Let V be a finite dimensional Hermitian vector space and K be a compact Lie subgroup of U(V) for which th representation of K on C[V] is multiplicity free. One obtains a canonical basis {pα} for the space C [VR]k of K-invariant polynomials on VR and also a basis {q's. The polynomial pα's yields the homogeneous component of highest degree in qα. The coefficient that express the qα's in terms of the pβ's are the generalized binomial coeffficients of Yan. The main result in this paper shows tht these numbers are rational.

We say that a subgroup H is an n-th maximal subgroup of G if there exists a chain of subgroups G = G0 > G1 > … > Gn = H such that each Gi is a maximal subgroup of Gi-1, i = 1, 2, …, n. The purpose of this note is to classify all finite simple groups with the property that every third maximal subgroup is nilpotent.

In this paper, we show the existence of solutions of functional equations fix ∈ Sx ∪ Tx and x = fix ∈ Sx ∪ Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to Kaneko concerning common fixed points of multivalued mappings weakly commuting with a single-valued mapping and satisfying a generalized contraction type. Some related results are also obtained.

In the present paper we prove that every finite dimensional non-atomic measure ν is open and monotone (viz. ν–1 preserves connected sets) relative to the usual Fréchet-Nikodým topology on its domain and the relative topology on its range. An arbitrary finite dimensional measure is found on the other hand to be biquotient.

Given a vector measure ν, we further investigate the properties of its integral map Tν: φ → ∫ φdν defined on the set of functions φ in L1(|ν|) for which φ(s) ∈ [0,1] |ν|-almost everywhere. When ν is finite dimensional, Tν is found to be always open. In general, when Tν is open, the set of extreme points of the closed convex hull of the range of ν is proved to be closed, and when ν and Tν are both open, the range of ν in itself is closed.

In the theory of multivalent functions there are several different levels of postulates for p-valency. Perhaps the most well-known is the class of mean p-valent functions in the sense of Spencer [8] (we shall refer to them as areally mean p-valent functions), whose basic properties are found, e.g., in Hayman [4]. Recently Eke [1, 2] extended to these functions a number of results which had been known for circumferentially mean p-valent functions.

Some results on fixed points of asymptotically regular mappings are obtained in complete metric spaces and normed linear spaces.

The structure of the set of common fixed points is also discussed in Banach spaces. Our work generalizes essentially known results of Das and Naik, Fisher, Jaggi, Jungck, Rhoades, Singh and Tiwari and several others.

Every directed graph defines a Hilbert space and a family of weighted shifts that act on the space. We identify a natural notion of periodicity for such shifts and study their C* -algebras. We prove the algebras generated by all shifts of a fixed period are of Cuntz-Krieger and Toeplitz-Cuntz-Krieger type. The limit C* -algebras determined by an increasing sequence of positive integers, each dividing the next, are proved to be isomorphic to Cuntz-Pimsner algebras and the linking maps are shown to arise as factor maps. We derive a characterization of simplicity and compute the K-groups for these algebras. We prove a classification theorem for the class of algebras generated by simple loop graphs.

We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package.

The purpose of this note is to characterize maximal right ideals of T0(G), the (left) mear-ring of transformations from a group (G, +) into itself which leave zero fixed. Using this characterization, we answer some questions raised by Heatherly (1972).

In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation:

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$N\geq 2$

Let G be a locally compact group with left invariant Haar measure m. Le H be a closed subgroup of G and K a compact group of G. Let R be the equivalence relation in G defined by (a, b)∈R if and if a = kbh for some k in K and h in H. We call E =G/R the double coset space of G modulo K and H. Donote by a the canonical mapping of G onto E. It can be shown that E is a locally compact space and α is continous and open Let N be the normalizer of K in G, i. e. .

In this paper we obtain asymptotics for the number of rooted 3-connected maps on an arbitrary surface and use them to prove that almost all rooted 3-connected maps on any fixed surface have large edge-width and large face-width. It then follows from the result of Roberston and Vitray [10] that almost all rooted 3-connected maps on any fixed surface are minimum genus embeddings and their underlying graphs are uniquely embeddable on the surface.

In 1943, Post conjectured that “monogenic normal systems are universal”, and in 1961 Minsky proved a stronger result “‘tag’ systems are universal” which implied the proof of Post's conjecture. The author had independently obtained a simple direct proof of Post's conjecture. The purpose of this note, then, is to present an exposition of Post's conjecture, and to show the full simplicity of its direct verification.

This paper presents a basis for the laws which hold in each of the finite simple groups, PSL(2, 2n), n> ≥ 2, thus partially solving a problem raised by Cossey, Macdonald and Street [3]. They considered the more general problem of finding bases for the laws which hold in PSL(2, pn), and succeded in finding a number of general laws, and in completing bases for pn ≤ 11. The solution of the general problem appears to be very difficult.

Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — αr)2 < c, r(q — βr)2 < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals α and β, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46¼ = 1/2.6043 ….

We give presentations for the groups PSL(2, pn), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2, 2n) = PSL(2, 2n), we show that these groups have deficiency greater than or equal to – 2. We give deficiency – 1 presentations for direct products of SL(2, 2n) for coprime ni. Certain new efficient presentations are given for certain cases of the groups considered.

We study the Witten multiple zeta function associated with the Lie algebra . Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w at least 2 is a finite ℚ-linear combination of alternating Euler sums of weight w and depth at most 2, except when the only nonzero argument is one of the two last variables, in which case ζ(w−1) is needed.

Rosay and Rudin introduced the notion of ‘tameness’ for discrete subsets of $\mathbf{C}^{n}$. We generalize the notion of tameness for discrete sets to arbitrary Stein manifolds, with special emphasis on complex Lie groups.

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the eigenvalue is specified, we conjecture that there is only one such representation. The representation we have found is described explicitly (modulo inversion of a certain rational map on Euclidean space) in terms of a positive definite function, and also by means of a quasi-invariant probability measure on the combinatorial boundary of the group.

If m and n are natural numbers satisfying 1 ≦ m < n let denote the least integer k such that the statement: ‘Every (0,1) matrix with n columns, with constant row-sum m, and with at least k distinct rows, has rank n’ is true. Then = +1 for m ≧ 2, n ≧ m2 + 2. Further, for 1 ≦ m < n.