It is well known that for any real number θ there are infinitely many positive integers n such that

Here ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

Much work has been done in the investigation of the properties of solutions of linear elliptic systems of partial differential equations. Among these systems, the class of Beltrami systems has been studied for many years and has been shown to be of fundamental importance. Another class, perhaps of equal importance, is the class defined by Bers (1), which the author has taken the liberty of calling Bers systems. Solutions of these systems will be called Beltrami and Bers functions respectively.

Since Beurling (1 ), the study of closed invariant subspaces of Hp(X) and Lp(X) on the unit circle X = {z:|z| = 1} has been done extensively and culminated into a very fine theory of generalized analytic functions ; cf. Hoffman (4), Srinivasan (7). Here we say that a space E of complex-valued functions on X is invariant if zE ⊆ E. Little is known, however, about the structure of closed invariant subspaces E of the space C(X) of continouus functions on X.

Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula

(1)

is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.

In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?

This paper deals with a problem raised in a paper by J. de Groot (1): Do there exist fields Ω whose full automorphism group is isomorphic to the additive group of integers Z?

The answer to this question is yes. In this paper we construct, given any subfield k of the complex numbers, extension fields Ω of k such that the automorphism group G(Ω/k) of Ω with respect to k is infinite cyclic. Fields having the infinite cyclic group as a full group of automorphisms are obtained by choosing the base field k in such a way that it does not contain any subfield k0 so that k possesses non-trivial automorphisms leaving k0 pointwise fixed.

A subset G of a non-empty partially ordered set C is called normal if it coincides with the set of all upper bounds of the set of lower bounds of G. This is equivalent to stipulating that G be the set of all upper bounds of some subset of C called a set of generators for G. When ordered by inclusion, the family of all normal subsets of C forms a complete lattice with maximum C and minimum empty or singleton. The meet operation is simply point set intersection; whence, the meet of a family Gi of normal subsets is the set of upper bounds of ∪ Fi where Fi generates Gi for each i. A normal subset is called proper if it is neither void nor C, and the proper normal subsets. of C form a boundedly complete lattice.

In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation

(1.1)

where A (r2), C(r2) are entire functions of the complex variable

Let n be a finite affine plane with n points on a line. We denote by G(II) the graph whose vertices are all points and lines of II, with two vertices adjacent if and only if one is a point, the other is a line, and the point and line are incident. Let L(11) denote the line graph of G(II), i.e., the vertices of L(II) are the edges of G(II), and two vertices of L(II) are adjacent if the corresponding edges of G(II) are adjacent. It is clear that L(II) is a regular, connected graph with n2(n + 1) vertices and valence 2n — 1.

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

1.1. Throughout this note, will denote an associative ring but we shall not require to possess a unit.

If A and B are subsets of , then A + B will denote the set {x + y| x ∊ A, y ∊ B}. Aτ will denote the set {u ∊ | au = 0 for all a ∊ A} .

Elements a and b will be said to be orthogonal if ab = ba = 0.

In this paper we consider finite graphs without loops and multiple edges. A graph is considered to be an ordered pair 〈G, *〉 where G is a finite set the elements of which are called the vertices of while * is a subset of [G]2 (where [G]2 is the set of all subsets of two elements of G). The elements of * are called the edges of . If {P, Q} ∊ *, we say that Q is adjacent to P. The degree of a vertex is the number of vertices adjacent to it. Let k be an integer. We say that is the complete k-graph if G has k elements and * = [G]2.

Si un réseau G de Rn contient Zn et est permis pour l'«octaèdre généralisé» Ωn d'équation

l'indice de Zn dans G est borné supérieurement par n! d'après les théorèmes généraux de Minkowski.

As is well known in the theory of graphs a tree is a connected graph without cycles. Many characterizing properties of trees are known (1), for example the cyclomatic number is equal to zero, which is also equal to p — 1, where p is the number of connected components of the graph. The graphs with cyclomatic number equal to p — 1 are defined here as tree-equivalent graphs. A tree is always a tree-equivalent graph but not conversely. The properties of tree-equivalent graphs are studied here.

The algebra of bounded analytic functions on the open unit disk D, usually written H∞ i is a commutative Banach algebra under the supremum norm. Since its compact maximal ideal space M (space of complex homomorphisms) is an extension space of the unit disk, there must be a continuous mapping form βD, the Stone-Čech compactification of D, onto M. R. C. Buck has remarked (4), that this mapping fails to be one-one, in the light of a classical theorem of Pick. If the points of βD are represented by filters of subsets of D, we can identify those filters which are sufficiently close in terms of the hyperbolic metric on D in an attempt to get a one-one correspondence between filters and points of M.

A category is made up of its class of objects , of a function M (or ) which assigns to each (A,B) ∊ × the class M(A, B) of all morphisms from A to B, of an operation μ which is a class of maps

A, B, C ∊ , and of a family of maps

In this work we obtain a simultaneous extension of Theorems 1.6 and 1.7 in Agmon and Nirenberg (1), together with a partial extension of the result on backward unicity for parabolic equations by Lions and Malgrange (4).

Let H be a Hilbert space. (·) and | | are the notations for the scalar product and the norm in this space. Consider in H a family B(t), 0 ≤ t ≤ T, of closed linear operators with dense domain DB(t) (varying) with t. Let L2(0, T, H) be the space of Bochner square-integrable vector-valued functions with values in H. Our main result is the following

A sequence is said to be well distributed (mod 1) (3, 4, 5) if the limit

(1.1)

exists, uniformly in p ≥ 0, for all intervals I in [0, 1], with length |I|, characteristic function where (x) is the fractional part of x. If (1.1) is true for p = 0 and all I in [0, 1] we say that is uniformly distributed (mod 1).

Lyapunov's second method is a well-known and powerful tool for studying the behaviour of solutions of a system of differential equations. One approach to the theory is the comparison method developed by Corduneanu (4). This approach has the advantage that it also leads to other results on asymptotic behaviour which originally appeared to be unrelated to Lyapunov's method. Some of these results have been obtained by the author in (2). The purpose of this paper is to make use of the comparison method to obtain some refinements of Lyapunov's theory.

The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).

This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.