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      Volume 129, Issue 1

      February 2019

    • General Editorial on Publication Ethics

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    • Some bounds for commuting probability of finite rings


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      Let $R$ be a finite ring. The commuting probability of $R$ is the probability that any two randomly chosen elements of $R$ commute. In this paper, we obtain some bounds for commuting the probability of $R$.

    • A simple expression for the remainder of divisor problem


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      By elementary methods, we obtain with ease a highly simple expression for $\Delta(x)$, the remainder term of Dirichlet’s divisor problem. Incidentally, we give an elegant expression for $\int^{1}_{0} \zeta(s_{1}, \alpha)(\zeta (s_{2}, \alpha) + \zeta(s_{2}, 1 − \alpha))d\alpha$ for Re $s_{1}$, Re $s_{2}$,Re $(s_{1} + s_{2})$ < 1. Here $\zeta(s, \alpha)$ is the Hurwitz zeta function.

    • On partial sums of arithmetical functions of two variableswith absolutely convergent Ramanujan expansions


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      In this article, we derive an asymptotic formula for the sums of the form $\sum_{n_{1},n_{2}\leq N} f (n_{1}, n_{2})$ with an explicit error term, for any arithmetical function $f$ of two variables with absolutely convergent Ramanujan expansion and Ramanujan coefficients satisfying certain hypothesis.

    • Blocking sets of tangent and external lines to a hyperbolic quadric in $PG(3, q), q$ even


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      Let $\mathcal{H}$ be a fixed hyperbolic quadric in the three-dimensional projective space $PG(3, q)$, where $q$ is a power of 2. Let $\mathbb{E}$ (respectively $\mathbb{T}$) denote the set of all lines of $PG(3, q)$ which are external (respectively tangent) to $\mathcal{H}$. We characterize the minimum size blocking sets of $PG(3, q)$ with respect to each of the line sets $\mathbb{T}$ and $\mathbb{E} \cup \mathbb{T}$.

    • Gromov hyperbolicity in lexicographic product graphs


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      If $X$ is a geodesic metric space and $x_{1}, x_{2}, x_{3} \in X$, a geodesic triangle $T = \{x_{1}, x_{2}, x_{3}\}$ is the union of the three geodesics $[x_{1}x_{2}], [x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $\delta$-hyperbolic (in the Gromov sense) if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\delta(X) = inf\{\delta \geq 0 : X$ is $\delta$-hyperbolic\}. In this paper, we characterize the lexicographic product of two graphs $G_{1} \circ G_{2}$ which are hyperbolic, in terms of $G_{1}$ and $G_{2}:$ the lexicographic product graph $G_{1} \circ G_{2}$ is hyperbolic if and only if $G_{1}$ is hyperbolic, unless if $G_{1}$ is a trivial graph (the graph with a single vertex); if $G_{1}$ is trivial, then $G_{1} \circ G_{2}$ is hyperbolic if and only if $G_{2}$ is hyperbolic. In particular, we obtain the sharp inequalities $\delta(G_{1}) \leq \delta(G_{1} \circ G_{2}) \leq \delta(G_{1}) + 3/2$ if $G_{1}$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

    • Solution of a general pexiderized permanental functional equation


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      A general solution of the pexiderized functional equation$$f (ux + vy, uy − vx, zw) = g(x, y, z) h(u, v,w)$$is determined without any regularity assumptions. This equation arises from identities satisfied by the permanent of certain symmetric matrices. The solution so obtained are applied to deduce a number of existing related functional equations.

    • Maximizing distance between center, centroid and subtree core of trees


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      For $n \geq 5$ and $2 \leq g \leq n−3$, consider the tree $P_{n−g,g}$ on $n$ vertices which is obtained by adding $g$ pendant vertices to one end vertex of the path $P_{n−g}$. We call the trees $P_{n−g,g}$ as path-star trees. The subtree core of a tree $T$ is the set of all vertices $v$ of $T$ for which the number of subtrees of $T$ containing $v$ is maximum. We prove that over all trees on $n \geq 5$ vertices, the distance between the center (respectively, centroid) and the subtree core is maximized by some path-star trees. We also prove that the tree $P_{n−g0,g0}$ maximizes both the distances among all path-star trees on $n$ vertices, where $g0$ is the smallest positive integer satisfying $2^{g0} + g0$ > $n − 1$.

    • On $n$-th class preserving automorphisms of $n$-isoclinism family


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      Let $G$ be a finite group and let $M$ and $N$ be two normal subgroups of $G$. Let $\rm{Aut}^{M}_{N}(G)$ denote the group of all automorphisms of $G$ which fix $N$ element-wise and act trivially on $G/M$. Let $n$ be a positive integer. In this article, we have shown that if $G$ and $H$ are two $n$-isoclinic groups, then there exists an isomorphism from $\rm{Aut}^{\gamma_{n+1}(G)}_{Z_{n} (G)} (G)$ to $\rm{Aut}^{\gamma_{ n+1}(H)}_{Z_{n} (H)} (H)$, which maps the group of $n$-th class preserving automorphisms of $G$ to the group of $n$-th class preserving automorphisms of $H$. Also, for a nilpotent group $G$ of class $(n + 1)$, if $\gamma_{n+1}(G)$ is cyclic, then we prove that $\rm{Aut}^{\gamma_{n+1}(G)}_{Z_{n} (G)} (G)$ is isomorphic to the group of inner automorphisms of a quotient group of $G$.

