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The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p, on Mn. In that case, if p ⊂ Rn+p, x ⊂ Mn, then Lp(x) = (d(x,p))2, where d is the Euclidean distance function.

1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3(q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

The concept of a “set of ρ-discs” for complex-valued functions defined in the open unit disc U was introduced in [4; 5, p. 265] and various results concerning the existence of such sets associated with spirals and Stolz angles were derived. These sets are non-Euclidean analogues to the classical cercles de remplissage of Milloux [7; 8]. In the present paper additional theorems about Stolz angle phenomena and the existence of ρ-discs in more general subsets of U are established and new proofs are indicated for some results of Seidel [9, Theorem 2 and Corollary 1].

1. Introduction. C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v = 2 (mod 4), and the skew C-matrices, of order v = 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, X)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tourna­ ments introduced by Szekeres [24] and by Reid and Brown [21]. In § 4 we introduce the notion of negacyclic C-matrices, analogous to the similar notion introduced by Berlekamp in the setting of coding theory (cf. [4, p. 211]). Eigenvalues of negacyclic matrices are characterized and standard forms are obtained. Negacyclic C-matrices are interpreted in § 5 as the matrices of a special class of the relative difference sets introduced by Butson [7]. Exploiting some results of Elliott and Butson [10], we obtain a "multiplier theorem" for negacyclic C-matrices, and adapting a result of [2], we show that any negacyclic C-matrix has a nontrivial multiplier. Necessary conditions for the existence of a negacyclic C-matrix of order v are obtained in § 6. The nonexistence of nega­ cyclic C-matrices of all orders v ^ 226, v ^ 1 + Ph, with p prime, has been verified. This leads to the conjecture that they do not exist, unless v = 1 + PkPaley [19] constructed C-matrices of all orders v = 1 + pk, p prime. In § 7 it is shown that every Paley matrix is equivalent to a negacyclic C-matrix, a fact

This paper contains in part the author’s Ph.D. dissertation written under the supervision of Professor F. Bagemihl at the University of Wisconsin-Milwaukee. This research was supported by grants from the National Science Foundation and the University of Wisconsin Graduate School.

In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.