In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,

1.1

and obtained the following infinite family of identities:

For A= 1,2, 3,…, let

1.2

where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then

1.3

As Eden noted, Fh(b;x) is the generating function of ph(m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct:

Let M(n) be the set of all the points (x1, x2,…, xn)∈En such that xi∈{0,1,2} for each i= 1, 2,…, n and let f(n) be the cardinality of a largest subset of M(n) containing no three distinct collinear points. L. moser [4] asked for a proof of the inequality

Let us consider the set Sn of those points (x1x2,…, xn)∈M(n) which satisfy |{i:Xi= 1}| = [(n +1)/3]. As Sn is a subset of the sphere with center at (1, 1,…, 1) and radius (n-[(n+1)/3])1/2, no three distinct points of Sn are collinear. Thus we have

1

Pn will denote a set of n points in the plane. A well known theorem of Gallai- Sylvester (see e.g. [4]) states that if the points of Pn do not all lie on a line then they always determine an ordinary line, i.e. a line which goes through precisely two of the points of Pn.

Using this theorem I proved that if the points do not all lie on a line, they determine at least n lines. I conjectured that if n>n0 and no n—1 points of Pn are on a line, they determine at least 2n-4 lines. This conjecture was proved by Kelly and Moser [3], who, in fact, proved the following more general result: Let Pn be such that at most n—k of its points are collinear.

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].

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.

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.

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

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.

We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].

As its name implies, this paper consists of observations on various topics in graph theory that stem from the concept of Hamiltonian cycle. We shall mainly adopt the notation and terminology of Harary [5]. However, we use vertices and edges for what are called "points" and "lines" in [5]. V(G), E(G) respectively will denote the sets of vertices and edges of graph G, and |X| will denote the cardinal of the set X.|V(G)| is the order of G, and |E(G)| the size of G. Throughout n is reserved for the order of G.

Throughout this paper H, H1 etc. will denote positive constants which will not necessarily be the same at different occurrences. If is a series, we shall use the notation For a real, define

1

Let {pn} be a sequence with p0>0 and pn≥0 for n > 0. Define

2

The following identities are immediate:

3

4

where

5

Almost tangent structures have been studied by Eliopoulos [1] and certain Riemannian structures subordinate to almost tangent structures have been studied by Closs [2]. In this paper we investigate those subordinate Riemannian structures for which the underlying almost tangent structure is without torsion and those for which the fundamental form is closed.

Similar studies have been carried out with respect to Riemannian structures subordinate to almost complex structures by Lichnerowicz [4] and with respect to Riemannian structures subordinate to almost product structures by Legrand [5].

Let A be a ring and s2(A) the set of idempotents in A. It is a familiar fact that s2(A) becomes a ring if we define 0, 1 and multiplication as in A, and a new negative and new addition by

1

R. A. Melter [3] made the surprising observation that s3(A), where in general sq(A)={x∈A:xq=x}, is a ring if 2 is a unit in A and we define 0,1 minus and multiplication as in A, and a new addition by

2

The nonobvious facts are that s3(A) is closed under ⊕ and that ⊕ is associative when applied to the elements of s3(A). The ⊕ in (1) is actually a formal group over A, and so is associative when applied to any elements of A. However in (2) (and similarly in other cases we shall define) the ⊕ is not a formal group, and the associative law depends on the fact that the elements involved are in s3(A).

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].

In the theory of deformations of compact complex manifolds, the hypothesis of constancy of the dimension of diverse structural cohomology groups pertaining to a fibre plays an important role (see, for instance, [3, Propositions 2.5 and 2.7, Theorems 2.2 and 2.3, and Definition 6.1]). This paper is the first of two devoted to the investigation of conditions under which constancy of the dimension of given cohomology groups is assured, and more generally, to the study of the variation of that dimension.

In [2] Griffiths introduces extendible forms in a holomorphic deformation. We consider in this paper a differentiate family, and besides extendible, also co-extendible and transportable forms (see §5), and deduce from their existence conclusions about the variation of the dimension of the corresponding structural cohomology groups. It is left to a subsequent paper to give more explicit conditions by means of cohomology operations, and to deal with some applications.

The author has defined [2] the approximately continuous Denjoy integral (AD-integral) which includes exactly the general Denjoy integral and the AP-integral defined by Burkill [2].

The aim of this paper is to give a constructive definition of the AD-integral.

The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.

The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.

For the Hermite polynomials Hn(z) defined by

1

it is easy to see from the Rodrigues formula that

2

where, as usual, Dz=d/dz.

In recent papers ([1], [2]) Carlitz has proved the following formulae:

3

4

where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).