Suppose that p > 1 and let Γ1, Γ2, …, Γp be Jordan arcs going from 0 to ∞ and having no other common point and numbered in anti-clockwise order around the origin. Let Γp+1 = Γ1 and let Dv be the domain bounded by Γv and Γv+1 for v = 1 to p.

Let f: M → N be a smooth map of a closed n-manifold into a p-manifold (n ≥ p) having only Morin singularities [17]. We study the topology of such a map and obtain a modulo 2 congruence formula involving the Euler characteristics of M, N, the singular sets and the regular fibres of f. We also consider applications of this formula to the existence problem of maps having only fold singular points. Stable maps into 3-manifolds are also studied and we obtain a modulo 2 congruence formula involving the swallow tails and the number of triple points of the discriminant set.

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finite p-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finite p-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a field Fp of p elements. It was immediately clear that the parameters Fp, ℚ of the McLain group G(Fp, ℚ) could be replaced by other linearly ordered, dense sets S and by other fields F without doing much harm to the construction. If F has characteristic 0, then G(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigated G(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group Aut G(Fp, S) in his dissertation at Cambridge in 1963.

Let p ≡ 1 (mod 3) or ≡ 1 (mod 5) be a rational prime and g a primitive root mod p. The non-zero residues g, g2,…,gp−1 (mod p) can then be divided into 3 or 5 classes , , , or , , , , respectively, by letting gν ∈ , , (respectively , , , , ) according as ν ≡ 0, 1, 2, (mod 3) (respectively 0, 1, 2, 3, 4 (mod 5)). Problems regarding the distribution of 1, 2, …, p − 1 amongst the 3 (respectively 5) classes are many. In this paper we consider the following problem in some detail (we shall state it here for the case p ≡ 1 (mod 3)). Let α, β, γ be typical members of , , respectively. Let the number of solutions of the congruences

be

The advent of quantum mechanics has provided a means of treating the interaction of atoms and the formation of molecules and has had very many successes in this field. In particular the phenomenon of the saturation of valence forces appears as a natural consequence of the theory. Nevertheless there are very many difficulties still to be overcome.

Let G = 〈x, t | w〉 be a one-relator group, where w is a word in x, t. If w is a product of conjugates of x then, associated with w, there is a polynomial Aw(X) over the integers, which in the case when G is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on w, that if all roots of Aw(X) are real and positive then G is bi-orderable, and that if G is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.

For a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

1. The object of this note is to show the relation between certain results obtained by Wiener and Paley*, and by L'evinson from the theory of Fourier transforms, and a theorem which I proved in a recent paper. We require a general theorem on the function of Phragmèn and Lindelöf:

Let Rc(n) be the number of representations of any non-negative integer n as the sum of c squares; i.e. Rc(n) is the number of different solutions (x1, x2, …, xc) of the equation

where the xμ are integers, and may be positive, negative, or zero. Two such solutions, (x1, x2, …, xc) and are only considered to be the same solution if

0. Introduction: In (14), higher order operations in K-theory, called Massey products, were introduced. These were motivated by the construction, in (7), of a spectral sequence in equivariant K-theory

Baumslag [Bm1] and Remeslennikov [R1] independently showed that the abstract group with finite presentation

〈a, x, y[mid ]ax =aay, [x, y] =1=[a, ay]〉.

is metabelian; the derived subgroup is free abelian of countably infinite rank. Both generalized this example to show that every finitely generated metabelian group embeds in a finitely presented metabelian group. Thomson [T] generalized this result still further to show that every finitely generated group of upper triangular matrices over an abstract ring A lies inside a finitely presented group of upper triangular matrices over some extension ring of A. For a valuable survey of these results, see [St].

Baumslag proved analogous theorems for Lie algebras in [Bm2]. In this paper we shall prove some analogous results for pro-p groups.

Notations. If G denotes an Abelian group and L a locally compact group topology on G, G(L) will denote the resulting topological group, and will denote the character group with the usual Pontrjagin topology (which is locally compact). Rn will denote the real n-dimensional vector group. Finally, for any set X, |X| will denote its cardinal number.

We study C1-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. In dimension 3, an important class of examples of such systems is given by those with a simple closed periodic curve tangent to the central bundle. We prove that there is a C1-open and dense subset of such diffeomorphisms such that every nonhyperbolic ergodic measure (i.e. with zero central exponent) can be approximated in the weak* topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent. Our method also allows to show how entropy changes across measures with central Lyapunov exponent close to zero. We also prove that any nonhyperbolic ergodic measure is in the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.

It is a theorem of Reidemeister [4] and Singer [6] that any two Heegaard splittings of a closed 3-manifold are stably equivalent. In this short note, we give this theorem a simple proof.

In previous papers [4, 6], B.-Y. Chen introduced a Riemannian invariant δM for a Riemannian n-manifold Mn, namely take the scalar curvature and subtract at each point the smallest sectional curvature. He proved that every submanifold Mn in a Riemannian space form Rm(ε) satisfies: δM[les ][n2(n−2)]/ 2(n−1)H2+[half](n+1)(n−2)ε. In this paper, first we classify constant mean curvature hypersurfaces in a Riemannian space form which satisfy the equality case of the inequality. Next, by utilizing Jacobi's elliptic functions and theta function we obtain the complete classification of conformally flat hypersurfaces in Riemannian space forms which satisfy the equality.

In this paper we consider the conjecture of Kervaire, that a nonsingular system of equations in one or more variables over an arbitrary group G has a solution in some overgroup of G.

A Banach space E is said to have the Banach-Saks property (BS) if every bounded sequence (xn) in E has a subsequence (x′n) with norm convergent Cesaro means; that is, there is x in E such that

If this occurs for every weakly convergent sequence in E it is said that E has the Weak Banach-Saks property (WBS) (also called Banach-Saks-Rosenthal property).

We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesàro averages of spherical averages in a Fuchsian group.

Let L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well-known Coates–Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen–Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h0(Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h0(Spec(L))(r), ), where L is totally real, r < 0 is odd and is a maximal order containing ℤ[]G, and will also provide some evidence for our conjecture.

In the theory of locales (or ‘pointless topology’), approximations to the Alexandrov-Hopf separation axioms have been considered by several authors [2,4,9,…]. The T1 axiom has proved more recalcitrant than the others; as yet, no-one has found a satisfactory lattice-theoretic condition of roughly the same strength.