Sets of independence are studied for compact abelian hypergroups and they are used, along with Riesz products, to investigate lacunarity questions on the dual object. It is shown that bounded Stechkin sets are always Sidon and that every bounded infinite subset of the dual contains an infinite Sidon set which is also a Λ set. Independent sets are shown to always be Sidon and a necessary condition for Sidonicity is provided. A result of Pisier is used to show that for compact non-abelian groups Sidon and central Λ are equivalent. Several applications are provided, primarily to questions regarding lacunarity on compact groups.

In the first paper of this series we studied on a compact regular contact manifold the integral of the Ricci curvature in the direction of the characteristic vector field considered as a functional on the set of all associated metrics. We showed that the critical points of this functional are the metrics for which the characteristic vector field generates a 1-parameter group of isometries and conjectured that the result might be true without the regularity of the contact structure. In the present paper we show that this conjecture is false by studying this problem on the tangent sphere bundle of a Riemannian manifold. In particular the standard associated metric is a critical point if and only if the base manifold is of constant curvature +1 or −1; in the latter case the characteristic vector field does not generate a 1-parameter group of isometries.

It is well-known that quasi-Frobenius rings are characterized by the property that all propective right modules are injective, as well as by the property that all injective right modules are projective. Similarly, either the property that every quasi-injective or that every quasi-injective is quasi-projective characterizes uniserial rings. Oshiro has given similar characterizations for generalized uniserial rings. The purpose of this paper is to characterize rings for which continuous right modules are discrete. We show that these rings are precisely the uniserial rings. The property that every discrete module is continuous is also investigated.

It is well-known that if G is a multigraph (that is, a graph with multiple edges), the maximum number of pairwise disjoint edges in G is ν(G) and its maximum degree is D(G), then |E(G)| ≤ ν [3D/2’. We extend this theorem for r-graphs (that is, families of r-element sets) and for r-multihypergraphs (that is, r-graphs with repeated edges). Several problems remain open.

We prove that the split integral group ring of a finite p-solvable group of finite representation type has a structure analogous to that of the p-modular semisimple deformation. The split integral deformation can be put in the same form as the p-modular deformation by an appropriate substitution for the parameter T. As an application we derive a simple formula for the matrix units in the semisimple group algebra over a nonmodular prime.

In this article we continue the study of weighted inductive limits of spaces of Fréchet-valued continuous functions, concentrating on the problem of projective descriptions and the barrelledness of the corresponding “projective hull”. Our study is related to the work of Vogt on the study of pairs (E, F) of Fréchet spaces such that every continuous linear mapping from E into F is bounded and on the study of the functor Ext1 (E, F) for pairs (E, F) of Fréchet spaces.

Epimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

Pfaffian graphs are those which can be oriented so that the 1-factors have equal sign, as calculated according to the prescription of Kasteleyn. We consider various operations on graphs and examine their effect on the Pfaffian property. We show that the study of Pfaffian graphs may be reduced to the case of subcubic graphs (graphs in which no vertex has degree greater than 3) or bricks (3-connected bicritical graphs).

Let R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where αj, are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) possesses an admissible solution w(z), then by some Möbius transformation u = (aw + b) / (cw + d) (ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τj (j = 1, … 4) are distinct constants, and σj (j = 1, … 4) are constants, not necessarily distinct.

We treat centralizer near-rings over ring modules in general, with particular emphasis on the case of free modules. Questions like the following are answered. When is the near-ring a nonring? When is the near-ring simple? What are its maximal and minimal left ideals? What is its subgroup structure? What is the radical? The cases where the ring concerned is a PID or a field are treated in some detail.

The structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.

Tame near-rings form an important class of near-rings. They have the common feature that all N-subgroups in a faithful N-group are ideals. Tame near-rings can be very close to and very far from rings. Most of the important classes of distributively generated near-rings and all 2-semisimple near-rings are examples of tame near-rings.

The study of the S-unit equation for algebraic numbers rests very heavily on Schmidt's Subspace Theorem. Here we prove an effective subspace theorem for the differential function field case, which should be valuable in the proof of results concerning the S-unit equation for function fields. Theorem 1 states that either has a given upper bound where are linearly independent linear forms in the polynomials with coefficients that are formal power series solutions about x = 0 of non-zero differential equations and where Orda denotes the order of vanishing about a regular (finite) point of functions ƒk, i: (k = 1, n; i = 1, n) or lies inside one of a finite number of proper subspaces of (K(x))n. The proof of the theorem is based on the wroskian methods and graded sub-rings of Picard-Vessiot extensions developed by D. V. Chudnovsky and G. V. Chudnovsky in their function field analogues of the Roth and Schmidt theorems. A brief discussion concerning the possibility of a subspace theorem for a product of valuations including the infinite one is also included.

Two-sided inequalties for the ratio of modified Bessel functions of first kind are given, which provide sharper upper and lower bounds than had been known earlier. Wronskian type inequalities for Bessel functions are proved, and in the sequel alternative proofs of Turan-type inequalities for Bessel and modified Bessel functions are also discussed. These then lead to a two-sided inequality for Bessel functions. Also incorporated in the discussion is an inequality for the ratio of two Bessel functions for 0 < x < 1. Verifications of these inequalities are pointed out numerically.