• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.87-98
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• 2014

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1445-1457
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• 2016

For a positive integer N divisible by 4, let ${\mathcal{O}}^1_N({\mathbb{Q}})$ be the ring of weakly holomorphic modular functions for the congruence subgroup ${\Gamma}^1(N)$ with rational Fourier coefficients. We present explicit generators of the ring ${\mathcal{O}}^1_N({\mathbb{Q}})$ over ${\mathbb{Q}}$ in terms of both Fricke functions and Siegel functions, from which we are able to classify all Fricke families of such level N.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.1
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• pp.9-16
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• 2014

Let $\mathbb{A}=\mathbb{F}_q[T]$ be a polynomial ring over $\mathbb{F}_q$. In this paper we determine an asymptotic mean value of quadratic Dirich-let L-functions L(s, ${\chi}_{{\gamma}D}$) at the central point s=$\frac{1}{2}$, where D runs over all monic square-free polynomials of even degree in $\mathbb{A}$ and ${\gamma}$ is a generator of $\mathbb{F}_q^*$.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.721-734
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• 2014

Let M be a module over a commutative ring R, and let T(M) be its set of torsion elements. The total torsion element graph of M over R is the graph $T({\Gamma}(M))$ with vertices all elements of M, and two distinct vertices m and n are adjacent if and only if $m+n{\in}T(M)$. In this paper, we study the basic properties and possible structures of two (induced) subgraphs $Tor_0({\Gamma}(M))$ and $T_0({\Gamma}(M))$ of $T({\Gamma}(M))$, with vertices $T(M){\backslash}\{0\}$ and $M{\backslash}\{0\}$, respectively. The main purpose of this paper is to extend the definitions and some results given in [6] to a more general total torsion element graph case.

• Journal of Biomedical Engineering Research
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• v.9 no.2
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• pp.215-224
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• 1988

Block copoly(L-lactide-${\gamma}$-benzyl-L-glutamate)was synthesized from L-lactide by cationic ring opening polymerization and ${\gamma}$-benzyl-L-glutamate N-carboxy anhydride by introducing amino group terminated poly(L-lactide). L-lactide was polymerized in the presence of stannous octate at $110^{\circ}C$ and ${\gamma}$-benzyl- L-glutamate was polymerized in the presence of NaH at room temperature. The synthesized monomers and copolymers were identified by IR and NMR. The Itermal properties of the copolymers were characterized by differential scanning calorimetry and thermogravimetry. The thermal stability and melting temperature(Tm) of the block copolymers were measured and discussed. The activation energies of thermal decomposition for the block copoly(L-lactide-${\gamma}$ benzyl-L-glutamate) were evaluated from the thermogravimetric data by Freeman and Carroll method.

• Journal of Radiation Industry
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• v.5 no.3
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• pp.249-252
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• 2011

Lignin is one of the natural macromolecules. Every year large amount of lignin arises from the cellulose production as a by-product worldwide. The use of lignin as a precursor to carbonaceous materials has gained interest due to its low cost and high availability. Therefore, we improved the properties of alkali-lignin by exposing to gamma ray in this study. The alkali-lignin is irradiated by Gamma ray irradiation with varying doses. The char yields of alkali-lignin were investigated by increasing up to 50 kGy. The cross-linking and bond scission of alkali-lignin occur simultaneously during gamma ray irradiation. The crosslinking was predominantly accelerated by gamma ray irradiation up to 50 kGy. Bond scission predominantly occurs between 50 and 500 kGy. ESCA analysis indicated that the alcoholic carbon increase up to 50 kGy. Solution viscosity was increased as absorbed dose increased up to 20 kGy. In addition, the aromatic ring was not influenced by irradiation at doses ranging from 20 to 500 kGy as shown in FT-IR results.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.563-580
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• 2019

In this paper, we study the strong metric dimension of zero-divisor graph ${\Gamma}(R)$ associated to a ring R. This is done by transforming the problem into a more well-known problem of finding the vertex cover number ${\alpha}(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring ${\mathbb{Z}}_n$ of integers modulo n and the ring of Gaussian integers ${\mathbb{Z}}_n$[i] modulo n. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.111-118
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• 1996

Throughout this paper Z and C will respectively denote the ring of rational integers and the complex number field.

• Archives of Pharmacal Research
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• v.4 no.2
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• pp.133-135
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• 1981

Reaction of unsaturated acids with 3-bromo-5, 5-dimenthylhydantoin in dry DMF at room temperature gives bromolactones in 58-91 % yields.

• Proceedings of the Korean Vacuum Society Conference
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• 2011.08a
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• pp.205-205
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• 2011

The self-assembly of ${\gamma}$-phenylalanine on Au(111) at 150 K was investigated using scanning tunneling microscopy (STM). Phenylalanine can potentially form two-dimensional (2D) molecular networks through hydrogen bonding (through the carboxyl and amino groups) and ${\pi}-{\pi}$ stacking interactions (via aromatic rings). We found that ${\gamma}$-phenylalanine molecules self-assembled on Au(111) surfaces into well-ordered structures such as ring-shaped clusters (at low and intermediate coverages) and 2D molecular domains (intermediate and monolayer coverages), whereas ${\alpha}$-phenylalanine molecules formed less-ordered structure on Au(111). The self-assembly of ${\gamma}$- but not ${\alpha}$-phenylalanine may be related to the flexibility of the carboxyl and amino groups in the molecule. Moreover, as expected, the 2D molecular network of ${\gamma}$-phenylalanine on Au(111) was mediated by a combination of hydrogen bonding and ${\pi}-{\pi}$ stacking interactions.