1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if

(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”

(ii) H= ⋂n∈N n-1Bn

(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.

A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

Quite recently Bekollé and Bonami [1] have characterized the weighted measures λ on the unit disk Δ for which the Bergman projection is bounded on Lp(Δ : λ), 1 < p < ∞. Our methods in [4] can be applied to even extend their result by replacing the unit disk with multiply connected domains. This is done via a rather interesting identity between the Bergman kernel and its “adjoint” [2]. As a corollary of our result we obtain a generalization of a result due to Shikhvatov [7].

Let D be a bounded plane domain and let λ be a positive locally integrate function in D. λ is said to belong to Mp(D) (1 < p < ∞) if it satisfies the Muckenhoupt condition:

where the supremum is taken over all sectors V ⊂ D, dσ is the area Lebesgue measure and |V| = σ(V).

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).

The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

Given a finite undirected graph G and A ⊆ E(G), G(A) denotes the subgraph of G having edge-set A and having no isolated vertices. For a partition {E1, E2} of E(G), W(G; E1) denotes the set V(G(E1)) ⋂ V(G(E2)). We say that Gis non-separable if it is connected and for every proper, non-empty subset A of E(G), we have |W(G; A)| ≧ 2. A split of a non-separable graph Gis a partition {E1, E2} of E(G) such that

|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.

Where {E1, E2} is a split of G, W(G; E2) = {u, v}, and e is an element not in E(G), we form graphs Gii= 1 and 2, by adding e to G(Ei) as an edge joining u to v.

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

K-coherence (K an integer ≧ –1), has been defined by W. R. R. Transue [3] in such a way that 0-coherence is connectedness and 1-coherence is unicoherence plus local connectedness. It is well-known (see, for instance, [5, p. 82]), that for metric spaces, unicoherence is cyclicly extensible and reducible; furthermore, this result has been generalized by Minear to locally connected spaces, [2, Theorems 4.1 and 4.3]. In this paper we show that for a (k – 1)-coherent and locally (k – 1)-coherent Hausdorff space M,k-coherence is cyclicly extensible and reducible.

In this paper, we study the following question: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to Tait [6], who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such that the length of a longest cycle in each member of the family is at most nc, where c = 1 – 2–17 and n is the number of vertices. The exponent c was subsequently reduced by Walther [8, 9] and by Grùnbaum and Walther [4].

In this paper we find a new bound for the function g(h), for which |A(G)|p ≧ pn whenever |G| ≧ pg(h) G a finite p-group. The existence of such a function was first conjectured by W. R. Scott in 1954, who proved that g (2) = 3. In 1956 Ledermann and Neumann proved that in the general case of finite groups g(h) ≦ (h – 1)3.-pn−1 + h[10]. Since then, J. A. Green, J. C. Howarth and K. H. Hyde have reduced this bound considerably. The best (least) bound to date for finite p-groups was obtained by K. H. Hyde [9]. He proved that for h ≧ 5 and g(h) = h + 1 for h ≦ 4. For finite non-abelian p-groups, we improve this bound to: for h ≧ 13, g(h) = 2h – 5 for 5 < h ≦ 8, g(h) = h for h ≦ 5 and for 8 < h ≦ 12 we prove that g(9) = 14, g(10) = 17, g(11) = 20, g(12) = 23.

Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.

THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose

1

Then is noncommutative Jordan.

The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.

Although the literature on splines has grown vastly during the last decade [11], the study of polynomial splines on the circle seems to have suffered neglect. The first to study the subject in depth seem to be Ahlberg, Nilson and Walsh [1]. Almost at the same time I. J. Schoenberg [8] studied the problem of interpolation at the roots of unity by splines and its relation to quadrature on the circle. For discrete polynomial splines on the circle we refer to [5]. M. Golomb [3] also considers interpolation by a class of “spline” functions in the complex plane but his point of view is based on minimum norm properties of spline functions. Perhaps the reason for this neglect may be attributed to the fact that one can pass from the circle to the line by means of the transformation z → exp 2-πix. This changes the problem on the circle into periodic interpolation on the line with the difference that instead of interpolation by piecewise polynomial, we now consider piecewise exponential polynomials with complex exponents.

If H is a closed subgroup of a locally compact group G, with G/H having finite G-invariant measure, then, as observed by Atle Selberg [8], for any neighborhood U of the identity in G and any element g in G, there is an integer n > 0 such that gn is in U·H·U. A subgroup satisfying this latter condition is said to be an S-sub group, or satisfies property (S). If G is a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: If H is a closed S-subgroup of G, then G/H is compact. Property (S) has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).

Let T be an abelian infinite countable group. We say that a property of subsets of T is divisible if it satisfies the following requirements. (We identify with the set of all subsets of T which satisfy .)

