0. Introduction

Morin [M] gave a local normal form for singular maps having almost maximal rank, where almost maximal means maximal minus 1. The aim of the present paper is to give a global version of his normal form. We concentrate here on the case of [sum ]1, 1, 0-singular maps. (For the definition see [Bo], [A–G–V], [G–G] and also here below.) The case of [sum ]1, 0 singular maps was considered by Haefliger in [Ha], see also [Sz1] and [Sz2]. For the motivation in finding such a global normal form see [Sz1], [Sz2], and the final remarks in this paper.

Let G be a connected locally finite simplicial graph with rk(π1(G))[ges ]2 and let T be the universal cover of G. Consider a π1(G)-invariant conformal density μ of dimension d on ∂T. The total mass function ϕμ of μ is defined on the set of vertices of G. Let |ϕμ| be its l2-norm. Let Ω be the geodesic flow space of G and mμ the invariant measure on Ω associated to μ. The main results of this paper are the following:

(i) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [les ]k. Then, if d>½log(k−1) and mμ(Ω)<∞, we have ϕμ∈l2(V) and

|ϕμ|2 [les ]Cmμ(Ω),

with C= (e2d−1)/ (e2d−k+1).

(ii) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [ges ]k. Then, if ϕμ∈l2(V), we have d>½log(k−1) and

Cmμ(Ω) [les ]|ϕμ|2.

Let S be an immersed compact surface into a Riemannian manifold M. We denote by H and K the mean curvature vector field of S and the sectional curvature function of M with respect to the tangent space of S and define

formula here

During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

For a general state space Markov chain on a space (X, [Bscr ](X)), the existence of a Doeblin decomposition, implying the state space can be written as a countable union of absorbing ‘recurrent’ sets and a transient set, is known to be a consequence of several different conditions all implying in some way that there is not an uncountable collection of absorbing sets. These include

([Mscr ]) there exists a finite measure which gives positive mass to each absorbing subset of X;

([Gscr ]) there exists no uncountable collection of points (xα) such that the measures Kθ(xα, ·)[colone ](1−θ)ΣPn(xα, ·)θn are mutually singular;

([Cscr ]) there is no uncountable disjoint class of absorbing subsets of X.

We prove that if [Bscr ](X) is countably generated and separated (distinct elements in X can be separated by disjoint measurable sets), then these conditions are equivalent. Other results on the structure of absorbing sets are also developed.

1. In the paper [Kai] it was observed that if A is a Banach algebra (over R or C) then the dual space is not only an A-A-bimodule, but is also injective as a left (or right) A-module. Furthermore, if M is a left (or right) Banach module over the unital Banach algebra A, then there is a natural bilinear map, there denoted TrA, from M × M′ to A′, defined by

formula here

(or 〈TrA(m, m′), a〉 = 〈m′, ma〉). The map TrA can be extended to the projective tensor product M[otimesas]M′, which is also an A–A–bimodule. It is easy to see that the map TrA is a bimodule homomorphism, so that the image is an A–A–submodule of A′. This module was denoted EA (M) in [Kai] and is in general not closed as a subspace of M′. It does, however, have a natural norm (as a quotient space of M[otimesas]M′) and the unit ball can be used to define a new norm ∥a∥new = sup {|〈e, a〉 | e∈ the unit ball of EA(M)} on A, and it is easy to see that this new norm is simply the operator norm of a as an operator on M. The conclusion is that if a′∈A′ is not only continuous with respect to the norm ∥a∥L(M) (which is of course in general smaller thatn the given norm on A) but also with respect to the weak topology on A given by the set of all functionals of the form (0·1), then a′ has a representation of the form

formula here.

1. Introduction

In a recent paper [3], an extended Liouville–Green formula

formula here

was developed for solutions of the second-order differential equation

formula here

Here γM(x) ∼Q−¼(x), QM(x) ∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.

The proportion ρk of gaps with length k between square-free numbers is shown to satisfy logρk=−(1+o(1))(6/π2) klogk as k→∞. Such asymptotics are consistent with Erdős's challenge to prove that the gap following the square-free number t is smaller than clogt/log logt, for all t and some constant c satisfying c>π2/12. The results of this paper are achieved by studying the probabilities of large deviations in a certain ‘random sieve’, for which the proportions ρk have representations as probabilities. The asymptotic form of ρk may be obtained in situations of greater generality, when the squared primes are replaced by an arbitrary sequence (sr) of relatively prime integers satisfying [sum ]r1/sr<∞, subject to two further conditions of regularity on this sequence.

