We study spectral and scattering properties of the Laplacian H^(σ)=-Δ in $L_2(\mathbf{R}^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function σ. For non-negative σ we prove that H^(σ) is unitarily equivalent to the Neumann Laplacian H⁽⁰⁾. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of H^(σ) under mild assumptions on σ.

The spectral order on Rⁿ induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^∞ of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$-orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

Exact propagators are obtained for the degenerate second order hyperbolic operators ∂²_t-t^2lΔ_x, l=1,2,..., by analytic continuation from the degenerate elliptic operators ∂²_t+t^2lΔ_x. The partial Fourier transforms are also obtained in closed form, leading to integral transform formulas for certain combinations of Bessel functions and modified Bessel functions.

We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between L₁ and the Morrey space $\mathcal{L}^{1,\lambda}$. The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.

The α-modulation spaces M^s,α_{p, q}(R^d), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x, D) with symbol in the Hörmander class S^b_ρ,0 extends to a bounded operator σ(x, D): M^s,α_{p, q}(R^d)→M^s-b,α_{p, q}(R^d) provided 0≤α≤ρ≤1, and 1< p, q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S^b_1,0 maps the Besov space B^s_{p, q}(R^d) into B^s-b_{p, q}(R^d).

Let $A=-(\nabla-i\vec{a})^2+V$ be a magnetic Schrödinger operator acting on L²(Rⁿ), n≥1, where $\vec{a}=(a_1,\cdots,a_n)\in L^2_{\rm loc}$ and 0≤V∈L¹_loc. Following [1], we define, by means of the area integral function, a Hardy space H¹_A associated with A. We show that Riesz transforms (∂/∂x_k-ia_k)A^-1/2 associated with A, $k=1,\cdots,n$, are bounded from the Hardy space H¹_A into L¹. By interpolation, the Riesz transforms are bounded on L^p for all 1< p≤2.

In this paper we generalize an interpolation result due to J.-O. Strömberg and A. Torchinsky to the case of one-sided Hardy spaces. This generalization is important in the study of the weak type (1,1) for lateral strongly singular operators. We shall need an atomic decomposition in which for every atom there exists another atom supported contiguously at its right. In order to obtain this decomposition we have developed a rather simple technique to break up an atom into a sum of others atoms.

We prove the existence of steady periodic capillary water waves on flows with arbitrary vorticity distributions. They are symmetric two-dimensional waves whose profiles are monotone between crest and trough.

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u), d(v)}≥n/2 for each pair of vertices u and v with distance d(u, v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u, v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x, y, u, w, z, v such that d(x, y)=d(y, u)=d(w, z)=d(z, v)=d(u, v)=2 and where x, y and u are three distinct vertices and w, z and v are also three distinct vertices (and possibly |{x, y}∩{w, z}| is 1 or 2), then G is Hamiltonian.

Dans ce papier, nous nous intéressons au problème suivant : soit a_N une suite de points d’une hypersurface algèbrique convergeant vers a∈H ; supposons que pour tout N, il existe un germe de disque holomorphe en a_N, γ_N, contenu dans H. Alors, d’après des résultats classiques (voir [5] et [6]) le point a n’est pas un point de 1-type fini et donc, il existe un germe de disque holomorphe en a vivant dans H (voir [5]). Dans ce qui suit, nous proposons une méthode effective pour reconstruire, à partir de la suite a_N et des γ_N, un germe d’ensemble analytique en a contenu dans H.