The basis number of a graph $G$ is defined to be the least integer $d$ such that there is a basis $ℬ$ of the cycle space of $G$ such that each edge of $G$ is contained in at most $d$ members of $ℬ$. We investigate the basis number of the lexicographic product of two circular ladders, two Möbius ladders, a circular ladder and a Möbius ladder, a Möbius ladder and a circular ladder, a ladder and a circular ladder, a circular ladder and a ladder, a Möbius ladder and a ladder, a ladder and a Möbius ladder, and two ladders.

Let $G$ be a graph of order $n$. A graph is linear forest if every component is a path. Let $S$ be a set of $m$ edges of $G$ that induces a linear forest. An edge $xy\in E(\text{G})$ is called an $S$-edge if $xy\in S$. An $S$-edge-length of a cycle in $G$ is defined as the number of $S$-edges that it contains. We prove that if the degree sum in $G$ of every pair of nonadjacent vertices of $G$ is at least $n+m$, then $G$ contains hamiltonian cycles of every $S$-edge-length between $0$ and $\left|S\right|$.

In this paper, we give an extension of the orthogonality results to dominant operators and $p$-hyponormal or log-hyponormal operators, also we will generalize some commutativity results.

We prove the $L^p$ boundedness of a class of parametric Marcinkiewicz integral operators ${ℳ}_{\mathrm{\Omega },h}^{\rho }$ when $h$ satisfies a certain integrability condition and $\text{Ω}$ belongs to the block space $B_q^{(0,-1/2)}({\mathbf{S}}^{n-1})$ for some , $n\ge 2$. Also, we obtain the $L^p$ boundedness for a class of rough parametric Marcinkiewicz integral operators ${ℳ}_{\mathrm{\text{Ω}},h,\lambda }^{*,\rho }$ and ${ℳ}_{\text{Ω},h,S}^{\rho }$ related to the Littlewood-Paley $g_{\lambda }^{*}$-function and the area integral $S$, respectively. Our results are essential improvement and extension of some previously known results.

We consider a manifold endowed with a metric tensor in its tangent bundle pulled back by its own projection. We shall give necessary and sufficient conditions for a vector field to be an infinitesimal isometry of a metric of this type in general and for some special classes. We also examine translations, i.e., the special class of Killing vector fields whose integral curves are geodesics of an associated Finsler manifold. As applications, we determine the Killing vector fields of Funk metrics, and we give a new proof for the fact that perturbing a Riemannian manifold by a one-form metrically equivalent to a Killing field yields a Randers manifold for which the original vector field is a Killing field as well.

Existence of a time-periodic solution to a non-linear wave equation with resonance is established by a variational method. We consider the $2\pi$-periodic weak solution to a wave equation $\square u(x,t)=h(x,t,u(x,t))$ of space dimension 1, where $h(x,t,\xi )$ is asymptotically linear in $\xi$ both as $\xi \to 0$ or $\xi \to \infty$, with the co-efficient as $\xi \to \infty$ belonging to $\sigma (\square )$. It was proved that there are some cases, where the difference of $h(t,x,\xi )$ from its linear approximation is not bounded, that guarantee the existence of a non-trivial weak solution. In this paper, we show that the restriction in our previous result imposed on these co-efficients can be further relaxed.