Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

Gnedenko and Kolmogorov [3, pp. 181-182] have shown that if Xn with law F(x) belong to the domain of normal attraction of a stable law of index 0<α<2, i.e. if partial sum Sn/an1/α converges in distribution to some stable law Vα, a>0 then there exist c1 and c2 such that

and

We call a ring R (associative and with identity) for which every quasi-injective right R-module is injective a QII-ring. Similarly R is called an SSI-ring when every semisimple right R-module is injective. Clearly every semisimple, artinian ring is a QII-ring and every QII-ring is an SSI-ring. One then asks whether these inclusions among classes of rings are proper. The purpose of this note is to point out an instance when SSI implies QII It is then easy to see that an example of Cozzens shows that the class of QII-rings properly contains the class of semisimple, artinian rings.

The generalized equations of bisymmetry, associativity and transitivity are, respectively,

1. (1) (x1y)2(z3u) = (x4z)5(y6u)

2. (2) (x1y)2z = x3(y4z)

3. (3) (x1z)2(y3z) = x4y.

The numbers 1, 2, 3,…, 6 represent binary operations and x, y, z and u are taken freely from certain sets.

We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.)

Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.

En étudiant certains cônes de fonctionnelles affines positives, nous avons été amenés à considérer les polynômes qui sont positifs sur l'intervalle [-1, 1]. En fait, la fonctionnelle affine

As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.

The Banach contraction principle has been generalized by Tan [6] to the mappings in separated locally convex spaces. We show that the result of Sehgal [5] and also of Holmes [3] can be generalized in the same way.

Throughout this note, we let X be a separated locally convex space, U a base for the closed absolutely convex neighborhoods of the origin O in X, K a nonempty subset of X, and Ta mapping from K to K.

The celebrated Fredholm theory of linear integral equations holds if the kernel K(x, y) or one of its iterates K(n) is bounded. Hilbert utilizing his theory of quadratic form was able to extend the theory to the kernels K(x, y) satisfying

a

b

where k is independent of u(x).

These theories were extended considerably by T. Carleman who deleted condition (b) above.

Equations involving this Carleman kernel have been found useful in connection with Hermitian forms, continued fractions, Schroedinger wave equations (see [1], [2]) and more recently in scattering theory in quantum physics, etc. [3]. See also [5] for a variety of applications and extensions.

Despite the abundance of generalizations of Carathéodory's theorem occurring in the literature (see [1]), the following simple generalization involving infinite convex combinations seems to have passed unnoticed. Boldface letters denote points of Rn and Greek letters denote scalars.

The following problem due to A. Boyd, has enjoyed a certain popularity in recent months with several mathematicians. A different solution to the one given here has been given independently by R. T. Bumby and J. Spencer.

The Problem, There are n ladies, and each one of them knows an item of scandal which is not known to any of the others. They communicate by telephone, and whenever two ladies make a call, they pass on to each other, as much scandal as they know at that time. How many calls are needed before all the ladies know all the scandal?

The purpose of this note is to prove the following:

Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.

Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite if

for every finitely nonzero sequence {fn} of vectors in H [1].

Let A be an algebra with 1 over a field F and let B be a fixed F-basis of A. Let F〈x〉=F〈x1,…, xn,…,〉 be the free algebra over F in noncommutative indeterminates x1,…, xn,…, and denote by AF〈x〉 the free product of A and F〈x〉 over F. The elements of AF〈x〉 of the form varies, repetitions allowed) form an F-basis of AF〈x〉. They will be referred to as basis monomials, and the involved in a particular basis monomial will be called the coefficients of that basis monomial.

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by

1

E0 = 1 and Er=0(r>m).

We define the rth symmetric mean by

2

where denote the binomial coefficient. If α1 α2,…,αm are positive reals then

we have two well-known inequalities

3

and

4

In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

In a recent note in the Bulletin, Murphy [5] gave a short proof that for complex m×n matrices A and B, r(A+B)=r(A)+r(B) if the rows of A are orthogonal to the rows of B and the columns of A are orthogonal to the columns of B. His proof was elegant and simple, an improvement on an earlier proof of the same result by Meyer [4].

The purpose of this note is to generalize a result of Gulliksen, Ribenboim and Viswanathan which characterized local group rings when both the ring and the group are commutative.

We assume throughout that all rings are associative with identity. If R is a ring we call R local if R/J(R) is a division ring where J(R) denotes the Jacobson radical of R. It is well known that R is local if and only if each element of R\J(R) is a unit. We need the following.

In [4] Hatcher introduced the notion of an identity in an arbitrary category and proved a characterization of quasivarietal subcategories which is similar to Birkhoff's characterization of varietal subcategories in universal algebra. The aim of this note is to show that the theorem of Hatcher as well as the categorical generalization of Birkhoff's theorem are special cases of a "relative" theorem, formulated with respect to a projective structure.

In this note we construct a pair of regular matrices T1 and T2 such that T1 is stronger than T2, and the T2-transforms of all bounded sequences are such that the sets of limit-points are connected, while there is a bounded sequence such that the set of limit-points of its T1-transform is not connected. This provides a negative answer to a question posed by Leviatan and Lorch [3, §6, (c)].

C(X) (resp. C*(X)) will denote as usual the ring of all (resp. all bounded) continuous functions into the real line R. Define C#(X) to consist of all f∊C(X) whose image M(f) in the residue class ring C(X)jM is real for every maximal ideal M in C(X). Then C# shares with C* the property of being an intrinsically determined subalgebra of C. Then C* shares with C* the property of being an intrinsically determined subalgebra of C. The compactification corresponding to C# (as uniformity determining subalgebra of C*) is thus also an intrinsically determined one. We show that this compactification is well known and "natural" in the cases of several elementary spaces X.