Dispersive and Strichartz estimates are obtained for solutions to the wave equation with a Laguerre potential in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the Schrödinger operator involved, we subsequently prove the dispersive estimate for the corresponding Schrödinger semigroup, obtain a Gaussian-type upper bound, establish Bernstein-type inequalities, and finally pass to the Müller–Seeger’s subordination formula. The desired Strichartz estimates follow by the established dispersive estimate and the standard argument of Keel–Tao.

By previous work of Giordano and the author, ergodic actions of $\mathbf{Z}$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in $\text{C}^{\ast }$-algebras and topological dynamics. Here we investigate how far from approximately transitive (AT) actions can be that derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, approximate transitivity arises. KIn addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

In this article, we study complete surfaces $\sum $, isometrically immersed in the product spaces ${{\mathbb{H}}^{2}}\,\times \,\mathbb{R}$ or ${{\mathbb{S}}^{2\,}}\times \,\mathbb{R}$ having positive extrinsic curvature ${{K}_{e}}$. Let ${{K}_{i}}$ denote the intrinsic curvature of $\sum $. Assume that the equation $a{{K}_{i\,}}\,+\,b{{K}_{e\,}}\,=\,c$ holds for some real constants $a\,\ne \,0$, $b\,>\,0$, and $c$. The main result of this article states that when such a surface is a topological sphere, it is rotational.

In this article we study the geometry of the eigenvarieties of unitary groups at points corresponding to tempered non-stable representations with an anti-ordinary (a.k.a evil) refinement. We prove that, except in the case where the Galois representation attached to the automorphic form is a sum of characters, the eigenvariety is non-smooth at such a point, and that (under some additional hypotheses) its tangent space is big enough to account for all the relevant Selmer group. We also study the local reducibility locus at those points, proving that in general, in contrast with the case of the eigencurve, it is a proper subscheme of the fiber of the eigenvariety over the weight space.

Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

On se propose d'étendre un résultat de Rorhlich [5] concernant les moyennes galoisiennes des valeurs en s = 1 des fonctions L associées à une forme modulaire de poids 2, tordue par certains caractères de Dirichlet. On considérera une représentation automorphe parabolique π de GL(n), n≦2, sur un corps de nombres K, et l'on s'intéressera aux valeurs des fonctions L(S,πχ) et de leurs dérivées L(m)(s, πχ), m ≧1, où χ parcourt certains caractères de Hecke de K, d'ordre fini, et où s appartient à la bande a < Re s < 1 — a, où a mesure la déviation de n par rapport à la conjecture de Petersson généralisée.

We show that a certain category 𝓖 whose objects are pairs G ⊃ H of groups subject to simple axioms is equivalent to the category of ≥ 2-dimensional vector spaces and injective semi-linear maps; and deduce via the "Fundamental Theorem of Projective Geometry" that the category of ≥ 2-dimensional projective spaces is equivalent to the quotient of a suitable subcategory of 𝓖 by the least equivalence relation which identifies conjugation by any element of H with the identity automorphism of G.

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.

In recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this by first showing that various results on operator ranges may be generalized to “measurable fields of operator ranges”.

Let (X, T, π) denote a flow, where X is a compact topological space metrizable by d, and T is a closed non-trivial subgroup of the reals under addition. T is recurrent if and only if for each and s > 0, there exists t > s such that x ∈ X implies . If T is almost-periodic, then T is both recurrent and distal. In §§ 4 and 5, it is shown that, under more stringent hypotheses, the recurrence of T is neither a necessary nor a sufficient condition for T to be distal. Let S be a closed non-trivial subgroup of T. It is shown in § 3 that T is recurrent if and only if S is recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).

We study the image of the model subspace ${{K}_{\theta }}$ under the composition operator ${{C}_{\varphi }}$, where $\varphi $ and $\theta $ are inner functions, and find the smallest model subspace which contains the linear manifold ${{C}_{\varphi }}{{K}_{\theta }}$. Then we characterize the case when ${{C}_{\varphi }}$ maps ${{K}_{\theta }}$ into itself. This case leads to the study of the inner functions $\varphi $ and $\psi $ such that the composition $\psi \,\text{o}\,\varphi $ is a divisor of $\psi $ in the family of inner functions.

This paper is a continuation of [1; 2]. In [2], I stated that I had been unable to construct examples of planes satisfying various conditions. Some of the examples that I have since constructed are given below. A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper. The relevant parts of [1] and [2] are [1, § 1, § 2 up to 2.4; 2, § 2]. We shall use the notation and terminology of [1; 2]; the axioms Cl*-C4* and C4** (referred to below) can all be found in [1].

We shall show here that spaces of dimension greater than 1 exist, both Archimedean and non-Archimedean, satisfying Cl*-C4*, in which not all points are isometric, and that C4** does not follow from Cl*-C4* in non- Archimedean geometries of dimension greater than 1.

In this paper we use a particular kind of weighted Sobolev space and pseudo-differential operators to study H3s propagation of singularities for the solution u ∊ Hs of the equations with second order.

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

Let $A={{({{a}_{j,k}})}_{j,k\ge 1}}$ be a non-negative matrix. In this paper, we characterize those $A$ for which ${{\left\| A \right\|}_{E,F}}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: ${{\ell }_{p}}$, $d(w,p)$, and ${{\ell }_{p}}(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:

(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that

(1)

The density u(x, t) of an ideal gas flowing through a homogeneous porous media satisfies the equation Here m > 1 is a physical constant and u also satisfies the initial condition If the initial data is not strictly positive it is necessary to work with generalized solutions of the Cauchy problem (1), (2) (see [1]). By a weak solution we shall mean a function u(x, t) such that for (in the sense of distributions) and

A Banach algebra P over C with identity element is called an N-algebra if any closed ideal in P is the intersection of maximal ideals. An example is given by the algebra of the continuous C-valued functions on a compact Hausdorff space X under the supremum norm; two others are discussed in § 3.

Let $D\subset \mathbb{C}$ be a domain with $0\in D$. For $R>0$, let $\widehat{\unicode[STIX]{x1D714}}_{D}(R)$ denote the harmonic measure of $D\cap \{|z|=R\}$ at $0$ with respect to the domain $D\cap \{|z|<R\}$ and let $\unicode[STIX]{x1D714}_{D}(R)$ denote the harmonic measure of $\unicode[STIX]{x2202}D\cap \{|z|\geqslant R\}$ at $0$ with respect to $D$. The behavior of the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, $\unicode[STIX]{x1D714}_{D}(R)\leqslant \widehat{\unicode[STIX]{x1D714}}_{D}(R)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0\in D$ and all $R>0$,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{D}(R)\geqslant C\widehat{\unicode[STIX]{x1D714}}_{D}(R)?\end{eqnarray}$$

$D$

The Lefschetz theorem on hyperplane sections, as proved by Andreotti and Frankel (1), depends upon the following result.

THEOREM. If M is a non-singular affine algebraic variety of real dimension 2k of complex n-space, then

This theorem, which is interesting in itself, has been strengthened by Milnor (7), who showed that M has the homotopy type of a k-dimensional CW-complex.

In this paper we generalize the above theorem in two directions. First, we replace complex n-space by some other complete simply connected Riemannian manifold which either has non-positive curvature or is a compact symmetric space. Secondly, we allow M and to be quasi-Kâhlerian (see below) instead of Kählerian.

We first introduce some notation. Let M and be C∞ Riemannian manifolds with M isometrically immersed in . Denote by 〈, 〉 the metric tensor of either M or .