• The Mathematical Education
• /
• v.47 no.1
• /
• pp.1-10
• /
• 2008

Some of school mathematics contents that have deep mathematical meanings are regarded as obvious and their importance is frequently overlooked. We first reexamined the mathematical meaning of pi as a constant. Then we indicated the educational implications of Archimedes' calculation method of pi and finally underlined the availability of pi as a valuable research topic in school mathematics.

• Research in Mathematical Education
• /
• v.18 no.1
• /
• pp.1-29
• /
• 2014

When taught the precise definition of ${\pi}$, students may be simply asked to memorize its approximate value without developing a rigorous understanding of the underlying reason of why it is a constant. Measuring the circumferences and diameters of various circles and calculating their ratios might just represent an attempt to verify that ${\pi}$ has an approximate value of 3.14, and will not necessarily result in an adequate understanding about the constant nor formally proves that it is a constant. In this study, we aim to investigate prospective teachers' conceptual understanding of ${\pi}$, and as a constant and whether they can provide a proof of its constant property. The findings show that prospective teachers lack a holistic understanding of the constant nature of ${\pi}$, and reveal how they teach students about this property in an inappropriate approach through a proving activity. We conclude our findings with a suggestion on how to improve the situation.

• Journal of the Korean Mathematical Society
• /
• v.49 no.3
• /
• pp.659-670
• /
• 2012

For any $K$ > $2/{\pi}$ we determine the optimal constant $L(K)$ for which the following holds. If $u$, $tilde{u}$ are conjugate harmonic functions on the unit disc with $\tilde{u}(0)=0$, then $$ {\int}_{-\pi}^{\pi}{\mid}\tilde{u}(e^{i\phi}){\mid}\frac{d{\phi}}{2{\pi}}{\leq}K{\int}_{-\pi}^{\pi}{\mid}u(e^{i{\phi}}){\mid}{\log}^+{\mid}u(e^{i{\phi}}){\mid}\frac{d{\phi}}{2{\pi}}+L(K).$$ We also establish a related estimate for orthogonal harmonic functions given on Euclidean domains as well as an extension concerning orthogonal martingales under differential subordination.

• Journal of the Korean Society for Precision Engineering
• /
• v.21 no.4
• /
• pp.79-87
• /
• 2004

The light pattern reflected from a machined surface contains some information like roughness and profile on the projected surface as expected in the Beckmann-Spizzichino model. In applying the theory into a real reliable measuring device, many parameters such as incident light power, wave length, spot size should be kept a constant optical value. However, the reflected light power is likely to change with the environmental noise, the variations of the light source, the reflectivity of the surface, etc. even though the incident light power is constant. In this study, a method for adjusting the incident light power to keep the reflected light power projected on a CMOS image sensor constant was proposed and a simple adjustment algorithm based on PI digital control was examined. Experiments verified that the proposed method made the surface roughness measurement better and more reliable even under variations of the height of light source.

• Proceedings of the International Microelectronics And Packaging Society Conference
• /
• 2001.11a
• /
• pp.171-177
• /
• 2001

The adhesion enhancement from inserting a RF bias-sputtered Cr layer between Cu and polyimide (PI) has been studied. The RF bias power applied in this study was ranged from 0 to 400 W. Without the RF bias, the peel strength, which measures the adhesion strength, was nearly o g/mm. As the RF power was increased, the peel strength rose up to ~130 g/mm at 200 W, which remained constant with further increase of the RF bias power. Cross-sectional transmission electron microscopy(TEM) was used to investigate the interfacial reaction between the Cr film and PI substrate during the bias sputtering. The Cr/PI interface without the application of RF dais showed a clean, sharp interface while the RF raised Cr/PI interface had about 10~30 nm thick atomistically mixed interlayer between the metal film and PI substrate. This interlayer appeared to have resulted from the implantation of high energy adatoms during the RF bias sputtering of Cr film. This mixed layer serves as an interlocking layer, which enhances adhesion between the metal and PI layers.

• Bulletin of the Korean Chemical Society
• /
• v.1 no.4
• /
• pp.123-126
• /
• 1980

Benzene solution of trans-1,2-bispyrazylethylene and electron deficient olefin, tetracyanoethylene, as a ${\pi}$-acceptor gave 1,2-bispyrazyl-3,3,4,4-tetracyanocyclobutane, a 2${\pi}$ + 2${\pi}$ cycloaddition product, on irradiation with 350 nm UV light. Fluorescence studies revealed the reaction to proceed through a singlet exciplex. The fluorescence of trans-1,2-bispyrazylethylene was quenched very efficiently by tetracyanoethylene with the quenching constant of 1.6 ${\times}$ 10$^{10}$M$^{-1}$s$^{-1}$ while electron rich olefin, tetramethylethylene, did not quench the fluorescence of bispyrazylethylene.

