• Journal of the Korean Mathematical Society
• /
• v.45 no.6
• /
• pp.1613-1622
• /
• 2008

Let G be a graph with vertex set V(G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V(G) such that $g(x)\;{\leq}\;f(x)$ for every vertex x of V(G). We use $d_G(x)$ to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that $g(x)\;{\leq}\;d_F(x)\;{\leq}\;f(x)$ for every vertex x of V(F). In particular, G is called a (g, f)-graph if G itself is a (g, f)-factor. A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {$F_1$, $F_2$, ..., $F_m$} be a factorization of G and H be a subgraph of G with mr edges. If $F_i$, $1\;{\leq}\;i\;{\leq}\;m$, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {$A_1$, $A_2$, ..., $A_m$} of E(H) with $|A_i|=r$ there is a (g, f)-factorization F = {$F_1$, $F_2$, ..., $F_m$} of G such that $A_i\;{\subseteq}E(F_i)$, $1\;{\leq}\;i\;{\leq}\;m$, then we say that G has (g, f)-factorizations randomly r-orthogonal to H. In this paper it is proved that every (0, mf - (m - 1)r)-graph has (0, f)-factorizations randomly r-orthogonal to any given subgraph with mr edges if $f(x)\;{\geq}\;3r\;-\;1$ for any $x\;{\in}\;V(G)$.

• Kyungpook Mathematical Journal
• /
• v.58 no.4
• /
• pp.747-759
• /
• 2018

A graph G = (V (G), E(G) of order n = |V (G)| and size m = |E(G)| is said to be odd harmonious if there exists an injection $f:V(G){\rightarrow}\{0,\;1,\;2,\;{\ldots},\;2m-1\}$ such that the induced function $f^*:E(G){\rightarrow}\{1,\;3,\;5,\;{\ldots},\;2m-1\}$ defined by $f^*(uv)=f(u)+f(v)$ is bijection. While a bipartite graph G with partite sets A and B is said to be bigraceful if there exist a pair of injective functions $f_A:A{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ and $f_B:B{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ such that the induced labeling on the edges $f_{E(G)}:E(G){\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ defined by $f_{E(G)}(uv)=f_A(u)-f_B(v)$ (with respect to the ordered partition (A, B)), is also injective. In this paper we prove that odd harmonious graphs and bigraceful graphs are equivalent. We also prove that the number of distinct odd harmonious labeled graphs on m edges is m! and the number of distinct strongly odd harmonious labeled graphs on m edges is [m/2]![m/2]!. We prove that the Cartesian product of strongly odd harmonious trees is strongly odd harmonious. We find some new disconnected odd harmonious graphs.

• Journal of the Korean Mathematical Society
• /
• v.45 no.3
• /
• pp.597-609
• /
• 2008

In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space Hand G, a group and let ${\alpha}$ : G${\rightarrow}$ AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product $\mathbb{M}=M{\times}{\alpha}$ G of M and G with respect to ${\alpha}$ is a von Neumann algebra acting on $H{\bigotimes}{\iota}^2(G)$, generated by M and $(u_g)_g{\in}G$, where $u_g$ is the unitary representation of g on ${\iota}^2(G)$. We show that $M{\times}{\alpha}(G_1\;*\;G_2)=(M\;{\times}{\alpha}\;G_1)\;*_M\;(M\;{\times}{\alpha}\;G_2)$. We compute moments and cumulants of operators in $\mathbb{M}$. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if $F_N$ is the free group with N-generators, then the crossed product algebra $L_M(F_n){\equiv}M\;{\times}{\alpha}\;F_n$ satisfies that $$L_M(F_n)=L_M(F_{{\kappa}1})\;*_M\;L_M(F_{{\kappa}2})$$, whenerver $n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}$.

• Communications of the Korean Mathematical Society
• /
• v.10 no.3
• /
• pp.681-688
• /
• 1995

Fixed-point theory has an extension to coincidences. For a pair of maps $f,g:X_1 \to X_2$, a coincidence of f and g is a point $x \in X_1$ such that $f(x) = g(x)$, and $Coin(f,g) = {x \in X_1 $\mid$ f(x) = g(x)}$ is the coincidence set of f and g. The Nielsen coincidence number N(f,g) and the Lefschetz coincidence number L(f,g) are used to estimate the cardinality of Coin(f,g). The aspherical manifolds whose fundamental group has a normal solvable subgroup of finite index is called infrasolvmanifolds. We show that if $M_1,M_2$ are compact connected orientable infrasolvmanifolds, then $N(f,g) \geq $\mid$L(f,g)$\mid$$ for every $f,g : M_1 \to M_2$.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.6
• /
• pp.1079-1089
• /
• 2009

In this paper, we shall show that for any entire function f, the function of the form $f^m(f^n$ - 1)f' has no non-zero finite Picard value for all positive integers m, n ${\in}\;{\mathbb{N}}$ possibly except for the special case m = n = 1. Furthermore, we shall also show that for any two nonconstant meromorphic functions f and g, if $f^m(f^n$-1)f' and $g^m(g^n$-1)g' share the value 1 weakly, then f $\equiv$ g provided that m and n satisfy some conditions. In particular, if f and g are entire, then the restrictions on m and n could be greatly reduced.

