We are here concerned with the problem of deciding when Σ∞n=1 f(n)/n ≠ 0, given that f is periodic and the series convergent. In particular, we consider

Conjecture A. Let p be a positive integer and f a (real-or complex-valued) number-theoretic function with period p.

Consider the following problem of Lagrange in the calculus of variations: relative to differentiable curves xi(t) satisfying xi(t0) = xi0 and xi(t1) = xi1 find a curve minimizing

1

We consider a system of differential equations

1

where 0 = (o, o) is an isolated singular point. Thus, there exists B > o such that S(0, B) contains only one singular point. Here, S(0, B) denotes a sphere centered at 0 with radius B. We shall denote the boundary of S(0, B) by ∂ S(0, B).

In [1] Dirac has determined the structure of all 3-connected graphs which do not contain two independent (i.e., disjoint) circuits. We shall here provide a short proof of this theorem by applying Tutte' s theory of 3-connected graphs [2].

By a group ring we mean in this paper a ring defined by a finite group G and an integral domain K:

such that A contains G and is freely generated by G over K, so that

The ring A = KG has a co-multiplication

defined by

so that A is a Hopf algebra.

Given an integer k ≤ 2 and a finite set M of rational integers. Let vi (i = 1, 2, …, n) be m-dimensional (column-)vectors with all components from M and such that the kn sums

1.1

are all different. Then we shall say that {v1, v2, …, vn} is a detecting set of vectors.

In the ranking of a collection of p objects by the method of paired comparisons, a measure of consistency is provided by the relative number of transitive (or consistent) triples and cyclic (or inconsistent) triples. This point of view was introduced by Kendall and Babington Smith [4]. They found a formula for the maximum number of cyclic triples, thereby determining the greatest inconsistency possible. The purpose of this note is to extend the result to obtain the maximum number of "strongly connected" collections of n objects among the given p objects.

E. Hewitt [1] used the Daniell approach to define a real-valued measure function on a σ-algebra of the real line. He began by defining an arbitrary non-negative linear functional I on L∞ ∞(R), (the space of all complex-valued continuous functions on the real line R which vanish off some compact subset of R).

Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.

For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).

A family of sets is said to possess property if there exists a set such that and F ⊄ B for each In [1], P. Erdos and A. Hajnal ask the following question: Does there exist for every positive integer k a finite family of finite sets satisfying

1. (i) |F|=k for each

1. (ii) | F∩ G| ≤ 1 for each F, , F ≠ G

1. (iii) does not possess property ?

A semi-lattice (Birkhoff, Lattice Theory, p. 18, Ex. 1) is an algebra <A,.> with a single binary operation satisfying: (1) x = xx, (2) xy = yx, and (3) (xy)z = x(yz). In this note we show that the three identities may be reduced to two but cannot be reduced to one.

It is easy to see that (2), (3) imply (4) (uv)((wx)(yz)) = ((vu)(xw))(zy). Setting w = y = u and x = z = v in (4) and using (1) we get uv = vu. Setting v = u, x = w, and z =y in (4) and using (1) we get u(wy) = (uw)y. And so (1) and (4) imply (2) and (3).