• The Transactions of the Korean Institute of Electrical Engineers
• /
• v.40 no.7
• /
• pp.696-705
• /
• 1991

This paper describes the procedures for pre-tuning and re-tuning the gains of PI controller based on output patterns -output error integral- of the unknown process which may not have any information, for example, system order, deadtime, time constant, etc. The key ideas of the proposed adaptive scheme are as follows. The scheme determines the initial gains by using ZNM (Ziegler-Nichols Method) with relay feedback, and then the adaptive algorithms by pattern recognition are introduced for re-runing the PI gains with on-line scheme whenever control conditions are changed. Because, among the various auto-tuning procedures, ANM with relay feedback has the difficulty in re-tuning with on-line and Bristol method has no comment on initial settings and has variables to pre-determine, which makes the algorithm comples, the proposed methods have the combined scheme with above two procedures to recover those problems. And this paper proposes a simple way to determine adaptive constant in Bristol method. To show the validity of the proposed method, an example is illustrated by computer simulation and a laboratory process, heat exchanger, is experimented.

• Journal of the Microelectronics and Packaging Society
• /
• v.29 no.2
• /
• pp.99-105
• /
• 2022

As the chip size gets smaller, the width of the electrode line is also fine, and the density of interconnections is increasing. As a result, RC delay is becoming a problem due to the difference in resistance between the capacitor layer and the electrical conductivity layer. To solve this problem, the development of electrodes with high electrical conductivity and dielectric materials with low dielectric constant is required. In this study, we developed low dielectric ink by mixing commercial PSR which protect PCB's circuits from external factors and PI with excellent thermal property and low-k characteristics. As a result, the ink mixture of PSR and PI 10:3 showed the best results, with a dielectric constant of about 2.6 and 2.37 at 20 GHz and 28 GHz, respectively, and dielectric dissipation was measured at about 0.022 and 0.016. In order to verify the applicability of future applications, various line-width transmission lines produced on Teflon were evaluated, and as a result, the loss of transmission lines using low dielectric ink mixed with PI was 0.12 dB less on average in S21 than when only PSR was used.

• Korean Journal of Materials Research
• /
• v.5 no.8
• /
• pp.1040-1044
• /
• 1995

Magnetic garnet films of (YSmLuCa)$_3$(FeGe)$\_$5/O$\_$12/ have been grown by the liquid phase eqitaxy method on the substrate of non-magnetic garnet Gd$_3$Ga$\_$5/O$\_$12/. The variation of Sm ion concentration were varied 0.3, 0.4, 0.6, mole/formula unit respectively. The magnetic properties of the samples for the bubble magnetic materials, such as, line width ΔH of ferromagnetic resonance (FMR), magnetic saturation induction 4$\pi$Ms, wall mobility u$\_$w/ uniaxial magnetic anisotropy energy Ku, were measured and discussed the relations between these properties. The line width ΔH decreases with increasing 4$\pi$Ms, and with decreasing Sm concentration. The anisotropy energy Ku increases not only with increasing Sm ion concentration, but also increasing 4$\pi$Ms. The value of wall mobility u$\_$w/ increase with increasing 4$\pi$Ms and decreases with increasing Sm concentration. We define a physical constant Eι from the fact that the product of 4$\pi$Ms and ΔH is constant with dimension of energy density. The Eι is dependent only on Sm concentration.density. The Eι is dependent only on Sm concentration.

• Transactions of the Korean Society of Mechanical Engineers
• /
• v.18 no.4
• /
• pp.1031-1038
• /
• 1994

Characteristics of pulsatile flow and heat transfer have been studied numerically in the constant heat flux curved tube with periodic pressure gradient. As the Womersley number increases, the phase difference between the pressure gradient and the cross section averaged axial velocity becomes larger. In case of the Womersley number $\beta = 2$, when cross section averaged axial velocity reaches periodic state with time, the reverse and the natural flow coexist at phase angle, $\lambda = 1.44\pi$ and $\lambda =1.96\pi$. For all the Womersley numbers of present investigation, the time variation of wall temperature near inner wall is higher than that of near outer wall, independent of phase angle.