• Communications of the Korean Mathematical Society
• /
• v.10 no.1
• /
• pp.235-249
• /
• 1995

Let K be a bounded linear operator from Hilbert space $H_1$ into Hilbert space $H_2$. When numerically solving the first kind equation Kf = g, one usually picks n orthonormal functions $\phi_1, \phi_2,...,\phi_n$ in $H_1$ which depend on the numerical method and on the problem, see Varah [12] for more details. Then one findes the unique minimum norm element $f_M \in M$ that satisfies $\Vert K f_M - g \Vert = inf {\Vert K f - g \Vert : f \in M}$, where M is the linear span of $\phi_1, \phi_2,...,\phi_n$. Such a solution $f_M \in M$ is called the M-solution of K f = g. Some methods for finding the M-solution of K f = g were proposed by Banks [2] and Marti [9,10]. See [5,6,8] for convergence results comparing the M-solution of K f = g with $f_0$, the least squares solution of minimum norm (LSSMN) of K f = g.

• Journal of KIISE:Computer Systems and Theory
• /
• v.31 no.1_2
• /
• pp.86-94
• /
• 2004

In this paper, we show that $ G(2^m, 4), m{\geq}3$with at most m-2 faulty elements has a fault-free cycle of length 1 for every ${\leq}1{\leq}2^m-f_v$ is the number of faulty vertices. To achieve our purpose, we define a graph G to be k-fault hypohamiltonian-connected if for any set F of faulty elements, G- F has a fault-free path joining every pair of fault-free vertices whose length is shorter than a hamiltonian path by one, and then show that$ G(2^m, 4), m{\geq}3$ is m-3-fault hypohamiltonian-connected.

• Journal of the Institute of Electronics Engineers of Korea SD
• /
• v.37 no.3
• /
• pp.20-26
• /
• 2000

In this work, the degradation of g$_{m}$ , f$_{T}$ and f$_{max}$ of RF-CMOS devices have been characterized at elevated temperature. Since MOS transistors in RF applications are usually in saturation region, a simple empirical model for temperature dependence of g$_{m}$ at any measurement bias has been suggested. Because f$_{T}$ and f$_{max}$ of CMOS devices are proportional to g$_{m}$, the temperature dependence of f$_{T}$ and f$_{max}$ could be obtained from the temperature dependence of g$_{m}$. It was found that the degradation of f$_{T}$ and f$_{max}$ at elevated temperature was due to the degradation of g$_{m}$. From the correlation between DC and RF performances of CMOS devices, we can predict the enhanced f$_{T}$ and f$_{max}$ performances at low temperature.

• Journal of Korean Society of Water and Wastewater
• /
• v.26 no.6
• /
• pp.825-831
• /
• 2012

The retrofitting of a thickening unit process is widely considered in municipal wastewater treatment plants in Korea to enhance the anaerobic digestion efficiency. The authors examined the effect of feed concentration (2-34.1 g VS/L) and feed to microorganism (F/M) ratio (0.50-1.35 g VS/g VS) on anaerobic batch digestion of sewage sludge. Methane yield over 90 mL $CH_4/g$ $VS_{feed}$ was found at a feed concentration in the range of 12-26 g VS/L and a F/M ratio below 0.6 g VS/g VS. A high F/M ratio decreased methane yield and rate with oragnic acid accumulation. As sudden increase of sewage sludge concentration prior to anaerobic digestion would jeopardize the digester performance due to the rasied F/M ratio, gradual increase of the sludge feed concentration or an additional biomass retention in the digester is recommended.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.2165-2182
• /
• 2017

Let G be a group and $\mathbb{C}$ the field of complex numbers. Suppose ${\sigma}:G{\rightarrow}G$ is an endomorphism satisfying ${{\sigma}}({{\sigma}}(x))=x$ for all x in G. In this paper, we first determine the central solution, f : G or $G{\times}G{\rightarrow}\mathbb{C}$, of the functional equation $f(xy)+f({\sigma}(y)x)=2f(x)+2f(y)$ for all $x,y{\in}G$, which is a variant of the quadratic functional equation. Using the central solution of this functional equation, we determine the general solution of the functional equation f(pr, qs) + f(sp, rq) = 2f(p, q) + 2f(r, s) for all $p,q,r,s{\in}G$, which is a variant of the equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) studied by Chung, Kannappan, Ng and Sahoo in [3] (see also [16]). Finally, we determine the solutions of this equation on the free groups generated by one element, the cyclic groups of order m, the symmetric groups of order m, and the dihedral groups of order 2m for $m{\geq}2$.