• Volume 125, Issue 4

November 2015,   pages  449-577l

• Bounds on the hop domination number of a tree

A hop dominating set of a graph 𝐺 is a set 𝐷 of vertices of 𝐺 if for every vertex of 𝑉 (𝐺) \ D, there exists $u \in D$ such that 𝑑(𝑢, 𝑣) = 2. The hop domination number of a graph 𝐺, denoted by 𝛾(𝐺), is the minimum cardinality of a hop dominating set of 𝐺. We prove that for every tree 𝑇 of order 𝑛 with 𝑙 leaves and 𝑠 support vertices we have (𝑛−𝑙−𝑠 +4)/3 ≤ 𝛾(𝐺) ≤ 𝑛/2, and characterize the trees attaining each of the bounds.

• On the order of magnitude of some arithmetical functions under digital constraint I

Let 𝑞 ≥ 2 be an integer and let 𝑆𝑞(𝑛) denote the sum of the digits in base 𝑞 of the positive integer 𝑛. We look for an estimate of the average of some multiplicative arithmetical functions under constraints on the arithmetical congruence of the integers and the sum of their digits.

• Yoneda algebras of almost Koszul algebras

Let 𝑘 be an algebraically closed field, 𝐴 a finite dimensional connected (𝑝, 𝑞)-Koszul self-injective algebra with 𝑝, 𝑞 ≥ 2. In this paper, we prove that the Yoneda algebra of 𝐴 is isomorphic to a twisted polynomial algebra $A^!$ [𝑡 ; 𝛽] in one indeterminate 𝑡 of degree 𝑞+1 in which $A^!$ is the quadratic dual of 𝐴, 𝛽 is an automorphism of $A^!$, and 𝑡𝑏 = 𝛽(𝑏)𝑡 for each $t \in A^!$. As a corollary, we recover Theorem 5.3 of [2].

• 𝑛-th Roots in finite polyhedral and centro-polyhedral groups

The probability that a randomly chosen element in a non-abelian finite group has a square root, has been investigated by certain authors in recent years. In this paper, this probability will be generalized for the 𝑛-th roots when 𝑛 ≥ 2 and it will be computed for every finite polyhedral group and all of the finite centro-polyhedral groups.

• Characterization of PGL(2, 𝑝) by its order and one conjugacy class size

Let 𝑝 be a prime. In this paper, we do not use the classification theorem of finite simple groups and prove that the projective general linear group PGL(2, 𝑝) can be uniquely determined by its order and one special conjugacy class size. Further, the validity of a conjecture of J. G. Thompson is generalized to the group PGL(2, 𝑝) by a new way.

• A new characterization of 𝐿2(𝑝) by NSE

In this paper we give a new characterization of simple group 𝐿2(𝑝) with 𝑝 a prime by both its order and 𝑛𝑠𝑒(𝐿2(𝑝)), the set of numbers of elements of 𝐿2(𝑝) with the same order.

• Regions of variability for a class of analytic and locally univalent functions defined by subordination

In this article, we consider a family $\mathcal{C}(A,B)$ of analytic and locally univalent functions on the open unit disc $\mathbb{D} = {z : |z| \lt 1}$ in the complex plane that properly contains the well-known Janowski class of convex univalent functions. In this article, we determine the exact set of variability of log$(f'(z_0))$ with fixed $z_0 \in \mathbb{D}$ and $f''(0)$ whenever $f$ varies over the class $\mathcal{C}(A,B)$.

• A three critical point theorem for non-smooth functionals with application in differential inclusions

A variety of three-critical-point theorems have been established for nonsmooth functionals, based on a minimax inequality. In this paper, a generalized form of a recent result due to Ricceri is introduced for non-smooth functionals and by a few hypotheses, without any minimax inequality, the existence of at least three critical points with a uniform bound on the norms of solutions, is obtained. Also, as an application, our main theorem is used to obtain at least three anti-periodic solutions for a second order differential inclusion.

• Existence of positive weak solutions for (𝑝, 𝑞)-Laplacian nonlinear systems

We mainly consider the existence of a positive weak solution of the following system\begin{equation*}\left\{\begin{matrix}-\Delta_p u + |u|^{p-2} u = \gamma [g (x) a(u)+ c(x) f (v)], \quad \text{ in } \Omega,\\-\Delta_q v + |v|^{q-2} v = \mu [g (x) b(v)+ c(x) h (u)], \quad \text{ in } \Omega,\\\hspace{3cm} u = v = 0, \hspace{3.8cm} \text{ on } \partial \, \Omega,\end{matrix}\right.\end{equation*}where $\Delta_p u = \text{ div}(|\nabla_u|^{p-2} \nabla_u), p, q &gt; 1$ and $\lambda, \, \mu$ are positive parameters, and $\Omega \subset R^N$ is a bounded domain with smooth boundary $\partial \Omega$ and $g, \, c$ are nonnegative and continuous functions and $f, h, a, b$ are $C^1$ nondecreasing functions satisfying $a(0), b(0) \geq 0$. We have proved the existence of a positive weak solution for $\lambda$, $\mu$ large when$$\lim\limits_{x \to \infty} \frac{f[M (h(x))^{\frac{1}{q-1}}]}{x^{p-1}} = 0$$for every $M &gt; 0$.

• Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions

In this article, we study the existence and multiplicity of non-negative solutions of the following p-fractional equation:\begin{equation*}\left\{\begin{matrix}-2 {\displaystyle\int}_{\mathbb{R}^n} \frac{|u(y) - u (x)|^{p-2} (u(y)-u(x))}{|x-y|^{n+p\alpha}} dy = \lambda h (x) |u|^{q-1} u + b (x)|u|^{r-1} u \text{ in } \Omega,\\u = 0 \quad \text{ in } \mathbb{R}^n \setminus \Omega, \quad u \in W^{\alpha,p} (\mathbb{R}^n)\end{matrix}\right.\end{equation*}where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with continuous boundary, $p \geq 2$, $n &gt; p \alpha$, $\alpha \in (0,1)$, $0 &lt; q &lt; p -1 &lt; r &lt; p^\ast - 1$ with $p^\ast = np (n -p\alpha)^{-1}$, $\lambda &gt; 0$ and $h, b$ are signchanging continuous functions. We show the existence and multiplicity of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists $\lambda_0$ such that for $\lambda \in (0, \lambda_0)$, it has at least two non-negative solutions.

• Disjoint hypercyclicity of weighted composition operators

In this paper, we discuss about disjoint hypercyclicity of weighted composition operators on some function spaces of analytic functions on a plane domain.

• On the (1, 1)-tensor bundle with Cheeger–Gromoll type metric

The main purpose of the present paper is to construct Riemannian almost product structures on the (1, 1)-tensor bundle equipped with Cheeger–Gromoll type metric over a Riemannian manifold and present some results concerning these structures.

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