Methods are used from descriptive set theory to derive Fubinilike results for the very general Method I and Method II (outer) measure constructions. Such constructions, which often lead to non-σ-finite measures, include Carathéodory and Hausdorff-type measures. Several questions of independent interest are encountered, such as the measurability of measures of sections of sets, the decomposition of sets into subsets with good sectional properties, and the analyticity of certain operators over sets. Applications are indicated to Hausdorff and generalized Hausdorff measures and to packing dimensions.

Let ℳ be the collection of all intersections of balls, considered as a subset of the hyperspace ℳ of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. It is proved that ℳ is uniformly very porous if and only if the space fails the Mazur intersection property.

For a completely regular space X, denote by Cp(X) the space of continuous real valued functions on X, endowed with the pointwise convergence topology. The spaces X and Y are t-equivalent if Cp(X) and Cp(Y) are homeomorphic. It is proved that, for metrizable spaces X, the countable dimensionality is preserved by t-equivalence. It is also shown that this relation preserves absolute Borel classes greater than 2 and all projective classes.

It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.

§1. Introduction. We study in this paper some properties of the Lusternik-Schnirelmann category of isolated invariant sets of continuous dynamical systems. There are several different definitions of this coefficient, although most of them agree in the important case of ANR's (Absolute Neighbourhood Retracts). We refer to the review articles [10] by R. H. Fox and [15, 16] by I. M. James for general information about this topological invariant. We shall use in this paper the definition of the Lusternik-Schnirelmann category of a compactum introduced by K. Borsuk in [4].

A method of finding critical points in a half space is developed. It is then applied to the study of semilinear boundary value problems, and used to determine conditions which lead to multiple non-trivial solutions.

§1. Introduction. The spectral function ρα(μ) (−∞<μ<∞) associated with the Sturm–Liouville equation

and a boundary condition

is a non-decreasing function of μ which is defined in terms of the Titchmarsh–Weyl function mα(λ) for (1.1) and (1.2).

§1. Introduction. We consider the spectral function ρα(λ) associated with the Sturm–Liouville equation

with the boundary condition

The integral representation for the solution of the 2-D Dirichlet problem for harmonic functions with boundary data on closed and open curves is obtained. The solution is expressed as a sum of potentials, the density of which obeys the uniquely solvable Fredholm integral equation of the second kind.

The problem of scattering of tidal waves by reefs and spits of arbitrary shape is reduced to a skew derivative problem for the two-dimensional Helmholtz equation in the exterior of open arcs in a plane. The resulting boundary-value problem is studied by potential theory and a boundary integral equation method. After some transformations, the skew derivative problem is reduced to a Fredholm integral equation of the second kind, which is uniquely solvable. In this way the solvability theorem is proved and an integral representation of the solution is obtained. A uniqueness theorem is also proved.

First, a special case of Knaster's problem is proved implying that each symmetric convex body in ℝ3 admits an inscribed cube. It is deduced from a theorem in equivariant topology, which says that there is no S4–equivariant map from SO(3) to S2, where S4 acts on SO(3) on the right as the rotation group of the cube, and on S2 on the right as the symmetry group of the regular tetrahedron. Some generalizations are also given.