Let (X, d1) and (Y, d2) be metric spaces. A mapping f:X→Y is said to be a Lipschitz mapping if there exists a real number λ such that

for each x,y∊X. We call λ a Lipschitz constant for f. If λ∊[0, 1), f is called a contraction mapping. Throughout this note CB(Y) denotes the set of closed and bounded subsets of Y equipped with the Hausdorff metric induced by d2.

An application of graph theory to automatic traffic control [2] gave rise to the problem of deciding which connected graphs have points of degree 2 which can be successively suppressed until only a single edge remains.

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.

Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

The purpose of this article is to present a new generalization of the classical Stone-Weierstrass theorem for commutative C*-algebras.

Under the assumption that B is a sub-C*-algebra of A separating the pure states of A and zero, Kaplansky has conjectured that B=A [4, p. 246]. He gave a proof for the case that A is postliminary ([4, Theorem 7.2]; see also [2, 11.1.8]). Glimm, Akemann, and Sakai have established the conjecture in the presence of various other additional hypotheses, most of which hold in the commutative case ([3], [1], [7]).

A hypersurface of a globally framed f-manifold (briefly, a framed manifold), does not in general possess a framed structure as one may see by considering the 4-sphere S4 in R5 or S5. For, a hypersurface so endowed carries an almost complex structure, or else, it admits a nonsingular differentiable vector field. Since an almost complex manifold may be considered as being globally framed, with no complementary frames, this situation is in marked contrast with the well known fact that a hypersurface (real codimension 1) of an almost complex manifold admits a framed structure, more specifically, an almost contact structure.

A linear connexion ∇ on a smooth manifold M with vanishing torsion and curvature is called flat. It is well known that such a connexion need not be the Levi-Civita connexion of a Riemannian (nor pseudo-Riemannian) structure (cf. [1]).

The following example with M homeomorphic to a cylinder is particularly simple. Using polar coordinates the punctured plane M=R2\{0} is covered by a smooth atlas with two charts

In recent years several attempts have been made to obtain estimates for the product of the primes less than or equal to a given integer n. Denote by the above-mentioned product and define as usual

Analysis of binomial and multinomial coefficients has led to results such as A(n)<4n, due to Erdôs and Kalmar (see [2]). A note by Moser [3] gave an inductive proof of A(n)<(3.37)n, and Selfridge (unpublished) proved A(n)<(3.05)n More accurate results are known, in particular those in a paper of Rosser and Schoenfeld [4] in which they prove Θ(n)< 1.01624n; however their methods are considerably deeper and involve the theory of a complex variable as well as heavy computations. Using only elementary methods we will prove the following theorem, which improves the results of [2] and [3] considerably.

In certain boundary value problems associated with two-parameter ordinary differential equations defined and having 2p, (p>1), turning points in a given interval, there arises certain equations in integers whose solutions determine the coefficients in the asymptotic expansions for the eigenvalues [1,2, 3, pp. 134-139].

It is shown that a 3rd-Engel group is an extension of a soluble group by a group of exponent 5.

In 1962, V. Ju. Sandberg gave a definition of a Lipschitz structure more general than that of a metric space. In [1], the author gave alternate definitions of a Lipschitz structure, one in terms of a family of pseudometrics and the other in terms of sequences of entourages. These definitions were proved equivalent to each other. In 1965, Sandberg gave conditions under which his Lipschitz structure was generated by a metric.

A graph is a collection of nodes some pairs of which are joined by a single edge. A k-path, or a path of length k, is a sequence of nodes {p1 p2,… Pk+1} such that Pi is joined to pi+1 for 1 ≤i≤k; we assume the nodes are distinct except that p1 and pk+1 may be the same in which case we call the path a k-cycle or a cycle of length k. (Notice that two nodes joined by an edge determine a 2-cycle according to this definition; it will also be convenient to regard a single node as a 1-cycle.) A spanning path or cycle is one that involves every node of the graph. One of the unsolved problems of graph theory is to characterize those graphs that have a spanning path or cycle.

The covering problem. Let Si(i = 1, 2,…,n) be given sets containing mi elements respectively and let

1

be their cartesian product. The elements of S(n) will be called vectors. The vector (x1 x2,…, xn) covers (y1 y2,…, yn) if xi =yi for at least n—1 values of i. A subset M of S(n) is said to be a covering (perfect covering) of S(n) if each member of S(n) is covered by at least (exactly) one member of M. A covering M is said to be linear if the sets Si are groups Gi and M is a subgroup of G(n) = S(n) Denote by σ(n; m1 m2,…, mn) the value of min |M| when M runs through all coverings of S(n) and by σ(n; m1 m2,…, mn) the value of min |M| when the sets Si are given groups Gi and M runs through all linear coverings of G(n).

King [5] devised two tests for determining when z = 1 is a singular point of the function f(z) defined by

1

having radius of convergence equal to one. The point z = 1 and radius of convergence one may be chosen without loss of generality.

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that

1.1

convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by

1.2

where then fi(xj) = δij and fi∊X* for each positive integer i.

A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).

Color the integers 1, 2,…, n red and blue. In this note we consider arithmetic sequences in which the discrepancy between red and blue integers is maximized.

More formally, we represent the coloring by a function λ:{1,…,n}→{+1, — 1}. Set

1

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that

1

is similar to a unitary operator.

The following problem is an extension of this result: If T and S are two operators such that

1. 1. sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)

2. 2. TS = ST

then there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?

Also, in [7] B. Sz.-Nagy has proved that every compact operator T such that

sup ‖Tn‖<∞ (n = 1, 2, 3,…)

is similar to a contraction.

In [2] Volger proved that the underlying functor of a category of set-valued models of an r-ary theory has a left adjoint. We want to show that his proof remains valid if instead of set valued models of an r-ary theory models of a finitary theory with values in an arbitrary cocomplete cartesian closed category are considered. As Volger for sets we show for any cocomplete cartesian closed category T C that for every finitary theory (S being a skeleton of the full subcategory of finite sets) the restriction of the left adjoint of on C(s) is a functor in ; here brackets around the exponent indicate as usual a restriction to functors which preserve finite products. We are very much indebted to the referee for pointing out that our proof of the last statement is only based on the properties of C mentioned above and the fact that S has and T preserves finite products. With this in mind and retaining only that part of the cartesian closedness which is relevant for the following considerations we can state the following.

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is

1. (i) abelian and the order of each element divides 4, or

2. (ii) abelian and the order of each element divides 6, or

3. (iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

Let D denote the open right half plane and

a Stolz domain in D with vertex at the origin. If h is a minimal harmonic function on D with pole at the origin then E⊂D is minimally thin at the origin iff where is the reduced function of h on E in the sense of Brelot. We now define

where s shall be fixed to be 1/e. For the set E∩In we shall let cn denote the outer ordinary capacity (see [1, pp. 320-321]), An the outer logarithmic capacity, and on the outer Green capacity with respect to D. If E⊂K, Mme. Lelong [3, p. 131] was able to prove that E is minimally thin at the origin Since one cannot easily relate the classical measure theoretic properties of a plane set with its Green capacity, it would appear desirable to find some other criteria for minimal thinness.

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.