• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.33-46
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• 2023

Geodesic orbit spaces are homogeneous Finsler spaces whose geodesics are all orbits of one-parameter subgroups of isometries. Such Finsler spaces have vanishing S-curvature and hold the Bishop-Gromov volume comparison theorem. In this paper, we obtain a complete description of invariant (α₁, α₂)-metrics on spheres with vanishing S-curvature. Also, we give a description of invariant geodesic orbit (α₁, α₂)-metrics on spheres. We mainly show that a Sp(n + 1)-invariant (α₁, α2)-metric on S⁴ⁿ⁺³ = Sp(n + 1)/Sp(n) is geodesic orbit with respect to Sp(n + 1) if and only if it is Sp(n + 1)Sp(1)-invariant. As an interesting consequence, we find infinitely many Finsler spheres with vanishing S-curvature which are not geodesic orbit spaces.

• Korean Journal of Crystallography
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• v.6 no.1
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• pp.27-35
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• 1995

The crystal structure of 1-tert-butoxycarbonyl-4-[N-(tert-butoxycarbonyl)-N-(ethoxycarbonylmethyl)amino]-3-phenylsulfonylpyrrolidind [C24H36O8N2S] has been from single crystal x-ray diffraction study ; C24H36O8N2S triclinic, p1, a=11.363(8)Å, b=11.589(6)Å, c=11.013(10)Å,α=95.32(6)°,β=98.64(7)°,γ=79.57(5)°,V=1406.8(18)Å3, t=293K, Z=2, CuKα(λ=1.5418Å). The molecular structure was solved by diredt method and refined by full-matrix least squares to a final R=9.78% for 3621 unique observed [F≥4σ(F)] reflections and 703 paramenters.

• East Asian mathematical journal
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• v.24 no.3
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• pp.251-261
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• 2008

Let {${\xi}_j;j\;{\geq}\;1$} be a centered strictly stationary random sequence defined by $S_0\;=\;0$, $S_n\;=\;\Sigma^n_{j=1}\;{\xi}_j$ and $\sigma(n)\;=\;33\sqrt {ES^2_n}$ where $\sigma(t),\;t\;>\;0$, is a nondecreasing continuous regularly varying function. Suppose that there exists $n_0\;{\geq}\;1$ such that, for any $n\;{\geq}\;n_0$ and $0\;{\leq}\;{\varepsilon}\;<\;1$, there exist positive constants $c_1$ and $c_2$ such that $c_1e^{-(1+{\varepsilon})x^2/2}\;{\leq}\;P\{\frac{{\mid}S_n{\mid}}{\sigma(n)}\;{\geq}\;x\}\;{\leq}\;c_2e^{-(1-{\varepsilon})x^2/2$, $x\;{\geq}\;1$ Under some additional conditions, we investigate some limsup results for the increments of partial sum processes of the sequence {${\xi}_j;j\;{\geq}\;1$}.

• Bulletin of the Korean Chemical Society
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• v.19 no.4
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• pp.457-462
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• 1998

A chiral pentadentate bis-amide ligand, 1,9-bis(S)-pyrrolidinyl-2,5,8-triazanonane-1,9-dio ne$(S,S-prodienH_2)$ has been synthesized from the reaction of bis(2-aminoethyl)amine(dien) and S-proline, and the structure of $[Co(S,S-prodien)H_2O]ClO_4$ has be en determined by single crystal X-ray diffraction. The geometrical structure of the Co(III) complex has been an αβ -form, where the dien moiety of ligand chelates to a facial in metal center, and the aqua ligand coordinates a cis site to the secondary nitrogen of dien. The Co-N(1), Co-N(3) distances of two amide moiety in S,S-prodien are shorter than the other Co-N(2), Co-N(4), and Co-N(5) distances because of the increased basicity of nitrogen in amide. The complex crystallizes in the monoclinic space group $P2_1$(#4), with a=7.838(1), b=12.675(1), c=9.710(1) Å, β=100.39(1) and z=2. Refinement gives the final R and $R_w$ values of 0.045 and 0.057, respectively for 2130 observed reflections. Based upon the CD and X-ray data, it is identified that the absolute configuration of the αβ -$[Co(S,S-prodien)H_2O]ClO_4$ has a Λ-form.

