The class of commutative rings known as Baer rings was first discussed by J. Kist [4], where many interesting properties of these rings were established. Not necessarily commutative Baer rings had previously been studied by I. Kaplansky [3], and by R. Baer himself [1]. In this note we show that commutative Baer rings, which generalize Boolean rings and p-rings, satisfy the Birkhoff conditions for a variety. Next we give a set of equations characterising this variety involving + and * as binary operations, – and as unary operations, and 0 as nullary operation. Finally we describe Baer-subdirectly irreducible commutative Baer rings and state the appropriate representation theorem.

The series is said to be summable |Eα| (0 < α < 1) if where Sv = a0 + a1 … + av, and . Since (see [2]), (1) is equivalent to .

We answer in the negative (by considering an example) the problem posed in exercise 6 of § 2.3, page 62 [1], namely: If a D-class D of a semigroup S is a subsemigroup of S, then is D necessarily bisimple? For any semigroup T, we let LT, RT, HT, DT and JT denote Green's relations on T.

It is well known that in the category of all groups and homomorphisms, every epimorphism is onto. This result does not hold for certain other categories of groups. The condition that an epimorphism θ:G → A is not onto is equivalent to the condition that θ(G) is a proper subgroup A with the property that any two homomorphisms α, β on A which agree elementwise on θ(G) must agree on A. Such a subgroup can be called dense (see e.g. [1]). Naturally the existence of such homomorphisms α and β depend on the particular class of groups that is available. We will choose to work within the context of varieties even though many of the results will hold true for more modest classes of groups.

If ρ is a congruence on a regular semigroup S, then the kernel of ρ is defined to be the set of ρ-classes which contain idempotents of S. Preston [7] has proved that two congruences on a regular semigroup coincide if and only if they have the same kernel: this naturally poses the problem of characterizing the kernel of a congruence on a regular semigroup and reconstructing the congruence from its kernel. In some sense this problem has been resolved by the author in [5]. Using the well-known theorem of M. Teissier (see for example, A. H. Clifford and G. B. Preston [1], Vol. II, Theorem 10.6), it is possible to characterize the kernel of a congruence on a regular semigroup S as a set A = {Ai: i ∈ I} of subsets of S which satisfy the Teissier-Vagner-Preston conditions:

By a topological semiring I mean a Hausdorff space together with two continuous associative operations, addition and multiplication, such that the multiplication distributes across the addition from both sides.

It is well known that a Boolean ring is isomorphic to a subdirect sum of two-element fields. In [3] a near-ring (B, +, ·) is said to be Boolean if there exists a Boolean ring (B, +, Λ, 1) with identity such that · is defined in terms of +, Λ, and 1 and, for any b ∈ B, b · b = b. A Boolean near-ring B is called special if a · b = (a ν x) Λ b, where x is a fixed element of B. It was pointed out that a special Boolean near-ring is a ring if and only if x = 0. Furthermore, a special Boolean near-ring does not have a right identity unless x = 0. It is natural to ask then whether any Boolean near-ring (which is not a ring) can have a right identity. Also, how are the subdirect structures of a special Boolean near-ring compared to those of a Boolean ring. It is the purpose of this paper to give a negative answer to the first question and to show that the subdirect structures of a special Boolean near- ring are very ‘close’ to those of a Boolean ring. In fact, we will investigate a class of near-rings that include the special Boolean near-rings and the Boolean semi- rings as defined in [8].

Let {Pn} be a given sequence of constants, real or complx, such that then , defines the sequence {tn} of (N, pn) means of Σnan. The series Σnan is said to be summable I|N, pn|, if {tn} ∈ BV, −|−tn−1| ∞.

Let R, μ and Mμ denote the set of real numbers, Lebesgue outer measure and the class of Lebesgue measurable subsets of R respectively. It is easy to prove that the complement Ec of E ∈ Mμ is a set of Lebesgue measure zero if the inequality holds for some δ > 0 and all intervals I of R. However, in Hewitt [1], raised a problem whether the result is still true if E is not a priori measurable set. In this paper, a negative answer to this question is given through a counter-example. Also, it is proved that for a given set E ∈ M,μ with μ(E) > 0 there is a non-measurable subset A of E satisfying μ(E)

The asymptotic behaviour, for large real positive values of the independent variable t, of solutions of the ordinary differential equation is considered. (The dots denote differentiation with respect to t.) It is shown that this asymptotic behaviour can be obtained by comparison with the differential equation For both equations when a2 < 2 the solutions are unbounded oscillatory functions but when a2 > 2 the solutions are essentially monotone. The critical case a2 = 2 has not been investigated here although the present results suggest that the solutions are probably monotone. The asymptotic estimates of the solutions of equation (1) are first obtained by heuristic means commonly used by Applied Mathematicians working on singular perturbation problems. A basis for a formal expansion is obtained in terms of a recurrence relation and the first few terms are displayed explicitly. These formal expansion estimates are used to guide a rigorous analysis which is essentially an extension of the techniques used in validating solutions obtained by the Langer related equation technique. For the oscillatory case, a pair of Volterra integral equations is used to justify the asymptotic representations for an arbitrary large interval for t but not including the point at infinity.

