Given positive integers k and n, let [Xfr ] be the class of all groups G such that γk(G) is locally nilpotent and [x1, x2, …, xk]n = 1 for any x1, x2, …, xk ∈ G. It is shown that [Xfr ] is a variety.

The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz system

x˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)

It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

Asymptotically, the solutions of Waring’s problem follow a limit law of which we are able to compute explicitly the limit density. In the special cases of sums of 3 and 4 squares where such a result is not possible, we establish a distribution result for slices of at least h0(n) consecutive integers ending at n, that is integers from n−h0(n)+1 to n, where h0(n) = nε for 4 squares and h0(n) = n¼+ε for 3 squares (ε > 0). We then deduce from this study the asymptotic behaviour of some kind of Riemann sums with an arithmetic constraint for which we point out an application related to the study of Schrödinger equation.

Let [Mfr ]g be the moduli space of smooth curves of genus g [ges ] 4 over the complex field [Copf ] and let [Tfr ]g ⊆ [Mfr ]g be the trigonal locus, i.e. the set of points [C] ∈ [Mfr ]g representing trigonal curves C of genus g [ges ] 4. Recall that each such curve C carries exactly one g13 (respectively at most two) if g [ges ] 5 (respectively g = 4). Let |D| be a g13 on C and suppose that it has a total ramification point at P (t.r. for short), i.e. that there is on C a point P such that 3P ∈ |D|. Such a P is a Weierstrass point whose first non-gap is three. In the present paper we study some sub-loci of [Tfr ]g related to curves possessing such points.

We introduce Γ-homology, the natural homology theory for E∞-algebras, and a cyclic version of it. Γ-homology specializes to a new homology theory for discrete commutative rings, very different in general from André–Quillen homology. We prove its general properties, including at base change and transitivity theorems. We give an explicit bicomplex for the Γ-homology of a discrete algebra, and elucidate connections with stable homotopy theory.

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.

Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.

On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

In this paper, we will prove that the highest Lyubeznik number λd,d(A) is one if A is a Cohen–Macaualy local ring containing a field, where d is the dimension of A.

Partial diagonal approximations are constructed for resolutions, due to Wall, of certain split extensions of cyclic groups by cyclic groups of order 2. These are used to determine the multiplicative structure in the cohomology of these groups arising from certain coefficient pairings.

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

In this paper, we construct Wakimoto modules for basic affine Lie superalgebras of type A(m−1, n−1)(1) and D(2, 1, a)(1). As an application, we compute the characters of irreducible highest weight modules at the critical level.

We prove that for any non-planar graph H, we can choose a two-colouring G of H such that G is intrinsically chiral, and if H is 3-connected and is not K3,3 or K5, then G is intrinsically asymmetric. No such asymmetric two-colouring is possible for K3,3 or K5.

Let L be a ring additively isomorphic to ℤd. The zeta function of L is defined to be

where the sum is taken over all subalgebras H of finite index in L. This zeta function has a natural Euler product decomposition:

These functions were introduced in a paper of Grunewald, Segal and Smith [5] where the local factors ζL[otimes ]ℤp(s) were shown to always be rational functions in p−s. The proof depends on representing the local zeta function as a definable p-adic integral and then appealing to a general result of Denef’s [1] about the rationality of such integrals. The proof of Denef relies on Macintyre’s Quantifier Elimination for ℚp [8] followed by techniques developed by Igusa [6] which employ resolution of singularities.

A generalized triangle group is a group with a presentation

GT(l, m, n; w) = 〈a, b | al = bm = wn = 1〉;

where l, m, n [ges ] 2, and w is a cyclically reduced word in the free product on {a, b}. In the setting of one-relator products of cyclic groups, Fine et al. [3] calculated their Euler characteristics for n [ges ] 3 under certain conditions on the defining word w.

Let R be a commutative Noetherian ring, I an ideal, M and N finitely generated R-modules. Assume V(I) [xcap ] Supp (M) [xcap ] Supp (N) consists of finitely many maximal ideals and let λ(Exti(N/InN, M)) denote the length of Exti(N/InN, M). It is shown that λ(Exti(N/InN, M)) agrees with a polynomial in n for n [Gt ] 0, and an upper bound for its degree is given. On the other hand, a simple example shows that some special assumption such as the support condition above is necessary in order to conclude that polynomial growth holds.

The problem of classifying line fields or, equivalently, Lorentz metrics up to homotopy is studied. Complete solutions are obtained in many cases, e.g. for all closed smooth manifolds N, orientable or not, of dimension n ≡ 0(4) and, in particular, in the classical space-time dimension 4.

Our approach is based on the singularity method which allows us to classify the monomorphisms u from a given (abstract) line bundle α over N into the tangent bundle. The analysis of the transition to the image line field u(α) then centers around the notion of ‘antipodality’.

We express our classification results in terms of standard (co-)homology and characteristic classes. Moreover, we illustrate them for large families of concrete sample manifolds by explicit bijections or by calculating the number of line fields.

We give a variation on the proof of Mostow's rigidity theorem, for certain hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie group-valued cocycles over piecewise smooth expanding Markov maps.

One can study the local topology of images of complex analytic maps and the restriction of such a map to a real subspace. Conditions are given for the topology of the image multiple point spaces of the complex and real maps to have the same homology or homotopy type.

We study the existence of complements to a partial tilting module over an arbitrary ring. As a consequence, we show that a finitely generated partial tilting module over an artin algebra has a (possibly infinitely generated) complement.

In this paper we prove a number of inequalities between the signature and the Betti numbers of a 4-manifold with even intersection form and prescribed fundamental group. Furthermore, we introduce a new geometric group invariant and discuss some of its properties.

In this paper we determine precisely the degrees r for which the Schur algebra S(2, r) is its own Ringel dual.