Let X be a separable and metrizable space containing uncountably many pairwise disjoint copies of the compactum K. We discuss the question whether X must contain K × 2ω.

Experiments indicate the importance of three-dimensional action during transition, while high-Reynolds-number-flow theory indicates a multi-structured type of analysis. In line with this, the three-dimensional nonlinear unsteady triple-deck problem is addressed here, for slower transition. High-amplitude/high-frequency properties show enhanced disturbance growth occurring downstream for single nonlinear oblique waves inclined at angles greater than tan−1 √2 (≈54.7°) to the free stream, in certain interesting special cases. The three-dimensional response there is very ‘spiky’ and possibly random, with sideband instabilities present. A second nonlinear stage, and then an Euler stage, are entered further downstream, although faster transition can go straight into these more nonlinear stages. More general cases are also considered. Sideband effects, sublayer bursting and secondary instabilities are discussed, along with the relation to experimental observations.

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.

It is proved that C(K) has no equivalent uniformly Gâteaux differentiable norm (UGD) when K-is an uncountable separable scattered com¬pact space. This result is applied to obtain an example of scattered compact K such that K′″ = ∅ and C(K) has no UGD renorming.

We shall study the properties of typical n-dimensional subspaces of , or equivalently, -typical n-dimensional quotients of , where the meaning of what is typical and what is not is defined in terms of the Haar measure μn,N on the Grassmann manifold Gn,N of all n-dimensional subspaces of .

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.

We prove a characterization of functions in B1/4(K)\C(K), where K is a compact metric space in terms of c0-spreading models, answering a Problem of R. Haydon, E. Odell and H. Rosenthal. Beginning with B1/4(K) we define a decreasing family (Vξ(K),║ · ║ξ)1≤ξ<ω1 of Banach spaces whose intersection is DBSC(K) and we prove an analogous stronger property for the functions in Vξ(K)\C(K). Defining the s-spreading model-index, we classify B1/4;(K) and we prove that s-SM[F]>ξ for every F∈ Vξ(K). Also we classify the separable Banach spaces by defining the c0-SM-index which measures the degree to which they have sequences with extending spreading models equivalent to the usual basis of c0. We give examples of Baire-1 functions and reflexive spaces with arbitrary large indices.

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 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}

Let Ω be a regular domain in the extended complex plane, i.e., it is a bounded domain and its boundary consists of a finite number of disjoint analytic simple closed curves. Let dm(z) be the Lebesgue area measure on Ω and let ds = dm(z)/ω(z) be the Poincare metric on Ω, a Riemannian metric of negative constant curvature. It may be proved that Ω(z) ≈ Euclidean distance from z to the boundary of Ω (see [8]).

We prove theorems relating descriptive set theory to nonreflexive Banach spaces. In Theorems 1, 2, and 3 X denotes a Banach space that is separable, but is not reflexive. JX denotes the cannonical embedding of X in X**.

The impaction on symmetrical obstacles placed in uniform streams of aerosols is investigated. The governing equations of motion are nonlinear differential equations involving a parameter called the Stokes number. The study differentiates between the critical value of the Stokes number on the centre-line, kcr, below which no particles reach the stagnation point in finite time, and the critical value of the Stokes number on the obstacle, Kcr, below which no particles may be deposited on the obstacle in finite time. Based on the properties of the centre-line fluid velocity of the potential and viscous flows past a variety of symmetrically shaped obstacles, upper and lower bounds of Kcr and Kcr are established. Furthermore, using a numerical procedure for solving nonlinear differential equations with unknown parameters the critical values of Kcr and Kcr are obtained.

In this paper, we consider an extended model of a coupled nonlinear reaction-diffusion equations with Neumann-Neumann boundary conditions. We obtain upper linear growth bound for one of the components. We also find the corresponding bound for the case of Dirichlet-Dirichlet boundary conditions.

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.

Kreck and Stolz recently exhibited exotic structures on a family of seven dimensional homogeneous spaces which are quotients of the compact Lie group SU3. We observe that there is an invariant obtained via the Pontrjagin–Thorn construction which detects these exotic structures in many cases.

We establish a generalization of the Cesari-Kannan existence result for problems of the type Lx = N(x), x∈X where X is a separable Hilbert functional space, L is a selfadjoint linear differential operator with nontrivial finite dimensional kernel and N:X→X is a bounded continuous nonlinear operator. This generalization leads to new results when the dimension of the kernel of L is greater than one. Applications to systems of second order ordinary differential equations are given.

Our aim in this paper is to establish new comparison principles. The equation

is compared with the equation

The equations considered can be in canonical or noncanonical forms.

We prove a result related to the Erdős-Ginzburg-Ziv theorem: Let p and q be primes, α a positive integer, and m∈{pα, pαq}. Then for any sequence of integers c= {c1, c2,…, cn} there are at least

subsequences of length m, whose terms add up to 0 modulo m (Theorem 8). We also show why it is unlikely that the result is true for any m not of the form pα or pαq (Theorem 9).

We shall give an explicit form of the Artin-Tschebotareff density theorem in function fields with several variable over finite fields. It may be an analogous prime number theorem in the higher dimensional case.

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.