Fibonacci algebras are groups equipped with an extra unary operation φ that satisfies a Fibonacci-type law. We described in an earlier paper the free objects in the resulting varieties, and here we do the same in the case when φ is assumed to be periodic. They turn out to be central extensions of Burnside groups with finite kernels whose orders can be expressed in terms of the resultants of certain polynomials.

The generalization of the Magnus embedding [7] proved by Smelkin [9] may bestated as follows. Let L be a free group freely generated by the set xi(i∈I), and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and ∏ is the V-freegroup with free generating set the symbols [g, xi] (g∈G, i∈I), then L/V(R) is embeddedin the semidirect product ∏ ⋊ G (where the action of G on ∏ is given by h · [g, xi] = [hg, xi], for h, g ∈ G).

The object of this paper is to remove the difficulty that arises in giving a general proof by projection methods of this theorem, without in any way interfering with the single-valuedness of the position of a radius vector tracing out angles from a given initial position, when the values of the trigonometrical ratios are given.

Implicit operations (new operations commuting with all old homomorphisms) on pseudovarieties have been shown to play an important role in the study of these classes. They may be used to axiomatize sub-pseudovariaties and to describe recognizable subsets of (relatively) free objects. This paper presents a case study for the pseudovariety CS consisting of all finite simple semigroups. Based on a result of profinite group theory, a structural description of semigroups of implicit operations on finite simple semigroups is used to deduce that CS is join-irreducible.

In this paper the factorization of arithmetical numbers of the form , where x; is a rational number such that kx is a perfect square, is investigated by means of a trigonometrical transformation. The number k will be taken to be prime for the present.

§ I. On an infinite series of Triad Circles derived from the inscribed circle. Determination of a direct relation between r and the three radii of the nth triad.

Each of the first triad touches two sides of ABC and the inscribed circle : generally, each circle of the mth triad touches two sides of ABC and also touches one of the circles of the (m – 1)th triad.

Let Γ be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of Γ can be simultaneously made parabolic in a group on the boundary of the quasi-conformal deformation space of Γ. We shall prove that for this, it is sufficient that the classes of the system are represented by disjoint primitive simple closed curves on the ideal boundary of H3/Γ.

The study of the topological properties of algebraic surfaces, considered as continua of four real dimensions, has thrown much light on the theory of the birational invariants of such loci. The results obtained for surfaces have been generalised to varieties of higher dimension by Hodge, and, particularly, by Lefschetz. Apart from this, little seems to be known about the general topological properties of algebraic loci of three (or more) dimensions, the detailed study of which seems to present considerable difficulty. In particular, apart from the general theorems of Lefschetz, nothing seems to be known about the cycles of three dimensions of an algebraic V3. The object of the present paper is to study these cycles on certain quite special V3, in the hope that some insight may be gained into the general theory.

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equation

and Pn(x) satisfies the differential equation

In [1] J. Ax studied a class of fields with similar properties as finite fields called pseudo-finite fields. One can prove that pseudo-finite fields are precisely the infinite models of the first-order theory of finite fields. Similarly a near-field F is called pseudo-finite if F is an infinite model of the first-order theory of finite near-fields. The structure theory of these near-fields has been initiated by U. Feigner in [5].

A harmonic morphism defined on $\mathbb{R}^3$ with values in a Riemann surface is characterized in terms of a complex analytic curve in the complex surface of straight lines. We show how, to a certain family of complex curves, the singular set of the corresponding harmonic morphism has an isolated component consisting of a continuously embedded knot.

AMS 2000 Mathematics subject classification: Primary 57M25. Secondary 57M12; 58E20

The concept of a pseudo-ring was introduced by Patterson (1). Briefly, a pseudo-ring is an algebraic system consisting of an additive abelian group A, a distinguished subgroup A*, and a multiplication operation A* × A→A under which A* is a ring and A a left A*-module. For convenience, we denote the pseudo-ring by = (A*, A). For the definitions of the various types of ideal, we refer the reader to (1).