Many authors have made interesting and important contributions to the study of vector fields or infinitesimal transformations on compact orientable Riemannian manifolds and Hermitian manifolds without boundary. Recently, Hsiung (6, 7, 8) has extended some of these results to compact orientable Riemannian manifolds with boundary. The purpose of this paper is to continue Hsiung's work by studying vector fields and infinitesimal transformations on almost-Hermitian manifolds with boundary.

In (1), M. Davis and M. Maschler define the kernel K of a characteristic-function game; they also prove, among other theorems, that K is a subset of the bargaining set M1(i) and that it is never void, i.e. that for each coalition structure b there exists a payoff vector x such that the payoff configuration (x, b) belongs to K. The main advantage of the kernel, as it seems to us, is that it is easier to compute in many cases than the bargaining set M1(i).

The problem treated here is an abstract version of the Cauchy problem for an equation of mixed type in the hyperbolic region with initial data on the parabolic line (cf. 2, 3, 5, 11, 13, 14, 15, 16, 21, 27). A more complete bibliography may be found in (3, 5, 18). We begin with the equation (6)

(1.1)

In this paper we prove the following characterization theorem (Section 3) :

Theorem 1. If each of g and m is a real-valued non-decreasing function on the number interval [a, b], then the following two statements are equivalent: (1) If R is a real-valued, refinement-unbounded (Section 3) function of subintervals of [a, b], then the integral (Section 2)

exists and is equal to g (b) — g (a), and

(2) g is absolutely continuous with respect to m.

It is well known that the number of normalized irreducible polynomials of degree m in a single indeterminate, with coefficients in GF(q), is given by

(1.1)

where μ(r) is the Möbius function. It follows from (1) that

(12)

It is well known that an ideal A in a. Dedekind domain has a prime radical if and only if A is a power of a prime ideal. The purpose of this paper is to determine necessary and sufficient conditions in order that a commutative ring with unit element have this property and to study the ideal theory in such rings. Domains with unit element having the above property possess many of the characteristics of Dedekind domains (however, they need not be Noetherian) and will be referred to in this paper as "almost Dedekind domains"—these domains are considered in Section 1.

If A is an algebra over a commutative ring with unity, Φ, then the Jacobson radical of the algebra A is equal to the Jacobson radical of A, thought of as a ring (1, p. 18, Theorem 1). The present note extends this result to all radical properties (in the sense of Kurosh 2) and allows ϕ to be any set of operators on A.

If A is a ring and Φ is an arbitrary set, we say that Φ is a set of operators for A if for any α in Φ and any x in A, the composition ax is defined and is an element of A, and if this composition satisfies the following two conditions:

We say that the restricted cancellation law for ideals (RCL) holds in the commutative ring R if from the equation AB = AC, where A, B, and C are ideals of R and AB ≠ (0), it follows that B = C. RCL is a weakened form of the cancellation law for ideals (CL) : If A, B, and C are ideals of R such that AB = AC and A ≠ (0), then B = C. A ring in which CL holds is an integral domain and in an integral domain, RCL is equivalent to CL.

Denote by {t(x)} some linear transform of the sequence

of the form

where x attains continuous or only integer values. The problem of estimating |t(x) — sm| as x and m tend to ∞ with some connection between them was considered first by H.

A dissection of the disk will be a cell complex (1, p. 39) K with polyhedron the closed disk B2. It will further be required that:

• (a) every edge of K be incident with two distinct vertices (called its ends) ;

• (b) no two edges have the same ends ; and

• (b) no two edges have the same ends ; and

The theory of non-negative matrices was initiated by Perron (7) and Frobenius (4). Wielandt in (11) gives an elegant exposition of the subject.

It is well known to workers in the field that if a matrix A has all its entries non-negative real numbers, then the pattern of zeros and non-zeros of A completely determines the pattern of zeros and non-zeros in every power of A. Ptak in (8) and Ptak and Sedlacek in (9) describe this behaviour in terms of some combinatorial constructs.

In (1) Baer introduced the concept of (C, γ)-transitivity and (C, γ)- homogeneity. A projective plane (see (5) for the requisite definitions and axioms) is (C, γ)-transitive if, given an ordered pair (P1, P2) of points collinear with C but distinct from C and not on γ, there is a collineation which maps P1 into P2 and leaves fixed every point on γ as well as every line through C. A projective plane is (C, γ)-homogeneous for a non-incident point-line-pair if it is (C, γ) transitive and there is a correlation which maps every line through C into its intersection with γ and every point on γ into its join with C.

Let Σ represent the class of functions

(1)

which are schlicht and regular, except for the pole at infinity, in |z| > 1. Further let Σ-1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2)

With every connected graph G there is associated a metric space M(G) whose points are the vertices of the graph with the distance between two vertices a and b defined as zero if a = b or as the length of any shortest arc joining a and b if a ≠ b. A metric space M is called a graph metric space if there exists a graph G such that M = M (G), i.e., if there exists a graph G whose vertex set can be put in one-to-one correspondence with the points of M in such a way that the distance between every two points of M is equal to the distance between the corresponding vertices of G.

A family of circuits of a graph G is said to be independent if no two of the circuits have a common vertex; it is called edge-independent if no two of them have an edge in common. A set of vertices will be called a representing set for the circuits (for the sake of brevity we shall call it a representing set), if every circuit of G passes through at least one vertex of the representing set. Denote by I(G) = k the maximum number of circuits in an independent family and by R(G) the minimum number of vertices of a representing set. Dirac and Gallai asked whether there is any relation between I(G) and R(G) (trivially R(G) ≥ I(G)).

In recent years a number of algebraic characterizations of the groups of Moebius transformations over finite fields have been given in the literature; see (1, 3, 6). H. W. E. Schwerdtfeger has noticed (4) that the group G of Moebius transformations over the real, complex, and certain other fields has the property:

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

Let R be a simply connected closed region in the Euclidean plane E2 whose boundary is a simple closed curve C. A triangular map, or simply "map," is a representation of R as the union of a finite number of disjoint point sets called cells, where the cells are of three kinds, vertices, edges, and faces (said to be of dimension 0, 1, and 2, respectively), where each vertex is a single point, each edge is an open arc whose ends are distinct vertices, and each face is a simply connected open region whose boundary consists of the closure of the union of three edges. Two cells of different dimension are incident if one is contained in the boundary of the other.

If f(z) is a non-constant, entire function, then Hadamard's three-circles theorem asserts that

is a convex, increasing function of log r. Hence, by well-known properties of logarithmically convex functions,

In this paper I shall prove the following theorem:

Theorem. Let ϕ(r) be increasing and convex in log r with

(This condition is imposed to exclude certain trivial cases.) Then there is an integral function f(z) such that