A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces Lpε (·) in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions pε converge uniformly to a limit function p0 and that p0 satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.

We show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker--Döring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

We solve a problem posed by Blasco, Bonilla and Grosse-Erdmann in 2010 by constructing a harmonic function on ℝN, that is frequently hypercyclic with respect to the partial differentiation operator ∂/∂xk and which has a minimal growth rate in terms of the average L2-norm on spheres of radius r > 0 as r → ∞.

Let $\tau_\varOmega$ denote the lifetime of Brownian motion in an open connected set $\varOmega\subset\mathbb{R}^m$. We obtain the asymptotic behaviour of the expected lifetime $\mathbb{E}_x^y[\tau_\varOmega]$ as $y\to x$, where the Brownian motion is conditioned to start at $x$ and to exit $\varOmega\setminus\{y\}$ at $\{y\}$.

We study homogenization processes for the heat equation in multilayers with interlayer conduction, modelled by Neumann transmission conditions. We establish the homogenized equations for three kinds of dependency of the interlayer conduction magnitude upon the interlayer distance.

We show that if L is a bounded (semi-)lattice and f is a range-closed idempotent residuated mapping on L then f is multiplicative if and only if it is decreasing. We then use this to prove that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a weakly multiplicative (right) Baer semigroup.

We construct multi-bump nodal solutions of the elliptic equation

in , when μ is large, under appropriate assumptions, for f superlinear and subcritical and such that the eigenvalues of the associated linearized operator on at zero, u ↦ u − λ(−Δ)−1(a+u), are positive. The solutions are of least energy in some Nehari-type set defined by imposing suitable conditions on orthogonal components of functions in .

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and pt ≧0 for t ≧1 and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions (including the positivity of the coefficients) have to be imposed only in an increasing sequence of annular regions surrounding the origin.

Comparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problems

and

with φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.

We use the theory of semiperfect rings to derive decomposition theorems for H- and co-H-spaces, generalizing the results of Wilkerson. The results are then used to prove reducibility of self-homotopy equivalences for arbitrary p-local H- and co-H-spaces.

It is proved that in the case ½<p<l the periodic Franklin system forms a Schauder basis for the real Hardy space Hp(T) defined on the one-dimensional torus.

In this note we prove the following

Theorem. The periodic Franklin system forms a Schauder basis in the real Hardyspace Hp(T) defined on the one-dimensional torus if ½<p< l.

To prove that certain standard 2-complexes are aspherical we explore a strategy that combines a well-known method based on a graph theoretic lemma of Stallings with a process of reversing the orientation of edges in spherical diagrams. We apply this strategy to labelled-oriented-tree complexes and, more generally, to labelled-oriented-graph (LOG) complexes and obtain classes of aspherical LOG complexes to which, traditionally (without the reversing of edges), Stallings's lemma could not be applied to prove asphericity. These classes contain examples whose asphericity, as far as we know, could not be established by any previous method.

This paper deals with two-point boundary-value problems for ordinary differential equations and the operators which they induce in the appropriate Hilbert space. The problems arenot required to be self-adjoint. No auxiliary condition such as Birkhoff-regularity is imposed. If T is such an operator, it may well have no meaningful spectral structure. It is shown, however, that when T is composed with its adjoint, the result is a non-negative self-adjoint differential operator. The eigenvalues and eigenfunctions of this composite operator are used to delineate the domain, action, range, and generalised inverse of T.

From the time of Galileo, experiment has been the core of Natural Science. Before him, of course, observation alone had in the development of astronomy played a fundamental part. Besides the great workers of the ancient civilisations, who knew the path of the sun amongst the fixed stars and could predict eclipses, and besides the fruits of Greek astronomy associated with the names of Hipparchus and Ptolemy, the more modern observational work of Tycho Brahe, analysed by Kepler, had vindicated the self-consistency of the Copernican theory of the solar system and had led to its remarkable refinement in the form of Kepler's three quantitative laws—the law of the ellipse, the law of areas, and the law connecting periodic times and major axes. This was a triumphant example of the execution of the programme then being put forward by Francis Bacon for discovering all natural laws—the method of induction from a number of instances. But it was reserved for Galileo to make a start with the process of ascertaining as far as might be, by controlled experiment, the particular nature of motion. The metaphysical questions associated with motion had not escaped the attention of the Greeks; but Zeno was apparently content with stating paradoxes, and did not resolve them. Galileo, first, experimented with moving bodies; and established that in falling they received equal increments of velocity in equal times—a kinematic theorem, like Kepler's laws. Huyghens was perhaps the first person to establish dynamical-theorems; that is to say, to infer a kinematic result from a stated physical principle—as, for example, his proof of the approximate isochromism of the pendulum based on the principle of vis viva, or, as we should now say, the conservation of energy. Huyghens, together with some of the early Restoration men of science in this country, dealt also with the collisions of bodies. The peerless Newton went further. Assuming outright three primitive “laws of motion,” he showed how the results of Galileo, Huyghens, and their contemporaries could be actually deduced; and by the addition of a fourth law, the law of universal gravitation, already conjectured by some thinkers, he arrived at the laws of Kepler as inferences. Not only so, but the four highly general and abstract laws introduced by Newton have been found sufficient to deduce an enormous complex of dynamical theorems, to express their relationships in the subsequent beautiful systems of Lagrange and of Hamilton, and to derive all but every detail in the motions both in the solar system and in distant binary stars. The basic principles laid down by Newton remained unaltered till our own day, when Einstein modified simultaneously the laws of motion, the law of gravitation, and the background of space and time which had been explicitly adopted by Newton as the scene in which his laws were to play their parts.

Stability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

This paper investigates the one-dimensional propagation of weak discontinuities, that is acceleration waves, in a homogeneous and isotropic half-space composed of an arbitrary number of non-linearly hyperelastic layers. The transmission and reflection coefficients are evaluated in terms of the initial condition at the boundary, and the steepening of the waves to form a shock is discussed. The results are specialised to the case of periodic layering.

We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation

\[ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in}\,{ {\mathbb{R}^{3}},} \]

$2 < p < \infty$

$q \ge 0$