Let $p$ be a prime number. We consider representations of $p\prime$- valenced Schurian schemes over a field of characteristic $p$, especially the case that the cardinality of the underlying set can be divided by $p$ and not by $p^2$. A typical example of such scheme is obtained by the following way. Let $G$ be a finite group of order $pq$, where $q$ is prime to $p$, and let $H$ be a $p\prime$-subgroup of $G$. Define the scheme by the action of $G$ on $H\backslash G$. In this case, we will show that the adjacency algebra is a direct sum of some Brauer tree algebras and simple algebras, and hence it has finite representation type.

Also we give some examples of the case that $G$ is the symmetric group of degree $p$ and $H$ is its Young subgroup.

This paper deals with weakly Ricci-symmetric lightlike hypersurfaces of indefinite Sasakian manifolds, tangent to the structure vector field. We prove that, in a weakly Ricci symmetric lightlike $\eta$-Einstein (or Einstein) hypersurface of an indefinite Sasakian manifold, the associated 1-forms $\alpha$ and $\beta$ satisfy $\alpha +\beta =0$ (Theorem 4). Also, we show that there exist no weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurfaces of indefinite Sasakian manifolds with cyclic parallel Ricci tensor if $\alpha +\beta +\gamma$ is not everywhere zero (Theorem 5). A particular case of weakly Ricci symmetric condition is studied and we prove that a special weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurface cannot be $\eta$-Einstein (or Einstein) and under certain condition, it cannot be $(D\perp \langle \xi \rangle ,D\text{'})$-mixed-totally geodesic (Theorem 7).

In this paper we give a sharp minimum degree condition for a $2$-connected star-free graph to have a $2$-factor containing specified edges. Let $G$ be a $2$-connected $K_{1,n}$-free graph with minimum degree $n+d$ and $I\subseteq E(G)$. If one of the followings holds, then $G$ has a $2$-factor which contains every edge in $I$: i) $n=3,d\ge 1,\left|I\right|\le 2$ and $|V(G)|\ge 8$ if $\left|I\right|=2$; ii) $n=4,d\ge 1,\left|I\right|\le 2$ and $|V(G)|\ge 11$ if $\left|I\right|=2$; iii) $n\ge 5,d\ge 1$ and $\left|I\right|\le 1$; iv) $n\ge 5,d\ge \lfloor (\sqrt{4n-3}+1)/2\rfloor$ and $\left|I\right|\le 2$.

Let $(R,\text{m})$ be a regular local ring of prime characteristic $p$ and $R\text{'}$ a Noetherian subring of $R$ such that $R^p\subset R\text{'}$. In the previous paper Furuya and Niitsuma [1], introducing the concept of $\text{m}$-adic $p$-basis, the first and the second authors proved the following theorem: The regular local ring $R$ has an $\text{m}$-adic $p$-basis over the subring $R\text{'}$ if and only if $R\text{'}$ is a regular local ring. In this paper, we generalize this result to semi-local rings.

In this paper we prove that the basis number of the Cartesian product of paths, cycles and theta graphs with Tenets is exactly 3. However, if we apply Theorem 4.1 of Ali and Marougi [3], which gives a general upper bound of the Cartesian product of disjoint connected graphs, we find that the basis number of these graphs is less than or equal to 4.

In the present investigation, we obtain some subordination and superordination results involving Hadamard product for certain normalized analytic functions in the open unit disk. Our results extend corresponding previously known results.

We study the propagation of singularities for a system of semilinear wave equations satisfying the null condition in one space dimension. We show that if a solution $(u,v)$ to the system is in $H_{loc}^s(\mathrm{\Omega })\cap H_{ml}^r(0,x_0,{\tau }_0,{\xi }_0)$, then $(u,v)\in H_{ml}^r(\mathrm{\Gamma })$ as long as , where $\mathrm{\Omega }\subset {ℝ}^2$ is an open set and $\mathrm{\Gamma }$ is a null bicharacteristic of $\square$ passing through $(0,x_0,{\tau }_0,{\xi }_0)$.

The cycle-complete graph Ramsey number $r\text{(}{C}_m\text{, }{K}_n\text{)}$ is the smallest integer $N$ such that every graph $G$ of order $N$ contains a cycle $C_m$ on m vertices or has independent number $\alpha (G)\ge n$. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that $r\text{(}{C}_m\text{,}{K}_n\text{)=(}m-1)(n-1)+1$ for all $m\ge n\ge 3$ (except $r(C_3,K_3)=6$). In this paper, we show that $r(C_6,K_8)\le 38$.

In this paper, testing for the equality of mean components and of two mean vectors in repeated measures with the intraclass correlation model are treated when the missing observations occur. We consider a new test statistic for the equality of mean components in one-sample problem. Further, we derive a new test statistic for the equality of two mean vectors. The distributions of the test statistics are given under the general case of missing observations. Finally, numerical examples by Monte Carlo simulation are conducted to illustrate power of the method proposed in this paper.

We consider a quarter-symmetric metric connection in a Kenmotsu manifold. We investigate the curvature tensor and the Ricci tensor of a Kenmotsu manifold with respect to the quarter-symmetric metric connection. We show that the scalar curvature of an $n$-dimensional locally symmetric Kenmotsu manifold with respect to the quarter-symmetric metric connection is equal to $n(1-n)$. Furthermore, we obtain the non-existence of generalized recurrent, $\phi$-recurrent and pseudosymmetric Kenmotsu manifolds with respect to quarter-symmetric metric connection.

We investigate the local orbit types of hyperbolic or elliptic orbits of the linear isotropy representations of exceptional semisimple (pseudo-Riemannian) symmetric spaces in terms of the restricted root systems with respect to maximal split abelian subspace.