置换群关于变换后基的强生成集的算法

本文给出置换群关于变换后的基的强生成集的一种算法及其理论证明,并讨论了既约强生成集和最小强生成集的问题。     

A projection method of estimation for a subfamily of exponential families

Summary This paper is concerned with estimation for a subfamily of exponential-type, which is a parametric model with sufficient statistics. The family is associated with a surface in the domain of a sufficient statistic. A new estimator, termed a projection estimator, is introduced. The key idea of its derivation is to look for a one-to-one transformation of the sufficient statistic so that the subfamily can be associated with a flat subset in the transformed domain. The estimator is defined by the orthogonal projection of the transformed statistic onto the flat surface. Here the orthogonality is introduced by the inverse of the estimated variance matrix of the statistic on the analogy of Mahalanobis's notion (1936,Proc. Nat. Inst. Sci. Ind.,2, 49–55). Thus the projection estimator has an explicit representation with no iterations. On the other hand, the MLE and classical estimators have to be sought as numerical solutions by some algorithm with a choice of an initial value and a stopping rule. It is shown that the projection estimator is first-order efficient. The second-order property is also discussed. Some examples are presented to show the utility of the estimator.     

Generalized Exponents and Forms

We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a linear character) using the theory of semi-invariant differential forms. Springer’s theory of regular numbers gives a formula when the group is generated by dim V reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.     

算子半稳定分布的分解

JAJTE于1977年提出了算子半稳定分布的概念,并给出了满的算子半稳定分布μ的刻画:存在c(0＜c＜1),h∈RN,和RN上非奇异线性算子B,使得μc=Bμ*δ(h)成立.作者利用μ的对称群给出μc=Bμ*δ(h)的B,再利用B的最小多项式具体给出μ的分解,把μ分解为简单的Poisson部分和Gauss部分.     

圈的自同结构群的阶

n个顶点的圈Cn的自同结构群的阶是2n.     

Characters of Fundamental Representations of Quantum Affine Algebras

We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra, in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions.     

Quantum automorphism groups of vertex-transitive graphs of order ≤ 11

We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups , symmetric groups S _n and quantum symmetric groups , by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.     

The Laplacian subalgebra of is a strongly singular masa

In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the -folded tensor product of the free group factor with itself (). The notions of strongly singular masas in type factors and that of asymptotic homomorphism were introduced by A. Sinclair and R. Smith. One of their first examples was the Laplacian subalgebra of the free group factor, generated by the sum of words of length 1 in . This subalgebra was known to be a singular masa. Using the results of A. Sinclair and R. Smith, we show that the unique trace-preserving conditional expectation onto the Laplacian subalgebra of is an asymptotic homomorphism, and hence the Laplacian subalgebra is a strongly singular masa for every .

     

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

Let W(x,y) = ax ³+ bx ⁴+ f ₅ x ⁵+ f ₆ x ⁶+ (3 ax ²)² y+ g ₅ x ⁵ y + h ₃ x ³ y ² + h ₄ x ⁴ y ² + n ₃ x ³ y ³+a ₂₄ x ² y ⁴+a ₀₅ y ⁵+a ₁₅ xy ⁵+a ₀₆ y ⁶, and X = , , where the coefficients are non-negative constants, with a > 0, such that X ²(x,x ²)−Y(x,x ²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (x _f ,y _f ) of Φ in the invariant set . 2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.