Arc-transitive cubic Cayley graphs on PSL(2,p)

For a prime p at least 5,let T=PSL(2,p).This paper gives a classification of the connected arc-transitive cubic Cayley graphs on T and a determination of the gener- ated pairs ((?),(?)) of T such that o((?))=2 and o((?))=3.     

On dynamics of Out(F n ) on PSL2(ℂ) characters

We introduce and study an open set of PSL₂(?) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.     

Strictly Cyclic Algebra of Operators Acting on Banach Spaces H p(β)

Let be a sequence of positive numbers and 1 p< . We consider the space H ^p() of all power series such that . We investigate strict cyclicity of the weakly closed algebra generated by the operator of multiplication by zacting on H ^p(), and determine the maximal ideal space, the dual space and the reflexivity of the algebra . We also give a necessary condition for a composition operator to be bounded on H ^p() when is strictly cyclic.     

PSL(n,F)中几类极大子群

<正> U是域F上n维向量空间,取定一组基{e_i;1≤i≤n}后可把U写成n维行向量空间.SL(n,F)是作用于U上的特殊线性群,PSL(n,F)是对应的射影群.为了叙述的方便,本文所有的命题都是对SL(n,F)叙述和证明的,但它们在对应的射影群PSL(n、F)中也显然都是成立的.     

The Crossing Number of C(8, 2)□P n

C(n, k) is a graph obtained from n-cycle by adding edges v _i v _i+k (i = 1, 2,...,n, i + k (mod n)). There are several known results on the crossing numbers of the Cartesian products of C(n, k) (n ≤ 7) with paths, cycles and stars. In this paper we extend these results, and show that the crossing number of the Cartesian product of C(8, 2) with P _n is 8n. Yuanqiu Huang: Research supported by NSFC (10771062) and New Century Excellent Talents in University (NCET-07-0276). Jinwang Liu: Research supported by NSFC (10771058) and Hunan NSFC(O6jj20053).     

RESEARCH ANNOUNCEMENTS Strong Discreteness of Subgroups of SL(2, Γ_n)

As in [1,6], let Г_n be the n-dimensional Clifford group, i.e., the set of all elements in ndimensional Clifford algebra An, which can be expressed as a finite product of non-zero-vectors and SL(2, Г_n) denote the group of n-dimensional Clifford matrices which satisfy:     

The λ- structure of the green ring Of GL(2,F p) in characteristic,IV

In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the $lambda;, and s-, structure of the Green ring of GL(2,F _p), and Sl(2,F _p). We shall now construct a subring of the Green ring which is invariant for the $lambda;, and s-, operations. It is generated by all the indecomposables with odd-dimensional composition factors. This sheds another light on the results in the previous sections. We shall also study a certain quotient of the Green ring, which is in fact the Green ring of a certain subgroup of GL(2,F _p) consisting of upper triangular matrices. The multiplication and the λ, s-, structure of this quotient Green ring is described. Moreover it is shown how this λ and s-, structure controls the deviation from being a λ, respectively s-, ring of the Green ring of any finite group with a normal Sylow subgroup of order p. The sequence of Adams operations for these groups is shown to be periodic, and the period reflects the internal p-structure of these groups.     

The Maximal Number of Iπ-Characters Over Normal Subgroups

Let G be a π-separable group for a set of primes, let N be a normal subgroup of G, and let θ be an I_π-character (i.e., irreducible π-partial character) of N. We obtain a necessary and sufficient condition for the number of I_π-characters of G over θ to take the possible maximum

π_|G:N|${\left|G:N\right|}_{\pi }$

. Some applications are given.     

The Stability in W s,p (Γ) Spaces of L 2-Projections on Some Convex Sets

ABSTRACT

For a polygonal open bounded subset of ?², of boundary Γ, we study stability estimates for the projection operator from L ¹(Γ) on a convex set K _h of continuous piecewise affine functions satisfying bound constraints. We establish stability estimates in L ^p (Γ) and in W ^s,p (Γ) for 1 ≤ p ≤ ∞ and 0 < s ≤ 1. This kind of result plays a crucial role in error estimates for the numerical approximation of optimal control problems of partial differential equations with bilateral control constraints.     

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 35)

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group PL(2, 3⁵).     

The number of circuits of length 4 in PSL(2,ℤ)-space

ABSTRACT

Professor Graham Higman has defined coset diagrams for the action of PGL(2,?) on the projective line over a finite field F_q, denoted by PL(F_q), where q is a prime power. These diagrams are composed of circuits. In this paper, we answer the question, for a fixed number of triangles T, how many distinct circuits of length 4, are evolved?     

Diagrams for certain quotients of PSL ( 2 , ℤ [ i ] )

Actions of the Picard group PSL(2, ?[i]) on PL(F _p ), where p ≡ 1(mod 4), are investigated through diagrams. Each diagram is composed of fragments of three types. A technique is developed to count the number of fragments which frequently occur in the diagrams for the action of the Picard group on PL(F _p ). The conditions of existence of fixed points of the transformations are evolved. It is further proved that the action of the Picard group on P L(F _p ) is transitive. A code in Mathematica is developed to perform the calculation.     

The representations of two types of functionals on L~∞(Ω,F) and L~∞(Ω,F,P)

This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.     

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.     

Sets of Type (a, b) From Subgroups of ΓL(1, pR)

In this paper k-sets of type (a, b) with respect to hyperplanes are constructed in finite projective spaces using powers of Singer cycles. These are then used to construct further examples of sets of type (a, b) using various disjoint sets. The parameters of the associated strongly regular graphs are also calculated. The construction technique is then related to work of Foulser and Kallaher classifying rank three subgroups of AL(1, p ^R). It is shown that the sets of type (a, b) arising from the Foulser and Kallaher construction in the case of projective spaces are isomorphic to some of those constructed in the present paper.     

The Central Limit Theorem on SO0(p, q)/SO(p)×SO(q)

We obtain a central limit theorem for the space SO ₀(p, q)/SO(p)×SO(q). To achieve this, we derive a Taylor expansion of the spherical function on the group SO ₀(p, q).