• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.487-494
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• 2016

Let p be an odd prime and c be a fixed integer with (c, p) = 1. For each integer a with $1{\leq}a{\leq}p-1$, it is clear that there exists one and only one b with $0{\leq}b{\leq}p-1$ such that $ab{\equiv}c$ mod p. Let N(c, p) denote the number of all solutions of the congruence equation $ab{\equiv}c$ mod p for $1{\leq}a$, $b{{\leq}}p-1$ in which a and $\bar{b}$ are of opposite parity, where $\bar{b}$ is defined by the congruence equation $b{\bar{b}}{\equiv}1$ mod p. The main purpose of this paper is using the mean value theorem of Dirichlet L-functions and the properties of Gauss sums to study the computational problem of one kind mean value function related to $E(c,p)=N(c,p)-{\frac{1}{2}}{\phi}(p)$, and give its an exact computational formula.

• Applied Chemistry for Engineering
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• v.21 no.2
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• pp.230-237
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• 2010

A homologous series of linear liquid crystal dimers, ${\alpha},{\omega}$-bis(4-nitroazobenzene-4'-carbonyloxy)alkanes (NATWESn, n = 2~8, 10, the number of methylene units in the spacer) have been synthesized, and the thermal behavior of the series has been investigated. All the dimers formed enantiotropic nematic phases. The nematic-isotropic transition temperatures of the dimers and their entropy variation at the phase transition showed a large odd-even effect as a function of n. This behavior was rationalized in terms of the change in the average shape of the spacer on varing the parity of the spacer. The thermal stability and degree of order in the nematic phase and the magnitude of the odd-even effect of NATWESn were very similar to those of the corresponding ether compounds, while they were significantly different from those of the monomesogenic compounds, 4-{4'-(nitrophenylazo)phenoxy}alkanoyl chlorides and the side-chain liquid-crystalline polymers, the poly[1-{4-(4'-nitrophenylazo) phenoxycarbonylalkanoyloxy}ethylene]s. The results were discussed in terms of the 'irtual trimer model'by Imrie.

• Applied Chemistry for Engineering
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• v.22 no.4
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• pp.358-366
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• 2011

A homologous series of linear liquid crystal dimers, the ${\alpha},{\omega}$-bis(4-cyano-azobenzene-4'-oxy)alkanes (CATWETn, where n, the number of methylene units in the spacer, is 2~10) were synthesized, and their thermotropic liquid crystalline phase behavior were investigated. The CATWETn with n of 3 and 6 exhibited monotropic nematic phases, whereas other derivatives showed enantiotropic nematic phases. The nematic-isotropic transition temperatures of the dimers and their entropy variation at the phase transition showed a large odd-even effect as a function of n. This phase transition behavior was rationalized in terms of the change in the average shape of the spacer on varying the parity of the spacer. The thermal stability and degree of order in the nematic phase and the magnitude of the odd-even effect of CATWETn were similar to those for the methoxy-, nitro-, and pentyl-substituted dimers, while they were significantly different from those for the monomesogenic compounds, 1-{4-(4'-cyanophenylazo)phenoxy}alkylbromides and the side-chain liquid-crystalline polymers, the poly[1-{4-(4'-cyanophenylazo)phenoxyalkyloxy}ethylene]s. The results were discussed in terms of 'virtual trimer model' by Imrie.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.24 no.1
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• pp.41-50
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• 2014

In this paper, we proposed an error detection in Gaussian normal basis multiplier over $GF(2^n)$. It is shown that by using parity prediction, error detection can be very simply constructed in hardware. The hardware overheads are only one AND gate, n+1 XOR gates, and one 1-bit register in serial multipliers, and so n AND gates, 2n-1 XOR gates in parallel multipliers. This method are detect in odd number of bit fault in C = AB.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.41-49
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• 2009

Let $\zeta$ be a fixed complex number. In this paper, we study the quantity $S(\zeta,\;n):=mas_{f{\in}{\Lambda}_n}\;|f(\zeta)|$, where ${\Lambda}_n$ is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given ${\zeta}\;{\in}\;{\mathbb{C}}$ and $n\;{\in}\;{\mathbb{N}}$, the quantity S($\zeta$, n) can be calculated. Then we compute the limit $lim_{n{\rightarrow}{\infty}}\;S(\zeta,\;n)/n$ for every ${\zeta}\;{\in}\;{\mathbb{C}}$ of modulus 1. It is equal to 1/$\pi$ if $\zeta$ is not a root of unity. If $\zeta\;=\;\exp(2{\pi}ik/d)$, where $d\;{\in}\;{\mathbb{N}}$ and k $\in$ [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin($\pi$/d)) and 1/(2d sin($\pi$/2d)) for d = 1, d even and d > 1 odd, respectively.

• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3478-3487
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• 2022

In this study, we incorporate an anisotropic scattering scheme involving spherical harmonics into the method of characteristics (MOC). The neutron transport solution in a light water reactor can be significantly improved because of the impact of an anisotropic scattering source with the MOC flat source approximation. Several problems are selected to verify the proposed scheme and investigate its effects and accuracy. The MOC anisotropic scattering source is based on the expansion of spherical harmonics with Legendre polynomial functions. The angular flux, scattering source, and cross section are expanded in terms of the surface spherical harmonics. Later, the polynomial is expanded to achieve the odd and even parity of the source components. Ultimately, the MOC angular and scalar fluxes are calculated from a combination of two sources. This paper presents various numerical examples that represent the hot and cold conditions of a reactor core with boron concentration, burnable absorbers, and control rod materials, with and without a reflector or baffle. Moreover, a small critical core problem is considered which involves significant neutron leakage at room temperature. We demonstrate that an anisotropic scattering source significantly improves solution accuracy for the small core high-leakage problem, as well as for practical large core analyses.

• Polymer(Korea)
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• v.32 no.5
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• pp.489-496
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• 2008

The thermotropic liquid crystalline behavior of a homologous series of poly[1-{4-(4' nitrophenylazo) phenoxycarbonylalkanoyloxy}ethylene]s (NAPEn, n = $2{\sim}8$,10, the number of methylene units in the spacer) have been investigated. All of the homologues formed monotropic nematic phases. The glass transition temperatures decreased with n. This is attributed to a plasticization of the backbone by the side chains. The isotropic-nematic phase transition temperatures decreased with increasing n up to 7 and showed the odd-even effect. However it became almost constant when n is more than 7. This behavior was rationalized in terms of the change in the average shape of the side chain on varing the parity of the spacer. This rationalization also accounts for the observed variation of the entropic gain for the clearing transition. The mesophase properties of NAPEn were entirely different from those reported for the polymers in which the azobenzene groups are attached to polyacrylate, polymathacrylate, and polystyrene backbones through polymethylene spacers. The results indicate that the mode of chemical linkage of the side group with the main chain plays an important role in the formation, stabilization, and type of mesophase.

• Applied Chemistry for Engineering
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• v.20 no.6
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• pp.603-611
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• 2009

Thermotropic liquid crystalline behavior of a homologous series of penta-O-4-{4'-(cyanophenylazo)phenoxy}alkyl-D-glucopyranoses(CAGETn, n = 2~10, the number of methylene units in the spacer) has been investigated. The CAGETn with n of 2 and 7 exhibited enantiotropic nematic phases whereas other derivatives showed monotropic nematic phases. This is the first report of glucose derivatives that form thermotropic nematic phases. The isotropic-nematic transition temperatures ($T_{iNS}$) of CAGETns and their entropy variation at $T_{iN}$ showed the odd-even effect as a function of n. This behavior was rationalized in terms of the change in the average shape of the side chains as the parity of the spacer is varied. This rationalization also accounts for the observed variation of nematic-crystalline phase transition temperatures ($T_{NkS}$) and associoated entropy change at $T_{Nk}$. The entropy change at $T_{iN}$ or $T_{Nk}$ reaches a mininum at n = 3, before it increases again for n = 4. This may be attributed to the difference in the arrangement of the side groups. The mesophase properties of CAGETns were entirely different from those reported for partially or fully alkylated glucopyranoses. This result suggests that the degree of substitution and chemical structure of the substituents play an important role in the formation of the mesophase structures in the liquid crystals.

• Polymer(Korea)
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• v.33 no.2
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• pp.144-152
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• 2009

The thermal and optical properties of poly {1-(cholesteryloxycarbonylalkanoyloxy) ethylene}s (PCALEn, n=2$\sim$8,10, the number of methylene units in the spacer) were investigated. All of the homologues formed monotropic cholesteric phases with left-handed helical structures. PCALEn with n=2 or 10, in constrast with PCALEn with $3{\leq}n{\leq}8$, did not display reflection colors over the full cholesteric range, suggesting that the helical twisting power of the cholesteryl group highly depends on the length of the spacer connecting the cholesteryl group to the polyethylene chain. The glass transition temperatures decreased with increasing n. The isotropic-cholesteric phase transition temperatures decreased with increasing n up to 7 and showed an odd-even effect. However it became almost constant when n is more than 7. This behavior is rationalized in terms of the change in the average shape of the side chain on varing the parity of the spacer. This rationalization also accounts for the observed variation of the entropy gain for the clearing transition. The thermal stability and degree of order in the mesophase and the temperature dependence of the optical pitch observed for PCALEn were significantly different from those reported for cellulose tri(cholesteryloxycarbonyl)alkanoates. The results were discussed in terms of the differences in the chemical structure and flexibility of main chain and the number of the mesogenic units per repeating unit.