Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form

(1.1)

with a continuous “kernel” φ: R → R,or

(1.2)

with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see [8]) and to the theory of fading memory in continuum mechanics [3]. However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form

(1.3)

where the kernel φ: R × T → R satisfies “Carathéodory conditions”.

Two basic unsolved problems of lattice theory are (1) the characterization of sublattices of free lattices and (2) the characterization of projective lattices. A solution to an important case of the first problem has been provided by Galvin and Jónsson [3], who characterize distributive sublattices of free lattices. In this paper, we solve the same case of the second problem by characterizing distributive projective lattices (Theorem 4.1). An interesting corollary is the verification for distributive lattices of the conjecture that a, finite lattice is projective if and only if it is a sublattice of a free lattice.

In this note we continue the study, initiated in [1], of the class S*(α) of functions

(1.1)

that are analytic and univalent in the unit disc U and satisfy the condition

(1.2)

S*(1) is the frequently studied class of univalent star-like functions. For each α, S*(α) is a subclass of the class K(α) of close-to-convex functions of order α introduced by Pommerenke [4]. Properties of the class S*(α) proved useful in studying the coefficient behaviour of bounded univalent functions that are analytic and map U onto a convex domain [1]. In this note we investigate the problem of determining

but we are able to give only a partial solution.

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.

Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided

(1)

and if > 0 is given, there exists Z0, |Z0| < 1, such that

Let ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

Interest in the Singer groups has arisen in various places. The name itself results from the connection Singer [7] made between these groups and perfect difference sets, and this is closely associated with the geometric property that a Singer group is regular on the points of a projective space. Some information about these groups appears in Huppert's book [3, p. 187]. Singer groups are frequently useful in constructing examples and counterexamples. Our aim in this paper is to make a systematic study of the Singer subgroups of the linear groups, with a particular view to analyzing the examples they provide of Frobenius regular groups. Frobenius regular groups are a class of permutation groups generalizing the Zassenhaus groups, and Keller [5] has shown recently that they provide a new characterization of A6 and M11.

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:

(1) s · g 1 g 2 = (s · g 1) · g 2;

(2) s · e = s;

(3) (s, g) → s · g is a Borel function from S × G to S.

If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).

Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].

Consider the following general problem. Let (G, S, μ) be a dynamical system.

To prove [2, Theorem 1], it is enough to remark that if x is a point or line of F4 which is fixed under H, then x is fixed under each subgroup of H. By Sylow's theorem, H has a subgroup H’ of order three. By [1, p. 420, Lemma 2.2], it follows that x is in the subplane of F4 generated by those elements of {A, B, C, D} which are fixed under H'. This subplane consists of a point only. Hence, x is in {A, B, C, D}, which is impossible. This argument can be used in many other cases.

Let Σ 4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S 4k+1k ≧ 1, and T: Σ4k+1 → Σ 4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ 4k+1 and the properties of the free involution T.

To be more specific, recall [5] that the h-cobordism classes of smooth manifolds homeomorphic (or, what is the same, homotopy equivalent) to S 4k+1, k ≧ 1, form a finite abelian group θ 4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP 4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP 4k+2 is either zero or cyclic of order 2. In the latter case the two distinct homotopy spheres are distinguished by the Arf invariant of the parallelizable manifolds they bound.

In this paper we investigate the properties of the product (or complete direct sum) of torsion Abelian groups. The chief results concern products of Abelian primary groups (p-groups). Given a set of p-groups, [Gλ], over an index set Λ, the product of these groups is written λλ∈ΔGλ, the torsion subgroup of the product of these p-groups is written T[λG λ], and the discrete direct sum of the p-groups is written Σ Gλ.

Definition. Σ Gλ is said to be an essentially bounded decomposition if and only if there exists an integer M > 0 such that MGλ = 0 for all but a finite number of Gλs; otherwise the decomposition is essentially unbounded.

