The line - graph of an ordinary graph G is that graph whose points can be put in one-to-one correspondence with the lines of G in such a way that two points of are adjacent if and only if the corresponding lines of G are adjacent. This concept originated with Whitney [ 5 ], has the property that its (point) chromatic number equals the line chromatic number of G, where the point (line) chromatic number of graph is the minimum number of colors required to color the points (lines) of the graph such that adjacent points (lines) are colored differently.

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given by

where the summation condition means sum over all positive integers d which divide n.

A fundamental construction of a category ζ((2), Appendice) is a triple (S, p, k), where S is a functor from ζ to itself and 2 where p:S2→S and k:1ζ→S are natural transformations such that

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity

1.1

By a (generalized) Schreier extension we mean a group G decomposed into a subinvariant series Gn ↣ Gn-1 ↣ Gn-2 … ↣ G1 ↣ G0 = Gn is anti-invariant in G, i. e. the only subgroup of G which is normal in Gn is the trivial one. ( “ ↣ ” denotes a group monomorphism, i. e. an injection homomorphism. ) As is well known, such groups G can be embedded into the repeated wreath product where Fi ≅ Gi / Gi+1 (cf. [ 2 ], notation of M. Hall [ 1 ], p. 81).

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).

It is assumed that the reader is acquainted with the first two parts of the present paper, [1] and [2], in which there was developed informally the theory of a certain class of hypothetical computing devices, the Q-machines. In the present part of the paper we develop a way of describing Q-programs and Q-computations; then, following the theorem in [2], we obtain some further connections between the arithmetical nature of a number x and the computational schémas for generating sequences of rational numbers converging to x; and finally, by means of the usual device of Goedel numbering we begin to examine some properties of programs which operate on programs.

Beineke and Harary [l] recently showed that the maximum number of strong tournaments with k nodes that can be contained in a tournament with n nodes is

if 3 ≤ k ≤ n. The object of this note is to obtain some additional results of this type. We will use essentially the same terminology as was used in [ l ], so we will not repeat the standard definitions here.

Homological monoids, as first defined by Hilton and Ledermann [ l ], are a generalization of abelian categories. It is known that if is an abelian category, so is here we prove the more general theorem that if is a homological monoid, so is . Our definition differs from that originally given by Hilton and Ledermann by the addition of a uniqueness condition in Axiom 1.

A graph is called Euler if it has no isolated vertices and every vertex has even or infinite degree. The graphs we consider may have multiple edges but no loops. Loosely speaking, we will be concerned with the conditional compactness of the set of all countable Euler subgraphs of a given graph.

Un demi-groupe d'applications d'un ensemble est dit séparateur si, pour tout triple d'éléments x, y, z de l'ensemble, avec x ≠ y, il existe une application α du demi-groupe telle que x.α ≠ z et y α ≠ z. Un tel demi-groupe d'applications est évidemment transitif. Tout demi-groupe isomorphe à un demigroupe séparateur d'applications est dit un demi-groupe séparateur.

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).

Brouwer's celebrated fixed point theorem states that every map of a closed n-cell into itself has a fixed point.

A similar theorem is here proved for coincidences between a pair of maps (f, g): In→In, where I denotes a closed n-cell (i.e. a homeomorph of the n-ball) and a coincidence is a point x∊In for which f(x)=g(x). That two arbitrary maps (f, g): In→In need not have a coincidence is shown by the pair f:In→y0, g:In→y1, where y0, y1∊In and y0≠y1. More generally, one can immediately construct a map g so that (f, g) is coincidence free if f is not surjective.

As is well known, any module M over a ring possesses injective hulls, i. e., injective essential extensions [3], unique up to isomorphisms which map M identically, but the various proofs for this all require some transfinite arguments and hence provide little indication as to how these hulls may actually be constructed for a given module.

Let Q(x) denote the number of squarefree integers ≤ x. Recently K. Rogers [ l ] has shown that Q(x) ≥ 53x / 88 for all x, with equality only at x = 176. Define R(x) to be Q(x) - 6/п2 X. (We observe that and ) Our objective will be to examine R(x). In particular, we show that for all x and observe that for x ≥ 8.

If

1.1

and

1.2

a ≥ 0, k=1, 2, 3, …; where is the Laguerre polynomial of order v, defined by

then we shall show that under certain conditions

1.3

Following the inversion theory, two representation theorems are given. The proofs of these theorems follow easily along the lines of Widder [4, Ch. VII] and are therefore omitted.

This is primarily an expository paper based on (and generalizing) some ideas of J. Pierpont [6], W.H. Young [8], R. L. Jeffery [4] and S. C. Fan [2]. Our aim is to give a simple and easily applicable theory of integration for arbitrary extended - real functions over arbitrary sets in a measure space. This will be achieved by using a generalized version of Pierpont1 s upper and lower integrals (with the upper integral playing the main role), and by appropriately defining the operations in the extended real number system, henceforth denoted by E*, so as to make it a commutative semigroup under addition and multiplication.

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.

If R is a ring and I is a right ideal of R then I is called faithful if R - I is a faithful right R-module, i.e. if { r ∊ R: Rr⊆ I} = (0). I is called irreducible [ 1 ] provided that if J1 and J2 are right ideals such that J1 ∩ J2 = I, then J1 or J2 = I. Let N(I){ r ∊ R: rI⊆ I} and [ I: a ] = { r ∊ R: ar⊆ I} for a ∊ R. We write (a)r for [(0): a ].

It is well known that Z(p∞) is isomorphic to each of its non-zero homomorphic images [3]. The aim of the present note is to generalize this fact about Z(p∞) to indecomposable infective modules over rings more general than the ring of integers which will include Dedekind domains as a special case.