If H, K are subgroups of a group G, then HK is a subgroup of G if and only if HK = KH. This condition certainly holds if H ≤ NG(K) or K ≤ NG(H). But the majority of groups can also be expressed as HK, where neither H nor K is normal. In this paper we consider groups G for which no subgroup G1 can be expressed as the product of non-normal subgroups of G1. Such a group is said to be equilibrated. Thus G is equilibrated if and only if either H ≤ NG(K) or K ≤ NG(H) whenever H, K and HK are subgroups of G.

Let L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

A quandle is the algebraic distillation of the second and third Reidemeister moves, on unframed links. Quandles have been studied by many people including Kauffman[K], Fenn and Rourke[F-R] and Joyce [J]. Joyce defines the fundamental quandle, which is a classifying invariant of irreducible, unframed links. Fenn and Rourke, who study generalised quandles or racks in [F—R], show that the natural map from the fundamental quandle of a knot K to the knot group is never injective if K is a connected sum. We now prove that the natural map from the fundamental quandle to the fundamental group of the complement of a link in S3 is injective if and only if the link is prime.

We show that certain natural aperiodic sum-free sets are incomplete, that is that there are infinitely many n not in S which are not a sum of two elements of S.

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard[3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

Erdös has asked whether the plane ℝ2, or more generally n-dimensional Euclidean space ℝn, can be partitioned into countably many sets none of which contains the vertices of an isosceles triangle. Assuming the Continuum Hypothesis (CH), Davies[2] (for n = 2) and Kunen[10] (for arbitrary n) proved that such partitions exist. Assuming Martin's Axiom, Erdös and Komjáth proved in [5] that such partitions exist for n = 2. We will prove here, without additional set-theoretic hypotheses, that there are such partitions in all dimensions.

Let ‖x‖ denote the usual Euclidean norm of a point x ∈ ℝn, so that ‖x − y‖ is the distance between x and y.

We investigate homomorphisms between finite oriented paths. We demonstrate the surprising richness of this perhaps simplest case of homomorphism between graphs by proving the density theorem for oriented paths. As a consequence every two dimensional countable poset is represented finite paths and their homomorphisms, and every finite dimensional poset is represented finite oriented trees and their homomorphisms. We then consider related problems of universal representability and extendability and on-line representability.

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H*(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗kN is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

The following problem was raised in the 1993 Miklós Schweitzer Mathematical Contest (see [4]). Let E be any connected set in the plane of diameter greater than 4, and let Z1, Z2, …, be any sequence of points on the plane. Then there is a point X ε E for which infinitely many of the products X¯Z¯1 · X¯Z¯n are greater than 1. Furthermore, the same is not necessarily true if the diameter of E is 4.

Let f: E → B be a fibration with fibre F over a connected space B. If F is homotopy equivalent to a finite complex, Becker and Gottlieb [2, 3] and others have constructed a transfer map

where for simplicity X+ denotes the suspension spectrum of the space obtained from X adding a disjoint basepoint. One key property of τ(f) is the fact that the composite map f+. τ(f): B+ → B+ induces a map on integral homology which is multiplication the Euler characteristic X(F).

In this paper, we are going to introduce an elliptic analogue of the classical Mahler measure of an integral polynomial. The measure is shown to vanish if and only if the roots of the polynomial are attached to division points of the curve. This is the exact analogue of the statement that the Mahler measure of an integral polynomial vanishes if and only if all of its roots are division points of the circle (in other words, roots of unity). The proof of this result exploits an integral representation for the local canonical heights on the elliptic curve. The integral is that of a simple polynomial function over the complete curve in the appropriate valuation. Attention then shifts to the calculation of local integrals of arbitrary rational functions on elliptic curves. Results are proved which show these integrals may be computed as effective limits of Riemann sums. Finally, consideration is given to analogous behaviour for abelian varieties. We give a method, in principle, for computing the global canonical height of a rational point on an abelian variety denned over an algebraic number field, once again exploiting the integral representation of the local heights. The methods used include recent inequalities for linear forms in elliptic and abelian logarithms.

