Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic $p$, and with coefficients in a field of odd characteristic $?\ne p$): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols.     

外FO-群

如果群 G 的所有真商群都是 FO-群,但是群 G 本身不是 FO-群,则称 G 为外 FO-群,本文将给出外 FO-群完全的结构描述．     

A FUNDAMENTAL SOLUTION FOR THE LAPLACE OPERATOR ON THE QUATERNIONIC HEISENBERG GROUP

In this paper, the author studies the Laplace operator on the quaternionic Heisenberg group, construct a fundamental solution for it and use this solution to prove the Lp-boundedness and the weak (1-1) boundedness of certain singular convolution operators on the quaternionic Heisenberg group.     

Fiber torsion sensor demodulated by a high-birefringence fiber Bragg grating

A new high-birefringence (Hi-Bi) fiber torsion sensor demodulated by a Hi-Bi fiber Bragg grating (FBG) is proposed in this study. The twisted Hi-Bi fiber sensor characteristics are analyzed using the Jones matrix. The intensity ratio from two reflected Bragg wavelengths is associated with the twist angle of the measured Hi-Bi fiber. It is found that the twist angle resolution is estimated at around 0.3° under ±0.1 dB readout from an optical spectrum analyzer if the polarization state of the light source is stable. The advantages of this new torsion sensor are: (1) insensitivity to intensity variations from the light source, (2) insensitivity to the torsion gauge length, and (3) absolute measurement in the twist angle. However, the polarization state of light in the proposed method needs to be controlled, and any birefringence change in the twisted Hi-Bi fiber needs to be prevented.     

级数小于1的亚纯函数的Borel例外值

证明了当超越亚纯函数的级小于1时，其Norel例外值最多只有一个．由于存在任何级的整函数，因此一个例外值总可达到，故所得结果不能再改进。     

A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o‐minimal structure, then G is divisible. Furthermore, if G is defined in an o‐minimal expansion of a field, k ∈ ? and p_k : G → G is the definable map given by p_k (x ) = x^k for all x ∈ G , then we have |(p_k )^–1(x )| ≥ k^r for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

Subgroup separability of graphs of abelian groups

In the present paper we give necessary and sufficient conditions for the subgroup separability of the fundamental group of a finite graph of groups with finitely generated abelian vertex groups.