Let G be a group and K be a complete ultrametric valued field. Let AP(G, K) be the algebra of the generalized almost periodic functions of G in K. We have shown in a previous paper that when AP(G, K) has an invariant mean, then any almost periodic linear representation is quasi-reducible. Here, we show that with the same hypothesis, any topologically irreducible almost periodic linear representation is finite dimensional; also, any almost periodic linear representation is the topological sum of irreducible representations. Furthermore, we obtain a Peter-Weyl theorem for the algebra AP(G, K).

We use the technical tools of Hopf algebra theory.

In this paper we investigate existence and nonexistence of positive solutions for inhomogeneous semilinear elliptic systems in $R^n$. A criteria of existence and nonexistence of positive solutions is given by the exponents of the rate of nonlinear terms at infinity.

Let ∧ denote a basic artin ring and r its radical. In most of this paper we assume that r2 = 0 and that Λ is a trivial extension Λ/r ⋉ r (see Section 1 for definition). Let P1 …, Pn be the non-isomorphic indecomposable projective (left) Λ-modules, and consider triples (Pi, Mi, ui), where the Mi, are (left) Λ-modules and ui:rPi → Mi/rMi isomorphisms. From this data we construct a new ring Г, which in “nice cases” has the property that r′3 = 0, Г/r′2 ≅ Λ, and r′ Qi ≅ Mi as (left) Λ-modules, where the Qi are the indecomposable projective (left) Г-modules and r′ is the radical of Г.

Two coupled bilinear equations are considered, and then two new coupled differential-difference systems are found. Also two special reductions of these two systems are studied. By using Hirota's method, Bäcklund transformation and superposition formulae, soliton solutions to these equations are presented.

A tensor is said to be isotropic relative to a group of transformations if its components are invariant under the associated group of coordinate transformations. In this paper we review the classification of tensors which are isotropic under the general linear group, the special linear (unimodular) group and the rotational group. These correspond respectively to isotropic absolute tensors [4, 8] isotropic relative tensors [4] and isotropic Cartesian tensors [3]. New proofs are given for the representation of isotropic tensors in terms of Kronecker deltas and alternating tensors. And, for isotropic Cartesian tensors, we provide a complete classification, clarifying results described in [3].

In the final section of the paper certain derivatives of isotropic tensor fields are examined.

We denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of a normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.

We study the automorphism group of the algebra $\co_q(M_n)$ of n × n generic quantum matrices. We provide evidence for our conjecture that this group is generated by the transposition and the subgroup of those automorphisms acting on the canonical generators of $\co_q(M_n)$ by multiplication by scalars. Moreover, we prove this conjecture in the case when n = 3.

By the group G(2 cos π/q) we mean the group of linear fractional transformations of the complex plane onto itself, generated by V(z)= — 1/z and U(z) = z+λq, where λq = 2 cos (π/q), qbeing a positive integer greater than 2. In this paper we shall be concerned only with the group given by q = 5, and we shall therefore omit the subscript 5 on the λ. We note that λ = λ5 satisfies the equation

x2–x–l=0;(1)

henceλ = (l + 5½)/2.

We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.

A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

The semigroups of unital extensions of separable C*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K-theoretic classification of both unital and non-unital extensions of C*-algebras, and in particular we obtain a complete K-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

Hirota representations of soliton equations have proved veryuseful. They produced many of the known families of multisoliton solutions, andhave often led to a disclosure of the underlying Lax systems and infinite sets ofconserved quantities.

A striking feature is the ease with which direct insight can be gained into thenature of the eigenvalue problem associated with soliton equations derivable from aquadratic Hirota equation (for a single Hirota function), such as the KdV equationor the Boussinesq equation. A key element is the bilinear Bäcklund transformation(BT) which can be obtained straight away from the Hirota representation of theseequations, through decoupling of a related “two field condition” by means of anappropriate constraint of minimal weight. Details of this procedure have beenreported elsewhere. The main point is that bilinear BT's are obtained systematically,without the need of tricky “exchange formulas”. They arise in the formof “Y-systems”, each equation of which belongs to a linear space spanned by a basisof binary Bell polynomials (Y-polynomials).

For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

The use of graphs in the study of groups is well-established. In this paper, we wish to indicate how certain graph-like objects may be used in a similar way. A diagram is a pseudograph which may have some free edges, i.e. edges with just one end.

Non-abelian homology of Lie algebras with coefficients in Lie algebras is constructed and studied, generalising the classical Chevalley-Eilenberg homology of Lie algebras. The relationship between cyclic homology and Milnor cyclic homology of non-commutative associative algebras is established in terms of the long exact non-abelian homology sequence of Lie algebras. Some explicit formulae for the second and the third non-abelian homology of Lie algebras are obtained.

There exist several different approaches to the problem of solving dual integral equations involving Bessel Functions [1, 2, 3, 4, 5, 6,7], and Erdelyi and Sneddon in a recent paper [8] have shown that the introduction of certain operators occurring in the theory of fractional integration enables the relationships between the various methods to be clearly demonstrated. For dual integral equations other than those involving Bessel Functions the operators introduced by Erdélyi and Sneddon are not always the appropriate ones to use and it seems to be of interest to consider this more general type of situation.

We give characterisations of the linear relations that are injective and open (respectively, almost open with dense range) in terms of the stability of the nullity (respectively, of the deficiency). Results of Mbekhta about bounded operators in Banach spaces (J. Operator Theory35 (1996), 191–201) are covered.

Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterexamples to several naturally raised situations in the process.

Let C denote Cayley's algebra defined over the field of rational numbers. This paper contains a simple characterization of arithmetics of C in terms of a given basis i0 = 1, i1, i2, …, i7. We deduce that certain of the arithmetics of C are isomorphic. The result that the maximal arithmetics are isomorphic is also contained in the work of van der Blij and Springer [2].

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.