A perturbation theorem for symmetric differential expressions is given and applied to even-order expressions. We assume the coefficients p0(t), pn(t) to be eventually positive and to have real powers of t as dominating terms. Then we are able to admit for the absolute values of the other coefficients a rate of growth depending on the growth of both of the coefficients p0, pn in order to obtain minimal deficiency indices.

Let G be a group and S a group of automorphisms of G. The simplicity of the near-ring, MS(G), of zero preserving functions on G which commute with the elements of S, is investigated. The relationship between simplicity, 2-primitivity and containment relations among the stabilizers of elements of G is explored.

This note pursues two aims: the first is historical and the second is factual.

1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.

2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.

A comprehensive appraisal of the title problem is presented in terms of a characterizing nondimensional co-ordinate ξ which is based upon the half excess of the momentum of the jet, J. Perturbation features of the problem appear as regular and singular boundary conditions in ξ upstream and downstream respectively. The conservation of momentum excess provides a monitor on the consistency of regular and singular perturbation series solutions. In particular the conservation constraint on the downstream singular perturbation solution confirms the inadequacy of expansions in inverse half powers of ξ and justifies formally the introduction of logarithmic terms.

The formulation provides the basis for a complete numerical integration over the semi-infinite region. Accordingly detailed knowledge of velocity excess along the axis of the jet is obtained and an undetermined coefficient in the asymptotic downstream perturbation solution may be estimated.

We investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2m, whose coefficients involve a real multiple ɑr of certain positive real powers β of the independent variable x. Information on the L2 nature is obtained by variation of parameters from Meijer function solutions of an associated homogeneous equation of hypergeometric type. When the coefficients of the differential expressions are positive, it is possible, by a suitable choice of ɑr, β and m, to obtain between m and 2m —1 linearly independent solutions in L2(0, ∞). This proves a conjecture of J. B. McLeod that the deficiency index can take values between m and 2m —1 for such operators.

The equation to be considered is

where pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.

The H-spaces considered have no homology p-torsion and are rationally equivalent as H-spaces to products of even dimensional Eilenberg–Maclane spaces. We obtain conditions which ensure that if the cohomology with coefficients in the ring of integers localized at the prime p is a polynomial algebra, then the Pontrjagin ring with these same coefficients is polynomial. A topological consequence is that BSUP has just one homotopy associative, homotopy commutative H-structure.

In this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.

It is shown that the spectrum of a linear operator T on a complete linear metric space X is a Borel set. If, in addition, X is locally convex or separable, then the spectrum of T is a Gδσ set.

In this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.

Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.

The statements of this paper are obtained by results recently proved elsewhere by the author.

Let τ denote the second-order elliptic expression

where the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

An asymptotic theory is developed for linear differential equations of odd order. The theory is applied to the evaluation of the deficiency indices N+ and N− associated with symmetric differential expressions of odd order. General conditions on the coefficients are given under which all possible values of N+ and N subject to | N+ − N | ≦ 1 are realized.

Let S be a symmetric subspace in a Hilbert space ℋ2 with finite equal deficiency indices and let S* be its adjoint subspace in ℋ2. We consider those self-adjoint subspace extensions ℋ of S into some larger Hilbert spaces ℋ2 = (ℋ × ℂm)2 which satisfy H⋂({0} × ℂm)2 = {{0,0}}. These extensions H are characterized in terms of inhomogeneous boundary conditions for S*; they are associated with eigenvalue problems for S* depending on λ-linear boundary conditions, which we also characterize.

For the non-linear problem

where f is a discontinuous function at 1, we show that the number of non-trivial positive solutions, for a given real number λ≧0, is related to the graph of a continuous function g. Then, by studying the function g it is possible in some special cases to give, for any λ≧0, the minimal or exact number of non-trivial positive solutions.

Consider solutions 〈H(x, ε), G(x, ε)〉 of the von Kármán equations for the swirling flow between two rotating coaxial disks

and

We also assume that |H(x, ε)|≦B√(ε) while |G(x, ε)|≦B. This work considers the shapes and asymptotic behaviour as ε→0+. We consider the kind of limit functions that are permissible. The only possible limits (interior) for G(x, ε) are constants. If that limit constant is not zero, then ε−½H(x, ε) will also tend to a constant.

Asymptotic estimates for the eigenvalues and eigenfunctions of a directed set of Sturm–Liouville operators are obtained. Particular attention is paid to the influence of the diffusion coefficient, as it becomes arbitrarily large.

Consider solutions 〈H(x, ε), G(x, ε)〉 of the von Kármán equations for the swirling flow between two rotating coaxial disks

and

We assume that |H(x, ε)| + |Hʹ(x, ε)| + |G(x, ε)|≦B. This work considers shapes and asymptotic behaviour as ε→0+. We consider the type of limit functions 〈H(x), G(x)〉 that are permissible. In particular, if 〈H(x, ε), G(x, ε)〉 also satisfy the boundary conditions H(0, ε)=H(1, ε)=0, Hʹ(0, ε)=Hʹ(1, ε)=0 then H(x) has no simple zeros. That is, there does not exist a point Z ε [0, 1] such that H(x)=0, Hʹ(z)≠0. Moreover, the case of “cells” which oscillate is studied in detail.

The polynomials which are orthogonal with respect to

when α> – 1, M>0 are considered when α<–1 and/or M<0. The Cauchy regularization of 〈·, ·〉 provides orthogonality and generates a Pontrjagin (Krein) space spanned by the polynomials. The polynomials are eigenfunctions associated with a self-adjoint, fourth order differential operator.

In this paper, two classes of linear bornological spaces are considered, the Kolmogorov spaces and the spaces of type b. These spaces satisfy conditions which are weakenings of the definition of infratopological linear bornological spaces. Various properties of these spaces are proved, and two examples are given, showing the independence of the two conditions introduced.

A sufficient condition on the angles of a bounded open subset of ℝn is given, ensuring the best regularity of solutions of a class of elliptic problems with non-linear mixed boundary conditions.