when\;{\epsilon}(k)=-k-1+\frac{1}{k}$$. So the bigger the parameter function ${\epsilon}(k_n)$ is, the larger the size of $E_{\epsilon}({\alpha})$ becomes.

• Journal of Korean Society for Quality Management
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• v.11 no.1
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• pp.10-17
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• 1983

This paper studies item-life test plans with the specified item mean life $T_1$ (MTBF) - Producer's risk ${\alpha}$ and item mean life $T_2$ (MTBF, $T_2$ < $T_1$) - Consumer's risk ${\beta}$ when the probability of item survival follows the Weibull distribution (known shape parameter) as a expansion of [1]. And Operating Characteristic Curves and Average Life-testing Times of item-life test plans are computed for this paper and [1]. Cost analysis procedures are same as [1]. These results are computed by using computer program written in Level II Basic for Apple II Plus Micro-computer. Both this paper and [6] reduce the life-testing time for Weibull distribution in comparision with Exponential distribution, but results of [6] were computed for different criterions from this paper.

• Membrane Journal
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• v.33 no.1
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• pp.13-22
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• 2023

Despite high capacity of lithium metal anode, its uncontrollable dendrite growth results in the poor electrochemical (EC) performance (low Coulomb efficiency and limited cycle stability) and unsafe operation. In this study, we demonstrated a lithium metal anode protected with BaTiO₃/PVDF based piezoelectric layer to enhance its EC performance by utilizing the locally polarized lithium metal after volume expansions. As-formed lithium metal electrode deposited with BTO@PVDF layer exhibited an enhanced Coulombic efficiency (> 98% for 100 cycles) and facilitated lithium ion diffusions (lithium diffusion coefficient: D_Li+), revealing the effectiveness of piezoelectric layer deposited lithium metal electrode approach.