The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

An example is given of an operator weight W that satisfies the dyadic operator Hunt–Muckenhoupt–Wheeden condition [Aopf ]d2 for which there exists a dyadic martingale transform on L2 (W) that is unbounded. The construction relates weighted boundedness to the boundedness of dyadic vector Hankel operators.

In 1998, the second author of this paper raised the problem of classifying the irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of Sn of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskii's problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p-power degree > 1 and all of the irreducible characters of G have degrees that are relatively prime to p or are powers of p?

The case of the double covers of the symmetric and alternating groups will be dealt with in a forthcoming paper; in particular, this completes the answer to Zalesskii's problem.

The paper is organized as follows. In Section 2, some results on hook lengths in partitions are proved. These results lead to an algorithm which allows us to show that every irreducible representation of Sn with prime power degree is labelled by a partition having a large hook. In Section 3, we obtain a new result concerning the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove Theorem 2.4, the main result. To do so, we combine the algorithm above with Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for large n. To complete the proof, we check that the algorithm terminates appropriately for small n (that is, those n [les ] 9.25 · 108) with the aid of a computer. In the last section we derive the classification of irreducible characters of An of prime power degree, and we solve Huppert's question for alternating groups.

We study, on the entire space ℝN(N [ges ] 1), the diffusive logistic equation

and its generalizations. Here p > 1 is a constant. Problem (1.1) plays an important role in understanding various population models and some other problems in applied mathematics. When λ = 1 and p = 2, it is also known as the Fisher equation and KPP equation, due to the pioneering works of Fisher [8] and Kolmogoroff, Petrovsky and Piscounoff [18].

The paper gives the first examples of C∞ measure preserving flows and diffeomorphisms on the torus that are partially mixing and have a simple spectrum.

Jordan superalgebras defined by brackets on associative commutative superalgebras are studied. It is proved that any such superalgebra is imbedded into a superalgebra defined by Poisson brackets. In particular, all Jordan superalgebras of brackets are i-special. The speciality of these superalgebras is also examined, and it is proved, in particular, that the Cheng–Kac superalgebra is special.

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

The object of the paper is to show that if f is a univalent, harmonic mapping of the annulus A(r, 1) = {z : r < [mid ]z[mid ] < 1} onto the annulus A(R, 1), and if s is the length of the segment of the Grötzsch ring domain associated with A(r, 1), then R < s. This gives the first, quantitative upper bound of R, which relates to a question of J. C. C. Nitsche that he raised in 1962. The question of whether this bound is sharp remains open.

A strictly singular non-compact operator on the Gowers and Maurey space GM is constructed.

It is proved that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X [oplus ] Y, then there exists a completely bounded automorphism τ: X [oplus ] Y → X [oplus ] Y with completely bounded inverse such that τZ = X0 [oplus ] Y0, where X0 and Y0 are completely complemented subspaces of X and Y, respectively. If X and Y are homogeneous, the existence is proved of such a τ under a weaker assumption that any operator from X to Y is strictly singular. An upper estimate is obtained for ∥τ∥cb∥τ−1∥cb if X and Y are separable homogeneous Hilbertian operator spaces. Also proved is the uniqueness of a ‘completely unconditional’ basis in X [oplus ] Y if X and Y satisfy certain conditions.

The paper studies the relation between the asymptotic values of the ratios area/length (F/L) and diameter/length (D/L) of a sequence of convex sets expanding over the whole hyperbolic plane. It is known that F/L goes to a value between 0 and 1 depending on the shape of the contour. In the paper, it is first of all seen that D/L has limit value between 0 and 1/2 in strong contrast with the euclidean situation in which the lower bound is 1/π (D/L = 1/π if and only if the convex set has constant width). Moreover, it is shown that, as the limit of D/L approaches 1/2, the possible limit values of F/L reduce. Examples of all possible limits F/L and D/L are given.

It is shown that there exists a non-separable reflexive Banach space on which every bounded linear operator is the sum of a scalar multiple of the identity operator and an operator of separable range. There is a strong sense that such a Banach space has as few operators as its linear and topological properties allow.

Let f(z) and g(z) be transcendental entire functions. Fuchs and Song proved that if (f(z); g(z)) parametrizes some complex algebraic curve, then f and g must have a transcendental common right factor. The paper proves this result by a different method that also allows a similar result to be proved for some transcendental curves. This result is then used to solve some factorization problems of entire functions.

Weak slice conditions are introduced and imbeddings of Sobolev spaces in various Lipschitz-type spaces are investigated.

It is proved that every solution of the Neumann initial-boundary problem

[formula here]

converges to some equilibrium, if the system satisfies (i) ∂Fi/∂uj [ges ] 0 for all 1 [les ] i ≠ j [les ] n, (ii) F(u * g(s)) [ges ] h(s) [midast ] F(u) whenever u ∈ ℝn+ and 0 [les ] s [les ] 1, where x * y = (x1y1, …, xnyn) and g, h : [0, 1] → [0, 1]n are continuous functions satisfying gi(0) = hi(0) = 0, gi(1) = hi(1) = 1, 0 < gi(s); hi(s) < 1 for all s ∈ (0, 1) and i = 1, 2, …, n, and (iii) the solution of the corresponding ordinary differential equation system is bounded in ℝn+. We also study the convergence of the solution of the Lotka–Volterra system

[formula here]

where ri > 0, α [ges ] 0, and aij [ges ] 0 for i ≠ j.

Let V and X be Hausdorff, locally convex, real, topological vector spaces with dim V > 1. It is shown that a map σ from an open, connected subset of V onto an open subset of X is homeomorphic and convexity-preserving if and only if σ is projective.

Let A be a Banach algebra, and let D: A → A* be a continuous derivation, where A* is the topological dual space of A. The paper discusses the situation when the second transpose D**: A** → (A**)* is also a derivation in the case where A** has the first Arens product.

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U ⊂ X be an open subset and f: U → X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K ⊂ U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

The aim of the paper is to establish a complete scale of isomorphisms for elliptic pseudodifferential boundary value problems generated by the operators of the Boutet de Monvel algebra. The result is an extension of the corresponding known results on elliptic differential boundary value problems. Because for any elliptic pseudodifferential boundary value problem there exists a parametrix belonging to the Boutet de Monvel algebra, the proof presented is much shorter than the known proof for differential problems. Systems that are elliptic in the sense of Douglis and Nirenberg are also considered.

The paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(an,0+an,1z+…+an,JzJ)zn, n [ges ] 0}. The primary focus is on the tridiagonal case where J = 1, and on how it compares with the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated, and aspects of this operator are discussed. In the diagonal case, Mz is a weighted unilateral shift. It is shown that in the tridiagonal case, this need not be so, and an example is given in which the commutant of Mz on a tridiagonal space is strikingly different from that on any diagonal space.