• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.349-355
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• 2016

For any integer $n{\geq}2$, each palindrome of n induces a circulant graph of order n. It is known that for each integer $n{\geq}2$, there is a one-to-one correspondence between the set of (resp. aperiodic) palindromes of n and the set of (resp. connected) circulant graphs of order n (cf. [2]). This bijection gives a one-to-one correspondence of the palindromes ${\sigma}$ with $gcd({\sigma})=1$ to the connected circulant graphs. It was also shown that the number of palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ is the same number of aperiodic palindromes of n. Let $a_n$ (resp. $b_n$) be the number of aperiodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). Let $c_n$ (resp. $d_n$) be the number of periodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). In this paper, we calculate the numbers $a_n$, $b_n$, $c_n$, $d_n$ in two ways. In Theorem 2.3, we $n_d$ recurrence relations for $a_n$, $b_n$, $c_n$, $d_n$ in terms of $a_d$ for $d{\mid}n$ and $d{\neq}n$. Afterwards, we nd formulae for $a_n$, $b_n$, $c_n$, $d_n$ explicitly in Theorem 2.5.

• Proceedings of the Korean Statistical Society Conference
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• 2002.05a
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• pp.49-54
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• 2002

For arbitrary random variables {$X_{n},n{\geq}1$}, the order of growth of the series. $S_{n}\;=\;{\sum}_{j=1}^n\;X_{j}$ is studied in this paper. More specifically, when the series S_{n}$ diverges almost surely, the strong law of large numbers $S_{n}/g_{n}^{-1}$($A_{n}{\psi}(A_{n}))\;{\rightarrow}\;0$ a.s. is constructed by extending the results of Petrov (1973). On the other hand, if the series $S_{n}$ converges almost surely to a random variable S, then the tail series $T_{n}\;=\;S\;-\;S_{n-1}\;=\;{\sum}_{j=n}^{\infty}\;X_{j}$ is a well-defined sequence of random variables and converges to 0 almost surely. For the almost surely convergent series $S_{n}$, a tail series strong law of large numbers $T_{n}/g_{n}^{-1}(B_{n}{\psi}^{\ast}(B_{n}^{-1}))\;{\rightarrow}\;0$ a.s., which generalizes the result of Klesov (1984), is also established by investigating the duality between the limiting behavior of partial sums and that of tail series. In particular, an example is provided showing that the current work can prevail despite the fact that previous tail series strong law of large numbers does not work.

• Korean Journal of Microbiology
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• v.35 no.1
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• pp.35-40
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• 1999

Effects of different molecular size fractions (100,000 nMW-0.1 $\mu\textrm{m}$m, 10,000 nMW-100,000 nMW and 1,000 nMW-'L0,000 nMW) of dissolved organic matter on the bacterial numbers and $\beta$-glucosidase activities in Lake Soyang were investigated. Even though the concentrations and characteristics of each fractions were different, bacterial growth curves of each fractioii were Lypical and similar. Each growth curve had highest peak of $1.1{\times}10^{7}$ cells $ml^{-1}$. But, the $\beta$-lucosidase activitics of each fraction were quite different. In high molecular weight fraction (HMW: 100,000 nMW-0.1 $\mu\textrm{m}$m), Vmax of Pglucosidase activity ranged from 550 to 1,160 nmol $1^{-1}$.$hi^{-1}$, but in low molecular weight f~action (1,000 nMW-10,000 nMW), that ranged from 1 to 14 om01 1$^[-1}$${-1}$

• Journal of the Korean Chemical Society
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• v.26 no.6
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• pp.403-406
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• 1982

Reactions of N-acyl-1,4-dimethoxy-2-naphthylamines with N,N-dimethylformamide dialkyl acetals afforded, in all cases, the same product, N,N-dimethyl-N'-2-(1,4-dimethoxy)naphthylurea. In these reactions, it was observed that N,N-dimethylformamide dialkyl acetals act as a dimethylamination agent when electron-deficient cites are present in the substrates.

