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.

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:

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)

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.

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.

When a disc rotates in a fluid at rest, fluid near the disc acquires azimuthal momentum because of the viscous torque of the disc and outwards radial momentum under the action of centrifugal forces. The resultant flow is essentially a swirling jet. Away from the disc continuity requires the existence of an axial flow towards the disc to compensate for the fluid which has been thrown outwards. If the disc is finite there is a discontinuity in the boundary conditions at the edge of the disc where the no-slip condition is suddenly replaced by a condition of zero stress in the plane of the disc. The flow discharged from the edge of the disc is essentially a wake embedded in a swirling radial jet. It appears that no investigation of this wake has yet been made.

Associated with each uniform space (S, U) there is a uniform lattice (P, V, ≦) called the scale of (S, U). The scale was first introduced by D. Bushaw (see [2] and [3]) for the purpose of studying stability in topological dynamics. Further properties of the scale were investigated in [4], and this paper pursues one of the research problems suggested in the latter paper.

The present paper on functions of bounded variation relative to a set has its point of departure in the work of R. L. Jeffery [10]. Below we recapitulate Jeffery's class U of functions of bounded variation relative to a set, we state and prove a number of preliminary lemmas and theorems, we introduce a suitable pseudo- metric space (X, d) of such functions, and the analogous space , and prove that (X, d) is separable, that every closed sphere in (X, d) is compact and that is complete. These results extend known results of C. R. Adams, and C. R. Adams and A. P. Morse for the space of usual BV functions.

In [1]R. G. Douglas proved that if A and B are continuous linear operators on a Hilbert space X, the following three conditions are equivalent:

J. C. Kelly [2] introduced the concept of a bitopological space. Lane [3], Patty [4] and Pervin [5] have continued his work. Our purpose in this paper is to identify the projective objects in a suitable category of bitopological spaces after the manner of Gleason [1] and generalize his theorem that in the category of compact Hausdoriff topological spaces, the projective spaces are precisely the extremally disconnected ones.

Let E be the set of idempotents of a regular semigroup; we prove that V(En) = En+1 (see below for the meaning of this notation). This generalizes a result of Miller and Clifford ([3], theorem 4, quoted as exercise 3(b), p. 61, of Clifford and Preston [1]) and the converse, proved by Howie and Lallement ([2], lemma 1.1), which together establish the case n = 1. As a corollary, we deduce that the subsemigroup generated by the idempotents of a regular semigroup is itself regular.

A semigroup is a nonvoid Hausdorff space together with a continuous associative operation. A semiring is a nonvoid Hausdorff space together with a couple of continuous associative operations, one of which (usually denoted as multiplication) distributes over the other (usually denoted as addition). If R is a semiring then an R-semimodule is a semigroup M under addition together with a continuous operation R × M → M which satisfies the associativity and distributivity conditions usually stipulated in the instance of an R-module. It is purpose of this paper to establish for semimodules certain propositions proved by Kaplansky [4], Pearson [8], Selden [9], Beidleman-Cox [1] and others.

The paper is closely related to [1] and [2]. A skew connection in a vector bundle E as defined here is a pseudo-connection (in the sense of [1]) which can be changed into a connection by transforming separately the bundle E itself and the bundle of its differentials, i.e. one-forms on the base with values in E. The properties of skew connections are thus expected to be only “algebraically” more complicated than those of connections; especially one can follow the pattern of [1], and prolong them to obtain higher order semi-holonomic and non-holonomic pseudo-connections. It is shown in this paper that under some circumstances the main theorem of [1] or [2] applies also to skew connections.

Let A be a class of rings, and let L(A) denote the lower radical class determined by A. In [3] Yu-Lee Lee showed that L(A) may b constructed in the following manner: Let H(A) be the class of all homomorphic images of rings in A.

We use the equivalence of the sequence space 12 and the function space L2 to obtain an orthogonal transformation related to two series inversion formulae due, respectively, to Titchmarsh [1, 2] and to Linfoot and Shepherd [3].

All rings considered are to be associative. For definitions not included in this paper see [2]. Let R be a ring and S a subring of R.

Finite groups in which normality is transitive have been studied by Best and Taussky, [1], Gaschütz, [3], and Zacher [16]. Infinite soluble groups in which normality is transitive have been studied by Robinson in [9]. A subgroup H of a group G is subnormal in G if H can be connected to G by a chain of r subgroups, in which each is normal in its successor, where r is a non-negative integer. The least such r is called the subnormal index of H in G (or the defect of H in G). Then groups in which normality is transitive are precisely those in which every subnormal subgroup has subnormal index at most one. Thus the structure of soluble groups in which every subnormal subgroup has subnormal index at most n (such a group is said to have bounded subnormal indices) has been dealt with by Robinson in [9] for the case where n is one. However Theorem D of [12] states that a soluble group of derived length n can be embedded in a soluble group in which the subnormal indices are at most n. Therefore we must impose further conditions on the groups if we hope to obtain any worthwhile results for the above problem with n greater than one.

Let E1 and E2 be real Banach spaces and let L(E1) and L(E2) be the Banach algebras of all continuous linear mappings on E1 and E2 respectively. It is a well- known result of M. Eidelheit [1] that L(E1) that L(E2) are isomorphic as rings if and only if E1 and E2 are topologically and algebraically isomorphic. It is easy to see that the essential part of his proof is the following fact.