We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with certain subgroups of the full symmetric group. The dimensions of these symmetry classes of tensors are also computed.

A large class of special Finsler manifolds can be endowed with Finsler connections whose “$h$-part” does not depend on the directions. We call these Finsler connections $h$-basic and present a systematic treatment of them, using (in a simplified form) the Frölicher-Nijenhuis calculus. We provide an axiomatic description of a distinguished class of $h$-basic Finsler connections, the class of Ichijyō connections. With the help of an Ichijyō connection we present new characterizations of generalized Berwald manifolds, as well as – in particular – of Berwald manifolds and locally Minkowski manifolds.

A new proof of the uniqueness of $OA(12,11,2,2)$ is proposed. It is shown that any $OA(12,11,2,2)$ can be presented as a juxtaposition of two unique $OA(12,5,2,2)$ and $OA(12,6,2,2)$.

We consider the initial value problem of a partial differential equation $\frac{\partial u}{\partial t}=c(x)\frac{\partial u}{\partial x}+g(x,u)$ in some function spaces $X$ on the interval $I$ of the real line. By using the representation formula of the solution to the equation, we define a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ of bounded linear operators on $X$. When $c(x)=\text{γ}x(\text{γ}\in ℝ),g(x,u)=h(x)u(h\in C(I,ℂ))$ and $I$ is $[0,1]$ or $[1,\infty )$, we give sufficient conditions for the semigroup to be chaotic by using the spectral property of its infinitesimal generator. When $c(x)=1$ and $g(x,u)=h(x)u$, we also give sufficient conditions for the semigroup to be chaotic by using the property of an admissible weight function.

We investigate some properties of lightlike submanifolds especially which are coisotropic and some totally umbilical screen distribution.