A theorem on connected graphs in which every edge belongs to a 1-factor

In this paper, we consider factor covered graphs, which are defined basically as connected graphs in which every edge belongs to a 1-factor. The main theorem is that for any two edges e and e′ of a factor covered graph, there is a cycle C passing through e and e′ such that the edge set of C is the symmetric difference of two 1-factors.

A note on extensions of Baer and P. P. -rings

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

On the Spectrum of almost periodic solutions of an abstract differential equation

If a linear operator A in a Banach space satisfies certain conditions, then the spectrum of any almost periodic solution of the differential equation u′ = Au + f is shown to be identical with the spectrum of f, where f is a Stepanov almost periodic function.

A generalization of Hermite's interpolation formula in two variables

Spitzbart [1] has considered a generalization of Hermite's interpolation formula in one variable and has obtained a polynomial p(x) of degree n + Σnj=0 = rj in x which interpolates to the values of a function and its derivatives up to order rj at xj, j = 0, 1,···n. Ahlin [2] has considered a bivariate generalization of Hermite's interpolation formula. He has developed a bivariate osculatory interpolation polynomial which agrees with f(x, y) and its partial and mixed partial derivatives up to a specified order at each of the nodes of a Cartesian grid. However, the above interpolation problem considered by Ahlin assumes that the values of partial and mixed partial derivatives of the same fixed order k – 1 are available at every point of the rectangular grid. It may also be observed that Ahlin's formula is essentially a Cartesian product of a special case of Spitzbart's formula in one variable.

On semigroups of endomorphisms of generalized Boolean rings

Magill in [4] first proved that two Boolean rings are isomorphic if and only if their respective endomorphism semigroups are isomorphic. His proof, however, relied on topological techniques. More recently Maxson has published a proof of the above using purely algebraic techniques [5]. In this paper, structure theorems are given which allow us to extend the above result to thepk-rings of Foster [1]. As a special case, the result is shown to apply also top-rings. An example is given to show that a further extension toJ-rings is impossible.

A note on group extensions

In [2] Hauptfleisch proved that if A, B, H, K are Abelian groups, φ:A → H and ψ:B → K are epimorphisms, then every central group extension G of H by K is homomorphic image of a central loop extension L of A by B. The aim of the present note is to prove (using almost the same argument as in [2])

Generalised Markovian control systems

The qualitative behaviour of control systems based on ordinary differential equations has been investigated with clarity and elegance using axiomatically defined General Control Systems. Here an attainablity set function, evolving in semigroup fashion, is the main entity of interest [1], [2], [3], [4].

A Contrast between complex and real-valued Taylor series

In this paper it is shown that if c is a point of the region of convergence of an analytic function f(z) = Σ∞ncnZn then in every neighborhood of c there exists a point e such that the value f(c) of the function f(z) is attained by some truncation Σkn=0cnZn off(z) at z = e, i.e., Σkn=0cnen = Σ∞n=0cncn. Also it is shown that the above does not hold in the case of real-valued functions of a real variable.

Semirings with a completely 0-simple additive semigroup

In a previous paper [2], we gave an explicit description of the structure of all semirings with a completely simple additive semigroup. The next step is then clearly to consider semirings with a completely 0-simple additive semigroup. We are able to classify these semirings according to the multiplicative nature of their additive zero. Let R be a semiring whose additive semigroup is completely 0-simple with zero ∞. First, if ∞ ∞ ≠ ∞, then the multiplication of R is trivial. Besides these trivial semirings, another class of semirings with a completely 0-simple additive semigroup can be easily obtained by adjoining an element ∞ which is together an additive zero and a multiplicative zero.