• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.467-480
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• 2009

In 1859, Bernhard Riemann, in his epoch-making memoir, extended the Euler zeta function $\zeta$(s) (s > 1; $s{\in}\mathbb{R}$) to the Riemann zeta function $\zeta$(s) ($\Re$(s) > 1; $s{\in}\mathbb{C}$) to investigate the pattern of the primes. Sine the time of Euler and then Riemann, the Riemann zeta function $\zeta$(s) has involved and appeared in a variety of mathematical research subjects as well as the function itself has been being broadly and deeply researched. Among those things, we choose to make a further investigation of the following subjects: Evaluation of $\zeta$(2k) ($k {\in}\mathbb{N}$); Approximate functional equations for $\zeta$(s); Series involving the Riemann zeta function.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.549-559
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• 2018

In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.545-555
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• 2004

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a proof of the classical Euler sum by following Lewin's method. We also consider some related formulas involving Euler sums, which are evaluatable from some known definite integrals.

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1163-1184
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• 2007

In this presentation dedicated to the tricentennial birth anniversary of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707-1783), we begin by remarking about the so-called Basler problem of evaluating the Zeta function ${\zeta}(s)$ [in the much later notation of Georg Friedrich Bernhard Riemann (1826-1866)] when s=2, which was then of vital importance to Euler and to many other contemporary mathematicians including especially the Bernoulli brothers [Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)], and for which a fascinatingly large number of seemingly independent solutions have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. We then investigate various recent developments on the evaluations and representations of ${\zeta}(s)$ when $s{\in}{\mathbb{N}}{\backslash}\;[1],\;{\mathbb{N}}$ being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for ${\zeta}(2n+1)(n{\in}{\mathbb{N}})$ which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that ${\zeta}(3)$ can be represented by means of series which converge much more rapidly than that in Euler's celebrated formula as well as the series used recently by Roger $Ap\'{e}ry$ (1916-1994) in his proof of the irrationality of ${\zeta}(3)$. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.

• Atmosphere
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• v.23 no.4
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• pp.401-412
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• 2013

East Asia experienced extremely cold weather in January 2011, while the previous December and the following February had normal winter temperature. In this study National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data are used to investigate the characteristic features observed in the meteorological fields such as temperature, sea-level pressure, geopotential height, and wind during this winter period. In January the planetary-wave pattern is dominated by stationary-wave form in the mid-to-high latitude region, while transient waves are significant in the previous month. To understand the planetary-wave features quantitatively, harmonic analyses have been done for the 500-hPa geopotential height field. In the climatological-mean geopotential heights the wave numbers 1, 2, and 3 are dominant during the whole winter. In January 2011 the waves of number 1, 2, and 3 are dominant and stationary as in the climatological-mean field. In December 2010 and February 2011, however, the waves of number 4, 5, and 6 play a major role and show a transient pattern. In addition to the distinctive features in each month the planetary-wave patterns dependent on the latitude are also discussed.

• Journal of Power Electronics
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• v.17 no.5
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• pp.1244-1255
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• 2017

The space vector pulse width modulation (SVPWM) technique is a popular PWM method for medium voltage drive applications. Conventional SVPWM (CSVPWM) and bus clamped SVPWM (BCSVPWM) are the most common SVPWM techniques. This paper evaluates the performance of various advanced BCSVPWM strategies in terms of their harmonic distortion and switching loss based on a uniform frame work. A uniform frame work, pulse number captures the performance parameter variations of different SVPWM strategies for various number of samples with heterogeneous pulse numbers. This work compares different advanced BCSVPWM techniques based on the modulation index and location of the clamping position (zero vector changing angle ) of a phase in a line cycle. The frame work provides a fixed fundamental frequency of 50Hz. The different BCSVPWM switching strategies are implemented and compared experimentally on a 415V, 2.2kW, 50Hz, 3-phase induction motor drive which is fed from an IGBT based 2 KVA voltage source inverter (VSI) with a DC bus voltage of 400 V. A low cost PIC microcontroller (PIC18F452) is used as the controller platform.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.58 no.3
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• pp.509-515
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• 2009

When a squirrel cage induction motor is loaded, the magnetic noise can increase depending on the load current. It is due to the variation of air gap harmonic fluxes from the rotor current induced by loading. This unfavorable noise can be anticipated by analysing the radial force waves in the air gap, the mode shapes of them, and stator core natural frequencies at each mode. With the experimental tests with the different rotor slot numbers, the variation of magnetic noise depending on the load current is studied and the method to reduce the magnetic noise is suggested with the newly developed magnetic noise analysis program.

• Journal of the Korean Institute of Telematics and Electronics
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• v.26 no.11
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• pp.1819-1828
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• 1989

In this paper, we determined the modulation index for minimization of speed ripple and position error using maximum speed ripple (SRF) and maximum position error(PEF) in current source TPWM inverter. Through computer simulation, we compared with total current harmonic distortion, SRF and PEF for square wave modulation method and TPWM method. As a result, it turns out that square wave modulation method is superior to TPWM method of 3 pulses per half cycle in speed ripple and position error contents. And TPWM is better than square wave method when pulse number is more than 5. Also, in these pulse numbers, moduladtion index of minimum speed ripple and munimum position error is 0.91.

• Proceedings of the KIEE Conference
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• 2008.10c
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• pp.9-11
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• 2008

When a squirrel cage induction motor is loaded, the electromagnetic noise can increase depending on the load current. It is due to the air gap harmonic fluxes from the rotor current induced during loading. This unfavorable noise can be anticipated by calculating the radial force waves in the air gap, vibration mode shapes of them, and stator core natural frequencies of each mode. With the experimental tests with the different rotor slot numbers, the variation of electromagnetic noise is studied between on-loaded and loaded running.

• Structural Engineering and Mechanics
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• v.5 no.6
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• pp.715-728
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• 1997

The aim of this paper is to investigate the secondary buckling behaviour and mode-coupling of spherical caps under uniformly external pressure. The analysis makes use of a rotational finite shell element on the basis of strain-displacement relations according to Koiter's shell theory (Small Finite Deflections). The post-buckling behaviours after a bifurcation point are analyzed precisely by considering multi-mode coupling between several higher order harmonic wave numbers: and on the way of post-buckling path the positive definiteness of incremental stiffness matrix of uncoupled modes is examined step by step. The secondary buckling point that has zero eigen-value of incremental stiffness matrix and the corresponding secondary mode are obtained, moreover, the secondary post-buckling path is traced.