We study some spectral properties of quasiperiodic Kronig--Penney model. We show the absence of point spectrum in some situations, and derive a relationship between the spectrum of this model and that of the corresponding tight-binding model.

We give some necessary and sufficient conditions for solvability of the matrix equation (*) $A^x + A^{y}=A^{z}$, with certain restrictions on integers $x, y, z$ and a matrix $A \in M_{k}(\bm{Z})$, by applying Ljunggen's result on trinomials. Moreover, we obtain full solution of (*) for the case $k=2$ by another technique.

A smooth surface $S$ in $\bf{R}^3$ is called parallel curved if there exists a plane in $\bf{R}^3$ such that at each point of $S$, there exists a principal direction parallel to the plane. For example, a plane, a cylinder and a round sphere are parallel curved. More generally, a surface of revolution is also parallel curved. The purposes of this paper are to study the behavior of the principal distributions on a real-analytic, parallel curved surface and to classify the connected, complete, real-analytic, embedded, parallel curved surfaces.

Under suitable assumption, we present a method to construct braided Hopf algebras (braided groups) $\bar{B}$ and $\underline{B}$ and in Yetter--Drinfel'd category ${}_H\mathscr{Y}\mathscr{D}^H_1$ and ${}_H\mathscr{Y}\mathscr{D}^H_2$ respectively. As applications, we study some special cases in both module and comodule form for $H$ quasitriangular and for $H$ coquasitriangular respectively. Finally, some examples are given.

We give a characterization of a totally umbilic submanifold $M^n$ with parallel mean curvature vector of a Riemannian manifold $\tilde{M}^{n+p}$, that is an extrinsic sphere $M^n$ of $\tilde{M}^{n+p}$, in terms of the extrinsic shape of circles on $M^n$ in the ambient manifold $\tilde{M}^{n+p}$. This characterization is an improvement of Nomizu and Yano's result ([2]).

In this paper we define conditions under which a tensor product $A \otimes H$, in a braided monoidal category, together with a crossed product structure $A \mathbin{\sharp_{\sigma}} H$ and a smash coproduct structure $A \propto H$ is a Hopf algebra. When $\sigma = \varepsilon_{H} \otimes \varepsilon_{H} \otimes \eta_{A}$, Radford's theorems characterizing the biproduct are obtained and when the antipode of $H$ is a $\sigma$-antipode we find an analogous result with the one due to Wang, Jiao and Zhao.

A glued Riemannian space is obtained from Riemannian manifolds $M_1$ and $M_2$ by identifying their isometric submanifolds $B_{1}$ and $B_{2}$. A curve on a glued Riemannian space which is a geodesic on each Riemannian manifold and satisfies certain passage law on the identified submanifold $B \mathrel{:=} B_{1} \cong B_{2}$ is called a $B$-geodesic. Considering the variational problem with respect to arclength $L$ of piecewise smooth curves through $B$, a critical point of $L$ is a $B$-geodesic. A $B$-Jacobi field is a Jacobi field on each Riemannian manifold and satisfies certain passage condition on $B$. In this paper, we extend Rauch's theorem which gives a comparison of the lengths of Jacobi fields along geodesics in different Riemannian manifolds to $B$-Jacobi fields along $B$-geodesics in different glued Riemannian spaces.

We introduce the notion of a quasi-triangular biFrobenius algeba and prove the $S^{2}$-formula and $S^{4}$-formula.

For any $CR$ orbifold $B$, of $CR$ dimension $n$, we build a vector bundle (in the sense of J. Girbau & M. Nicolau, [13]) $T_{1,0}(B)$ over $B$, so that $T_{1,0}(B)_{p} \approx \bm{C}^{n}/G_{x}$ at any singular point $p=\varphi(x)\in B$ (and the portion of $T_{1,0}(B)$ over the regular part of $B$ is an ordinary $CR$ stmcture), hence study the tangential Cauchy--Riemann equations on orbifolds. As an application, we build a two-sided parametrix for for the Kohn--Rossi laplacian $\Box_{\Omega}$ (on the domain $\Omega$ of a local uniformizing system $\{\Omega, G, \varphi\}$ of $B$) inverting $\Box_{\Omega}$ over the $G$-invariant $(0, q)$-forms $(1 \leq q \leq n-1)$ up to (smoothing) operators of type $1$ (in the sense of G. B. Folland & E. M. Stein, [12]).

A prehomogeneous vector space is called simple if the group under consideration is the product of a simple algebraic group and some multiplicative groups. The explicit forms of the $b$-functions of regular simple prehomogeneous vector spaces have been determined except for two spaces. In this paper, we shall calculate the $b$-functions explicitly for the remaining cases.