• Communications of the Korean Mathematical Society
• /
• v.19 no.2
• /
• pp.321-328
• /
• 2004

In this paper, we will prove the following: Let D be a nonempty of a normed linear space X and T : D -> X be a nonexpansive mapping. Let ${x_n}$ be a sequence in D and ${t_n}$, ${s_n}$ be real sequences such that (i) $0\;{\leq}\;t_n\;{\leq}\;t\;<\;1\;and\;{\sum_{n=1}}^{\infty}\;t_n\;=\;{\infty},\;(ii)\;(a)\;0\;{\leq}\;s_n\;{\leq}\;1,\;s_n\;->\;0\;as\;n\;->\;{\infty}\;and\;{\sum_{n=1}}^{\infty}\;t_ns_n\;<\;{\infty}\;or\;(b)\;s_n\;=\;s\;for\;all\;n\;{\geq}\;1\;and\;s\;{\in}\;[0,1),\;(iii)\;x_{n+1}\;=\;(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n)\;for\;all\;n\;{\geq}\;1.$ Then, if the sequence {x_n} is bounded, then $lim_{n->\infty}\;$\mid$$\mid$x_n-Tx_n$\mid$$\mid$\;=\;0$. This result improves and complements a result of Deng [2]. Furthermore, we will show that certain conditions on D, X and T guarantee the weak and strong convergence of the Ishikawa iterative sequence to a fixed point of T.

• Bulletin of the Korean Chemical Society
• /
• v.24 no.5
• /
• pp.583-592
• /
• 2003

The gas phase identity nucleophilic substitution reactions of halide anions (X = F, Cl, Br) with cyclopentadienyl halides (1) are investigated at the B3LYP/6-311+G**, MP2/6-311+G** and G2(+)MP2 levels involving five reaction pathways: σ-attack $S_N2$, β-$S_N$2'-syn, β-$S_N$2'-anti, γ-$S_N$2'-syn and γ-$S_N$2'-anti paths. In addition, the halide exchange reactions at the saturated analogue, cyclopentyl halides (2), and the monohapto circumambulatory halide rearrangements in 1 are also studied at the same three levels of theory. In the σ-attack $S_N2$ transition state for 1 weak positive charge develops in the ring with X = F while negative charge develops with X = Cl and Br leading to a higher energy barrier with X = F but to lower energy barriers with X = Cl and Br than for the corresponding reactions of 2. The π-attack β-$S_N$2' transition states are stabilized by the strong $n_C-{\pi}^{*}_{C=C}$ charge transfer interactions, whereas the π-attack γ-$S_N$2' transition states are stabilized by the strong $n_C-{\sigma}^{*}_{C-X}$ interactions. For all types of reaction paths, the energy barriers are lower with X = F than Cl and Br due to the greater bond energy gain in the partial C-X bond formation with X = F. The β-$S_N$2' paths are favored over the γ-$S_N$2' paths only with X = F and the reverse holds with X = Cl and Br. The σ-attack $S_N2$ reaction provides the lowest energy barrier with X = Cl and Br, but that with X = F is the highest energy barrier path. Activation energies for the circumambulatory rearrangement processes are much higher (by more than 18 kcal $mol^{-1}$) than those for the corresponding $S_N2$ reaction path. Overall the gas-phase halide exchanges are predicted to proceed by the σ-attack $S_N2$ path with X = Cl and Br but by the β-$S_N$2'-anti path with X = F. The barriers to the gas-phase halide exchanges increase in the order X = F < Br < Cl, which is the same as that found for the gas-phase identity methyl transfer reactions.

• Journal of the Korean Chemical Society
• /
• v.24 no.1
• /
• pp.1-7
• /
• 1980

Solvolysis rate constants for methylchloroformate, methylthionochloroformate, methylthiolchloroformate and methyldithiochloroformate have been determined conductometrically in methanol, ethanol and ethanol-water mixtures and activation parameters have been derived. Results show that methylchloroformate solvolyzes through $S_N2$ process while methyldithiochloroformate solvolyzes by $S_N1$ process in all the solvent systems. The rate of hydrolysis decreased in the order, $CH_3S(CS)Cl>CH_3S(CO)Cl>CH_3O(CS)Cl>CH_3O(CO)Cl$ which corresponds to the order of decreasing $S_N1$ character. In methanol, $CH_3S(CS)Cl$ solvolyzed via the $S_N1$ mechanism while the others solvolyzed via the $S_N2$ process. In ethanol, however, $S_N2$ character was dominant for all the compounds, except methyldithiochloroformate, for which $S_N1$ character was still strong enough to accelerate the rate of ethanolysis. In ethanol-water mixtures, $CH_3S(CS)Cl$ and $CH_3S(CS)Cl$ solvolyzed via $S_N2$ process in ethanol-rich region while the $S_N1$ character increased greatly in water-rich region for the solvolysis of these compounds. The order of $S_N1$ character for solvolysis in water-rich region was the same as the order of hydrolysis rate.

