In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings.

We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup ft with preorder defined by α≺β if β is in the ideal generated by α, and we use the direct sum to construct an Ω-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in Ω when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring S[ft] correspond bijectively to pairs (Φ, Q) with Φ a prime ideal of Ω and Q a prime ideal of S. The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when Ω is an infinite free band, by showing that S[Ω] is prime whenever S is.

When Ω satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of Ω, and also to the direct product of the component rings.

If the group of inner automorphisms of a semigroup S of transformations of a finite n-element set contains an isomorphic copy of the alternating group Altn, then S is an Sn-normal semigroup and all the automorphisms of S are inner.

In this note we consider the Dirichlet problem Δu + f(x, u)=0 in Ω, u = 0 on ∂Ω here Ω is a bounded domain in ℝn(n≧3), with smooth boundary ∂Ω. We prove the existence of strong solutions to the previous problem, which are positive if f satisfies a suitable condition. As a consequence we find that the problem with , may have positive solutions even if g is not a lower-order perturbation of Next We examine the case .

Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.

The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halicioğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.

In this paper we generalize the folding process initiated by Stallings for graphs to a class of generalized covering spaces. These spaces are called pinched coverings or pinched cores, depending on the particular situation. We then apply our generalized folding process to manipulate these spaces into actual coverings. By using elementary homotopy arguments, we can calculate the fundamental groups of these spaces. As a corollary to our main result we obtain a generalization of a result due to Gersten concerning monomorphisms between free products of groups.

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.

For distributive double p-algebras, a close connection is established between being congruence coherent and congruence regular. Every congruence regular distributive double p-algebra is congruence coherent. Even though every congruence coherent distributive double p-algebra that has either a non-empty core or finite range is congruence regular, an example is given that is congruence coherent but not congruence regular.

The initial-boundary value problem for the nonlinear heat equation u1 = Δu + λf(u) might possibly have global classical unbounded solutions, for some “critical” initial data . The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;λ) for some values of λ We find, for radial symmetric solutions, that u*(r, t)→w(r) for any 0<r≤l but supu*(·, t) = u*(0, t)→∞, as t→∞. Furthermore, if , where is some such critical initial data, then û = u(x, t; û0) blows up in finite time provided that f grows sufficiently fast.

In this paper we consider the quartic diophantine equation 3(y2 – 1) = 2x2(x2 – 1) in integers x and y. We show that this equation does not have any other solutions (x, y) with x≧0 than those given by x = 0,1,2,3,6,91. Two approaches are emphasized, one based on diophantine approximation techniques, the other depends on the structure of certain quartic number fields.

If G is a hyperfinite locally soluble group and A an artinian ZG-module then Zaĭcev proved that A has an f-decomposition. For G being a hyper-(cyclic or finite) locally soluble group, Z. Y. Duan has shown that any periodic artinian ZG-module A has an f-decomposition. Here we prove that: if G is a hyper-(cyclic or finite) group, then any artinian ZG-module A has an f-decomposition.

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

In this paper we present an existence result for a class of nonlinear evolutions inclusions. A result on the compactness of the solution set for a differential inclusion is also established.

Let G be a Moore group. Then, for each f∈L1(G), the convolution operator Lf: L1(G)→L1(G) is decomposable. On the other hand, there is a discrete probability measure µ on a compact group G such that Lµ: Ll(G)→Ll(G) fails to be decomposable.

The aim of this work is to study the existence of solutions for a mathematical model of the displacement of a piezoviscous lubricant between two elastic surfaces. As we deal with a rolling ball contact problem, the deformations are modelled by the linear Hertzian theory. The fluid pressure behaviour is governed by the classical Reynolds equation for thin film displacement. The relevant aspect of cavitation in lubrication is described by means of the Elrod Adams model which leads to a mathematical free boundary problem.

The two main original features of the model problem in relation to previous works are: the supply of lubricant coming from a groove that is transversal to the direction of fluid displacement and the consideration of a piezoviscous law of Barus. Mathematically, the first one leads to a mixed Dirichlet-Neumann problem for the Reynolds equation and the second one involves an additional nonlinearity in a diffusion type term.

We consider nearly-Euclidean balls of the shape

where ε is a small positive number, and n is even. If ε is small enough, then the maximum lattice-packing density of this body is essentially greater than the Minkowski-Hlawka bound for large n.

We prove that for a function f(z) transcendental and meromorphic in the plane and not of the form exp(az + b), we have either N(r, 1/ff″)≠0(T(r, f′/f)) or .