There are several ways of shortening a curve on a surface in order to establish the existence of closed geodesics. These methods are often useful in the study of much more general spaces, e.g. locally compact length spaces (see [1] chapter 10). Recently, J. Hass and P. Scott in [6] have introduced a new curve flow, the polygonal flow, which takes into consideration the complexity of the resulting geodesic. They construct the polygonal flow on surfaces so that the number of self-intersection points of a curve does not increase throughout the flow. In this paper we are concerned with ideal polyhedra with finitely many cells of dimension 2. These spaces consist of finite ideal hyperbolic triangles glued together by isometries along their sides.

This paper studies the spectral properties of a class of probability measures on the circle. The key aim is to describe the local structure of the maximal ideal space on the L-subspaces generated by the measures and hence the spectral properties of these measures. In particular we give a necessary and sufficient condition for such measures to belong to M0 (T).

This paper presents the asymptotic expansion, with respect to the large real parameter λ, of the singular Fourier integral

formula here

where 0 < b [les ] + ∞, the symbol fp designates an integration in the finite part sense of Hadamard and complex function K(x, u) belongs to a specific set of pseudo-functions which may present a singular behaviour of logarithmic nature at the endpoints.

Asymptotic analysis and distribution theory are two very useful branches of Applied Mathematics. It is only recently, however, that the close ties between these theories have been studied in detail [5, 6, 11, 17].

Among the various approaches to the distributional theory of asymptotic expansions [6] gives an analysis based on the moment asymptotic expansion.

It has been shown that univalent circle packings filling the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result.

We also establish a circle packing analogue of Liouville's theorem: if f is a circle packing map whose domain packing is infinite, univalent, and has recurrent tangency graph, then the ratio map associated with f is either unbounded or constant.

1. Introduction

Along with the prevalence of probabilistic methods in approximation theory the study of approximation of operators by other operator sequences, or the study of limiting properties of operator sequences, has attracted more and more attention. It was shown by Stancu [15] in the 1960s that Szász operators and Baskakov operators are limiting operators of Stancu–Mühlbach operator sequences in an appropriate sense. Later, Butzer and Hahn [4, 5, 11] established rates of convergence of some probabilistic type operators towards their limiting operators as applications of their general approximation theorems in probability theory. In 1988, Khan [13] proved that Szász operators are limiting operators of Bleimann–Butzer–Hahn operator sequences. Recently de la Cal, Luquin, Adell and San Miguel investigated the limiting properties of a series of probabilistic type operator sequences and exhibited that their limiting operators might be Bernstein, Szász, Baskakov, gamma or beta operators [1, 2, 6, 7, 8].

Levinson's theorem with all its ramifications is a well-established tool in the spectral analysis of ordinary differential operators (see in particular §§3·10, 11 and 4·3, 4 of Eastham's book [5]). In this note we should like to draw attention to a result that describes the solutions of asymptotically constant linear systems under weaker assumptions less precisely than the Levinson theorem. This result can be called the Perron–Lettenmeyer–Hartman–Wintner theorem after the contributions of these authors in [11, 10, 6, 8]. (Note that the Hartman–Wintner theorem that is discussed in [5, p. 17] is a substantially different version of this theorem.)

A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian evolution equation on one or more space dimensions. This generalization, called multi-symplectic structures, is shown to be natural for dispersive wave propagation problems. Application of the abstract properties of the multi-symplectic structures framework leads to a new variational principle for space-time periodic states reminiscent of the variational principle for invariant tori, a geometric reformulation of the concepts of action and action flux, a rigorous proof of the instability criterion predicted by the Whitham modulation equations, a new symplectic decomposition of the Noether theory, generalization of the concept of reversibility to space-time and a proof of Lighthill's geometric criterion for instability of periodic waves travelling in one space dimension. The nonlinear Schrödinger equation and the water-wave problem are characterized as Hamiltonian systems on a multi-symplectic structure for example. Further ramifications of the generalized symplectic structure of theoretical and practical interest are also discussed.

We consider the diffraction of the dominant acoustic wave mode which propagates out of the mouth of a semi-infinite waveguide made of a soft and hard half plane. This semi-infinite waveguide is symmetrically located inside an infinite waveguide whose infinite plates are soft and hard. The whole system constitutes a trifurcated waveguide. Another trifurcated waveguide is obtained by interchanging the infinite plates. A closed form solution of the resulting matrix Wiener–Hopf equation is obtained for each configuration. Thus we present exact closed-form solutions to two new waveguide trifurcation problems.

The purpose of this short note is to reformulate Theorem 4·2 in [1]. This theorem is not correct in the part concerning maximality of the norm in Orlicz spaces. The theorem below provides a correct version of this statement. We wish to point out that all other theorems in [1&rsqb; are independent of Theorem 4·2.

The proof of Lemma 2·4 in the paper above is not correct because of mistakes in computations (4)–(8) using Araki operations.