In this paper, we establish the notion of strong (r, s)-stability concerning closed space-like hypersurfaces immersed with higher-order mean curvatures linearly related in the de Sitter space . In this setting, we prove that totally umbilical round spheres of are strongly (r, s)-stable. Afterwards, we obtain sufficient geometric conditions that guarantee that a closed strongly (r, s)-stable space-like hypersurface in must be a totally umbilical round sphere.

In this paper, we consider the energy equality of the 3D Cauchy problem for the magneto-hydrodynamics (MHD) equations. We show that if a very weak solution of MHD equations belongs to $L^{4}(0,\,T;L^{4}(\mathbb {R}^{3}))$, then it is actually in the Leray–Hopf class and therefore must satisfy the energy equality in the time interval $[0,\,T]$.

We prove that the Lorentz sequence spaces do not have the property B of Lindenstrauss. In fact, for any admissible sequences w, v ∈ c0 \ l1, the set of norm-attaining operators from the Orlicz space hϕ(w) (ϕ is a certain Orlicz function) into d(v, 1) is not dense in the corresponding space of operators. We also characterize the spaces such that the subset of norm-attaining operators from the Marcinkiewicz sequence space into its dual is dense in the space of all bounded and linear operators between them.

We construct non-trivial laminates distributed continuously along a smooth closed ‘evolute’ curve by approximating an auxiliary (‘involute’) curve by a polygon with rank-1 sides. We give an explicit example in which the evolute has no rank-1 connections. Using these techniques, given a finite set of matrices B1,…,Br and any other matrix A, we show how to construct a laminate with positive mass at each Bi, any proportion less than 1 of its support on the Bi, and centre of mass at A.

We use Melnikov function techniques together with geometric methods of bifurcation theory to study the interactions of forcing, damping and detuning on resonant periodic orbits for single and coupled forced Van der Pol oscillators. For a coupled pair the local bifurcation geometry is almost everywhere described in terms of the singularities of a line congruence in three dimensions.

We use spectral sequence techniques to compute centralizers of elements within graded Lie algebras, and the methods are then applied to the calculation of unique normal forms of elements within one-parameter matrix Liealgebras. A finiteness criterion for unique normal forms is presented.

Let Ω be a bounded domain in Rn, n ≥ 3 and 0 ∈ Ω. It is known that the heat problem ∂u/∂t + Lλ*u = 0 in Ω × (0, ∞), u(x, 0) = u0 ≥ 0, u0 ≢ 0, where Lλ* := −Δ − λ*/|x|2, λ* := ¼(n − 2)2, does not admit any solutions for any t > 0. In this paper we consider the perturbation operator Lλ*q := −Δ − λ*q(x)/|x|2 for some suitable bounded positive weight function q and determine the border line between the existence and non-existence of positive solutions for the above heat problem with the operator Lλ*q. In dimension n = 2, we have similar phenomena for the critical Hardy–Sobolev operator L* := −Δ − (1/4|x|2)(log R/|x|)−2 for sufficiently large R.

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod 2 cohomology (as a graded vector space, a ring and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct ‘exotic’ conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such.

We characterize the weights w0, w1, w2 such that the weighted bilinear gradient inequality

holds for all functions , with a positive constant K independent of f1 and f2, for all possible values of q, p1 and p2 with 1 < q, p1, p2 < ∞.

We introduce what are called regular materials for which, by definition, the corresponding solution of the classical periodic homogenization problem remains bounded in . We give examples of two types of such materials depending on whether the coefficients representing them belong to W1,∞ or not. A complete characterization is obtained in the former case.

We show that if (Ω, Σ, μ) and (Ω′, Σ′, μ′) are probability spaces, then every regular operator T : Lp(μ) → Lq(μ′), 1 < p < ∞, 1 ≤ q < ∞, is thin if and only if it is strictly singular. We also show that if 0 ≤ S ≤ T : Lp(μ) → Lq(μ′), then T thin implies S is thin. We extend these results to some Köthe function spaces.

A bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

A recent paper of Ockendon et al. discusses the Fanno model for quasi-one-dimensional flow of gas in a tube, in situations where the flow is turbulent and the tube is long enough for wall drag to be important. Based on appropriate scalings and with associated boundary conditions they derive equations for similarity solutions and make predictions about travelling and evolving waves. In this paper the existence, uniqueness and asymptotic behaviour of these wave forms is proved rigorously. Techniques include shooting methods (both one- and two-parameter), appropriate changes of variables, and comparison techniques.

We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

AC rings (rings satisfying Auslander's condition) have recently been studied by Christensen and Holm, among others. We will employ tilting theory to study Auslander's condition. In particular, we prove that the class of AC Artin algebras is closed under tilting equivalences.

In this paper, we study under what boundary conditions the inequality

$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$

The confluent hypergeometric function Φ2(β,β′, γ, x, y) satisfies a system of partial differential equations which possesses the singular loci x = 0, y = 0, x − y = 0 of regular type and x = ∞, y = ∞ of irregular type. Near x = ∞ (|y| is bounded) and near y = ∞ (|x| is bounded), asymptotic expansions and Stokes multipliers of linearly independent solutions of the system are obtained. By a connection formula, the asymptotic behaviour of Φ2(β,β′, γ, x, y) itself is also clarified near these singular loci.

The relationship between the modulation of electron beams at optical frequencies (the Schwarz-Hora effect) and at microwave frequencies is discussed. At optical frequencies the interaction between the modulating field and the electron beam must be described quantum mechanically, although the field itself may be described classically; in the microwave case the process may be described entirely classically. The interaction modifies the state functions of the individual electrons, but the observable modulation of the total electron beam results from the coherence of the modulating field. The main features of the Schwarz-Hora effect result from ‘single photon’ processes, but the beam modulation in the klystron is a ‘multi-photon’ process.

The exponential decay of the depth of modulation with distance from the interaction region, observable in the optical frequency case but not in the klystron, is not an inherently quantal effect. The periodic variation of the depth of modulation with distance along the beam, observed in the optical frequency case, is an essentially quantum mechanical effect, and is different, in its origins and in its dependence on the modulation frequency, from the space-charge waves which are observed on a klystron beam.

Liouville type theorems are obtained for the solutions to elliptic equations of the form Δ2u −q(x)Δu + p(x)f(u)=0 by means of two subharmonic functionals and Green type inequalities.