The local controlled generalized H-Bézier model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is owed to its good geometric properties, e.g., symmetry and shape adjustable property. In this paper, some geometric continuity conditions for the generalized cubic H-Bézier model are studied for the purpose of constructing shape-controlled complex curves and surfaces in engineering. Firstly, based on the linear independence of generalized H-Bézier basis functions (GHBF), the conditions of first-order and second-order geometric continuity (namely, G¹ and G² continuity) between two adjacent generalized cubic H-Bézier curves are proposed. Furthermore, following analysis of the terminal properties of GHBF, the conditions of G¹ geometric continuity between two adjacent generalized H-Bézier surfaces are derived and then simplified by choosing appropriate shape parameters. Finally, two operable procedures of smooth continuity for the generalized H-Bézier model are devised. Modeling examples show that the smooth continuity technology of the generalized H-Bézier model can improve the efficiency of computer design for complex curve and surface models.

Let $\mathcal{A}$ be a unital ∗-algebra containing a nontrivial projection. In this paper, we show that every nonlinear mixed bi-skew Jordan-type higher derivation on $\mathcal{A}$ is an additive higher ∗-derivation. Moreover, we apply the above result to prime ∗-algebras, factor von Neumann algebras and standard operator algebras.

Let $\mathcal{A}$ be a unital associative ∗-algebra over the complex field $\mathbb{C}$ containing a nontrivial projection and $\xi$ be a nonzero scalar. In this paper, we give a characterization of $\xi$-bi-skew Lie triple derivation on $\mathcal{A}$. As applications, we apply the above result to prime ∗-algebras, factor von Neumann algebras and standard operator algebras.

Let $\mathcal{A}$ be a unital ∗-algebra containing a nontrivial projection P. In this paper, we introduce the notion of a nonlinear mixed skew Jordan-type derivation (that is, $p_n\left(S_1,S_2,S_3,\cdots ,S_n\right)=S_1\diamond S_2•S_3•\cdots •S_n$, where $S\diamond K=S^{*}K+K^{*}S$ and $S•K=SK+K{S}^{*}$ for all $S,K\in \mathcal{A}$) and show that if a map $\mathsf{\Phi }$: $\mathcal{A}\to \mathcal{A}$ satisfies $\mathsf{\Phi }\left(p_n\left(S_1,S_2,\cdots ,S_n\right)\right)={\sum }_{k=1}^n{p}_n(S_1,S_2,\cdots S_{k-1},\mathsf{\Phi }\left(S_k\right),S_{k+1},\cdots ,S_n)$, for all $S_1,S_2,\cdots ,S_n\in \mathcal{A}$ and all $n\ge 3$, then $\mathsf{\Phi }$ is an additive ∗-derivation. Furthermore, we generalize this result to higher derivable maps on $\mathcal{A}$. As applications, the above result is applied to several special classes of unital ∗-algebras such as prime ∗-algebras, factor von Neumann algebras and standard operator algebras.