In this paper we give an algorithm for computing the 2-Selmer group of an elliptic curve

which has complexity O(LD(0·5),c1)), where D is the absolute discriminant of the curve. Our algorithm is unconditional but the complexity estimate assumes the GRH and a standard conjecture on the distribution of smooth reduced ideals. This improves on the corresponding algorithm of Birch and Swinnerton-Dyer, which has complexity of O(√D).

Let h be the second fundamental form of a compact submanifold M of the Cayley projective plane CaP2. We determine all compact totally complex submanifolds of complex dimension n in CaP2 satisfying |h|2 ≤ n.

Let G be a group. The rth term LrG of the lower central series of G is the subgroup generated by the r-fold commutators

where , and for r > 1,

In this paper elementary properties are established for matrices whose coordinates are elements of a lattice L. In particular, many of the results of Luce [4] are extended to the case where L is an orthomodular lattice, a lattice with an orthocomplementation denoted by in which a ≦ b ⇒ a ∨(a′ ∧ b) = b. Originally, these were called orthocomplemented weakly modular lattices, Foulis [2]. In Theorem 1 a characterization is given of the nucleus with respect to matrix multiplication, which is in general nonassociative. Matrices with A-1 = transpose (A) are characterized in Lemma 8. Theorems 3 and 4 respectively, give partial characterizations of zero divisors and inverses.

It is well known that, provided that the indexing set I is not too large, the product

of a family of bornological locally convex topological vector spaces Eαis bornological. Products of bornological spaces were first studied by Mackey [3]. He reduced the problem to the study of R1, showing that this space is bornological if and only if I satisfies a certain condition, related to a problem in measure theory posed by Ulam [5]. We shall therefore call it the Mackey-Ulam condition on I. A similar study of the spaces R1 is to be found in the paper [4] by Simons; see also [1, Ch. IV, §6, exercise 3].

In this note we show that any derivation of a simple C*-algebra, whose range is not dense, is closable. We also derive a necessary and sufficient condition for a *-derivation of a C*-algebra, which is defined on the domain of a closed *- derivation, to be closed.

1. Dual integral equations of the form

where f(x) and g(x) are given and Ψ(x) is the unknown, have been increasingly studied in recent years; the first solutions were given for the case g(x) ≡ 0 by Titchmarsh [1] (for 0 < α < 2) and Busbridge [2] (for — 2 < α < 0). An interesting and much simpler method of solving the equations in the same case, g ≡ 0, was given by Gordon [3]. He also showed that the problem of solving the general equations (1) and (2) can be reduced to a problem in which g ≡ 0. He did not pursue this idea as far as finding and simplifying the solution of (1) and (2) but this has been done recently (see [4]) and Noble [5] used a similar idea in treating the case f ≡ 0.

Consider the following hyperlogistic equation

where r, K, τj ∈ (0, ∝), and αj = pj/qj are rational numbers with qj odd, pj and qj, are co-prime, 1 ≤ j ≤ m, and .

1. Every family of subnormal operators in a Hilbert space fulfils the Halmos-Bram condition on a suitable dense subset of its domain [2], [3]. In [2] and [4] it is shown that the generalized commutation relation implies the Halmos-Bram condition for one operator. In this paper it is proved that the generalized commutation relation implies the Halmos-Bram condition for infinite families of operators (in a special case Jorgensen proved it in a different way for finite families of operators, see [2]) and as an example of the application of this property it is shown that every family of generalized creation operators in the Bargmann space of an infinite order, indexed by mutually orthogonal vectors from I2 is subnormal. See [1] for the definitions.

The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = ℤ/2ℤ⊕ℤ/2ℤ is not a direct summand of R, where ℤ denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i. e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2. 6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2. 5 and 2. 7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

Let R be a ring with identity and let Eij ∈ Mn(R) be the usual n X n matrix units, where n ≥ 2 and 1≤i, j≤N. Let En(R) be the subgroup of GLn(R) generated by all Tij(q where r ∈ R and i ≄ j. For each (two-sided) R-ideal q let En(R, q) be the normal subgroup of En(R) generated by Tij(q), where q ∈ q. The subgroup En(R, q) plays an important role in the theory of GLn(R). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GLn(R), when R ∈ and n≥3, contains the subgroup En(R, q0), where q0 is the R-ideal generated by αij, rαij-αjjr (i ≄ j, r ∈ R), for all (αij) ∈ S. (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, En(R, q) has a simple set of generators when n ≥ 3. Let Ên(R, q) be the subgroup of En(R, q) generated by Tij(r)Tij(q)Tij(−r), where r ∈ R, q ∈ q. Then Ên(R, q) = En(R, q), for all q, when R ∈ and n ≥ 3.(See [13, Lemma 8].)

Given any finite group G, we wish to determine a relationship between the highest power of a prime p dividing the order of G, denoted by |G|p, and |A(G)|p, where A(G) is the automorphism group of G. It was shown by Herstein and Adney [8] that |A(G)|p ≧ p whenever |G|p= ≧P2. Later Scott [16] showed that A(G)p≧P2. For the special case where G is abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow p-subgroup of G is abelian, and gave an example where |G|p= p4 and |A(G)|P =p2. We are able to show in Theorem 4.5 that, if |G|p = ≧ p5, then |A(G)| = ≧ p3.

Throughout V will be a finite dimensional vector space over a field k and T(V) will denote the tensor algebra over V. For simplicity the symbol ⊗ will be omitted in the writing of the elements of T(V). Let be a basis of V ordered by Xi<Xi+1 for all i. Then we order the non-commutative monomials and 1 ≤ is ≤ n for s = 1,…, l} lexicographically from the left. D. Anick [1, p. 652] defines the high term of an element b in T(V) to be the highest monomial appearing in b. As a consequence of [1,3.2], if the set of the high terms of homogeneous relations is combinatorically free in the sense of no overlap ambiguities, then the connected algebra has global dimension 2. The purpose of this note is to prove this result and more for quadratic algebras under other hypotheses on the relations.

In [2], R. C. James proved that a weakly closed subset X of a real Banach space is weakly compact if and only if each continuous linear form attains its supremum on X. He also extended the result to the locally convex case, and, in [5], J. D. Pryce gave a simplified proof of the general result that is recorded below for reference in the sequel.

Apart from simple Ore extensions such as the Weyl algebras, the best known example of a primitive Ore extension is the universal enveloping algebra U(g) of the 2-dimensional solvable Lie algebra g over a field k of characteristic zero, see [4, p. 22]. U(g) is a polynomial algebra over k in two indeterminates x and y with multiplication subject to the relation xy – yx = y, and may be regarded either as an Ore extension of k [x] by the k-automorphism which maps x to x – 1 or as an Ore extension of k[y] by the derivation yd/dy. The argument suggested in [4, p. 22] to prove the primitivity of U(g) can easily be generalised [6] to show that, if α is an automorphism of the ring R then the following conditions are sufficient for R[x, α] to be primitive: (i) no power αs, s ≧ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are necessary and sufficient for the simplicity of the skew Laurent polynomial ring R[x, x–1, α] but are not necessary for the primitivity of R[x, α] (the ordinary polynomial ring D[x] over a division ring D not algebraic over its centre is easily seen to be primitive).

Tricomi [2] has shown that if φi ∈ Lpi (− ∞, ∞), i = 1, 2, where 1 < pi < ∞, (pi)–l + (P2)–1 < 1, and if H denotes the Hilbert transformation, that is

where the symbol (P) denotes that the integral is taken in the Cauchy principal value sense, then