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.

The ventricular rhythm differs from the auricular, firstly, on the occurrence of a ventricular extra-systole, and, secondly, in cases where there is allorhythmia due to depression of conductivity of the heart muscle. When conductivity is depressed, the contraction stimulus at regular or irregular intervals may be blocked at the auriculo-ventricular muscle bridge; so that whereas some stimuli induce both auricular and ventricular systole, others induce auricular systole alone, the ventricular systole being missed. That condition, incomplete heart-block, is of not infrequent occurrence in the human heart, and has been graphically recorded by Mackenzie (1), Finkelnburg (2), and other writers.

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

In this paper we prove, on one hand, extrapolation from infinity for the one-sided classes , and on the other hand, the BMO-boundedness for one-sided singular integrals. We also provide several applications of our results.

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.

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

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.

This paper commences with the following extract from the author's private journal, of date January 6, 1858:—

“Let be rectangular components of an impulsive force and an impulsive couple applied to a solid of invariable shape, with or without inertia of its own, in a perfect liquid, and let u, ν, ω, ѽ, ρ, σ, be the components of linear and angular velocity generated.

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 $\mathbb{O}(3)$.

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.

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.

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.

This paper is concerned with the existence results for generalized transition waves of space periodic and time heterogeneous lattice Fisher-KPP equations. By constructing appropriate subsolutions and supersolutions, we show that there is a critical wave speed such that a transition wave solution exists as soon as the least mean of wave speed is above this critical speed. Moreover, the critical speed we construct is proved to be minimal in some particular cases, such as space-time periodic or space independent.

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.

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.

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.

We extend a result of J. F. Toland concerning bifurcation from infinity and we made some applications to variational inequalities.

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.

When we consider that the Phlogistic Theory formed, as it were, the central point round which the facts of chemistry first crystallized into regular scientific form, and that for more than a hundred years it was recognised by all as the foundation of the science, we might reasonably suppose that it should contain at least some germ of truth.

I think I shall be able to show in the following note that not only is this the case, but that the theory itself, as stated by its founders, Beccher and Stahl, is, if not strictly true, a very close approximation to what we now recognise as truth.