This paper is concerned with the blowup of positive solutions of the semilinear heat equation

with zero boundary conditions. The domain Ω is supposed to be smooth, convex and bounded. We first show that, under the assumption that the initial data are uniformly monotone near the boundary, solutions that exist on the time interval (0, T form a compact family in a suitable topology. We then derive some localisation properties of these solutions. In particular, we discuss a general criterion, independent of the initial data, which in some cases ensures single-point blowup.

By using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the full C*-algebra C*ℐ and the reduced C*-algebra C*(ℐ) with particular emphasis on the case of the free monogenic inverse semigroupℑℐt.

We consider a mathematical model for the motion of a marked monomer in a system of reacting polymers at equilibrium. A well-posed integro-differential initial value problem for the probability of finding the marked monomer in a molecule of a given length is formulated. We prove exponential convergence of the probability to a unique equilibrium distribution. A quite complete spectral analysis is carried out for a self adjoint operator, which is a perturbation of a multiplication operator by an integral operator and is related to the generator of the time evolution.

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.

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.

We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda > 0$, there is a travelling wave solution to fKdV and fNLS $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

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.

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.

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.

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 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.

We consider the positive solutions to singular boundary-value problems of the form where λ > 0, β ∈ (0,1) and Ω is a bounded domain in ℝN, N ≥ 1, with smooth boundary ∂Ω. Here, we assume that f: [0, ∞) → (0, ∞) is a C1 non-decreasing function and f(s)/sβ is decreasing for s large. We establish the uniqueness of the positive solution when λ is large.

In this paper, we deal with the nonlinear Schrödinger–Poisson system

where λ > 0, V and Q are radial functions, which can be vanishing or coercive at ∞. With assumptions on f just in a neighbourhood of the origin, existence and multiplicity of non-trivial radial solutions are obtained via variational methods. In particular, if f is sublinear and odd near the origin, we obtain infinitely many solutions of (SP)λ for any λ < 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.

The purpose of this paper is to prove strong-type inequalities with one-sided weights for commutators (with symbol b ∈ BMO) of several one-sided operators, such as the one-sided discrete square function, the one-sided fractional operators, or one-sided maximal operators given by the convolution with a smooth function. We also prove that b ∈ BMO is a necessary condition for the boundedness of commutators of these one-sided operators.

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

Balanus nubilis, one of the sessile Cirripedes, grows to a considerable size, a large specimen measuring some 10 to 11 cm. across the base and an equal distance in height. During a visit to Nanaimo Biological Station an opportunity was taken to study in this giant form a few features pertaining to the mechanism of movement of the operculum, which is attached within the cup of the shell by the opercular membrane. Peripherally the membrane is attached close to the downwardly directed sharp edge of an overhanging shelf formed by the inner wall of the shell; centrally it is attached to the free edge of the operculum on its basal aspect.

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.

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.

What appears to be an entirely new foot-print has lately been found in the red sandstone of this district. I have seen many of the foot-prints from the various quarries in the neighbourhood, but have not before observed this one, nor is it like any figured in Sir William Jardine's splendid work on the “Ichnology of Annandale.” The foot-prints in question were found in a bed about 20 feet from the surface, at Locharbrigg's Quarry, three miles from Dumfries. They exhibit the usual large hind and smaller fore foot; the impression of the hind foot measures ′·5 × 2′·6, the fore foot 2′·3 × 1′·9; the stride of the animal appears to have been about 10′. The impression of the hind foot does not interfere with that of the fore foot, as is the case with several of the foot-prints figured in Sir William Jardine's work, the interval between them being 2′: the hind foot, therefore, must have been put down in the rear of the fore foot when the animal was walking. The impression of the foot shows five toes, the thumb being placed far back. The most characteristic features in these foot-prints are the well-developed claws, and the oblique position of the toes, i.e., they are placed to march one behind the other. In almost all the foot-prints I have seen, where the toes can be made out, the middle one appears the most prominent; this foot-print is markedly distinct in this respect.