Realizations of Symmetric Maps by Symmetric Polyhedra

Representations by polyhedra of several regular maps of positive genus are developed. The groups of symmetries of some of these polyhedra are proper subgroups of their groups of combinatorial symmetries, but we show that faithful representations can be obtained if appropriately general polyhedra are admitted. The general definition of polyhedra, which may be useful in other contexts as well, is also discussed, as are some of the reasons for its absence from the literature. Received February 3, 1997, and in revised form August 5, 1997.     

具有对称性的Copula

首次引入左边块对称和右边块对称的概念,利用Copula研究了其联合分布函数的特点,并分析了这两种对称特性与其它对称特性的联系．最后,提出几个值得进一步讨论的问题     

Symmetric spectra

The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. Our method can be used more generally to invert a monoidal functor, up to homotopy, in a way that preserves monoidal structure. Symmetric spectra were discovered at about the same time as the category of -modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.

     

Symmetric Jacobians

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ? such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.     

Symmetric Framelets

We study tight wavelet frames associated with symmetric compactly supported refinable functions, which are obtained with the unitary extension principle . We give a criterion for the existence of two symmetric or antisymmetric compactly supported framelets. All refinable masks of length up to 6 satisfying this criterion are found.     

Polynomial Rings Over Symmetric Rings Need Not be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.     

基于对称化原理的对称区域分裂法

本文利用对称化原理,讨论了一种只需在子区域上计算两个完全独立子问题就可得到原问题解的对称区域分裂法,并用此方法求解线性算子方程和线性透射问题.此方法可作为并行算法在MIMD计算机上使用.     

Polynomial Rings over Symmetric Rings Need Not Be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.     

Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempts to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed.