On Abelian Automorphism Groups of Hypersurfaces

Given integers $d\ge 3$ and $N\ge 3$. Let $G$ be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree $d$ in the complex projective space $\mathbb{P}^{N-1}$. Suppose $G\subset PGL(N, \mathbb{C})$ can be lifted to a subgroup of $GL(N,\mathbb{C})$. Suppose moreover that there exists an element $g$ in $G$ such that $G/\langle g\rangle$ has order coprime to $d-1$. Then all possible $G$ are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given $(d,N)$. In particular, we show (Proposition 5.1) that the order of an automorphism of a smooth cubic fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers is achieved by a unique (up to isomorphism) cubic fourfold.Comment: 14 pages. Theorem 4.3 is restated and a gap in its original proof is fixed. To appear in Israel Journal of Mathematic

Pedagogical Thoughts on Liszt\u27s Six Concert Etudes

This paper focuses on suggestions for learning or teaching each of these concert etudes, dealing with the structure of themes, keys, dynamics, programmatic interpretation, and specific technical issues. Several performing elements are discussed in detail, including melody and bass lines, musical phrasing, rhythmic pulse, tempo rubato, layered dynamics, and pedaling

Comparison study of constitutive models for overconsolidated clays

Widely distributed in natural deposits, the overconsolidated (OC) clays have attracted extensive experimental investigations on their mechanical behaviors, especially in the 1960s and 1970s. Based on these results, numerous constitutive models have also been established. These models generally fall into two categories: one based on the classical plasticity theory and the other the bounding surface (BS) plasticity theory, with the latter being more popular and successful. The BS concept and the subloading surface (SS) concept are the two major BS plasticity theories. The features of these two concepts and the representative models based on them are introduced, respectively. The unified hardening (UH) model for OC clays is also based on the BS plasticity theory but distinguishes itself from other models by the integration of the reference yield surface, unified hardening parameter, potential failure stress ratio, and transformed stress tensor. Modification is made to the Hvorslev envelop employed in the UH model to improve its capability of describing the behaviors of clays with extremely high overconsolidation ratio in this paper. The comparison among the BS model, SS model, and UH model is performed. Evidence shows that all these three models can characterize the fundamental behaviors of OC clays, such as the stress dilatancy, strain softening and attainment of the critical state. The UH model with the revised Hvorslev envelop has the fewest parameters which are identical to those of the modified Cam-Clay model