In this paper we show the Parseval formula (Theorem 2.5) for the generator of wave equations with some dissipative terms of rank one. Moreover we resolve the solutions for that equations into several modes by using the Parseval formula.

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to a subgraph of $G$ isomorphic to $H$. We say that $G$ is $H$-magic if there is a total labeling $f:V\cup E\to \{1,2,3,\dots ,|V|+|E\left|\right\}$ such that for each subgraph $H^{\prime }=(V^{\prime },E^{\prime })$ of $G$ isomorphic to $H$, ${\displaystyle {\sum }_{v\in V^{\prime }}f(v)+}{\displaystyle {\sum }_{e\in E^{\prime }}f(e)}$ is constant. When $f(V)=\{1,2,\dots ,|V\left|\right\}$, then $G$ is said to be $H$-supermagic. In this paper we show that a chain of any 2-connected simple graph $H$ is $H$-supermagic.

For any ordinal $\omega \le \lambda \le {\omega }_1$ and any natural we prove that a $\lambda$-large subgroup $L$ of a primary $C_{{\omega }_1}$-group $A$ is $n$-summable if and only if $A$ is $n$-summable. This strengthens a classical result due to Linton (Pac. J. Math., 1978) and a recent author’s result (Algebra Colloq., 2009) as well.

Integrability and symmetry reductions of the Kawahara equation are investigated. Exact traveling wave solutions, soliton solutions and Galilean invariant solutions are constructed. Auto-Bäcklund transformations and multiplication law of solutions are deduced.

From the definition of the Moyal $\ast$-product in terms of projective limits of the ring of polynomials of vector fields, the Moyal configuration space of Schwartzian functions, equipped with the $\ast$-product, is built as a formal power series ring with elements assimilated to free indeterminates. We then define the projector on the ideal depending on a fixed indeterminate, which allows to use the definition of algebraic derivations with respect to any order of field derivative. As a consequence and in a direct manner, Euler-Lagrange equations of motion, in the framework of both the noncommutative scalar and gauge induced Dirac fields, are deduced from the nonlocal Lagrange function. A connection of this theory to a generalized Ostrogradski's formalism is also discussed here.

We classify $N(k)$-contact metric manifolds satisfying the conditions $\sout{Z}(\xi ,X)\cdot C_0=0$, $C_0(\xi ,X)\cdot \sout{Z}=0$ and $C_e(\xi ,X)\cdot \sout{Z}=0$, where $\sout{Z},C_0$ and $C_e$ denote the concircular curvature tensor, the contact conformal curvature tensor and the extended contact conformal curvature tensor, respectively.

We give an elementary proof of Frank’s Theorem stating that a (finite, undirected, nonempty) multigraph has $k$ edge disjoint spanning trees if and only if it can be obtained from $K_1$ by repeatedly (i) adding an edge or (ii) chosing a sequence $\sigma$ of $k$ vertices and pairwise distinct non-loop edges, deleting the edges of $\sigma$, and adding a new vertex plus one edge to each vertex of $\sigma$ plus one edge to each end of every edge of $\sigma$.

Let $A$ be a basic self-injective Nakayama algebra over an algebraically closed field. In this paper, we investigate the ring structure of the orbit algebra $\mathbb{A}\left({\tau }_{A^e};A\right)={\oplus }_{i\ge 0}{\underset{\_}{\text{Hom}}}_{A^e}({\tau }_{A^e}^i(A),A)$, where $A^e$ is the enveloping algebra of $A$ and ${\tau }_{A^e}$ is the Auslander-Reiten translation for $A^e$.

For a reduced plane curve $C$ and a line $L$ in ${\mathbb{P}}^2$, we put ${ℂ}_L^2:={\mathbb{P}}^2-L$, and $C_L:=C-(C\cap L)$. If $C$ and $L$ intersect transversaly and ${\pi }_1({\mathbb{P}}^2-C,b_0)$ is abelian, it is known that ${\pi }_1({ℂ}_L^2-C_L)$ is also abelian. In this article, we study ${\pi }_1({ℂ}_L^2-C_L)$ and the Alexander polynomial for the case when a quartic curve $C$ and a line $L$ do not intersect transversaly.

We give in this paper sufficient conditions for $r$-lightlike submanifolds $M$ of dimension $m$, which is not totally geodesic in an $(m+n)$-dimensional semi-Riemannian manifold of constant curvature $c$ to admit a reduction of codimension. We consider proper $r$-lightlike, coisotrope and totally lightlike submanifolds, generalizing thus previous results on isotropic submanifolds [1] as well as in the Riemannian case developed in [2, 5, 10].