We study the variation of the zeros of the Hermite function Hλ(t) with respect to the positive real variable λ. We show that, for each non-negative integer n, Hλ(t) has exactly n + 1 real zeros when n < λ ≤ n + 1, and that each zero increases from – ∞ to ∞ as λ increases. We establish a formula for the derivative of a zero with respect to the parameter λ; this derivative is a completely monotonic function of λ. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of Hλ(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.

We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

Presented herein is a new method for analysing the long-time behaviour of solutions of nonlinear, dispersive, dissipative wave equations. The method is applied to the generalized Korteweg–de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time becomes unboundedly large. In circumstances to be spelled out presently, we establish the existence of a universal asymptotic structure that governs the final stages of decay of solutions. The method entails a splitting of Fourier modes into long and short wavelengths which permits the exploitation of the Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approach applies particularly well to cases where the damping increases in strength sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the situation wherein the nonlinearity is quasilinear with regard to the dissipative mechanism.

We show that if the leading coefficient p in a Sturm-Liouville expression is negative on a set E with positive Lebesgue measure, then the minimal operator (and hence any self-adjoint realization of the Sturm-Liouville expression) is not bounded below. The case when E contains an interval follows from standard methods but these methods fail when E contains no interval, e.g. when E is a Cantor-like set. This is the case considered here.

This paper is concerned with the existence of minimizers for functionals having a double-well integrand with affine boundary conditions. Such functionals are related to the so-called Kohn–Strang functional, which arises in optimal shape design problems in electrostatics or elasticity. They are known to be not quasiconvex, and therefore existence of minimizers is, in general, guaranteed only for their quasiconvex envelopes. We generalize the previous results of Allaire and Francfort, and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers. Our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory. We also prove by a general argument that their rank-one convexifications coincide with their quasiconvexifications.

We deal with the large-time behaviour of scalar hyperbolic conservation laws with source terms

which are often called hyperbolic balance laws. Fan and Hale have proved existence of a global attractor for this equation with x ∈ S1. consists of spatially homogeneous equilibria, a large number of rotating waves and of heteroclinic orbits between these objects. In this paper, we solve the connection problem and show which equilibria and rotating waves are connected by a heteroclinic orbit. Apart from existence results, our approach via generalized characteristics also gives geometric information about the heteroclinic solutions, e.g. about the shock curves and their strength.

Developing an approach of Repnitskii and Volkov, we focus on properties of semigroups of order-preserving mappings on finite chains; in particular, we show that the class of all these semigroups has no finite quasi-identity basis (although it has an infinite one).

In this paper we consider ‘slowly’ oscillating perturbations of almost periodic Duffing-like systems, i.e. systems of the form ü = u − (a(t) + α(wt))W′(u), t ∈ ℝ, u ∈ ℝN, where W ∈ C2N(ℝN, ℝ) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if w > 0 is small enough, then the system admits a multibump dynamics. As a consequence we get that the system ü = u − a(t)W′(u), t ∈ ℝ, u ∈ ℝN, admits multibump solutions whenever a belongs to an open dense subset of the set of positive almost periodic continuous functions.

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies, from the case of the Dirichlet Laplacian of a region Ω ⊆ ℝN to that of the higher-order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.

We define uniformly spread sets as point sets in d-dimensional Euclidean space that are wobbling equivalent to the standard lattice ℤd. A linear image ϕ(ℤd) of ℤd is shown to be uniformly spread if and only if det(ϕ) = 1. Explicit geometrical and number-theoretical constructions are given. In 2-dimensional Euclidean space we obtain bounds for the wobbling distance for rotations, shearings and stretchings that are close to optimal. Our methods also allow us to analyse the discrepancy of certain billiards. Finally, we take a look at paradoxical situations and exhibit recursive point sets that are wobbling equivalent, but not recursively so.

In this paper we study a moving boundary problem for an anisotropic two-phase Hele–Shaw flow. Using a regularization technique, we prove existence of a local solution. Under suitable conditions on the initial free boundary we obtain a global solution and study its asymptotic behaviour.

For hyperbolic systems in one spatial dimension ∂tu + C∂xu = f(u), u(t, x) ∈ ℝd, we study sequences of oscillating solutions by their Young-measure limit, μ, and develop tools to study the evolution of μ directly from the Young measure, v, of the initial data. For d ≤ 2 we construct a flow mapping, St, such that μ(t) = St(v) is the unique Young-measure solution for initial value v. For d ≥ 3 we establish existence and uniqueness of Young measures that have product structure, that is the oscillations in direction of the Riemann invariants are independent. Counterexamples show that neither μ nor the marginal measures of the Riemann invariants are uniquely determined from v, except if a certain structural interaction condition for f is satisfied. We rely on ideas of transport theory and make use of the Wasserstein distance on the space of probability measures.

Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

In this note, we present some Liouville type theorems about the non-negative solutions to some indefinite elliptic equations.

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expression

acting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

For certain classes of groups, it is shown that there are restrictions on the type of action a group in the class can have on a Λ-tree, where Λ is an arbitrary ordered abelian group, generalizing results by other authors in the case Λ = ℝ. The main classes considered are locally nilpotent, polycyclic by finite, locally (polycyclic by finite) and locally (hyperabelian by finite). The arguments involve an investigation of the relation between the type of action a group has on a Λ-tree and the type of action of its subgroups by restriction.

We outline a completely probabilistic study of travelling-wave solutions of the FKPP reaction-diffusion equation that are monotone and connect 0 to 1. The necessary asymptotics of such travelling-waves are proved using martingale and Brownian motion techniques. Recalling the connection between the FKPP equation and branching Brownian motion through the work of McKean and Neveu, we show how the necessary asymptotics and results about branching Brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed.