Existence and properties of incompressible surfaces in 3-dimensional manifolds are surveyed. Some conjectures of Waldhausen and Thurston concerning such surfaces are stated. An outline is given of the proof that such surfaces can be pulled back by non-zero degree maps between 3-manifolds. The effect of surgery on immersed, incompressible surfaces and on hierarchies is discussed. A characterisation is given of the immersed, incompressible surfaces previously studied by Hass and Scott, which arise naturally with cubings of non-positive curvature.

Given a group G and a finite generating set G, we take pG: G → Z to be the function which counts the number of geodesics for each group element g. This generalizes Pascal's triangle. We compute pG for word hyperbolic and describe generic behaviour in abelian groups.

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. We study the orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0. The methods are essentially number-theoretic.

1991 Mathematics subject classification (Amer. Math. Soc): primary 17B50; secondary 12E20.

A new arrow notation is used to describe biordered sets. Biordered sets are characterized as biordered subsets of the partial algebras formed by the idempotents of semigroups. Thus it can be shown that in the free semigroup on a biordered set factored out by the equations of the biordered set there is no collapse of idempotents and no new arrows.

We investigate a locally full HNN extension of an inverse semigroup. A normal form theorem is obtained and applied to the word problem. We construct a tree and show that a maximal subgroup of a locally full HNN extension acts on the tree without inversion. Bass-Serre theory is employed to obtain a group presentation of the maximal subgroup as a fundamental group of a certain graph of groups associated with the D-structure of the original semigroup.

Let T be a continuous t-norm (a suitable binary operation on[0, 1]) and Δ + the space of distribution functions which are concertratede on [0,∞. theτT product of any F, G in Δ+ is defined at any real x by , and the pair (Δ+, τT) forms a semigroup. Thus, given a sequence {Fi} in Δ+, the n-fold product τT(F1 … Fn) is well-defined for each n. Moreover, that resulting sequence {τT(F1, …, Fn)} is pointwise non-increasing and hence has a weak limit. This paper establishes a convergence theorem which yields a representation for this weak limit. In addition, we prove the Zero-One law that, for Archimedean t-norms, the weak limit is either identically zero or has supremum 1.

In this paper, we focus on the multivariate Hausdorff operator of the form

$$\begin{eqnarray}\mathbf{H}_{\unicode[STIX]{x1D6F7}}(f)(x)=\int _{(0,+\infty )^{n}}{\displaystyle \frac{\unicode[STIX]{x1D6F7}\big(\frac{x_{1}}{t_{1}},\frac{x_{2}}{t_{2}},\ldots ,\frac{x_{n}}{t_{n}}\big)}{t_{1}t_{2}\cdots t_{n}}}f(t_{1},t_{2},\ldots ,t_{n})\,\mathbf{dt},\end{eqnarray}$$

$\mathbf{dt}=dt_{1}\,dt_{2}\cdots \,dt_{n}$

$\mathbf{dt}=d_{q}t_{1}\,d_{q}t_{2}\cdots d_{q}t_{n}$

$q$

$L^{p}$

$p\in \mathbb{R}\backslash \{0\}$

In [7] Lyndon introduced the concept of inner rank for groups. He defined the inner rank of an arbitrary group G to be the upper bound of the ranks of free homomorphic images of G. Both Lyndon and Jaco have shown that the inner rank of the fundamental group of a closed 2-manifold with Euler characteristic 2 − p, p ≧ 0, is [p/2] where [p/2] is the greatest integer ≦ p/2. The proof given by lyndon [8] uses algebraic techniques; whereas, the proof by Jaco [4] is geometrical.

In continuation of the two previous papers (10; 11), this paper was originally written at the Indian Institute of Technology, Kharagpur and revised at the University of Sydney under the advice of Prof. T. G. Room. Although the altitudes of a general simplex S(A) in n-space (n > 2) do not concur as they do for a triangle (n = 2), yet we observe that its Monge point, M (1; 5), is an appropriate analogue of the orthocentre of a triangle such that M coincides with its orthocentre when it is orthogonal (or orthocentric). In consistency with the previous papers (10; 11; 13; 15) we shall call M as the S-point of S(A) and denote it as S as explained in § 1.2. The altitudes of S(A) are all met by the (n − 2)-spaces normal to its plane faces at their orthocentres, each parallel to of them, thus indicating the associated character of the altitudes as discussed separately in 2 other papers (12; 16). Before we introduce an orthogonal simplex and develop its properties in regard to its γ-altitudes and associated hyperspheres, we come across a number of intermediate ones of special interest. Two special types are treated here and the other two are developed in 2 other papers (13; 15).

We obtain analogues, in the setting of semigroups with zero, of McAlister's convering theoren and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroups is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all E*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical E*-unitary semigroups recently considered by Gomes and Howie.

Let Bn(x) denote the number of 1’s occurring in the binary expansion of an irrational number x>0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting numbers such as , e or π: their conjectural simple normality in base 2 is equivalent to Bn(x)∼n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x,y>0, which we prove to be sharp up to a multiplicative constant. As a by-product, we provide an answer to a question raised by Bailey et al. (D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, ‘On the binary expansions of algebraic numbers’, J. Théor. Nombres Bordeaux16(3) (2004), 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides a nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.

We describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.

The set E of idemponents of a semigroup S can be partially ordered by defining e ≦f and only if ef = fe = e (e,f ∈ E). If E = {ei: i = i = 0,1…} and under this ordering e0 > e1 > e2… then we call S an ω-semigroup. Munn [10] has given a complete classification of simple regular ω-semigroups in terms of groups and group homomorphisms. Let ∧0(S) denote the set of congruences on a simple regular w-semigroup S consisting of those congruences which either are idempotent-separating or are group congruences on S. It is evident that ∧0(S) is a sublattice of the lattice of all congruences on S.

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

The Redheffer matrix An = (aij)n×n defined by aij = 1 when i|j or j = 1 and aij = 0 otherwise has many interesting number theoretic properties. In this paper we give fairly precise estimates for its eigenvalues in punctured discs of small radius centred at 1.

The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

We prove certain Lp estimates (1<p<∞) for nonisotropic singular integrals along surfaces of revolution. The singular integrals are defined by rough kernels. As an application we obtain Lp boundedness of the singular integrals under a sharp size condition on their kernels. We also prove a certain estimate for a trigonometric integral, which is useful in studying nonisotropic singular integrals.