Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.

For a series Σan with partial sums An=a0 + a1 + … + an(n ≥ 0), supposed to be real in this note, we define, in a generally accepted notation ([2], pp. 7, 9, 94–98), the following transforms:

(C, α) sequence-transform,

(H, k)sequence-transform (k = 0, 1, 2, …):

(A: C, α)fnunction-transform,

Following the notation in Faudree and Schelp [3], we write G → (F, H) to mean that every 2-coloring of E(G), the edge set of G, contains a green (the first color) F or a red (the second color) H. Then the Ramsey number r(F, H) of two graphs F and H with no isolated vertices has been defined as the minimum p such that Kp → (F, H).

The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequality

Later Green [2] improved this estimate to

In the Present paper this will be revised, for sufficiently large h, to

We give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g ≥ 3. The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3,r) for certain values of r.

The aim of this paper is to explore some facets of the geometry of generic isotopies of plane curves. Our major tool will be the paper of Arnol'd [1] on the evolution of wavefronts. The sort of questions one can ask are: in a generic isotopy of a plane curve how are vertices created and destroyed? How does the dual evolve? How can the Gauss map change? In attempting to answer these questions we are taking advantage of the fact that these phenomena are all naturally associated with singularities of type Ak. Now the bifurcation set of an Ak+1 singularity and the discriminant set of an Ak singularity coincide. So we can apply Arnol'd's results on one parameter families of Legendre (discriminant) singularities (e.g. the duals) to get information on one parameter families of Lagrange (bifurcation) singularities (e.g. the evolutes). For bifurcation sets of functions with singularities other than those of type Ak one runs up against problems with smooth moduli—see [4].

The commutator [a, b] of two elements a and b in a group G satisfies the identity

ab = ba[a, b].

The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.

The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation

(A) [g, x]Y = [g, y]−1[g, xy]

holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that

{g}G = {g, K(g)}.

The purpose of this note is to establish the following

Theorem. The centre of a (left) hereditary local ring is either afield or a one-dimensional regular local ring.

Before starting the proof, it is necessary to explain the terminology. A ring R with an identity element is called a left local ring if the elements of R which do not have left inverses form a left ideal I. In these circumstances (see [1, Proposition 2.1, p. 147]), I is necessarily a two-sided ideal and it consists precisely of all the elements of R which do not have right inverses. Furthermore, every element of R which is not in I possesses a two-sided inverse. Thus there is, in fact, no difference between a left local ring and a right local ring and therefore one speaks simply of a local ring. In addition, I contains every proper left ideal and every proper right ideal. We may therefore describe I simply as the maximal ideal of R.

Let N be a homomorphically closed class of associative rings. Put N1 = Nl = N and, for ordinals a ≥ 2, define Nα (Nα) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in Nβ for some β<α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical class IN (lower left strong radical class IsN) determined by N. The chain {Nα} is called Kurosh's chain of N. Suliński, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lN ≠ Nk for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = Nn+l ≠ Nn. Some results concerning the termination of the chain {Nα} were obtained in [2,4]. In this paper we present some classes N with Nα = Nα for all α Using this and Beidar's example we prove that for every natural number n ≥ 1 there exists an N such that Nα = Nα for all α and Nn ≠ Nn+i = Nn+2. This in particular answers Question 6 of [4].

In [1, Theorem 3.3], E. Bishop proved that an operator S on a Hilbert space ℋ is subnormal if and only if there is a net of normal operators {Nα} that converges to S strongly (that is, ‖(Nα–S) f‖→ 0 for every f in ℋ). The proof that such a net exists if S is subnormal is not so difficult; in fact, a sequence of normal operators converging strongly to S can be found. Bishop's proof of the converse, however, is rather complicated and involves, among other things, some complicated arguments using operator-valued measures. The purpose of this note is to provide an easier proof of this part of the theorem. Our interest in finding such a proof was aroused by Paul Halmos.

We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).

We show that every Hilbert C$^*$-module $E$ is a JB$^*$-triple in a canonical way, establish an explicit expression for the holomorphic automorphisms of the unit ball of $E$, discuss the existence of fixed points for these automorphisms and give sufficient conditions for $E$ to have the density property.

Compact hyperbolic surfaces of given genus g containing discs of the maximum radius have been studied from various points of view. In this paper we connect these different approaches and observe some properties of the Fuchsian groups uniformizing both compact and punctured extremal surfaces. We also show that extremal surfaces of genera g=2,3 may contain one or several extremal discs, while an extremal disc is necessarily unique for g \ge 4. Along the way we also construct explicit families of extremal surfaces, one of which turns out to be free of automorphisms.

Denote by $(R,\cdot)$ the multiplicative semigroup of an associative algebra $R$ over an infinite field, and let $(R,\circ)$ represent $R$ when viewed as a semigroup via the circle operation $x\circy=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $R$. Namely, we prove that the following conditions on $R$ are equivalent: the semigroup $(R,\circ)$ satisfies an identity; the semigroup $(R,\cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $R$ satisfies the Engel condition. When $R$ is finitely generated these conditions are each equivalent to $R$ being upper Lie nilpotent.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25

We prove under quite general assumptions the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Neumann boundary conditions.

We characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also, studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.

It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k).

We characterize and classify the “regular classes of heaps” introduced by the author using ideas of Fan and of Stembridge. The irreducible objects fall into five infinite families with one exceptional case.

We prove that there are only $O(H^{3+\epsilon})$ quartic integer polynomials with height at most $H$ and a Galois group which is a proper subgroup of $S_4$. This improves in the special case of degree four a bound by Gallagher that yielded $O(H^{7/2} \log H)$.

We propose a differential-geometric classification of the four-component hyperbolic systems of conservation laws which satisfy the following properties: (a) they do not possess Riemann invariants; (b) they are linearly degenerate; (c) their rarefaction curves are rectilinear; (d) the cross-ratio of the four characteristic speeds is harmonic. This turns out to provide a classification of projective congruences in ${\mathbb P}^5$ whose developable surfaces are planar pencils of lines, each of these lines cutting the focal variety at points forming a harmonic quadruplet. Symmetry properties and the connection of these congruences to Cartan's isoparametric hypersurfaces are discussed.