    • Augmentation quotients for Burnside rings of some finite $p$-groups


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      Let $G$ be a finite group, $\Omega(G)$ be its Burnside ring and $\Delta(G)$ the augmentation ideal of $\Omega(G)$. Denote by $\Delta^{n}(G)$ and $\mathcal{Q}_{n}(G)$ the $n$-th power of $\Delta(G)$ and the $n$-th consecutive quotient group $\Delta^{n}(G)/\Delta^{n+1}(G)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for $\Delta^{n}(\mathcal{H})$ and determine the isomorphism class of $\mathcal{Q}_{n}(\mathcal{H})$ for each positive integer $n$, where $\mathcal{H} = \langle g, h| g^{p^{m}} = h^{p} = 1, h^{−1}gh = g^{p^{m−1}+1}\rangle$, $p$ is an odd prime

    • Monomial ideals induced by permutations avoiding patterns


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      Let $S$ (or $T$ ) be the set of permutations of $[n] = \{1, . . . , n\}$ avoiding123 and 132 patterns (or avoiding 123, 132 and 213 patterns). The monomial ideals $I_{S} = \langle\rm{x}^\sigma = \prod^{n}_{i=1}x^{\sigma(i)}_{i} : \sigma \in S\rangle$ and $I_{T} = \langle\rm{x}^{\sigma} : \sigma \in T \rangle$ in the polynomial ring$R = k[x_{1}, . . . , x_{n}]$ over a field $k$ have many interesting properties. The Alexander dual $I^{[n]}_{S}$ of $I_{S}$ with respect to $\bf{n} = (n, . . . , n)$ has the minimal cellular resolution supported on the order complex $\Delta(\Sigma_{n})$ of a poset $\Sigma_{n}$. The Alexander dual $I^{[n]}_{T}$ also has the minimalcellular resolution supported on the order complex $\Delta(\tilde{\Sigma}_{n})$ of a poset $\tilde{\Sigma}_{n}$. The number of standard monomials of the Artinian quotient $\frac{R}{I^{[n]}_{S}}$ is given by the number of irreducible(or indecomposable) permutations of $[n + 1]$, while the number of standard monomials of the Artinian quotient $\frac{R}{I^{[n]}_{T}}$is given by the number of permutations of $[n + 1]$ having no substring $\{l, l + 1\}$.

    • A criterion for quasinormality in $\mathbb{C^{n}}$


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      In this article, we give a Zalcman type renormalization result for thequasinormality of a family of holomorphic functions on a domain in $\mathbb{C^{n}}$ that takes values in a complete complex Hermitian manifold.

    • Nambu structures and associated bialgebroids


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      We investigate higher-order generalizations of well known results for Liealgebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu–Poisson manifold (of order $n$ > 2) gives rise to a special bialgebroid structure which is referred to as a weak Lie–Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu–Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie–Filippov bialgebroid over a point.

    • On contraction of vertices of the circuits in coset diagrams for $PSL(2,\mathbb{Z})$


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      Coset diagrams for the action of $PSL(2,\mathbb{Z})$ on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field $F_{q}$ , denoted by $PL(F_{q})$, vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for $PL(F_{q})$ is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by $(xy)^{m_{1}} (xy^{−1)m_{2}}$, that is, $(m_{1},m_{2})$. Let $\alpha$ be the homomorphic image of $(m_{1},m_{2})$ obtained by contracting a pair of vertices $v$, $u$ of $(m_{1},m_{2})$. If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from $\alpha$. In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of $(m_{1},m_{2})$?We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.

    • A note on the high power diophantine equations


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      In this paper, we solve the simultaneous diophantine equations $x^{\mu}_{1} + x^{\mu}_{2} + . . . + x^{\mu}_{n} = k \cdot (y^{\mu}_{1} + y^{\mu}_{2} + . . . + y^{\mu}_{\frac{n}{k}}),\mu = 1, 3$, where $n \geq 3$ and $k \neq n$ is a divisor of $n(\frac{n}{k} \geq 2)$, and we obtain a nontrivial parametric solution for them. Furthermore, we present a method for producing another solution for the above diophantine equation (DE) for the case $\mu = 3$, when a solution is given.We work out some examples and find nontrivial parametric solutions for each case in nonzero integers. Also we prove that the other DE $\sum^{n}_{i=1} pi \cdot x^{a_{i}}_{i} = \sum^{m}_{j=1} qj \cdot y^{b_{j}}_{j}$, has parametric solution and infinitely manysolutions in nonzero integers with the condition that there is an $i$ such that $pi = 1$ and $(a_{i}, a_{1} \cdot a_{2} . . . a_{i-1} \cdot a_{i+1} . . . a_{n} \cdot b_{1} \cdot b_{2} . . .b_{m}) = 1$, or there is a $j$ such that $qj = 1$ and $(b_{j} , a_{1} · · · a_{n} \cdot b_{1} · · · b_{j−1} \cdot b_{j+1} · · · b_{m}) = 1$. Finally, we study the $\rm{DE} x^{a} + y^{b} = z^{c}$.

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