In this paper we propose a general setting in which to study the radical theory of group graded rings. If is a radical class of associative rings we consider two associated radical classes of graded rings which are denoted by G and ref. We show that if is special (respectively, normal), then both G and ref are graded special (respectively, graded normal). Also, we discuss a graded version of the ADS theorem and the termination of the Kurosh lower graded radical construction.

Here we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups , with respect to generators {x, y}. This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group

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The authors begin by presenting a brief survey of the various useful methods of solving certain integral equations of Fredholm type. In particular, they apply the reduction techniques with a view to inverting a class of generalized hypergeometric integral transforms. This is observed to lead to an interesting generalization of the work of E. R. Love [9]. The Mellin transform technique for solving a general Fredholm type integral equation with the familiar H-function in the kernel is also considered.

This paper gives variants of results from classical algebraic geometry and commutative algebra for quadratic algebras with conjugation. Quadratic algebras are essentially two-dimensional algebras with identity over commutative rings with identity, on which a natural operation of conjugation may be defined. We define the ring of conjugate polynomials over a quadratic algebra, and define c-varieties. In certain cases a close correspondence between standard varieties and c-varieties is demonstrated, and we establish a correspondence between conjugate and standard polynomials, which leads to variants of the Hilbert Nullstellensatz if the commutativering is an algebraically closed field. These results may be applied to automated Euclidean geometry theorem proving.

A continuous selection and a coincidence theorem are proved in H-spaces which generalize the corresponding results of Ben-El-Mechaiekh-Deguire-Granas, Browder, Ko-Tan, Lassonde, Park, Simon and Takahashi to noncompact and/or nonconvex settings. By applying the two theorems, some intersection theorems concerning sets with H-convex sections are obtained which generalize the corresponding results of Fan, Lassonde and Shih-Tan to H-spaces. Some applications to minimax principle are given.

A lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.

Local uniqueness of solutions of the characteristic Cauchy problem is shown for operators which are perturbations of operators which already have such a uniqueness.

A Bochner-Martinelli-Koppelman type integral formula with weight factors is derived on complete intersection submanifolds of domains of Cn.

In this paper we extend the notion of perfect, θ-continuous, irreducible and θ-perfect mappings to bitopological spaces. The main result is the following: the (small) image of an (i, j)-canonical open sets is an (i, j)-canonical open set under a pairwise θ-closed irreducible surjective mapping. Also we extend the notion of θ-proximity spaces to quasi θ-proximity spaces and point out the interrelation between it and separated quasi-proximity spaces by means of a pairwise θ-perfect irreducible mappings.

We consider the existence of multiple positive solutions of a nonlinear two-point boundary value problem by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the number of positive solutions and find their Conley indices and thus determine their stabilities.

The group near-ring constructed from a right near-ring R and a group G is studied in the special case where the near-ring is distributively generated. In particular, results concerning homomorphisms of near-rings or of groups and the augmentation ideal are obtained which resemble closely those obtained for group rings.

A group is said to be factorizable if it has a finite number of abelian subgroups, H1, H2, … Hn, such that G = H1H2 … Hn. It is shown that, if G is a factorizable or connected locally compact group, then every derivation from L1 (G) to an arbitrary L1 (G)-bimodule X is continuous.

Our set-up will consist of the following: (i) a graph with vertex set V and edge set E; (ii) for each vertex ∈ V a non-trivial group Gv given by a presentation (xν; rν); (iii) for each edge e = {u, ν} ∈ E a group Ge given by a presentation (xu, xv; re) where re consists of the elements of ru ∪ rv, together with some further words on xu ∪ xv. We let G = (x; r) where x = ∪v∈v xv, r = ∪e∈E re. Ouraim is to try to describe the structure of G in terms of the groups Gv, (v ∈ V), Ge (e ∈ E). Under suitable conditions the natural homomorphisms Gv, → G (ν ∈ V), Ge → Ge (e ε E) are injective; and there is a short exact sequence (where, for any group H, IH is the augmentation ideal). Some (co)homological consequences of these resultsare derived.

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruence relations of K. In addition, if L has more than one element, then we show how to construct K so that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.

We study the bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the exact number of steady-state solutions which are totally ordered in an order interval. We are then able to find their Conley indices and thus determine their stabilities.

Special radical classes of near-rings are defined and investigated. It is shown that our approach, which differs from previous ones, does cater for all the well-known radicals of near-rings. Moreover, most of the desirable properties from their ring theory counterpart are retained. The relationship between the special radical of a near-ring and the corresponding matrix near-ring is given.

A composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

In this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

Let 1≤p <∞ and 1/p+1/q = 1. If φ ∈ Lq, we denote by Tφ the functional defined on the Hardy space Hp by . A function f in Hp, which satisfies Tpφ(f) = ‖Tpφ‖ and ‖f‖p ≤ 1, is called an extremal function. Also, φ is called an extremal kernel when ‖φ‖q =‖Tpφ‖. In this paper, using the results in the case of p = 1, we study extremal kernel and extremal functions for p > 1.

In this paper we characterize the ideals of the centralizer near-ring N = MR(R2), where R is a commutative principle ideal ring. The characterization is used to determine the radicals Jυ(N) and the quotient structures N/ Jv(N), v = 0, 1, 2.