We study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

In a previous paper by the author [3], duality theorems for finitely generated vector lattices and lattice-ordered Abelian groups are described. In particular, the category of finitely generated semi-simple vector lattices is shown to be equivalent to a geometrical category V whose objects are topologically closed cones in Euclidean space, and whose morphisms, called ll-maps\ form a special subclass of the class of piece wise homogeneous linear maps between such cones. Under this categorical duality, finitely generated projective vector lattices and closed polyhedral cones correspond; indeed, the category of finitely generated projective vector lattices is equivalent to the dual of a category whose objects are Euclidean closed polyhedral cones and whose morphisms consist of all piecewise homogeneous linear maps between such cones.

In this paper, we study the boundary version of the classical Cartan theorem. We show that for some weakly pseudoconvex domains, when a holomorphic self-mapping has a sufficiently high order of contact (which depends only on the geometric properties of the domains) with the identical map at some boundary point, then it must coincide with the identity.

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:

with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:

(1.1)

Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion

(1.2)

Let S denote the class of functions of the form

that are analytic and univalent in the unit disk Δ = {z:|z| < 1}, with S*(α) and K(α) designating the subclasses of S that are, respectively, starlike of order a and convex of order α, 0 ≦ α < 1. If f(z) and g(z) are analytic in Δ, we say that f(z) is subordinate to g(z), written f ≺ g, if there exists a Schwarz function w(z), w(0) = 0 and |w(z)| < 1 in Δ, such that f(z) = g(w(z)). A function f(z) = z + … is said to be in S*[A, B] if

(1)

and in K[A, B] if

(2)

All graphs considered in this paper are finite, undirected, loopless and without multiple edges.

The vertex set and the edge set of a graph G will be denoted by V(G) and E(G)y respectively. Thus we have

This paper deals with processes of combining convex bodies in Euclidean n-space which are, in a sense, dual to the process of Minkowski addition and some of its generalizations.

All the convex bodies considered will have a common interior point Q. Variables x and y denote vectors drawn from Q; we shall speak of their terminal points as the points x and y. Unit vectors will be denoted by u; ||x|| signifies the length of x. Convex bodies will be symbolized by K with distinguishing marks. ∂K means the boundary of K. λK will mean the image of K under a homothetic transformation in the ratio λ : 1. The centre of the homothety will always be Q.

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.

We are given two identical symmetrical bodies (e.g., cubes) with independent random orientations; then we can always, in several ways, turn one of these bodies about some axis through its centre of gravity, so as to bring it into the same orientation as the other body. The smallest angle of rotation needed will be called the disorientation, d, of the two bodies, and we shall be concerned with the distribution of d under these conditions.

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

In recent years a number of inequalities have appeared which involve rearrangements of vectors in Rn and of measurable functions on a finite measure space. These inequalities are not only interesting in themselves, but also are important in investigations involving rearrangement invariant Banach function spaces and interpolation theorems for these spaces [2; 8; 9].

We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.

We consider parabolic operators in self-adjoint form

(1)

as well as the non-self-adjoint operator

(2)

where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are

• (i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and

• (ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that

1.1

A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence

1.2

with

1.3

Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by

1.4

with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.

Modern physical theory, both classical and quantal, faces serious difficulties which arise from the divergence of certain integrals. Perhaps the best known of these “infinities” is the self-energy of the point electron. Most of the simpler devices used to eliminate the infinities, such as the introduction of a finite electron radius, are non-relativistic and must therefore be rejected. Relativistic theories1 which do avoid some or all of the infinities are very complicated and often suffer from difficulty in physical interpretation.

The motivation for this paper was the work of Thurston and Jørgensen on volumes of hyperbolic 3-manifolds. They prove, among other things, that the set of all volumes of complete hyperbolic 3-manifolds is well-ordered. In particular, there is a hyperbolic 3-manifold which has minimum volume among all complete hyperbolic 3-manifolds. Further, there is a minimum volume member in the collection of complete hyperbolic 3-manifolds with one cusp; and similarly for n cusps. Computer studies to date show that the manifold obtained by performing (5,1) Dehn surgery on the figure-eight knot in the 3-sphere is the leading candidate for the minimum volume hyperbolic 3-manifold. Its volume is about 0.98. The leading one-cusp minimum volume candidate is the figure-eight knot complement in the 3-sphere. Its volume is about 2.03.

We apply the expression for the norm of a function in the weighted Lorentz space, with respect to the distribution function, to obtain as a simple consequence some weighted inequalities for integral operators.

1. Introduction. The number of connected linear graphs having V vertices labelled 1, … , V and λ (unlabelled) lines is found below. Similar formulas are found for graphs in which slings, lines “in parallel,” or both are allowed and for directed graphs with or without slings or parallel lines. Some of these graphs are also counted when the lines are labelled and the vertices are unlabelled.

A major goal of this paper is to give a proof of the following isotropy criterion: Let X = (X,G) be a space of orderings in the terminology of [9] or [10], and let f be a form defined over G.Then f is anisotropic over X if and only if f is anisotropic over some finite subspace of X.This is the content of Theorem 1.4, and generalizes [1, Corollary 3.4]. Moreover, in view of the known structure of finite spaces (see [9]), this has, essentially, the strength of [2, Satz 3.9] or [12, Theorem 8.12]. The technique used to prove this criterion is roughly patterned on that of [6], and yields some interesting by-products: An interesting invariant of a space of orderings called the chain length is introduced (Definition 1.1) and spaces of orderings with finite chain length are classified (Theorem 1.6).