To a real reductive group G there is attached a family of (real) groups, each of lower dimension but sharing Cartan subgroups with G (cf. [8]). The purpose of these groups is to provide “building blocks” (in a specific sense (cf. [11])) for analysis on G. Their définition is via an L-group construction; the connected component of the identity, LH0, in the L-group of such a group H is naturally a subgroup of LG0 but the requirement that H “share” Cartan subgroups with G precludes defining LH the full L-group of H, as a subgroup of LG. Nevertheless, the principle of functoriality in the L-group suggests that the embeddings of LH in LG will play a role in analysis. In this paper, we study the embeddings of LH in LG in order to resolve a problem about the normalization of orbital integrals.

The first part of this paper, which will be referred to by I, appeared in Volume 30 of this journal. The present paper will use the same bibliography as I.

Theorem 1 in I shows that the knowledge of all continuous linear functionals in o[A]p is essential in determining convergence and summability factors for strong summability. So far, Theorem 7 in I was for a general A the only tool in deciding whether a given sequence ∈ generates such a functional. We mentioned in a remark following Theorem 7 the difficulties in verifying the conditions of this theorem (two parameters are involved). In the present paper, we study continuous linear functionals in o[A]1 in more detail, and we obtain in a corollary to Theorem 22 a condition which appears to be a more satisfactory answer to the question, whether a given sequence ∈ generates a continuous linear functional in o[A]1.

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.

A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].

In a survey of methods in enumerative map theory [14], W. T. Tutte pointed out that little has been done towards enumerating unrooted maps other than plane trees. A notable exception is to be found in the work of Brown, who took the initial step in this direction by enumerating non-separable maps up to sense-preserving homeomorphisms of the plane [2]. He then took a further step, allowing sense-reversing homeomorphisms, by counting triangulations and quad-rangulations of the disc [3, 4]. In all these problems, however, there is a fixed outer region of the plane. This can be considered as a certain type of rooting of a planar map, which is normally regarded as lying on the sphere or closed plane. It is our object here to find an expression for the number of unrooted planar maps in a given set, in terms of the numbers of maps in that set which have been rooted in a special way.

Let H be a separable complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Let π be the quotient mapping from B(H) onto the Calkin algebra B(H)/K(H), where K(H) denotes all compact operators on B(H). An operator T ∈ B(H) is said to be convexoid[14] if the closure of its numerical range W(T) coincides with the convex hull co σ(T) of its spectrum σ(T). T ∈ B(H) is said to be essentially normal, essentially G1, or essentially convexoid if π(T) is normal, G1 or convexoid in B(H)/K(H) respectively.

In the theory of enumeration, the part devoted to the counting of planar maps gives rather surprising results. Of special interest to the combinatorialists is the conspicuous feature of counting numbers associated with families of maps as discussed in the papers of Tutte, Brown and Mullin. These formulas are by no means easy to prove; this is also a part of their charm. We are convinced therefore that these maps possess deep combinatorial properties. We discuss such properties in this paper.

The main result of this article is the construction of a bisection between planar maps and a special family of trees: their vertices are labeled with numbers that differ by at most 1 on adjacent vertices. These trees are said to be well labeled. Combinatorialists usually claim that theorems specifying the “geometry” of objects lead to enumerating formulas. Our result obeys this general rule: we can indeed obtain a new proof of the formula counting the number of rooted planar maps with m edges. Four different proofs of this formula are known to date.

Let σ4 denote the group of all permutations of {a, b, c, d}. It has 24 elements, partitioned into five conjugacy classes: (1) the identity 1; (2) 6 transpositions: (ab), …, (cd); (3) 8 elements of order 3: (abc), …, (bcd); (4) 6 elements of order 4: (abcd), …, (adcb); (5) 3 elements of order 2: x = (ab)(cd), y = (ac)(bd), z = (ad)(bc).

In this paper, we study the differentiate actions of σ4 on odd-dimensional homotopy spheres modelled on the linear actions, with the fixed point set of each transposition a codimension two homotopy sphere.

A simple (2n – l)-knot is a differentiate embedding of a homotopy sphere K2n–l into a homotopy sphere Σ2n+1 such that πj(Σ – K) = πj(S1) for j < n.

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

In this note we provide alternative and unified proofs for two theorems on the homotopy groups of spaces of (continuous) maps into Eilenberg-MacLane spaces. The first theorem is due to Thorn, and independently Federer, and deals with spaces of maps into Eilenberg- MacLane spaces of type (π, n) for n ≧ 1 with π abelian. The second theorem is due to Gottlieb and deals with spaces of maps into Eilenberg- MacLane spaces of type (π, 1) with π nonabelian. As a main tool we shall use the homotopy sequences for certain fibrations of spaces of maps.

