A class of Noetherian semigroup algebras K[S] is described. In particular, we show that, for any submonoid S of the semigroup Mn of all monomial n × n matrices over a polycyclic-by-finite group G, K[S] is right Noetherian if and only if S satisfies the ascending chain condition on right ideals. This is then used to prove that every prime homomorphic image of a semigroup algebra of a finitely generated Malcev nilpotent semigroup S satisfying the ascending chain condition on right ideals is left and right Noetherian.

This paper is concerned with the existence and qualitative property of solutions for a Hénon-like equation

where Ω = {x ∈ ℝN : 1 < |x| < 3} with N ≥ 4, 2* = 2N/(N − 2), τ > 0 and ε > 0 is a small parameter. For any given k ∈ ℤ+, we construct positive solutions concentrating simultaneously at 2k different points for ε sufficiently small, among which k points are near the interior boundary {x ∈ ℝN : |x| = 1} and the other k points are near the outward boundary {x ∈ ℝN : |x| = 3}. Moreover, the 2k points tend to the boundary of Ω as ε goes to 0.

Sufficient conditions are obtained for a ring R, faithfully graded by a bisimple inverse semigroup S, to be (a) prime and (b) right primitive, these conditions being on the subring RG consisting of all elements of R with support contained in G, a maximal subgroup of S. Earlier results on semigroup rings arise as special cases.

In this paper, the electrostatic potential of a point charge in a Reisser-Nordström gravitational field is found in closed form by using the theory of Hadamard's elementary solution of a partial differential equation of elliptic type.

We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2 Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2 P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y + Q(x)P(x))2 – Q(x)2 = 0 of hyperelliptic type.

Weighted Poincaré inequalities are established in any bounded domain D in ℝn (n ≧ 2), and their connection with the Minkowski content of ∂D is explored.

In this paper we study the Morse functions on Grassmann manifolds. It is shown that many well-known calibrations can be used to construct degenerate Morse functions on Grassmann manifolds. In the study we use Clifford algebras and Lie groups.

We compute the heat content asymptotics of odd dimensional hemispheres in closed form.

Large time behaviour of solutions to a damped quasi-linear wave equation are studied. Conditions are obtained which guarantee the global existence of a classical solution. The asymptotic behaviour of this solution is studied in the case of a unique equilibrium solution and in the case of multiple equilibria. The results are applied to various special examples.

Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.

We provide a general procedure for computing the algebraic K-theory of finitely generated virtually free groups. The procedure describes these groups in terms of the algebraic K-theory of various finite subgroups and various Farrell Nil groups. We illustrate this process by carrying out the computation for several interesting classes of examples. The first two classes serve as a check on the method and show that our algorithm recovers results that already exist in the literature. The last two classes of examples yield new computations.

Using an algebraic approach to the nth order linear differential equations (see [3] for n = 2 and [13 and 14] for n ≧ 2) it is shown the essence of the relation between the limit circle classification and boundedness of solutions of y″ = q(t)y. On this example it is demonstrated that if a problem can be formulated in the rank of the approach, then it is only a technical matter to answer it, e.g. the relationship studied here is based on a standard fact from the theory of functions.

We establish an existence theorem for a stationary semiconductor model which takes into account the current generated by the gradient of the temperature.

The purpose of this paper is to give universal bounds for the order of the Postnikov k-invariants of infinite loop spaces. This is done by giving universal bounds for the order of the k-invariants of m-connected r-fold loop spaces in dimensions ≦ r + 2m. An application of the result provides information on the Hurewicz homomorphism between the algebraic K-theory of aring and the homology of its general linear group.

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions'.

We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group $\mathbb{O}(3)$.

The author has recently proposed a new algorithm for the solution of the Lyapunov matrix equation of stability theory. This algorithm is based on a formula for the solution of a special case of the equation. This formula is established in the present paper by means of a geometric interpretation. The key ideas are the uses of shift operators and non-orthogonal projections in infinite-dimensional Hilbert space.

A Banach algebra A is said to be topologically nilpotent if sup {‖x1x2…xn‖1/n: xi ∈ A, ‖xi‖ ≦ 1 (1 ≦ i ≦ n)} tends to zero as n → ∞. A Banach algebra A is uniformly topologically nil if sup {‖xn‖ 1/n: x ∈ A, ‖x‖ ≦ 1} tends to zero as n → ∞. These notions are equivalent for commutative algebras and a topological version of the Nagata-Higman Theorem gives a partial result for the non-commutative case. Topologically nilpotent algebras have a strong non-factorisation property and this yields theorems of the type “factorisation implies the existence of arbitrarily slowly decreasing powers”. Extensions of topologically nilpotent algebras by topologically nilpotent algebras are topologically nilpotent.

Sufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability of

implies the asymptotic stability of

then these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example

We prove a Carleman estimate for elliptic second-order partial differential expressions with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function u ∈ W2,2 with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular, we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and Lipschitz constant.

A local homeomorphism from a compact, connected manifold with boundary to a simply connected manifold without boundary is shown to be one-to-one if it is one-to-one on each component of the boundary.