Complex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.

Examples are constructed of stably finite, imitai, separable C* -algebras A of real rank zero such that the partially ordered abelian groups K0(A) do not satisfy the Riesz decomposition property. This contrasts with the result of Zhang that projections in C* -algebras of real rank zero satisfy Riesz decomposition. The construction method also produces a stably finite, unital, separable C* -algebra of real rank zero which has the same K-theory as an approximately finite dimensional C*-algebra, but is not itself approximately finite dimensional.

We characterize, by means of congruence identities, all varieties having a weak difference term, and all neutral varieties. Our characterization of varieties with a difference term is new even in the particular case of locally finite varieties.

In this paper we study in which rings a non-zero idempotent element can be presented as a sum of two nilpotent elements.

This paper continues our investigation of backward continued fractions, associated with the generalized Renyi maps on [0,1). We first show that the dynamics of the shift map on a specific class of shift invariant spaces of nonnegative integer sequences exactly models the maps Tu for u € (0,4). In the second part we construct a new family of explicit invariant measures for certain values of the parameter u.

We show that bounded John domains and bounded starshaped domains with respect to a point satisfy the following inequality

where F: [0, ∞) → [0, ∞) is a continuous, convex function with F(0) = 0, and u is a function from an appropriate Sobolev class. Constants b and K do depend at most on D. If F(x) = xp, 1 ≤ p < ∞, this inequality reduces to the ordinary Poincaré inequality.

Let λ be a map from the additive Euclidean n-group Rn into the space R of real numbers and define a multiplication * on Rn by v * w = (λ(w))v. Then (Rn, + , *) is a topological nearring if and only if λ is continuous and λ(av) = aλ(v) for every v € Rn and every a in the range of λ. For any such map λ and any topological space X we denote by Nλ(X, Rn) the nearring of all continuous functions from X into (Rn, +, *) where the operations are pointwise. The ideals of Nλ(X, Rn) are investigated in some detail for certain λ and the results obtained are used to prove that two compact Hausdorff spaces X and Y are homeomorphic if and only if the nearrings Nλ(X, Rn) and Nλ(Y, Rn) are isomorphic.

Let R be a left and right perfect right self-injective ring. It is shown that if the radical of R is countably generated as a left ideal then R is quasi-Frobenius. It is also shown that the same conclusion can be drawn if r(A ∩ B) = r(A) + r(B) for all left ideals A and B of R.

As it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant —n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.

The generalized binomial function can be obtained as the solution of the equation y = 1 +zyα which satisfies y(0) = 1 where α ≠ 1 is assumed to be real and positive. The technique of Lagrange inversion can be used to express as a series which converges for |z| < α-α|a — l|α-1. We obtain a representation of the function as a contour integral and show that if α > 1 it is an analytic function in the complex z plane cut along the nonnegative real axis. For 0 < α < 1 the region of analyticity is the sector |arg(—z)| < απ. In either case defined by the series can be continued beyond the circle of convergenece of the series through a functional equation which can be derived from the integral representation.

Among the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

Let d ≥ 2, and K ⊂ ℝd be a convex body with 0 ∈ int K. We consider the intersection body IK, the cross-section body CK and the projection body ΠK of K, which satisfy IK ⊂ CK ⊂ ΠK. We prove that [bd(IK)] ∩ [bd(CK)] ≠ (a joint observation with R. J. Gardner), while for d ≥ 3 the relation [CK] ⊂ int(ΠK) holds for K in a dense open set of convex bodies, in the Hausdorff metric. If IK = c ˙ CK for some constant c > 0, then K is centred, and if both IK and CK are centred balls, then K is a centred ball. If the chordal symmetral and the difference body of K are constant multiples of each other, then K is centred; if both are centred balls, then K is a centred ball. For d ≥ 3 we determine the minimal number of facets, and estimate the minimal number of vertices, of a convex d-polytope P having no plane shadow boundary with respect to parallel illumination (this property is related to the inclusion [CP] ⊂ int(ΠP)).

The mutual injective hull of an arbitrary family of modules is constructed. Applications to the calculations of quasi-injective and π-injective hulls of direct sums are given.

An A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

We study the pure infiniteness of C* -crossed products by endomorphisms and automorphisms. Let A be a purely infinité simple unital C*-algebra. At first we show that a crossed product A × p N by a corner endomorphism p is purely infinite if it is simple. From this observation we prove that any simple C*-crossed products A ×αZ by an automorphism α is purely infinite. Combining this with the result in [Je] on pure infiniteness of crossed products by finite groups, one sees that if α is an outer action by a countable abelian group G then the simple C*-algebra A ×α G is purely infinite.

We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt} is a path of such operators, we can associate to {Bt} an integer, sf({Bt}), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {Bt} the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0.

We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.

We show that every Turaev-Viro invariant for 3-manifolds is a sum of three new invariants and discuss their properties. We also find a solution of a conjecture of L. H. Kauffman and S. Lins. Tables of the invariants for closed orientable 3-manifolds of complexity ≤ 3 are presented at the end of the paper.

Let denote the convex set of all positive linear maps from the matrix algebra Mn(ℂ) into itself. We construct a join homomorphism from the complete lattice of all faces of into the complete lattice of all join homomorphisms between the lattice of all subspaces of ℂn . We also characterize all maximal faces of .

The Abbena-Thurston manifold (M,g) is a critical point of the functional where Q is the Ricci operator and R is the scalar curvature, and then the index of I(g) and also the index of — I(g) are positive at (M,g).

Let R be a prime ring with involution and d, δ be derivations on R. Suppose that xd(x)—δ(x)x is central for all symmetric x or for all skew x. Then d = δ = 0 unless R is a commutative integral domain or an order of a 4-dimensional central simple algebra.