• Journal of the Korean Crystal Growth and Crystal Technology
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• v.15 no.2
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• pp.86-91
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• 2005

The layered structure of organic-inorganic perovskite hybrids, $(C_nH_{2n+1}NH_3)_2CuC1_4$ (n = 6, 8, 10, 12) have synthesized. In $(C_nH_{2n+1}NH_3)_2CuC1_4$ compounds, the long-chain protonated alkylammonium ions as tilted bilayer type are inserted into perovskite-type layers of corner sharing $CuCl_6$ octahedron. Three solid phases have been characterized in the perovskite layered compound $(C_nH_{2n+1}NH_3)_2CuC1_4$ using HT-XRD and DSC. The $(C_nH_{2n+1}NH_3)_2CuC1_4$ compounds shows solid-solid phase transitions with stepwise increasing of the layer distance. Three different structures are explained by the conformational change of the long-chain protonated alkylammonium ions.

• Journal of the Korean Chemical Society
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• v.15 no.5
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• pp.228-235
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• 1971

The solubilities of n-butyl bromide in nitrobenzene and in 1,2,4-trichlorobenzene have been measured at $19^{\circ},\;25^{\circ},\;and\;40^{\circ}C$ in the presence and absence of gallium bromide. When gallium bromide does not exist in the system, the solubility of n-butyl bromide in nitrobenzene is greater than in 1,2,4-trichlorobenzene, indicating a stronger interaction of n-butyl bromide with nitrobenzene than with 1,2,4-trichlorobenzene. In the presence of gallium bromide, complex of n-butyl bromide with gallium bromide, 1:1 complex, $n-C_4H_9Br{\cdot}GaBr_3$, is formed in the solution. The instability constant K of the complex was evaluated. $n-C_4H_9Br{\cdot}GaBr_3{\rightleftharpoons}n-C_4H_9Br+\frac{1}{2}Ga_2Br_6$ The changes of enthalpy, free energy and entropy for the dissociation of the complex were also calculated.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.377-384
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• 1998

Let ${X_n,n \geq 1}$ be a sequence of stochatically dominated pairwise independent random variables. Let ${a_n, n \geq 1}$ and ${b_n, n \geq 1}$ be seqence of constants such that $a_n \neq 0$ and $0 < b_n \uparrow \infty$. A strong law large numbers of the form $\sum^{n}_{j=1}{a_j X_i//b_n \to 0$ almost surely is obtained.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.9-20
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• 2008

Let ${X_i{\mid}i{\geq}1}$ be a strictly stationary sequence of associated random variables with mean zero and let ${\sigma}^2=EX_1^2+2\sum\limits_{j=2}^\infty{EX_1}{X_j}$ with 0 < ${\sigma}^2$ < ${\infty}$. Set $S_n={\sum\limits^n_{i=1}^\{X_i}$, the precise asymptotics for ${\varepsilon}^{{\frac{2(r-p)}{2-p}}-1}\sum\limits_{n{\geq}1}n^{{\frac{r}{p}}-{\frac{1}{p}}+{\frac{1}{2}}}P({\mid}S_n{\mid}{\geq}{\varepsilon}n^{{\frac{1}{p}}})$,${\varepsilon}^2\sum\limits_{n{\geq}3}{\frac{1}{nlogn}}p({\mid}Sn{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ and ${\varepsilon}^{2{\delta}+2}\sum\limits_{n{\geq}1}{\frac{(loglogn)^{\delta}}{nlogn}}p({\mid}S_n{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ as ${\varepsilon}{\searrow}0$ are established under the suitable conditions.

• Archives of Pharmacal Research
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• v.17 no.1
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• pp.39-41
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• 1994

Various N,N'-disubstituted ureas 5 were easily prepared from the corresponding primary amide 1 by tratment with N-Bromophthalimide $(NBP)-AgOAc-RNH$_{2}$ 4 in dry N,N-dimrthylformamide (DMF). This reaction envolved the intemediate formation of isocyanate 3 from amide 1 via Hofmann rearrangement by treatment with AgOAc and NBP and nucleophilic addition of amine 4 to this isocyante 3. This method is simple enough to be applied to the synthesis of various N,N'-disubstututed ureas scale conviently.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.205-215
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• 2009

Sufficient conditions for the existence of at least one solution of the boundary value problems for higher order nonlinear difference equations $\{{{{{\Delta^n}x(i-1)=f(i,x(i),{\Delta}x(i),{\cdots},\Delta^{n-2}x(i)),i{\in}[1,T+1],\atop%20{\Delta^m}x(0)=0,m{\in}[0,n-3],}\atop%20\Delta^{n-2}x(0)=\phi(\Delta^{n-1}(0)),}\atop%20\Delta^{n-1}x(T+1)=-\psi(\Delta^{n-2}x(T+1))}\$. are established.