We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

Let Mn, n ≥ 3, be a complete oriented minimal hypersurface in Euclidean space Rn+1. It is shown that, if the total scalar curvature on M is less than the n/2 power of 1/2Cs, where Cs is the Sobolev constant for M, and the square norm of the second fundamental form is a L2 function, then M is a hyperplane.

Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let Nθ(R) = card{RΩθ∩ℤ2}, where Ωθ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, Nθ(R) = |Ω|R2 + O(R2/3), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Ω under the Gauss map is also obtained.

We prove nonlinear stability of planar shock fronts for certain relaxation systems in two spatial dimensions. If the subcharacteristic condition is assumed and the initial perturbation is sufficiently small and the mass carried by the perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is, in general, non-zero and governed by a similarity solution to the heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞(R2) without imposing extra decay rates in space for the initial perturbations. The proofs are based on an elementary weighted energy analysis to the error equation.

Let

where φ denotes Euler's function. In this memoir we study the set w of sigmaphi numbers, that is, those composite natural numbers n which satisfy

The smallest such number is 65, and they appear to be moderately frequent. There are 290 sigma-phi numbers not exceeding 105 and 1,231 not exceeding 106. By comparison, we observe that the number of primes in these ranges is 9,592 and 78,498, respectively. Since the primes also satisfy the relationship (1.2) a sigma-phi number can be thought of as a kind of pseudo-prime. The motivation for studying sigma-phi numbers is that they should have similar properties to Carmichael numbers but be easier to study. A Carmichael number is a number n such that the least common multiple of the φ(pk) with pk||n divides n-1, i.e., by Korseldt's criterion, a number for which p-1||n-1 whenever p|n. The number of Carmichael numbers not exceeding 105 and 106 is 16 and 43, respectively. It seems that the counting functions for Carmichael and sigma-phi numbers have somewhat similar growth rates. The counts above are skewed by the fact that there are many sigma-phi numbers with exactly two prime factors but there are no Carmichael numbers of this kind.

In this paper we continue the investigation begun in [11]. Let λ1…., λs and μ1, …, μs be real numbers, and define the forms

Further, let τ be a positive real number. Our goal is to determine conditions under which the system of inequalities

has a non-trivial integral solution. As has frequently been the case in work on systems of diophantine inequalities (see, for example, Brüdern and Cook [6] and Cook [7]), we were forced in [11] to impose a condition requiring certain coefficient ratios to be algebraic. A recent paper of Bentkus and Gotze [4] introduced a method for avoiding such a restriction in the study of positivede finite quadratic forms, and these ideas are in fact flexible enough to be applied to other problems. In particular, Freeman [10] was able to adapt the method to obtain an asymptotic lower bound for the number of solutions of a single diophantine inequality, thus finally providing the expected strengthening of a classical theorem of Davenport and Heilbronn [9]. The purpose of the present note is to apply these new ideas to the system of inequalities (1.1).

This paper presents existence criteria for continuous and discrete boundary value problems on the infinite interval, using the notion of upper and lower solution.

A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X, Y with complex coefficients is irreducible if the degrees of f and g are coprime. Panaitopol and Stefǎnescu generalized this result, by giving an irreducibility condition for a larger class of polynomials called “generalized difference polynomials”. This paper gives an irreducibility criterion for more general polynomials, of which the criterion of Panaitopol and Stefǎnescu is a special case.

We consider a phase-field model for diffusion-induced grain boundary motion. The model couples a parabolic variational inequality to a degenerate diffusion equation. Using a regularization technique, we prove an existence theorem for the resulting system. We also obtain a uniqueness result, provided the solution has some additional regularity.

The shape stability of the reaction interface for reactive flow in a porous medium is investigated. Previous work showed that the Reaction-Infiltration Instability could cause the reaction zone to lose stability when the Peclet number exceeded a critical value. The new feature of this study is to include a velocity-dependent hydrodynamic dispersion. A mathematical model for this phenomenon is given in the form of a moving free-boundary problem. The spectrum of the linearized problem is obtained, and the related analysis and numerical calculations show that the onset of the instability is not eliminated by the new dispersive terms. The details of analysis show that the instability is reduced especially by the transverse dispersion.

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.

In this paper, we develop a mathematical model to describe interactions between tumour cells and a compliant blood vessel that supplies oxygen to the region. We assume that, in addition to proliferating, the tumour cells die through apoptosis and necrosis. We also assume that pressure differences within the tumour mass, caused by spatial variations in proliferation and degradation, cause cell motion. We couple the behaviour of the blood vessel into the model for the oxygen tension. The model equations track the evolution of the densities of live and dead cells, the oxygen tension within the tumour, the live and dead cell speeds, the pressure and the width of the blood vessel. We present explicit solutions to the model for certain parameter regimes, and then solve the model numerically for more general parameter regimes. We show how the resulting steady-state behaviour varies as the key model parameters are changed. Finally, we discuss the biological implications of our work.

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation

[formula here]

with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution

[formula here]

of the complex Hamilton–Jacobi equation

[formula here].