The class of nilpotent semigroups was introduced, via a semigroup identity, independently in [8] and [9], cf. [11], Every nilpotent cancellative semigroup S was shown to have a group of classical fractions G which is nilpotent and of the same nilpotency class as S. Groups, and linear groups in particular, satisfying certain related semigroup identities, introduced in [19], have been recently studied in [1], [15] and [18]. In particular, finitely generated residually finite groups satisfying a semigroup identity must be almost nilpotent, [18]. On the other hand, it was recently shown in [14] that a finitely generated linear semigroup S ⊆ Mn(K), over a field K, with no free non-commutative subsemigroups satisfies an identity and for every maximal subgroup H of Mn(K) the subgroup gp(S ∩ H) of H generated by S ∩ H is almost nilpotent. A natural question that arises here is to decide which of these semigroups are nilpotent. Because of the powerful classical theory of nilpotent linear groups, cf. [21], one can also ask whether such semigroups can be approached via group theoretical methods. We note that the very special case of nilpotent connected algebraic monoids has been recently considered in [4]. In [5] the structure of semigroup algebras of nilpotent semigroups was studied, in particular via prime Goldie homomorphic images, leading naturally to nilpotent subsemigroups of the matrix monoids Mn(D) over division rings D.

Let X be an algebraic (complete) variety over a fixed algebraically closed field k. To every Cartier divisor D on X, we can associate the graded k-algebra . As is known, for a semi-ample divisor D, R(X, D) is a finitely generated k-algebra (see [21] or [9]), while this property is no longer true for arbitrary nef and big divisors (see [21]).

It is one of the first theorems proved in the theory of modular forms [1, p. 78] that every element in SL (2, ℤ), and hence in SL(2, ), can be written as a product of the two generators

While this is quite an easy result, it required more advanced tools such as spectral analysis on Riemann surfaces [2] to show that there is in fact always such a product of length O(log p). This result is not constructive and therefore immediately [2, p. 102] raises the following:

Problem. Is there an algorithm polynomial in log p for constructing a monomial in S and T, of degree O(log p), whose value is

In this note, we construct such an algorithm. We will carry out all the details only for the particular element

but it will become obvious that our method works for all elements in SL(2, ).

A braided surface of degree m is a compact oriented surface S embedded in a bidisk such that is a branched covering map of degree m and , where is the projection. It was defined L. Rudolph [14, 16] with some applications to knot theory, cf. [13, 14, 15, 16, 17, 18]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R4 is a closed oriented surface F embedded in R4 such that and pr2 │F: F → S2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R4. It is related to 2-knot theory, cf. [8, 9, 10]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [5, 8, 9, 14, 16], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [1].

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

Let p be an odd prime number, and let M2 be the extra-special p-group of order p5 and of exponent p2. For every p-group K, we denote by H*(K) the mod p cohomology of K. The purpose of this paper is to calculate the mod p cohomology groups of M2 and of Cp2* M2 (the central product of the cyclic group Cp2 of order p2 and M2).

Throughout this paper, let A be a Noetherian local ring with maximal ideal m and M a finitely generated A-module with d = dimAM ≥ 1. Denote by N the set of all positive integers.

Let x = (x1, …, xd) be a system of parameters (s.o.p) for M and let

We consider the following two problems: (i) When is the length of Koszul homology

a polynomial in n for all k = 0, …, d and n1; …, nd sufficiently large (n ≫ 0)?

(ii) Is the length of the generalized fraction in a polynomial in n for n ≫ 0?

In this paper, a distance formula to a Douglas algebra is established. We use this distance formula to show that the distance of an L∞ function to the intersection of arbitrary Douglas algebras is equivalent to the supremum of the distance of that function to these algebras. As an application, we prove that the Bourgain algebra of the intersection of two Douglas algebras is equal to the intersection of the Bourgain algebras of these two algebras.