Many transformation semigroups S over a set X have the inner-automorphism property: their automorphisms are precisely those mappings of the form g-1 · g: α → g-1αg(α∈S) where g is a fixed permutation of S. For example, the full transformation semigroup, any two sided ideal of this semigroup, and, with some exceptions for small cardinals, the alternating group and the symmetric group itself. See Malcev (1952) and Scott (1965, Chapter 11). In this paper we determine all transformation semigroups over finite X with this property. In fact, we do more: we characterize all transformation semigroups invariant under the mappings g-1 · g. We refer to these as -normal transformation semigroups.

In earlier work Borrego, Cohen and DeVun (1971a, b) topological semigroups which were uniquely representable in terms of two subsets were investigated. In this note we extend the definition and prove similar results for a larger class of semigroups.

Generalized Ramsey theory for graphs was formulated and developed in the previous papers in this series. We extend the area here by introducing generalized Ramsey numbers for higher dimensional simplicial complexes. In particular we calculate explicitly the Ramsey numbers for several small “pure 2-complexes”, or more briefly plexes, in which each edge is contained in some 2–call.

The Hamiltonian system Y′ = BY + CZ, Z′ = – AY – B*Z is considered where the coefficients are continuous on I = [a, ∞, C = C* ≧ 0, and A = A* ≦ 0. A solution (Y, Z) satisfying Y*Z = Z*Y is defined to be principal (coprincipal) provided that (i) Y−1 exists on I (Z−1 exists on I) and (ii) as t→∞ ( as t → ∞). Three conditions are given which are separtely equivalent to the condition that a solution is principal iff it is coprincipal. For a self-adjoint scalar operator L of order 2n, this problem is related to the deficiency index problem and to a problem of Anderson and Lazer (1970) which concerns the number of lnearly independent solutions of L (y) =0 satisfying y(k) ∈ (a, ∞) (k = 0, …, n).

Let G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

A new arithmetic invariant E(f) is defined for integral binary quadratic forms f. It has the property that, denoting by fm the norm-form of a quadratic number field Q(m), E(fm) < 1 if and only if Q(m) has class number one.

An attempt is made to characterise the zeros of some Hilbert spaces of analytic functions means of their kernel functions. Results on the zeros of functions in DΦ and their uniqueness s are included, in particular we give an affirmative answer to a question of Shapiro and Shield

In this paper we investigate some new sequence spaces which naturally emerge from the concept of almost convergence. Just as ordinary, absolute and strong summability, it is expected that almost convergence must give rise to almost, absolutely almost and strongly almost summability. Almost and absolutely almost summable sequences have been discussed by several authors. The object of this paper is to introduce the spaces of strongly almost summable sequences which happen to be complete paranormed spaces under certain conditions. Some topological results, characterisation of strongly almost regular matrices, uniqueness of generalized limits and inclusion relations of such sequences have been discussed.

In this short note, two results on a solid, pointed, closed cone C in Rn will be given: first, C is polyhedral iff it has a finite number of maximal faces; second, for any face F of C, C* ∩ F┴ is a face of its dual cone C* of dimension n – dim F.

For positive integers a, b and n define the combinational expression We give geometric and probabilistic interpretations of these expressions (and their multidimensional extensions) and find new, simple proofs of the convolution identities known to hold for such expressions.

Shrinkande and Bhagwan Das (1970) showed how to extend a (4t – 1, 4t) row-orthogonal matrix with entries ± 1 to a Hadamard matrix of order 4t. Using a slightly different approach we consider extensions of (4t – k, 4t) row-orthogonal matrix to a Hadamard matrix of order 4t.

(Point, closed subset)-separation axioms and closed subsets separation axioms for topological spaces will be uniformly defined. Then it is shown that a subcategory of TOP is bireflective in TOP if and only if Ob consists of all separated spaces for some (point, closed subset)-separation axiom. A characterization theorem on subcategories of all separated spaces for closed subsets separation axioms is also given by using the category SEP of all separation spaces and the embedding functor G: TOP → SEP. As an application we have that a T1-space is normal if and only if it is embedded in a product space of the unit intervals in SEP.

Let L = HK be a semidirect product of a normal locally finite π′-group H by a locally finite π′-group K, where π, is a set of primes. Suppose CK(H) = 1 and L is Sylow π-sparse (which in the countable case just says that the Sylow π-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K—this had previously been obtained under the assumption that L is locally soluble. The answer is the same—essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called “pinched” groups.

An (r, 1) system is pair (V, F) where V is a v-set and F is a family of non-null subsets of V (b in number) which satisfy the following.

(i) Every pair of distinct members of V occur in precisely one member of F;

(ii) Every member of V occurs in precisely r members of F.

A pseudo parallel complement PPC(n, α) is an (n+1, 1) system with ν = n2 – αn and b ≦ n2 + n – α in which there are at least n – α blocks of size n. A pseudo intersection complement PIC (n, α) is an (n+1, 1) system with ν=n2 – αn + α – 1 b ≦ n2+n – α in which there are at least n – α+1 blocks of size n-1. It is shown that for α ≧ 4, every PIC(n, α) can be embedded in a PPC(n, α – 1) and that for n > (α4 + 2α3 + 2α2 + α)/2 every PPC(n, α) can be embedded in a projective plant of order n. The latter generalizes a result of Totten (who proves the result for α = 1).

Let G be a finitely generated metabelian group whose derived group G′ has finite rank. It is shown that G can be embedded in a finitely presented metabelian group H with H′ of finite rank.