In his fundamental papers [7,8], Pelczynski introduced properties (u), (V), and (V*) as tools as study the structure of Banach spaces. Let X be a Banach space. It is said that X has property (u) if, for every weak Cauchy sequence (xn) in X, there exists a weakly unconditionally Cauchy (wuC) series in X such that the sequence is weakly null. It is said that X has property (V) if, for every Banach space Z, every unconditionally converging operator from X into Z is weakly compact; equivalently, whenever K is a bounded subset of X* such that for every wuC series in X, then K is relatively weakly compact. A Banach space X is said to have property (V*) if whenever K is a bounded subset of X such that 0 for every wuC series in X*, then K is relatively weakly compact. Some well-known results which shall be needed later are contained in the following.

We use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.

1. Consider the following dilatation equation

where |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfy

where is the Fourier transform of f, and

which immediately leads to

Let k be an algebraic number field and Ck its ideal class group (in the wider sense). Suppose K is a finite extension of k. Then we say that an ideal class of k capitulates in K if this class is in the kernel of the homomorphism

induced by extension of ideals from k to K (See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields, with distinct primes Pi ≡ 1 (mod 4), for which all the ideal classes of the 2-class group, Ck,2 (the 2-Sylow subgroup of Ck), capitulate in an unramified quadratic extension of k. In these examples, Ck,2 is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

Recently several authors have studied dualizing Goldie dimension of a module: spanning dimension in [2], codimension in [13], corank in [16] and also [9,17,12, 5,11, 6, 4, 7] ([13] may be read in comparison with the others). In the present note we prove the equality corank RP = corank SS, where P is a quasi-projective left R-module and S is its endomorphism ring. This result is an answer to the question [12, p. 1898] and an extension of [3, Corollary 4.3] which shows the above equality for a Σ-quasi-projective left R-module P.

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

With much sadness we note the death of John Leech, on 28 September 1992. Perhaps best known for his discovery of the “Leech Lattice” (which provides the best known sphere-packing in 24 dimensions), John will also be remembered for his contributions to the use of computers in mathematics, and to computational algebra in particular.

In this paper we prove that the weighted group algebra L1 (G, w) is semi-regular if and only if G is either abelian or discrete.

In the present note, Σr denotes the class of all right pure semisimple rings (= right pure global dimension zero). It is known that if R ∈ Σr, then R is right artinian and every indecomposable right R-module is finitely generated. The class Σr is not closed under ultraproducts [4]. While Σr is closed under elementary descent (i.e. if S ∈ Σr and R is an elementary subring of S then R ∈ σr) [4], it is an open question whether right pure-semisimplicity is preserved under the passage to ultrapowers [4, Prob. 11.16]. In this note, this question is answered in the affirmative.

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 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 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**.

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.

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.