• The Korean Journal of Mycology
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• v.17 no.2
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• pp.76-81
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• 1989

The mitotic nuclear divisions in hyphae and chromosome number in 10 strains of Fusarium oxysporum were studies with the aid of Giemsa-HCl techniques. The chromosome number of fungi was ranged from 4 to 8. Of the 10 strains (F. oxysporum f. sp. lycoperici, F. oxysporum Kangnung D2) are n=4; two (F. oxysporum Sachun3, F. oxysporum S Kohung D2) n=5; five (F. oxysporum S Kohung 3, F. oxysporum CS Hongchun D16, F. oxysporum S Bosung 5, F. oxysporum SSunchun4 and F. oxysporum S Haenam 4) n=7 and one (F. oxysporum from the Australia) are n=8. These results along with my previous papers indicate that the basic chromosome number of the F. oxysporum may be n=4 and may have been evolutionary modification within this fugal group through diploidy and aneuploidy. As additional strains are studied, the chromosome number should help to reveal steps possible phylogenetic relationship within the group as well as more clearly defining taxonomic group and variation factors.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• Journal of the Korea Society of Computer and Information
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• v.20 no.3
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• pp.69-74
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• 2015

This paper offers solutions to unresolved $43{\leq}R(5,5){\leq}49$ and $102{\leq}R(6,6){\leq}165$ problems of Ramsey's number. The Ramsey's number R(s,t) of a complete graph $k_n$ dictates that n-1 number of incidental edges of a arbitrary vertex ${\upsilon}$ is dichotomized into two colors: (n-1)/2=R and (n-1)/2=B. Therefore, if one introduces the concept of distance to the vertex ${\upsilon}$, one may construct a partite graph $K_n=K_L+{\upsilon}+K_R$, to satisfy (n-1)/2=R of {$K_L,{\upsilon}$} and (n-1)/2=B of {${\upsilon},K_R$}. Subsequently, given that $K_L$ forms the color R of $K_{s-1)$, $K_S$ is attainable. Likewise, given that $K_R$ forms the color B of $K_{t-1}$, $K_t$ is obtained. By following the above-mentioned steps, $R(s,t)=K_n$ was obtained, satisfying necessary and sufficient conditions where, for $K_L$ and $K_R$, the maximum distance should be even and incidental edges of all vertices should be equal are satisfied. This paper accordingly proves R(5,5)=43 and R(6,6)=91.

• Journal of the Korean Mathematical Society
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• v.36 no.3
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• pp.633-648
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• 1999

In this paper, for any two bipartite ordered sets P and Q, we define the convolution P * Q of P and Q. For dim(P)=s and dim(Q)=t, we prove that s+t-(U+V)-2 dim(P*Q) s+t-(U+V)+2, where U+V is the max-mn integer of the certain realizers. In particular, we also prove that dim(P)=n+k- {{{{ { n+k} over {3 } }}}} for 2 k n<2k and dim(Pn ,k)=n for n 2k, where Pn,k=Sn*Sk is the convolution of two standard ordered sets Sn and Sk.

• Bulletin of the Korean Chemical Society
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• v.30 no.7
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• pp.1535-1538
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• 2009

We present calculations for $S_N$2/E2 reactions in protic solvents (t-butyl alcohol, ethylene glycol). We focus on the role of the hydroxyl (-OH) groups in determining the $S_N$2/E2 rate constants. We predict that the ion pair E2 mechanism is more favorable than the naked ion E2 reaction in ethylene glycol. E2 barriers are calculated to be much larger (~ 9 kcal/mol) than $S_N$2 reaction barriers in protic solvents, in agreement with the experimental observation [Kim, D. W. et al. J. Am. Chem. Soc. 2006, 128, 16394] of no E2 products in the reaction of CsF in t-butyl alcohol.

• East Asian mathematical journal
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• v.24 no.4
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• pp.415-419
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• 2008

Let M be a generic submanifold of $S^n\;{\times}\;S^n$. If M admits an Sasakian structure, then M is a Brieskorn manifold.