Abstract. For large real positive values of the parameter k asymptotic representations of integrals of the form , where f and g are analytic functions, can be obtained by using methods such as steepest desents. Here methods are considered for obtaining estimates, for fixed values of k, of the minimum errors achievable when such asymptotic representations are appropriately curtailed. A priori criteria are derived for the optimum point at which to curtail such asymptotic representations are appropriately curtailed. A priori criteria are derived for the optimum point at which to curtail such asymptotic representations. Both the curtailment points and the minimum errors are related to the distance between certain marked points on the path of integration and the singular points of f(u) and the zeros of g(u). The analysis permits the determination of errors whose presence is not indicated by the numerical behaviour of the asymptotic representations. It is also capable of extension to complex parameters k and to the derivation of asymptotic representations for the most significant errors. It can therefore be used to extend the domain of k for which asymptotic representations are available.

Let A and B denote subsets of an abelian group (G, +). The pair (A, B) is called a complementing pair for A + B provided each element of the sum is uniquely represented. A complementing pair (A, B) for C is denoted (A, B) ~ C.

The purpose of this paper is to provide a proof for a result announced in [3]. The result arose from a search for just-non-Cross varieties (recall that a Cross variety is one which can be generated by a finite group, and a just-non-Cross variety is a non-Cross variety every proper subvariety of which is Cross). For the motivation for this search, we refer the reader to [12]: for related results, see [1], [12], [13].

Suppose that Ф is a topological space equipped with a Hausdorff topology and that T is a continuous function mapping Ф into Ф. We discuss the existence and uniqueness of fixed points of T and the convergence of the Picard sequence of iterates, from the viewpoint of the existence of a homotopy with special properties.

Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

Generalized integers are defined in {2] as follows: Suppose there is given a finite or infinite sequence {p} of real numbers which are called generalized primes such that 1 < p1 < p2 < …. Form the set {l} of all possible p-products, i.e. the products of the form where are integers ≧ 0 of which all but a finite number are 0. Call these numbers generalized integers and suppose that no two generalized integers are equal if their α's are different. Then arrange {l} in an increasing sequence I = l1 < l2 < l3 < … < ln < ….

The Laplace-Stieltjes Transform m(s) of the matrix renewal function M(t) of a Markov Renewal process is expanded in powers of the argument s, in this paper, by using a generalized inverse of the matrix I–P0, where P0 is the transition probability matrix of the imbedded Markov chain. This helps in obtaining the values of moments of any order of the number of renewals and also of the moments of the first passage times, for large values of t, the time. All the results of renewal theory are hidden under the Laplacian curtain and this expansion helps to lift this curtain at least for large values of t and is thus useful in predicting the number of renewals.

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, …) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where bj, (j = 0, 1, …) is the probability that j individuals enter the population at each generation. Defining Xn (n = 0, 1, …) to be the population size at the n- th generation, it is known that {Xn} defines a Markov chain on the non-negative integers.

A self-mapping T of a subset C of a normed linear space is said to be non-expansive provided ║Tx — Ty║ ≦ ║x – y║ holds for all x, y ∈ C. There has been a number of recent results on common fixed points of commutative families of nonexpansive mappings in Banach spaces, for example see DeMarr [6], Browder [3], and Belluce and Kirk [1], [2]. There have also been several recent results concerning common fixed points of two commuting mappings, one of which satisfies some condition like nonexpansiveness while the other is only continuous, for example see DeMarr [5], Jungck [8], Singh [11], [12], and Cano [4]. These results, with the exception of Cano's, have been confined to mappings from the reals to the reals. Some recent results on common fixed points of commuting analytic mappings in the complex plane have also been obtained, for example see Singh [13] and Shields [10].

In 1957 Kurzweil [1] proved some theorems concerning a generalized type of differential equations by defining a Riemann-type integral, but he did not study its properties beyond the needs of that research. This was done by R. Henstock [2, 3], who named it a Riemann-complete integral. He showed that the Riemann-complete integral includes the Lebesgue integral and that the Levi monotone convergence theorem holds. The purpose of the present paper is to give a necessary and sufficient condition for a function to be Riemann-complete integrable and to establish a termwise integration theorem for a uniformly convergent sequence of Riemann-complete integrable functions.