• Smart Structures and Systems
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• v.16 no.5
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• pp.891-918
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• 2015

In many of microdevices a part of a microbeam is covered by a piezoelectric layer. Depend on the application a DC or AC voltage is applied between upper and lower side of the piezoelectric layer. A common method in many of previous works for evaluating the response of these structures is discretizing by Galerkin method. In these works often single mode shape of a uniform microbeam i.e. the microbeam without piezoelectric layer has been used as comparison function, and so the convergence of the solution has not been verified. In this paper the Galerkin method is used for discretization, and a comprehensive analysis on the convergence of solution of equation that is discretized using this comparison function is studied for both clamped-clamped and clamped-free microbeams. The static and dynamic solution resulted from Galerkin method is compared to the modal expansion solution. In addition the static solution is compared to an exact solution. It is denoted that the required numbers of uniform microbeam mode shapes for convergence of static solution due to DC voltage depends on the position and thickness of deposited piezoelectric layer. It is shown that when the clamped-clamped microbeam is coated symmetrically by piezoelectric layer, then the convergence for static solution may be obtained using only first mode. This result is valid for clamped-free case when it is covered by piezoelectric layer from left clamped side to the right. It is shown that when voltage is AC then the number of required uniform microbeam shape mode for convergence is much more than the number of required mode in modal expansion due to the dynamic effect of piezoelectric layer. This difference increases by increasing the piezoelectric thickness, the closeness of the excitation frequency to natural frequency and decreasing the damping coefficient. This condition is often indefeasible in microresonator system. It is concluded that discreitizing the equation of motion using one mode shape of uniform microbeam as comparison function in many of previous works causes considerable errors.

• 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.41 no.2
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• pp.257-267
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• 2004

In this paper, we introduce, for each approximate distribution $\~{T}$ of [a, b], the approximate Henstock-Stieltjes integral with value in Banach spaces. The Henstock integral is a special case of this where $\~{T}\;=\;\{(\tau,\;[a,\;b])\;:\;{\tau}\;{\in}\;[a,\;b]\}$. This new concept generalizes Henstock integral and abstract Perron-Stieltjes integral. We establish a uniform convergence theorem for approximate Henstock-Stieltjes integral, which is an improvement of the uniform convergence theorem for Perron-Stieltjes integral by Schwabik [3].

• Journal of applied mathematics & informatics
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• v.37 no.1_2
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• pp.123-131
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• 2019

In this paper, we introduce a uniform mesh method for a Maxwell's equation with discontinuous coefficients. We observe optimal O(h) order for the electric field and O(h) order for the curl.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.185-201
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• 2014

In this paper, we construct a numerical method to solve singularly perturbed one-dimensional parabolic convection-diffusion problems. We use Euler method with uniform step size for temporal discretization and exponential-spline scheme on spatial uniform mesh of Shishkin type for full discretization. We show that the resulting method is uniformly convergent with respect to diffusion parameter. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. The obtained numerical results show that the method is efficient, stable and reliable for solving convection-diffusion problem accurately even involving diffusion parameter.

• Journal of Internet Computing and Services
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• v.21 no.3
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• pp.21-28
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• 2020

Fireworks Algorithm (FWA) is a relatively novel swarm-based metaheuristic algorithm for global optimization. To solve the low-efficient local searching problem and convergence of the FWA, this paper presents a Variable Amplitude Coefficient Fireworks Algorithm with Uniform Local Search Operator (namely VACUFWA). Firstly, the explosive amplitude is used to adjust improving the convergence speed dynamically. Secondly, Uniform Local Search (ULS) enhances exploitation capability of the FWA. Finally, the ULS and Variable Amplitude Coefficient operator are used in the VACUFWA. The comprehensive experiment carried out on 13 benchmark functions. Its results indicate that the performance of VACUFWA is significantly improved compared with the FWA, Differential Evolution, and Particle Swarm Optimization.

• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.121-126
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• 1991

In this paper, we show that uniform integrability is equivalent to convergence to a ${\mu}$-integrable function f in $L_1$ for ${\mu}$-integrable functions in the sense of the integral defined by Lewis.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.75-93
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• 2008

The rate of convergence of the distribution of order statistics to the corresponding extreme-value distribution may be characterized by the uniform and total variation metrics. de Haan and Resnick [4] derived the convergence rate when the second order generalized regularly varying function has second order derivatives. In this paper, based on the properties of the generalized regular variation and the second order generalized variation and characterized by uniform and total variation metrics, the convergence rates of the distribution of the largest order statistic are obtained under weaker conditions.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.479-486
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• 2006

Let $Z_n(s,\;f)=n^{-1}\;{\sum}^{ns}_{i=1}(f(X_i)-Pf)$ be the sequential empirical process based on the independent and identically distributed random variables. We prove that convergence problems of $sup_{(s,\;f)}|Z_n(s,\;f)|$ to zero boil down to those of $sup_f|Z_n(1,\;f)|$. We employ Ottaviani's inequality and the complete convergence to establish, under bracketing entropy with the second moment, the almost sure convergence of $sup_{(s,\;f)}|Z_n(s,\;f)|$ to zero.

• ETRI Journal
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• v.31 no.6
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• pp.778-783
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• 2009

To fabricate uniform and reliable thin gold stripes that provide low-loss optical waveguides, we developed a novel liftoff process placing an additional $SiN_x$ layer under conventional photoresists. By patterning a photoresist and over-etching the $SiN_x$, the photoresist patterns become free-standing structures on a lower-cladding. This leads to uniform metal stripes with good reproducibility and effectively removes parasitic structures on the edge of the metal stripe in the image reversal photolithography process. By applying the newly developed process to polymer-based gold stripe waveguide fabrication, we improved the propagation losses about two times compared with that incurred by the conventional image-reversal process.