Let Tt, t > 0, be a strongly continuous semigroup of positive linear contractions on the L 1-space of a σ-finite measure space . We denote the integral ∫0 t T s ƒ ds, ƒ ∈ L 1, by S 0t ƒ, which is denned as the limit of Riemann sums, in the norm topology of L 1. It is easy to see that, given ƒ ∈ L 1 +, there exists a function F on the product space X× (0, ∞), measurable with respect to the usual product σ-field, such that for every t ≧ 0, ∫0tF(·, s) ds gives a representation of S 0t ƒ. We write S 0t ƒ(x) for ∫0t F(x, S) ds, with a fixed choice of F.

In this paper we study incompressible and injective (see § 2 for definitions) surfaces embedded in M 2 × S 1, where M 2 is a surface and S 1 is the 1-sphere. We are able to characterize embeddings which are incompressible in M 2 × S 1 when M 2 is closed and orientable. Namely, a necessary and sufficient condition for the closed surface F to be incompressible in M 2 × S 1, where M 2is closed and orientable, is that there exists an ambient isotopy ht , 0 ≦ t ≦ 1, of M 2 × S 1onto itself so that either

(i) there is a non-trivial simple closed curve J ⊂ M 2 and h 1(F) = J × S 1, or

(ii) p\h 1 (F) is a covering projection of h 1 (F) onto M 2, where p is the natural projection of M 2 × S 1onto M 2.

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.

Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.

1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

Let [3, p. 170, (16)]

(1.1)

denote the Jacobi polynomial of order (α, β), α, β > – 1, and let g(k, m, n; α, β) be denned by

(1.2)

where Rn (α, β)(x) = Pn (α, β)(x)/Pn (α, β)(1). It is well known [1; 2; 4; 5; 6] that the harmonic analysis of Jacobi polynomials depends, at crucial points, on the answers to the following two questions.

Question 1. For which (α, β) do we have

(1.3)

Question 2. For which (α, β) do we have

(1.4)

where G depends only on (α, β)?

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.

In § 2, a fibre-type construction for graphs is given which, applied to block designs with λ = 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande [22], in which the number of points in the intersection of any pair of blocks attains only two values.

1.1. Let Σ an be an infinite series and sn its nth partial sum. Let (pn) be a sequence of positive numbers such that

If the sequence

(1)

is of bounded variation, that is, Σ |tn – tn –1| < ∞, then the series Σ an is said to be absolutely (R, pn, 1) summable or |R, pn, 1| summable.

Let ƒ be an integrable function with period 2π and let its Fourier series be

(2)

Dikshit [4] (cf. Bhatt [1] and Matsumoto [7]) has proved the following theorems.

THEOREM I. Suppose that (i) the sequence (pn/Pn) is monotone decreasing, (ii) mn > 0, (iii) the sequence (mnpn/Pn) decreases monotonically to zero, and (iv) the series Σ mnPn/Pn) is divergent.

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.

In this paper we settle (with a counterexample) the question raised by Clifford and Preston in [2, p. 275], concerning maximal group homomorphic images of semigroups. We also consider the question in a more general context and characterize all such examples. The notation and definitions follow [1; 2].

By a type of semigroups we mean a class of semigroups, closed under isomorphisms and containing the one-element semigroup. If S is any semigroup and is a type, then a semigroup S* is defined, in [1, p. 18], to be a maximal homomorphic image of S having type if

(i) ,

(ii) S* is a homomorphic image of S, and

(iii) whenever and T is a homomorphic image of S, then there exists a homomorphism from S* onto T.

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.

In this paper we explore the extent to which embedding and isomorphism questions about a Noether lattice ℒ can be reduced to questions about simpler structures associated with ℒ.

In § 1, we use a variation of Dilworth's congruence approach [2] to associate a collection of semi-local Noether lattices with a given Noether lattice ℒ. We show that these semi-localizations determine ℒ to within isomorphism (Corollary 1.5); thus embedding and isomorphism questions about ℒ are largely reduced to the semi-local case.

In § 2, we consider the influence on a semi-local Noether lattice ℒ of the substructure ∂ ℒ consisting of all elements, all of whose associated primes are maximal. Here we find that if ∂ ℒ can be embedded in a semi-local Noether lattice ℒ *, then ℒ can be embedded in an extension of ℒ *.