Let A be a product of symmetric matrices, A = RQ, with R non-singular, and let v be an eigenvector of A. For certain R and Q, a convenient formula for the expression (R−1v)tv is obtained. This expression occurs in the diagonalization of A and, in the particular case where A is associated with the quasi-derivative formulation of higher-order differential equations, the expression occurs in the asymptotic theory of solutions of the differential equation.

This note complements an earlier paper of the same title. Let G be a discontinuous group of homeomorphisms of a connected, locally path connected, Hausdorff space X, and let ∏:X → X/G denote the associated projection. We work relative to a G-invariant subgroup H of the fundamental group of X and investigate the quotient group ∏1(X/G)/∏*(H). By choosing H appropriately, we can calculate ∏1(X/G) and show that ∏1(X/G)/∏*(∏1(X)) is isomorphic to G/F, where F is the normal subgroup of G generated by those elements which have fixed points. In a final section, we give analogous results for actions of a compact Lie group.

For a symmetric linear relation S with a directing mapping, the notion of a spectral function is defined by means of a Bessel–Parseval inequality, and a description of all such spectral functions is given. As an application, we describe the set of all spectral functions of a canonical regular first order differential system.

Let the values of F be convex compact subsets of Rn and let F be upper semicontinuous with respect to x. There are two ways known of replacing F by a more regular map so that the set of solutions of (2) remains unchanged. We prove that both ways lead to the same more regular map and extend the results to the case where Rn is replaced by a separable Banach space.

We describe the structure of regular semigroups in which each element is dominated, in the Nambooripad order, by a unique maximal element that is ℋ-equivalent to a mididentity. These are precisely direct products of a rectangular band and a uniquely unit regular semigroup.

Let G be a locally compact abelian group and let Γ be the dual of G. Let A, B be Banach spaces and Lp(G,A) the Bochner-Lebesgue spaces. We prove that the space of bounded linear translation invariant operators from L1(G, A) to LX(G, B) can be identified with the space of bounded convolution invariant (in some sense) operators and also with the space of a(A, B)-valued “weak regular” measures with the relation Tf = f *μ. (A. The existence of a function m∈ L∞ (Γ,α(A,B)), such that is also proved.

The present paper studies some asymptotic (including oscillatory) properties of the solutions of operator-differential inequalities of the form

where

(the latter symbol denotes the space of locally summable functions).

As an application of the results obtained, theorems are proved for the asymptotic behaviour of the solutions of certain classes of functional-differential and integro-differential neutral-type equations.

This article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.

This work provides an account of the asymptotic behaviour in terms of Abelian theorems for the Mellin and inverse Mellin transforms in a distributional setting.

We study the evaluation of multilinear forms under arbitrary rearrangements of the entries of increasing n-tuples x(1), x(2),…, x(m), and we show that the difference of two such multilinear forms under certain circumstances can be written as a sum of obviously definite forms.

The global topological structure of the space of configurations of a non-rotating elastic string under compression and tension is studied. The part of the string under tension is specified by a measurable subset of the interval. The set of such intervals, with the Hausdorff topology, is considered a parameter space for the equation satisfied by the string, and the solutions are shown to form an infinitedimensional continuum over this parameter space. A new global topological theorem is needed, since the parameter space is not Euclidean. The topological theorem is based on the fixed-point transfer.

A result for the Erdélyi-Kober operators, mentioned briefly by Buschman, is discussed together with a second related result. The results are proved rigorously by means of an index law for powers of certain differential operators and are shown to be valid under conditions of great generality. Mellin multipliers are used and it is shown that, in a certain sense, the index law approach is equivalent to, but independent of, the duplication formula for the gamma function. Various statements can be made concerning fractional integrals and derivatives which produce, as special cases, simple instances of the chain rule for differentiation and changes of variables in integrals.

This paper deals with some multiplicity results for elliptic problems with jumping nonlinearities. Our results are concerned with the case in which only one eigenvalue of the linear problem is jumped and it is simple. The main tool used is the Leray–Schauder topological degree. We consider a parametrized problem and prove the existence of two or three distinct solutions for suitable values of the parameter.

Uniqueness of non-negative solutions conjectured in an earlier paper by Shivaji is proved. Our methods are independent of those of that paper, where the problem was considered only in a ball. Further, our results apply to a wider class of nonlinearities.

In this note, we study the well-posedness of the exterior traction value problem for linear anisotropic non-homogeneous elastostatics. We prove existence and continuous dependence upon the data. In particular, in the isotropic homogeneous case, provided the body force is “simple”, we show that solutions tend to zero uniformly at large spatial distances.

We examine the case of plane, time-harmonic acoustic waves in two dimensions, scattered by an obstacle on the surface of which an impedance boundary condition is imposed. A stable method is developed for solving the inverse problem ofdetermining both the shape of the scatterer and the surface impedance from measurements of the asymptotic behaviour of the scattered waves at low frequencies. We accomplish this by minimizing an appropriate functional over a compact set of admissible boundary curves and admissible impedances.

In this paper, we study the local structure of the secant mapping of a pair of disjoint curves. We show that for generic curves, the secant map and unit secant maps are locally stable. If we allow our curves to coincide, we can define anew unit secant map to be the natural unit tangent map near the diagonal. This is, for a generic curve, a locally stablemap away from the diagonal. Along the diagonal, it is locally stable as a ℤ2 symmetric germ (the ℤ2 symmetry originating with reflection in the diagonal).

A one-phase Stefan problem can be reduced to an equivalent variational inequality by using the Baiocchi-Duvaut transformation. In this paper, we study the variational inequality by formulating it as a set-valued partial differential equation. The existence of solutions is proved by applying a generalized Schauder fixed fixed point theorem for set-valued mappings. Uniqueness and regularity of solutions are also obtained. In §3, we regard the boundary value Neumann data as boundary controls and combine both the variational inequality and the classical approaches to study the effects of controls on the free boundary and the state (i.e. temperature). In §4, we further use the theory to study an optimal “ice-melting” problem. Our results show that if the controls have fixed total input heat flux and are constrained in magnitude, then the optimal control is “bang-bang”. If the admissible controls are not constrained in magnitude, then the optimal control is a Dirac delta type distribution which is no longer admissible. In the last section, our existence theory is combined with the finite difference method and non-linear programming techniques to obtain numerical solutions.

Comparison functions are constructed for the problem of minimizing

over maps u: D(⊆ℝ2)→ℝ2 with det≥0, subject to the constraint u= f on ∂D, D the unit disk. This is accomplished for maps / which are reparameterizations of ∂D or which are “graph-like” maps. Estimates involving half derivative boundary norms are obtained.