1. (i) ∅ ∉ and T ∈

2. (ii) A ∈ and B ⊃ A implies B ∈

3. (iii) A ∈ and A = B1 ∪ B2 implies that either B1 or B2 is in

The Picard group Γ = PSL2 (Z [i]) is the group of linear fractional transformations

with ad – bc = ± 1 and a, b, c, d Gaussian integers.

Γ is of interest as an abstract group and in automorphic function theory. In an earlier paper [1], a decomposition of Γ as a free product with amalgamated subgroup was given and this was utilized to investigate Fuchsian subgroups. Karrass and Solitar used a similar decomposition to characterize abelian and nilpotent subgroups. Maskit [6], Mennicke [7] and Fine [2], used Γ to generate faithful representations of Fundamental Groups of Riemann Surfaces while more recently Wielenberg [10] represented certain knot and link groups as subgroups of Γ. In this paper, we will examine the structure of the congruence subgroups of Γ. Our technique will be to use the decomposition cited above [1], together with the Karrass-Solitar subgroup structure theory for free products with amalgamations [3]. Finally, we give a conjecture and some results concerning Fuchsian subgroups which are contained in congruence subgroups.

Consider the basic problem in the calculus of variations, that of minimizing

1.1

over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that

1.2

along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that

1.3

a relation which always evokes classical Hamiltonian mechanics and conservation laws.

We denote by the class of functions of the form

that are regular in Δ = {z:|;z| < 1} and satisfy Re h(z) > 0 there. For 0 ≦ r < 1, we write

We note that, for , the inequality

is classical.

We investigate here classes of operators T between Banach spaces E and F, which have factorization of the form

where L is a Banach lattice, V is a p-convex operator, U is a q-concave operator (definitions below) and jF is the cannonical embedding of F in F”. We show that for fixed p, q this class forms a perfect normed ideal of operators Mp, q, generalizing the ideal Ip,q of [5]. We prove (Proposition 5) that Mp, q may be characterized by factorization through p-convex and q-concave Banach lattices. We use this fact together with a variant of the complex interpolation method introduced in [1], to show that an operator which belongs to Mp, q may be factored through a Banach lattice with modulus of uniform convexity (uniform smoothness) of power type arbitrarily close to q (to p). This last result yields similar geometric properties in subspaces of spaces having G.L. – l.u.st.

Physicists have long been using Racah's [7] 6-j symbols as a representation for the addition coefficients of three angular momenta. Racah himself discovered a series representation of the 6-j symbol which can be expressed as a balanced 4F3 series of argument 1, that is, a generalized hypergeometric function such that the sum of the 3 denominator parameters exceeds that of the 4 numerator parameters by 1. What Racah does not seem to have realized or, perhaps, cared to investigate, is that his 4F3 functions, with variables and parameters suitably identified, form a system of orthogonal polynomials in a discrete variable. The orthogonality of 6-j symbols as an orthogonality of 4F3 polynomials was recognized much later by Biedenharn et al. [3] in some special cases. Recently J. Wilson [13, 14] introduced a very general system of orthogonal polynomials expressible as balanced 4F3 functions of argument 1 orthogonal with respect to an absolutely continuous measure and/or a discrete weight function. Wilson's polynomials contain Racah's 6-j symbols as a special case. These polynomials might rightfully be credited to Wilson alone, but justice might be better served if we call them Racah-Wilson polynomials.

It has been claimed that a sheaf of abelian groups on a Hausdorff space in which the compact open sets form a basis is injective in the category of all such sheaves whenever its group of global elements is divisible (Dobbs [1]). The purpose of this note is to present an optimal counterexample to this by showing, more generally, that on any nondiscrete T0-space there exists a sheaf of the type in question which is not injective.

Recall that a sheaf A of abelian groups on a space X assigns to each open set U in X an abelian group AU and to each pair U, V of open sets in X such that V ⊆ U a group homomorphism, denoted s ⟿ s|V, satisfying the familiar sheaf conditions ([3, p. 246]) which make A a special type of contravariant functor from the category given by the inclusion relation between the open sets of X into the category Ab of abelian groups, and that a map between sheaves A and B of abelian groups is a natural transformation h:A → B, with component homomorphisms hu:AU → BU. In the following, AbShX will be the category with these A as objects and these h:A → B as maps (= morphisms).

B. Banaschewski [1] has produced a counterexample to [2, Theorem 6]. As noted in [1, Remark 1], our error occurs in the final paragraph of the purported proof of [2, Theorem 6], for P need not be a subpresheaf of I. Accordingly, it remains an open problem to find an analogue of [2, Proposition 1] in the context of Boolean spaces. We hope that attacks on this problem will be facilitated by the (valid) initial three paragraphs of the argument given for [2, Theorem 6].

The following alterations to [2] are in order. Example 5, being a corollary of Theorem 6, remains doubtful, although the special case noted on pp. 1034-1035 is not affected. In Corollary 7, the assertion that j preserves infectives remains doubtful, although the proof for divisibility of j(M)(G) is valid.