Dimension prints were developed in 1988 to distinguish between different fractal sets in Euclidean spaces having the same Hausdorff dimension but with very different geometric characteristics. In this paper we compute the dimension prints of some fractal sets, including generalized Cantor sets on the unit circle S1 in ℝ2 and the graphs of generalized Lebesgue functions, also in ℝ2. In this second case we show that the dimension print for the graphs of the Lebesgue functions can approach the maximal dimension print of a set of dimension 1. We study the dimension prints of Cartesian products of linear Borel sets and obtain the exact dimension print when each linear set has positive measure in its dimension and the dimension of the Cartesian product is the sum of the dimensions of the factors.

In this paper, we first give a characterization of a regular endomorphism of a graph. Then, we explicitly describe its inverses through idempotents.

We prove that the relation type of all high powers of an ideal in a Noetherian ring is either one or two. It is one exactly when some power of the ideal is locally principal.

For any fixed positive real number ε, any integer b≥2 and any dε{0, 1,…, b−1}, the set of Borel's simply normal numbers to base b in [0, 1] is partitioned into a countable number of sets in eight different ways according to the largest place and the number of places at which the proportion d's to that place in the b-adic expansion of such a number exceeds or is not less than b−1 – ε, or is less than or does not exceed b−1 – ε. For selected values ε, the Lebesgue measures of the sets in these decompositions are given explicitly.

In this paper, we will denote by convex figure a compact convex subset of the n-dimensional Euclidean space ℝn, and by convex body a convex figure with non-empty interior. The principal kinematic formula in integral geometry gives the measure of the set of congruent convex bodies intersecting with a fixed convex body. Specifically, let K, L be two convex bodies in ℝn and G(n) the group of special motions in ℝn. Each element, g: ℝn → ℝn, of G(n) can be represented by

where b∈ℝn and e is an orthogonal matrix of determinant 1. Let μ be the Haar measure on G(n) normalized as follows: Let μ:ℝn × SO(n) → G(n) be defined by φ(t, e)x = ex + t, xεℝn, where SO(n) is the rotation group of ℝn. If v is the unique invariant probability measure on SO(n), η is the Lebesgue measure on ℝn, then μ is chosen as the pull back measure of η⊗v under φ−1. If Wi(K), Wi(L) are the quermassintegrals of K, L, i= 0, 1,…, n, the principal kinematic formula states that

where ωn is the volume of the unit n–ball.

Let X be an infinite set and T(X) be the full transformation semigroup on X. In [4] and [6] Howie gives a description of the subsemigroup of T(X) generated by its idempotents. In order to do this he defines, for α in T(X),

and refers to the cardinals s(α) = |S(α)|, d(α) = |Z(α)| and |c(α) = |C(α)| as the shift, the defect, and the collapse of α respectively. Then putting

he proves that the subsemigroup of T(X) generated by its idempotents is . Furthermore, both F and Q are generated by their idempotents

Deformation spaces of quasi-Fuchsian groups provide the simplest nontrivial examples of deformation spaces of Kleinian groups. Their understanding is of interest with respect to the study of more general Kleinian groups. On the other hand, these spaces contain subspaces isomorphic to Teichmüller spaces, and are often useful for the study of properties of Teichmüller space. A recent example of this is the study of the Teichmüller space of the punctured torus by Keen and Series [KS].

The nonlinear interaction equations describing vortex-Rayleigh wave interactions in highly curved boundary layers are derived. These equations describe a strongly nonlinear interaction between an inviscid wave system and a streamwise vortex. The coupling between the two structures is quite different from that found by Hall and Smith [13] in the absence of wall curvature. Here the vortex is forced over a finite region of the flow rather than in the critical layer associated with the wave system. When the interaction takes place the wave system remains locally neutral as it moves downstream and its self interaction drives a vortex field of the same magnitude as that driven by the wall curvature. This modification of the mean state then alters the wave properties and forces the wave amplitude to adjust itself in order that the wave frequency is constant. Solutions of the interaction equations are found for the initial stages of the interaction in the case when the wave amplitude is initially small. Our analysis suggests that finite amplitude disturbances can only exist when the vortex field is nonzero at the initial position where the interaction is stimulated.

A compact space K is said to have the Namioka Property, or to belong to the class *, if, for every Baire space B and every separately continuous function Ψ:B × K → ℝ, there is dense δ subset H of B such that Ψ is (jointly) continuous at all points of H × K. Although the terminology is more recent, the idea of looking at properties of this kind goes back to Namioka's paper [6] on separate and joint continuity. Talagrand [8] gave the first example of a compact space that is not in * and it is now known [4] that there are even examples of scattered compact spaces that are not in *. On the other hand, many good classes of compact spaces have been shown to be contained in *, probably the most general being the class of continuous images of Valdivia compacts [2]. The aim of this note is to prove the following stability result: a compact space which is a countable union of closed subsets with the Namioka Property does itself possess that property.

Let X be a completely regular space, and Cb(X) the space of all bounded continuous real valued functions on X equipped with the metric associated to the uniform norm. For f∈Cb(X) and γ∈¡ we use the following standard notations: inf(f) = infx∈Xf(x) and {f<γ} = {x∈X:f(x)<γ}