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Bull. Korean Math. Soc. 2017; 54(6): 1873~2210
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2017-11-30
Enumeration of graphs with given weighted number of connected components
Joungmin SongAbstract : We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of $q$-partite graphs of given order, size and number of connected components.
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2017-11-30
Enumeration of optimally labelled graphs of bandwidth 2
Gab-Byung Chae, MinSeok Cheong, Sang-Mok KimAbstract : An optimally labelled graph of bandwidth 2 is an ordered pair $(G, f)$ where $G$ is a simple graph with $bw(G)=2$ and $f : V(G) \rightarrow [n]$ is a bijection such that $bw(G, f)=2$. In this paper, the number of optimally labelled graphs of bandwidth two of order $n$ is enumerated by counting linear forests.
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2017-11-30
A nonstandard finite difference method applied to a mathematical cholera model
Shu Liao, Weiming YangAbstract : In this paper, we aim to construct a nonstandard finite difference (NSFD) scheme to solve numerically a mathematical model for cholera epidemic dynamics. We first show that if the basic reproduction number is less than unity, the disease-free equilibrium (DFE) is locally asymptotically stable. Moreover, we mainly establish the global stability analysis of the DFE and endemic equilibrium by using suitable Lyapunov functionals regardless of the time step size. Finally, numerical simulations with different time step sizes and initial conditions are carried out and comparisons are made with other well-known methods to illustrate the main theoretical results.
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2017-11-30
Rings with ideal-symmetric ideals
Juncheol Han, Yang Lee, Sangwon ParkAbstract : Let $R$ be a ring with identity. An ideal $N$ of $R$ is called $ideal$-$symmetric$ (resp., $ideal$-$reversible$) if $ABC \subseteq N$ implies $ACB \subseteq N$ (resp., $AB \subseteq N$ implies $BA \subseteq N$) for any ideals $A, B, C$ in $R$. A ring $R$ is called $ideal$-$symmetric$ if zero ideal of $R$ is ideal-symmetric. Let $S(R)$ (called the $ideal$-$symmetric$ $radical$ of $R$) be the intersection of all ideal-symmetric ideals of $R$. In this paper, the following are investigated: (1) Some equivalent conditions on an ideal-symmetric ideal of a ring are obtained; (2) Ideal-symmetric property is Morita invariant; (3) For any ring $R$, we have $S(M_{n}(R)) = M_{n}(S(R))$ where $M_{n}(R)$ is the ring of all $n$ by $n$ matrices over $R$; (4) For a quasi-Baer ring $R$, $R$ is semiprime if and only if $R$ is ideal-symmetric if and only if $R$ is ideal-reversible.
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2017-11-30
Rational homology balls in 2-handlebodies
Heesang Park, Dongsoo ShinAbstract : We prove that there are rational homology balls $B_p$ smoothly embedded in the $2$-handlebodies associated to certain knots. Furthermore we show that, if we rationally blow up the $2$-handlebody along the embedded rational homology ball $B_p$, then the resulting $4$-manifold cannot be obtained just by a sequence of ordinary blow ups from the $2$-handlebody under a certain mild condition.
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2017-11-30
Ding projective dimension of Gorenstein flat modules
Junpeng WangAbstract : Let $R$ be a Ding-Chen ring. Yang \cite{Yang2012} and Zhang \cite{Zhang2015} asked whether or not every $R$-module has finite Ding projective or Ding injective dimension. In this paper, we give a new characterization of that all modul\-es have finite Ding projective and Ding injective dimension in terms of the relationship between Ding projective and Gorenstein flat modules. We also give an example to obtain negative answer to the above question.
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2017-11-30
An extension of the extended Hurwitz-Lerch Zeta Functions of Two Variables
Junesang Choi, Rakesh K. Parmar, Ram K. SaxenaAbstract : We aim to introduce a further extension of a family of the extended Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate several interesting properties of the extended function such as its integral representations which provide extensions of various earlier corresponding results of two and one variables, its summation formula, its Mellin-Barnes type contour integral representations, its computational representation and fractional derivative formulas. A multi-parameter extension of the extended Hurwitz-Lerch Zeta function of two variables is also introduced. Relevant connections of certain special cases of the main results presented here with some known identities are pointed out.
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2017-11-30
On $t$-almost Dedekind graded domains
Gyu Whan Chang, Dong Yeol OhAbstract : Let $\Gamma$ be a nonzero torsionless commutative cancellative \linebreak monoid with quotient group $\langle \Gamma \rangle$, $R = \bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a graded integral domain graded by $\Gamma$ such that $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, $H$ be the set of nonzero homogeneous elements of $R$, $C(f)$ be the ideal of $R$ generated by the homogeneous components of $f \in R$, and $N(H) = \{f \in R \mid C(f)_v = R\}$. In this paper, we introduce the notion of graded $t$-almost Dedekind domains. We then show that $R$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain and $R_H$ is a $t$-almost Dedekind domains. We also show that if $R = D[\Gamma]$ is the monoid domain of $\Gamma$ over an integral domain $D$, then $R$ is a graded $t$-almost Dedekind domain if and only if $D$ and $\Gamma$ are $t$-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if $\langle \Gamma \rangle$ satisfies the ascending chain condition on its cyclic subgroups, then $R = D[\Gamma]$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain.
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2017-11-30
Volume mean operator and differentiation results associated to root systems
Chaabane RejebAbstract : Let $R$ be a root system in $\mathbb{R}^d$ with Coxeter-Weyl group $W$ and let $k$ be a nonnegative multiplicity function on $R$. The generalized volume mean of a function $f\in L^1_{loc}(\mathbb{R}^d,m_k)$, with $m_k$ the measure given by $dm_k(x):=\omega_k(x)dx:=\prod_{\alpha\in R}|\mathop{\langle\alpha,x\rangle}|^{k(\alpha)}dx$, is defined by: $\forall\ x\in \mathbb{R}^d$, $\forall\ r>0$, $M_B^r(f)(x):=\frac{1}{m_k[B(0,r)]}\int_{\mathbb{R}^d}f(y)h_k(r,x,y)\omega_k(y)dy$, where $h_k(r,x,\cdot)$ is a compactly supported nonnegative explicit measurable function depending on $R$ and $k$. In this paper, we prove that for almost every $x\in\mathbb{R}^d$, $\lim_{r\rightarrow0}M_B^r(f)(x)=f(x)$.
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2017-11-30
Weak amenability of the Lau product of Banach algebras defined by a Banach algebra morphism
Mohammad RamezanpourAbstract : Let $A$ and $B$ be two Banach algebras and $T:B\to A$ be a bounded homomorphism, with $\|T\|\leq 1$. Recently, Dabhi, Jabbari and Haghnejad Azar (\textit{Acta Math. Sin. $($Engl. Ser.$)$} \textbf{31} (2015), no. 9, 1461--1474) obtained some results about the $n$-weak amenability of $A\times_T B$. In the present paper, we address a gap in the proof of these results and extend and improve them by discussing general necessary and sufficient conditions for $A\times_T B$ to be $n$-weakly amenable, for an integer $n\geq0$.
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