• Honam Mathematical Journal
• /
• v.39 no.2
• /
• pp.297-305
• /
• 2017

Let ${\alpha}$ > 0 be a real number and $(s_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$. We investigate Dirichlet series of the form ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-s}$. To do this, a generalization of Euler's totient function is required. For a real ${\alpha}$ > 0 and a positive integer n, an arithmetic function ${\varphi}{\alpha}(n)$ is defined to be the number of positive integers m for which gcd(m, n) = 1 and 0 < m/n < ${\alpha}$. Under a condition Re(s) > 1, this paper establishes an identity ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-S}=1+{\sum}^{\infty}_{n=1}{\varphi}_{\alpha}(n)({\zeta}(s)-{\zeta}(s,1+n^{-1}))n^{-s}$.

• Bulletin of the Korean Chemical Society
• /
• v.22 no.4
• /
• pp.421-423
• /
• 2001

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.33 no.9C
• /
• pp.669-675
• /
• 2008

For an odd prime p, n=4k and $d=((p^2k+1)/2)^2$, there are $(p^{2k}+1)/2$ distinct decimated sequences, s(dt+1), $0{\leq}l<(p^{2k}+1)/2$, of a p-ary m-sequence, s(t) of period $p^n-1$. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+l) takes the values in $\{-1,-1{\pm}\sqrt{p^n},-1+2\sqrt{p^n}\}$ and their, cross-correlation distribution is also derived.

• Journal of the Korean Chemical Society
• /
• v.26 no.5
• /
• pp.310-319
• /
• 1982

Cobalt(III) complexes with sexidentate ligands, N,N,N',N'-tetrakis(2-amino-ethyl)-1,2-ethanediamine (ten), -1,3-propanediamine (ttn), -1,4-butanediamine (ttmd), -(R,R)-and -(R,S)-2,4-pentanediamine (tptn) were prepared, and the characterization of d-d absorption band on the variation of chelate ring size and conformation of these complexes were studied by means of electronic spectra. The first d-d absorption bands of $[Co(L)]^{3+}$ complexes are shifted to smaller wave numbers in the order. ttn > (R,R)-tptn > ten > ttmd${\simeq}$(R,S)-tptn for (L). The UV, $^{13}C$ NMR, and Circular Dichroism studies indicate that the R,S-tptn ligand of $[Co(R,S-tptn)]^{3+}$ complex coodinates to cobalt(Ⅲ) ion as a sexidentate with one methyl group in axial position.

• Journal of the Korean Mathematical Society
• /
• v.43 no.4
• /
• pp.783-801
• /
• 2006

Let A be the mod p Steenrod algebra for p an arbitrary odd prime and S the sphere spectrum localized at p. In this paper, some useful propositions about the May spectral sequence are first given, and then, two new nontrivial homotopy elements ${\alpha}_1{\jmath}{\xi}_n\;(p{\geq}5,n\;{\geq}\;3)\;and\;{\gamma}_s{\alpha}_1{\jmath}{\xi}_n\;(p\;{\geq}\;7,\;n\;{\geq}\;4)$ are detected in the stable homotopy groups of spheres, where ${\xi}_n\;{\in}\;{\pi}_{p^nq+pq-2}M$ is obtained in [2]. The new ones are of degree 2(p - 1)($p^n+p+1$) - 4 and 2(p - 1)($p^n+sp^2$ + sp + (s - 1)) - 7 and are represented up to nonzero scalar by $b_0h_0h_n,\;b_0h_0h_n\tilde{\gamma}_s\;{\neq}\;0\;{\in}\;Ext^{*,*}_A^(Z_p,\;Z_p)$ in the Adams spectral sequence respectively, where $3\;{\leq}\;s\;<\;p-2$.

• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.707-719
• /
• 2020

In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.

• Journal of the Korean Chemical Society
• /
• v.45 no.2
• /
• pp.131-137
• /
• 2001

The protonation and stability constants of complexation of Zn(II), Cd(II) and Pb(II), Hg(II) ion with sulfur-containing podand ligands such as tri(phenylthio-2-ethy1)amine (Podand N$_1$S$_3$)tetra(phenylthio-2-ethyl)ethylenediamine (Podand N$_2$S$_4$) , tris(1-benzylaminoethyl)amine (Podand N$_4$) have been determined by potentiometric titration in 95% methanol at variable temperatures, From these studies, we observed that podand N$_4$ ligand showed the largest protonation constant. Protonation constant, stability constant, enthalpy, entropy of Zn(II), Cd(II) and Pb(II), Hg(II),ions increased with the following order podand N$_1$S$_3$$_2$S$_4$$_4$.