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      Volume 128, Issue 1

      February 2018

    • Editorial

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    • On 3-way combinatorial identities


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      In this paper, we provide combinatorial meanings to two generalized basic series with the aid of associated lattice paths. These results produce two new classes of infinite 3-way combinatorial identities. Five particular cases are also discussed. These particular cases provide new combinatorial versions of Göllnitz–Gordon identities and Göllnitz identity. Seven $q$-identities of Slater and five $q$-identities of Rogers are further explored using the same combinatorial object. These results are an extension of the work of Goyal and Agarwal (Utilitas Math. 95 (2014) 141–148), Agarwal and Rana (Utilitas Math. 79 (2009) 145–155), and Agarwal (J. Number Theory 28 (1988) 299–305).

    • No hexavalent half-arc-transitive graphs of order twice a prime square exist

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      A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. Let $p$ be a prime.Wang and Feng (Discrete Math.310 (2010) 1721–1724) proved that there exists no tetravalent half-arc-transitive graphs of order $2p^{2}$. In this paper, we extend this result to prove that no hexavalent half-arc-transitive graphs of order $2p^{2}$ exist.

    • Alternating groups as a quotient of $PSL (2,\mathbb{Z}[i])$


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      In this study, we developed an algorithm to find the homomorphisms of the Picard group $PSL(2, Z[i])$ into a finite group $G$. This algorithm is helpful to find a homomorphism (if it is possible) of the Picard group to any finite group of order less than 15! because of the limitations of the GAP and computer memory. Therefore, we obtain only five alternating groups $A_n$, where $n$ = 5, 6, 9, 13 and 14 are quotients of the Picard group. In order to extend the degree of the alternating groups, we use coset diagrams as a tool. In the end, we prove our main result with the help of three diagrams which are used as building blocks and prove that, for $n \equiv$ 1, 5, 6(mod 8), all but finitely many alternating groups $A_n$ can be obtained as quotients of the Picard group $PSL(2, Z[i])$. A code in Groups Algorithms Programming (GAP) is developed to perform the calculation.

    • Positive integer solutions of certain diophantine equations


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      In this study, the diophantine equations $x^{2} − 32B_{n}xy − 32y^{2} = \pm32^{r}, x^{4} − 32B_{n}xy − 32y^{2} = \pm32^{r}$ and $x^{2} − 32B_{n}xy − 32y^{4} = \pm32^{r}$ are considered and determined when these equations have positive integer solutions. Moreover, all positive integer solutions of these diophantine equations in terms of balancing and Lucas-balancing numbers are also found out.

    • A proof of the Anderson–Badawi rad$(I)^{n} \subseteq I$ formula for $n$-absorbing ideals


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      In [1], Anderson and Badawi conjectured that rad$(I)^{n} \subseteq I$ for every $n$ absorbing ideal $I$ of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.

    • Codismantlability and projective dimension of the Stanley–Reisner ring of special hypergraphs


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      In this paper, we generalize the concept of codismantlable graphs tohypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley–Reisner ring of special hypergraphs in some sense.

    • Frobenius splitting of projective toric bundles

      HE XIN

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      We prove that the projectivization of the tangent bundle of a nonsingulartoric variety is Frobenius split.

    • A generalization of zero divisor graphs associated to commutative rings


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      Let $R$ be a commutative ring with a nonzero identity element. For a natural number $n$, we associate a simple graph, denoted by $\Gamma^{n}_{R}$, with $R^{n}\backslash\{0\}$ as the vertex set and two distinct vertices $X$ and $Y$ in $R^{n}$ being adjacent if and only if there exists an $n\times n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that $X^{T} AY = 0$ or $Y^{T} AX = 0$, where, for a matrix $B$, $B^{T}$ is the matrix transpose of $B$. If $n = 1$, then $\Gamma^{n}_{R}$ is isomorphic to the zero divisor graph $\Gamma(R)$, and so $\Gamma^{n}_{R}$ is a generalization of $\Gamma(R)$ which is called a generalized zero divisor graph of $R$. In this paper, we study some basic properties of $\Gamma^{n}_{R}$. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

    • Some aspects of shift-like automorphisms of $\mathbb{C}^{k}$


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      The goal of this article is two fold. First, using transcendental shift-like automorphisms of $\mathbb{C}^{k} , k \geq 3$ we construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of $\mathbb{C}^{k} , k \geq 3$ whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon–Esterle in $\mathbb{C}^{2}$. The second example shows the existence of a Fatou–Bieberbach domain in $\mathbb{C}^{k} , k \geq 3$ that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay–Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.

    • Global weighted estimates for second-order nondivergence elliptic and parabolic equations


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      In this paper, we obtain the global weighted $L^{p}$ estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain $L^{p}$ -type regularity estimates for such equations.

    • Properties of singular integral operators $S_{\alpha,\beta}$


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      For $\alpha, \beta \in L^{\infty}(S^{1})$, the singular integral operator $S_{\alpha,\beta}$ on $L^{2}(S^{1})$ is defined by $S_{\alpha,\beta} f := \alpha P f + \beta \mathcal{Q} f$, where $P$ denotes the orthogonal projection of $L^{2}(S^{1})$ onto the Hardy space $H^{2}(S^{1})$, and $\mathcal{Q}$ denotes the orthogonal projection onto $H^{2}(S^{1})^\bot$ . In a recent paper, Nakazi and Yamamoto have studied the normality and self adjointness of $S_{\alpha,\beta}$ . This work has shown that $S_{\alpha,\beta}$ may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of $S_{\alpha,\beta}$.

    • Sharp Adams-type inequality invoking Hardy inequalities


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      We establish a sharp Trudinger–Moser type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of [8].

    • Rotationally symmetric extremal pseudo-Kähler metrics of non-constant scalar curvatures


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      In this paper, we explicitly construct some rotationally symmetric extremal (pseudo-)Kähler metrics of non-constant scalar curvature, which depend on some parameters, and on some line bundles over projective spaces. We also discuss the phasechange phenomenon caused by the variation of parameters.

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