Article ID 0000 February 2019
General Editorial on Publication Ethics
Article ID 0001 February 2019 Research Article
Some bounds for commuting probability of finite rings
DHIREN KUMAR BASNET JUTIREKHA DUTTA
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 $R$.
Article ID 0002 February 2019 Research Article
A simple expression for the remainder of divisor problem
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.
Article ID 0003 February 2019 Research Article
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.
Article ID 0004 February 2019 Research Article
Blocking sets of tangent and external lines to a hyperbolic quadric in $PG(3, q), q$ even
BINOD KUMAR SAHOO BIKRAMADITYA SAHU
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 $\mathbb{E} \cup \mathbb{T}$.
Article ID 0005 February 2019 Research Article
Gromov hyperbolicity in lexicographic product graphs
WALTER CARBALLOSA AMAURIS DE LA CRUZ JOSÉ M RODRÍGUEZ
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.
Article ID 0006 February 2019 Research Article
Solution of a general pexiderized permanental functional equation
VICHIAN LAOHAKOSOL WUTTICHAI SURIYACHAROEN
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.
Article ID 0007 February 2019 Research Article
Maximizing distance between center, centroid and subtree core of trees
DHEER NOAL SUNIL DESAI KAMAL LOCHAN PATRA
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 $2^{g0} + g0$ > $n − 1$.
Article ID 0008 February 2019 Research Article
On $n$-th class preserving automorphisms of $n$-isoclinism family
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 $G$.
Article ID 0009 February 2019 Research Article
Augmentation quotients for Burnside rings of some finite $p$-groups
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
Article ID 0010 February 2019 Research Article
Monomial ideals induced by permutations avoiding patterns
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 $\{l, l + 1\}$.
Article ID 0011 February 2019 Research Article
A criterion for quasinormality in $\mathbb{C^{n}}$
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.
Article ID 0012 February 2019 Research Article
Nambu structures and associated bialgebroids
SAMIK BASU SOMNATH BASU APURBA DAS GOUTAM MUKHERJEE
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.
Article ID 0013 February 2019 Research Article
On contraction of vertices of the circuits in coset diagrams for $PSL(2,\mathbb{Z})$
QAISER MUSHTAQ ABDUL RAZAQ AWAIS YOUSAF
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.
Article ID 0014 February 2019 Research Article
A note on the high power diophantine equations
MEHDI BAGHALAGHDAM FARZALI IZADI
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 $\rm{DE} x^{a} + y^{b} = z^{c}$.
Volume 130, 2020
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