• Volume 128, Issue 2

April 2018

• A GENERALIZATION OF TOTAL GRAPHS

Let $R$ be a commutative ring with nonzero identity, $L_{n}(R)$ be the set of all lower triangular $n \times n$ matrices, and $U$ be a triangular subset of $R^{n}$, i.e., the product of any lower triangular matrix with the transpose of any element of $U$ belongs to $U$. The graph $GT^{n}_{U}(R^{n})$ is a simple graph whose vertices consists of all elements of $R^{n}$, and two distinct vertices $(x_{1}, . . . , x_{n})$ and $(y_{1}, . . . , y_{n})$ are adjacent if and only if $(x_{1} + y_{1}, . . . , x_{n} + y_{n}) \in U$. The graph $GT^{n}_{U}(R^{n})$ is a generalization for total graphs. In this paper, we investigate the basic properties of $GT^{n}_{U}(R^{n})$. Moreover, we study the planarity of the graphs $GT^{n}_{U}(U)$, $GT^{n}_U(R^{n}\backslash U)$ and $GT^{n}_{U}(R^{n})$.

• 2-Domination number of generalized Petersen graphs

Let $G = (V, E)$ be a graph. A subset $S \subseteq V$ is a $k-dominating$ set of $G$ if each vertex in $V − S$ is adjacent to at least $k$ vertices in $S$. The $k-domination number$ of $G$ is the cardinality of the smallest $k$-dominating set of $G$. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs $P(5k + 1, 2)$ and $P(5k+2, 2)$, for $k > 0$, is $4k+2$ and $4k+3$, respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs $P(2k, k)$ and $P(5k+4, 3)$. Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs $P(5k + 1, 3)$, $P(5k + 2, 3)$ and $P(5k + 3, 3)$.

• Fourth power diophantine equations in Gaussian integers

In this paper, we examine a class of fourth power diophantine equations of the form $x^{4} + kx^{2} y^{2} + y^{4} = z^{2}$ and $ax^{4} + by^{4} = cz^{2}$, in the Gaussian integers, where $a$ and $b$ are prime integers.

• Character average of fourth power of Dirichlet L-series at unity

For a Dirichlet character modulo an integer $q \geq 3$, we use a highly simple elementary method to give an asymptotic formula for $\sum_{\chi \neq \chi 0(\rm mod q)} |L(1, \chi)|^{4}$, where $\chi 0(\rm mod q)$ is the principal character. This result seems to be new.

• On a generalization of semisimple modules

Let $R$ be a ring with identity. A module $M_{R}$ is called an $r$-semisimple module if for any right ideal $I$ of $R$, $M I$ is a direct summand of $M_{R}$ which is a generalization of semisimple and second modules. We investigate when an $r$-semisimple ring is semisimple and prove that a ring $R$ with the number of nonzero proper ideals $\leq 4$ and $J (R) = 0$ is $r$-semisimple. Moreover, we prove that $R$ is an $r$-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever $R$ is an $r$-semisimple ring.

• A note on generalized skew derivations on Lie ideals

Let $\mathcal{R}$ be a prime ring, $\mathcal{Z(R)}$ its center, $\mathcal{C}$ its extended centroid, $\mathcal{L}$ a Lie ideal of $\mathcal{R, F}$ a generalized skew derivation associated with a skew derivation $d$ and automorphism $\alpha$. Assume that there exist $t \geq 1$ and $m, n \geq 0$ fixed integers such that $\mathcal{vu = u^{m}F(uv)^{t}u^{n}}$ for all $\mathcal{u, v \in L}$. Then it is shown that either $\mathcal{L}$ is central or char$({\mathcal R}) = 2,{\mathcal R} \subseteq {\mathcal M}_{2}({\mathcal C)}$, the ring of $2 \times 2$ matrices over $\mathcal{C, L}$ is commutative and ${\mathcal u}^{2} \in \mathcal{Z(R)}$, for all $\mathcal{u \in L}$. In particular, if $\mathcal{L = [R, R]}$, then $\mathcal{R}$ is commutative.

• Positive solutions with single and multi-peak for semilinear elliptic equations with nonlinear boundary condition in the half-space

We consider the existence of single and multi-peak solutions of thefollowing nonlinear elliptic Neumann problem \begin{align*} &\left\{ \begin{array}{1} -\Delta u + \lambda^{2} u = Q(x)|u|^{p-2}u & {\rm in}\quad {\mathbb R}^N_+,\\ \frac{\partial u}{\partial n} = f(x,u) & {\rm on} \quad \partial {\mathbb R}^N_+, \end{array} \right. \end{align*} where $\lambda$ is a large number, $p \in (2, \frac{2N}{N−2})$ for $N \geq 3, f (x, u)$ is subcritical about $u$ and ${\mathcal Q}$ is positive and has some non-degenerate critical points in ${\mathbb R}^{N}_{+}$. For $\lambda$ large, we can get solutions which have peaks near the non-degenerate critical points of ${\mathcal Q}$.

• Root and critical point behaviors of certain sums of polynomials

It is known that no two roots of the polynomial equation $$\prod^{n}_{j=1}(x-rj)+\prod^{n}_{j=1}(x+rj)=0$$,

where 0 < $r_{1}\leq r_{2}\leq \cdots\leq r_{n}$, can be equal and the gaps between the roots of (1) in the upper half-plane strictly increase as one proceeds upward, and for 0 < $h$ < $r_{k}$, the roots of $$(x-r_{k}-h)\prod^{n}_{{j=1}\atop{j\neq k}}(x-r_{j})+(x+r_{k}+h)\prod^{n}_{{j=1}\atop{j\neq k}}(x+r_{j})=0$$

and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (1) and (2) are located.

• Weighted local Hardy spaces associated with operators

Let L be a self-adjoint positive operator on $L^{2}({\mathbb R}^{n})$. Assume that the semigroup $e^{−tL}$ generated by $−L$ satisfies the Gaussian kernel bounds on $L^{2}({\mathbb R}^{n})$. In this article, we study weighted local Hardy space $h^{1}_{L,w}({\mathbb R}^{n})$ associated with $L$ in terms of the area function characterization, and prove their atomic characters. Then, we introduce the weighted local BMO space $bmo_{L,w}({\mathbb R}^{n})$ and prove that the dual of $h^{1}_{L,w}({\mathbb R}^{n})$ is $bmo_{L,w}({\mathbb R}^{n})$. Finally a broad class of applications of these results is described.

• Pair frames in Hilbert $C^{\ast}$-modules

In this paper, we introduce pair frames in Hilbert $C^{\ast}$-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert $C^{\ast}$-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert $C^{\ast}$-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.

• A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracyof the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional orderVan der Pol oscillator (VPO) equation.

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