2. Basic notation and some preliminary remarks. For any pair of connected CW-complexes X and Y with base points, we denote by M(X, Y), respectively F(X, Y), the space of free maps, respectively based maps, of X into Y.

An orthogonal design in order n of type (u1, …, ut) on the commuting variables x1, …, xt is an n × n matrix X with entries 0, ±x1, …, ±xt such that

In [5] Geramita and Wallis show that if n = 24a+b·n0, where n0 is odd and 0 ≦ b > 4, then t ≧ ρ(n) = 8a + 2b. The result is essentially Radon's limit on the number of anti-commuting, real, anti-symmetric, orthogonal matrices in order n. Garamita and Pullman show that this limit is sharp for orthogonal designs: i.e., given n, there exists an orthogonal design in order n with ρ(n) variables [6].

Two orthogonal designs, X and F, are called amicable if XYt = YXt.

Such pairs of orthogonal designs are especially useful in generating new orthogonal designs [5] or [6].

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.

In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions.

Quite some years ago a number of mathematicians were interested in obtaining explicit expressions for the bounds in Schottky's and Landau's theorems, specifically numerically évaluable bounds of a particular form. The best bounds of this type in Schottky's theorem were given by the author [3]. For Landau's theorem the chosen form is as follows. Let F(Z) be regular in |Z| < 1, omit the values 0 and 1 and have Taylor expansion about Z = 0

Then

Using the same method employed for Schottky's theorem the author showed that one can take K = 5.94. By a slight modification of the author's method Lai [6] gave the further value K = 4.76.

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.

1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

This paper extends several graph-theoretic results to matroids. The main result of Tutte's paper [10] which introduced the theory of n-connection for matroids was a generalization of an earlier result of his [9] for 3-connected graphs. The latter has since been strengthened by Halin [3] and in Section 3 of this paper we prove a matroid analogue of Halin's result. Tutte used his result for 3-connected graphs to deduce a recursive construction of all simple 3-connected graphs having at least four vertices. In Section 4 we generalize this by giving a recursive construction of all 3-connected matroids of rank at least three. Section 2 contains a generalization to minimally n-connected matroids of a result of Dirac [2] for minimally 2-connected graphs.

Each variety of lattice ordered groups determines a variety of groups, namely the variety of groups generated by the groups i n . In this paper a completely new and different correspondence between varieties of groups and varieties of lattice ordered groups is developed. It is known that the variety of representable lattice ordered groups is defined by the law z+ Λ u-1z-u = 1. Here we consider the varieties defined by laws of this form where u is restricted to lie in some fully invariant subgroup of the free group Fx on a countable set X. All the varieties considered contain the variety of representable l-groups and therefore the free group with appropriate ordering.

Let be a finite family of finite sets with the property that |A ∩ B| ≦ 1 whenever A and B are distinct members of . It is a conjecture of Erdös, Faber, and Lovász ([1] a 50 pound problem and [2] a 100 dollar problem) that there is an n-coloring of (i.e., a function such that A ∩ B = ∅ whenever A and B are distinct members of with f(A) = f(B). They actually state the conjecture in a different form. They actually state the conjecture in a different form. Namely, if n is a positive integer and is a family of n sets satisfying (1) |Bp| = n for each p and (2) |Bp ∩ Bq| ≦ 1 when p ≠ q, then there is an n-coloring of the elements of so that each set Bp gets all n colors. The equivalence of the two forms is easily seen by interchanging the roles of elements and sets.

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such that

The sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.

In [1, b.2, VIII, 128] Pólya and Szegö give the following interesting result of A. Cohn:

THEOREM 1. If a prime p is expressed in the decimal system as

then the polynomial irreducible inZ[x].

The proof of this result rests on the following theorem of Pólya and Szegö [1, b.2, VIII, 127] which essentially states that a polynomial f(x) is irreducible if it takes on a prime value at an integer which is sufficiently far from the zeros of f(x).

THEOREM 2. Let f(x) ∊ Z[x] be a polynomial with the zeros α1, α2, …, αn. If there is an integer b for which f(b) is a prime, f(b – 1) ≠ 0, and for 1 ≦ i ≦ n, then f(x) is irreducible inZ[x].

In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.

Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy–l is fixed point free on for all x ≠ y in M.

Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy–1 is fixed point free on for all x ≠ y in M.

A subset A of a topological space X is an approximateretract of X if for every neighborhood U of A in X there is a retract R of X such that A ⊂ R ⊂ U. A compactum X is an absolute approximate retract (AAR-space) if whenever X is embedded as a subset of a compactum Z, then X is an approximate retract of Z. These concepts were first defined in [2] where it is shown that every AAR-space is a contractible Peano continuum. In [3] an example is given to show that there exists a contractible LC∞ compactum which is not an AAR-space.