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.

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.

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.

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.

Following on from the work of T. P. Speed [1], we will deal with some obvious conjectures, using particularly results 3.3 and 8.5 of [1].

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

Let M be an n-dimensional (n = 2m + 1, m ≦ 1) real differentiable manifold. if on M there exist a tensor field , a contravariant vector field ξi and a convariant vector field ηi such that then M is said to have an almost contact structure with the structure tensors (φ,ξ, η) [1], [2]. Further, if a positive definite Riemannian metric g satisfies the conditions then g is called an associated Riemannian metric to the almost contact structure and M is then said to have an almost contact metric structure. On the other hand, M is said to have a contact structure [2], [4] if there exists a 1-form η over M such that η ∧ (dη)m ≠ 0 everywhere over M where dη means the exterior derivation of η and the symbol ∧ means the exterior multiplication. In this case M is said to be a contact manifold with contact form η. It is known [2, Th. 3,1] that if η = ηidxi is a 1-form defining a contact structure, then there exists a positive definite Riemannian metric in gij such that and define an almost contact metric structure with and ηi where the symbol ∂i standing for ∂/∂xi.

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

In 1942, Burgers [1] observed that in cylindrical polar coordinates, the steady Navier-Stokes equation governing viscous incompressible fluid motion can be reduced to a set of ordinary differential equations if the velocity components vr, vo and vz are assumed to have a special form.

The set E of idemponents of a semigroup S can be partially ordered by defining e ≦f and only if ef = fe = e (e,f ∈ E). If E = {ei: i = i = 0,1…} and under this ordering e0 > e1 > e2… then we call S an ω-semigroup. Munn [10] has given a complete classification of simple regular ω-semigroups in terms of groups and group homomorphisms. Let ∧0(S) denote the set of congruences on a simple regular w-semigroup S consisting of those congruences which either are idempotent-separating or are group congruences on S. It is evident that ∧0(S) is a sublattice of the lattice of all congruences on S.

We obtain the enumerator by node-valencies of planted plane trees, whose square gives the enumerator of rooted plane trees. We also study the enumeration by number of nodes and black-node-valencies of bichromatic rooted plane trees, encountering a remarkably simple inversion formula. Finally, we remark that these bichromatic trees are in 1–1 correspondence with solutions to the weak lead ballot problem.

P. M. Cohn [7] calls a submodule P of the left A-module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E. This definition of purity, which Cohn [7] has shown to be equivalent to the usual one when A is a PID (= Principal Ideal Domain), was studied in [9] and [10]. Here we show that the ring A is (von Neumann) regular if every left (or right) ideal is pure. This leads us to define regular modules as modules all of whose submodules are pure. The ring A is then regular if all its left (or right) A-modules are regular. A regular socle, analogous to the usual socle is defined. For commutative A, some localization theorems are proved, and used to settle a conjecture of Bass [1] concerning commutative perfect rings.

Mrowka and Engleking [1] have recently introduced a notion more general than that of compactness. Perhaps the most convenient direction at departure is the following: for spaces X and E, X is said to be E-compact if X is topologically embeddable as a closed subset of a product Em for some cardinal m, in which case we write X ⊂cl Em. More generally, X is said to be E-completely regular if X is topologically embeddable in a product Em for some m. For example, if we take E to be the unit interval I, we obtain the class of compact spaces and completely regular spaces, respectively, as is well-known. The question then arises, of course, given a space E, what spaces are compact with respect to it? A related question, to which we address ourselves in this note, is the following. Denote by K[E] all those topological spaces which are E-compact. Then we ask: are there very many distinct E-compact classes? It will develop that there are indeed quite a large number of such classes.

In this paper we shall deal with a probability space (S, Σ, P), a separable Banach space X having its strong dual X* and a strictly stationary random sequence defined as in [7], where are X-valued, Gelfand-Pettis (weakly) integrable [6], [9], and strongly measurable random variables. In the case when Yk's are Bochner (strongly) integrable random variables one can find the ergodic theorem for such a sequence and, with respect to strong convergence in X, in the papers [7], [8].

A Sasakian space [1]Mn (n = 2m + 1) is a Riemannian n-space with a positive definite metric tensor gij and a unit Killing vector field η which satisfies where the comma denotes covariant differentiation with respect to the metic tensor. In a recent paper [2] M. C. Chaki and A. N. Roy Chowdhury studied conformally recurrent spaces of second order, or briefly conformally 2-recurrent spaces, that is, non-flat Riemannian spaces Vn (n > 3) defined by where is the conformal curvature tensor: and alm is a tensor not identically zero.