Let p > 0 be prime, and let K be any field of characteristic p. Let X and Y be independent (commuting) variables over K. A polynomial f(X) ∈ K[X] is said to be additive (over K) if and only if f(X+Y) = f(X)+f(Y) in K[X, Y]. It is elementary (and classical) that the additive f(X) over K are precisely the polynomials of the type Σj=0dajXpj, where d [ges ] 0 and a0, …, ad∈K. Now let k [ges ] 2, and let X, Y, T0, …, Tk−1 be algebraically independent over K. We put q = ph (h ∈ ℕ) and define

formula here

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

In this paper, we introduce the notion of Jacobi polynomials for codes, and establish some fundamental features of it. This notion comes out of considerations on the various invariants of the codes (e.g. [22–24]); it has to do with Jacobi theta-series of positive definite quadratic forms (cf. [9], p. 82). Thus the name ‘Jacobi polynomials for codes’ traces this fact.

Let V be a 4-dimensional complex space. A congruence Y is an integral surface of the Grassmann variety G = Gr(2, 4) of 2-dimensional subspaces V2 of V (we denote by Vi a subspace of V of dimension i). They have been extensively studied by both classical and modern geometers. We bring to their study the tool of Chow forms, characterizing them by differential equations, following the program of M. Green and I. Morrison [3]. The first results in this direction are due to Cayley ([1], [2]) and are rederived in [3]. Our results share much of the geometrical flavour of Cayley's.

We study algebraic models of the Segal category of Riemann surfaces [11]. These are 2-categories based on the 1+1 dimensional cobordism category enriched by diffeomorphisms of surfaces (or isotopy classes of such). In particular, we show that one of these models is a symmetric strict monoidal strict 2-category. The main technical tools are quotient constructions on 2-categories.

In this paper we give for any r, n, 0 [les ] r [les ] n, a Quillen's model structure to the category of simplicial groups where the weak equivalences are those morphisms f[bull] such that πq(f[bull]) is an isomorphism for r [les ] q [les ] n. This is carried out by studying the cases r = 0 and n → ∞ previously and, in each one of them, we make explicit some constructions for the associated homotopy theories, such as the cylinder and path objects and the loop and suspension functors, and we also relate the simplicial homotopy relation to the homotopy relation obtained from these structures.

Real-valued function spaces CoL lack almost transitive norms. If there is a complex-valued CoL with almost transitive norm then there is also one with transitive norm. We give fairly restrictive topological conditions for the complex-valued CoL to have transitive norm.

During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

For a general state space Markov chain on a space (X, [Bscr ](X)), the existence of a Doeblin decomposition, implying the state space can be written as a countable union of absorbing ‘recurrent’ sets and a transient set, is known to be a consequence of several different conditions all implying in some way that there is not an uncountable collection of absorbing sets. These include

([Mscr ]) there exists a finite measure which gives positive mass to each absorbing subset of X;

([Gscr ]) there exists no uncountable collection of points (xα) such that the measures Kθ(xα, ·)[colone ](1−θ)ΣPn(xα, ·)θn are mutually singular;

([Cscr ]) there is no uncountable disjoint class of absorbing subsets of X.

We prove that if [Bscr ](X) is countably generated and separated (distinct elements in X can be separated by disjoint measurable sets), then these conditions are equivalent. Other results on the structure of absorbing sets are also developed.

This paper studies the spectral properties of a class of probability measures on the circle. The key aim is to describe the local structure of the maximal ideal space on the L-subspaces generated by the measures and hence the spectral properties of these measures. In particular we give a necessary and sufficient condition for such measures to belong to M0 (T).

Asymptotic analysis and distribution theory are two very useful branches of Applied Mathematics. It is only recently, however, that the close ties between these theories have been studied in detail [5, 6, 11, 17].

Among the various approaches to the distributional theory of asymptotic expansions [6] gives an analysis based on the moment asymptotic expansion.

Levinson's theorem with all its ramifications is a well-established tool in the spectral analysis of ordinary differential operators (see in particular §§3·10, 11 and 4·3, 4 of Eastham's book [5]). In this note we should like to draw attention to a result that describes the solutions of asymptotically constant linear systems under weaker assumptions less precisely than the Levinson theorem. This result can be called the Perron–Lettenmeyer–Hartman–Wintner theorem after the contributions of these authors in [11, 10, 6, 8]. (Note that the Hartman–Wintner theorem that is discussed in [5, p. 17] is a substantially different version of this theorem.)

A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian evolution equation on one or more space dimensions. This generalization, called multi-symplectic structures, is shown to be natural for dispersive wave propagation problems. Application of the abstract properties of the multi-symplectic structures framework leads to a new variational principle for space-time periodic states reminiscent of the variational principle for invariant tori, a geometric reformulation of the concepts of action and action flux, a rigorous proof of the instability criterion predicted by the Whitham modulation equations, a new symplectic decomposition of the Noether theory, generalization of the concept of reversibility to space-time and a proof of Lighthill's geometric criterion for instability of periodic waves travelling in one space dimension. The nonlinear Schrödinger equation and the water-wave problem are characterized as Hamiltonian systems on a multi-symplectic structure for example. Further ramifications of the generalized symplectic structure of theoretical and practical interest are also discussed.

The purpose of this short note is to reformulate Theorem 4·2 in [1]. This theorem is not correct in the part concerning maximality of the norm in Orlicz spaces. The theorem below provides a correct version of this statement. We wish to point out that all other theorems in [1] are independent of Theorem 4·2.